{"id":941,"date":"2026-10-05T02:57:26","date_gmt":"2026-10-05T02:57:26","guid":{"rendered":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/?p=941"},"modified":"2026-10-05T03:17:07","modified_gmt":"2026-10-05T03:17:07","slug":"3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers","status":"publish","type":"post","link":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/","title":{"rendered":"*3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers"},"content":{"rendered":"<div id=\"pl-941\"  class=\"panel-layout\" ><div id=\"pg-941-0\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-941-0-0\"  class=\"panel-grid-cell\" ><div id=\"panel-941-0-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"0\" ><div class=\"textwidget custom-html-widget\"><div id=\"FILE1-CH3-GANIT\" style=\"max-width:860px;margin:12px auto;border:3.5px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:8px 10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:6px 18px;border-radius:20px;font-weight:bold;font-size:11px;\">Ch3 - \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 - \u092d\u093e\u0917 1\/4 - \u092a\u0930\u093f\u091a\u092f<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"ch3g-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - \u092a\u0930\u093f\u091a\u092f - \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;\">\u092a\u0930\u093f\u091a\u092f - \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0915\u094d\u092f\u093e? - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0926\u0941\u0928\u093f\u092f\u093e \u091a\u0932\u0924\u0940 \u0939\u0948\u0964 \u0917\u093f\u0928\u0924\u0940 \u0938\u0947 \u0936\u0941\u0930\u0942 \u0939\u0941\u0908 \u0925\u0940\u0902\u0964 \u092b\u093f\u0930 \u0927\u0940\u0930\u0947 \u0927\u0940\u0930\u0947 \u0928\u090f \u092a\u094d\u0930\u0915\u093e\u0930 \u092c\u0928\u0947\u0964 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0947 \u092a\u0942\u0930\u094d\u0923 \u092b\u093f\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092b\u093f\u0930 \u092a\u0930\u093f\u092e\u0947\u092f\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:15px;font-weight:bold;\">\u092a\u094d\u0930\u093e\u0915\u0943\u0924 N : 1,2,3,4... \u0917\u093f\u0928\u0924\u0940 \u0935\u093e\u0932\u0940<\/div><div style=\"font-size:15px;font-weight:bold;\">\u092a\u0942\u0930\u094d\u0923 W : 0,1,2,3... N + 0<\/div><div style=\"font-size:15px;font-weight:bold;\">\u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 Z : ...-2,-1,0,1,2... W + \u090b\u0923<\/div><div style=\"font-size:13px;\">N \u092e\u0947\u0902 0 \u0928\u0939\u0940\u0902 W \u092e\u0947\u0902 \u0939\u0948 Z \u092e\u0947\u0902 \u090b\u0923 \u092d\u0940 \u0939\u0948<\/div><\/div>\u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u092c\u0938\u0947 \u091b\u094b\u091f\u0940 1 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u092a\u0942\u0930\u094d\u0923 \u0938\u092c\u0938\u0947 \u091b\u094b\u091f\u0940 0 \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092e\u0947\u0902 \u090b\u0923 \u092d\u0940 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0905\u092d\u093e\u091c\u094d\u092f \u092d\u093e\u091c\u094d\u092f - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u091c\u093f\u0938\u0915\u0947 \u0926\u094b \u0917\u0941\u0923\u0928\u0916\u0902\u0921 1 \u0914\u0930 \u0938\u094d\u0935\u092f\u0902 \u0935\u0939\u0940 \u0905\u092d\u093e\u091c\u094d\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u091c\u0948\u0938\u0947 2,3,5,7,11,13\u0964 \u091c\u093f\u0938\u0915\u0947 \u0926\u094b \u0938\u0947 \u091c\u094d\u092f\u093e\u0926\u093e \u0917\u0941\u0923\u0928\u0916\u0902\u0921 \u0935\u0939\u0940 \u092d\u093e\u091c\u094d\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u091c\u0948\u0938\u0947 4,6,8,9,10\u0964 1 \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#e3f2fd;border:2px solid #2196f3;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">\u0905\u092d\u093e\u091c\u094d\u092f : 2,3,5,7,11,13... 2 \u0939\u0940 \u0938\u092e \u0905\u092d\u093e\u091c\u094d\u092f<\/span><br><span>\u092d\u093e\u091c\u094d\u092f : 4,6,8,9,10... 1 \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3g-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;\">\u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094d\u092f\u093e? - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u091c\u094b p\/q \u0930\u0942\u092a \u092e\u0947\u0902 \u0932\u093f\u0916\u0940 \u091c\u093e \u0938\u0915\u0947 \u0935\u0939\u0940 \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 q \u0936\u0942\u0928\u094d\u092f \u0928\u0939\u0940\u0902 \u0939\u094b\u0928\u093e \u091a\u093e\u0939\u093f\u090f\u0964 p,q \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u094b\u0928\u0947 \u091a\u093e\u0939\u093f\u090f\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:18px;font-weight:bold;\">\u092a\u0930\u093f\u092e\u0947\u092f Q : p\/q , q\u22600 , p,q \u2208 Z<\/div><div>\u091c\u0948\u0938\u0947 1\/2 , 3 , -4 , 0 , 0.5 , 0.333...<\/div><div>0 \u092d\u0940 \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u0948 0\/1 \u0930\u0942\u092a \u092e\u0947\u0902<\/div><\/div>\u0939\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u091c\u0948\u0938\u0947 3=3\/1\u0964 \u0926\u0936\u092e\u0932\u0935 \u091c\u094b \u0938\u092e\u093e\u092a\u094d\u0924 \u092f\u093e \u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b \u0935\u0939 \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0915\u094d\u092f\u093e? - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u091c\u094b p\/q \u092e\u0947\u0902 \u0928 \u0932\u093f\u0916\u0940 \u091c\u093e \u0938\u0915\u0947 \u0935\u0939\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u0940 \u0939\u0948\u0964 \u0926\u0936\u092e\u0932\u0935 \u0928 \u0938\u092e\u093e\u092a\u094d\u0924 \u0928 \u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#ffebee;border:2px solid #d32f2f;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">\u0905\u092a\u0930\u093f\u092e\u0947\u092f : \u221a2 , \u221a3 , \u221a5 , \u03c0 , 0.101101110...<\/span><br><span>p\/q \u092e\u0947\u0902 \u0928\u0939\u0940\u0902 \u0932\u093f\u0916 \u0938\u0915\u0924\u0947<\/span><\/div>\u221a2=1.414213... \u0915\u092d\u0940 \u0938\u092e\u093e\u092a\u094d\u0924 \u0928\u0939\u0940\u0902 \u0915\u092d\u0940 \u0926\u094b\u0939\u0930\u093e\u0924\u093e \u0928\u0939\u0940\u0902\u0964 \u0907\u0938\u0932\u093f\u090f \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0939\u0948\u0964 \u03c0 \u092d\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0939\u0948\u0964<\/div><\/div>\n\n<div id=\"ch3g-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0914\u0930 \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:6px 10px;font-weight:bold;\">\u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u0902\u0916\u094d\u092f\u093e - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u092a\u0930\u093f\u092e\u0947\u092f \u0914\u0930 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u092e\u093f\u0932\u0915\u0930 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u092c\u0928\u0924\u0947 \u0939\u0948\u0902\u0964 \u0938\u092d\u0940 \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u091c\u094b \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u092a\u0930 \u0906 \u0938\u0915\u0947\u0902 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0939\u094b\u0924\u0940 \u0939\u0948\u0902\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:16px;font-weight:bold;\">\u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 R = \u092a\u0930\u093f\u092e\u0947\u092f Q + \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><div>N \u2282 W \u2282 Z \u2282 Q \u2282 R<\/div><div>\u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u092c\u0938\u0947 \u091b\u094b\u091f\u093e \u092a\u0930\u093f\u0935\u093e\u0930<\/div><div>\u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u092c\u0938\u0947 \u092c\u0921\u093c\u093e<\/div><\/div>\u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092a\u0942\u0930\u094d\u0923 \u0939\u0948\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0942\u0930\u094d\u0923 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0939\u0948\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u0948\u0964 \u092a\u094d\u0930\u0924\u094d\u092f\u0947\u0915 \u092a\u0930\u093f\u092e\u0947\u092f \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0939\u0948\u0964<div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u092a\u0930 \u0928\u093f\u0930\u0942\u092a\u0923 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u092a\u0930 \u092c\u0940\u091a \u092e\u0947\u0902 0 \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0926\u093e\u090f\u0901 \u0927\u0928 \u092c\u093e\u090f\u0901 \u090b\u0923\u0964 \u092a\u0930\u093f\u092e\u0947\u092f \u0915\u094b \u092c\u093f\u0902\u0926\u0941 \u0938\u0947 \u0926\u093f\u0916\u093e\u0924\u0947 \u0939\u0948\u0902\u0964 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u221a2 \u0915\u094b \u092a\u093e\u0907\u0925\u093e\u0917\u094b\u0930\u0938 \u0938\u0947 \u0926\u093f\u0916\u093e\u0924\u0947 \u0939\u0948\u0902\u0964 1 \u0915\u093e \u0935\u0930\u094d\u0917 1 \u090a\u092a\u0930 \u0932\u0947 \u091c\u093e\u0913 \u0924\u094b \u221a2 \u092c\u0928\u0924\u093e \u0939\u0948\u0964<div style=\"background:#e3f2fd;border:2px solid #2196f3;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">\u221a2 \u0930\u0947\u0916\u093e \u092a\u0930 : 1 \u0907\u0915\u093e\u0908 \u0938\u092e\u0915\u094b\u0923 \u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u0930\u094d\u0923 \u221a2<\/span><br><span>\u0915\u0902\u092a\u093e\u0938 \u0938\u0947 \u0930\u0947\u0916\u093e \u092a\u0930 \u0915\u093e\u091f\u094b<\/span><\/div>\u0907\u0938\u0940 \u0924\u0930\u0939 \u221a3 \u221a5 \u092c\u0928\u093e \u0938\u0915\u0924\u0947 \u0939\u0948\u0902\u0964<\/div><\/div>\n\n<div id=\"ch3g-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - \u0926\u0936\u092e\u0932\u0935 \u0914\u0930 Trick<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#f3e5f5;border-left:4px solid #6a1b9a;padding:6px 10px;font-weight:bold;\">\u0926\u0936\u092e\u0932\u0935 \u092a\u094d\u0930\u0915\u093e\u0930 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u0926\u0936\u092e\u0932\u0935 \u0926\u094b \u092a\u094d\u0930\u0915\u093e\u0930 \u0915\u093e \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u0938\u092e\u093e\u092a\u094d\u0924 \u0914\u0930 \u0905\u0938\u092e\u093e\u092a\u094d\u0924\u0964 \u0905\u0938\u092e\u093e\u092a\u094d\u0924 \u092e\u0947\u0902 \u0906\u0935\u0930\u094d\u0924\u0940 \u0914\u0930 \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:14px;font-weight:bold;\">\u0938\u092e\u093e\u092a\u094d\u0924 : 0.5 , 0.25 , 0.125 => \u092a\u0930\u093f\u092e\u0947\u092f<\/div><div style=\"font-size:14px;font-weight:bold;\">\u0905\u0938\u092e\u093e\u092a\u094d\u0924 \u0906\u0935\u0930\u094d\u0924\u0940 : 0.333... , 0.666... => \u092a\u0930\u093f\u092e\u0947\u092f<\/div><div style=\"font-size:14px;font-weight:bold;color:#d32f2f;\">\u0905\u0938\u092e\u093e\u092a\u094d\u0924 \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 : 0.101101110... => \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><div style=\"font-size:13px;\">\u221a2=1.414... \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><\/div>\u0938\u092e\u093e\u092a\u094d\u0924 \u0914\u0930 \u0906\u0935\u0930\u094d\u0924\u0940 \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#e0f2f1;border-left:4px solid #009688;padding:6px 10px;font-weight:bold;margin-top:8px;\">Trick \u092f\u093e\u0926 - \u0939\u0947\u0921\u093f\u0902\u0917 - Most Important<\/div><div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:15px;font-weight:bold;\">N=1,2,3... W=0,1,2... Z=...-1,0,1...<\/div><div style=\"font-size:15px;font-weight:bold;\">Q=p\/q q\u22600 , 0 \u092a\u0930\u093f\u092e\u0947\u092f 0\/1<\/div><div style=\"font-size:15px;font-weight:bold;\">\u0905\u092a\u0930\u093f\u092e\u0947\u092f \u221a2 \u221a3 \u221a5 \u03c0 \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940<\/div><div style=\"font-size:15px;font-weight:bold;\">R=Q+\u0905\u092a\u0930\u093f\u092e\u0947\u092f , N\u2282W\u2282Z\u2282Q\u2282R<\/div><div style=\"font-size:13px;margin-top:4px;background:#fff;padding:3px;border-radius:5px;\">\u0905\u092d\u093e\u091c\u094d\u092f 2 \u0939\u0940 \u0938\u092e , 1 \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f<\/div><\/div>\u092f\u0939\u0940 \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u092e\u0947\u0902 \u092a\u0915\u094d\u0915\u093e \u092a\u0942\u091b\u0924\u093e \u0939\u0948\u0964<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"ch3gp\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"ch3gi\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-size:15px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"ch3gn\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('ch3g-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('ch3gi').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('ch3gn').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('ch3gp').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-941-1\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-941-1-0\"  class=\"panel-grid-cell\" ><div id=\"panel-941-1-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"1\" ><div class=\"textwidget custom-html-widget\"><div id=\"FILE2-CH3-GANIT\" style=\"max-width:860px;margin:12px auto;border:3.5px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:8px 10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:6px 18px;border-radius:20px;font-weight:bold;font-size:11px;\">Ch3 - \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 - \u092d\u093e\u0917 2\/4 - \u092b\u0949\u0930\u094d\u092e\u0942\u0932\u0947 \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"ch3g2-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u0930\u093f\u0935\u093e\u0930 \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 1 - N W Z Q R \u0938\u0902\u092c\u0902\u0927 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u0938\u092d\u0940 \u092a\u0930\u093f\u0935\u093e\u0930 \u090f\u0915 \u0926\u0942\u0938\u0930\u0947 \u092e\u0947\u0902 \u0938\u092e\u093e\u090f \u0939\u0948\u0902\u0964 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u092c\u0938\u0947 \u0905\u0902\u0926\u0930 \u0939\u0948\u0964 \u0935\u093e\u0938\u094d\u0924\u0935\u093f\u0915 \u0938\u092c\u0938\u0947 \u092c\u093e\u0939\u0930 \u0939\u0948\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:16px;font-weight:bold;\">N = {1,2,3...}<\/div><div style=\"font-size:16px;font-weight:bold;\">W = {0} \u222a N = {0,1,2...}<\/div><div style=\"font-size:16px;font-weight:bold;\">Z = { ...-2,-1,0,1,2...} = W \u222a \u090b\u0923<\/div><div style=\"font-size:16px;font-weight:bold;\">Q = {p\/q , q\u22600} Z \u2282 Q<\/div><div style=\"font-size:16px;font-weight:bold;\">R = Q \u222a \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><div style=\"font-size:14px;margin-top:4px;background:#fff;padding:3px 8px;border-radius:5px;\">N \u2282 W \u2282 Z \u2282 Q \u2282 R<\/div><\/div><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 2 - \u0905\u092d\u093e\u091c\u094d\u092f \u092d\u093e\u091c\u094d\u092f \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u0905\u092d\u093e\u091c\u094d\u092f \u0915\u0947 \u0920\u0940\u0915 \u0926\u094b \u0917\u0941\u0923\u0928\u0916\u0902\u0921 \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 1 \u0914\u0930 \u0938\u094d\u0935\u092f\u0902\u0964 \u092d\u093e\u091c\u094d\u092f \u0915\u0947 \u0926\u094b \u0938\u0947 \u091c\u094d\u092f\u093e\u0926\u093e \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<div style=\"background:#e3f2fd;border:2px solid #2196f3;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">\u0905\u092d\u093e\u091c\u094d\u092f : \u0917\u0941\u0923\u0928\u0916\u0902\u0921 2 = 1 \u0914\u0930 \u0938\u094d\u0935\u092f\u0902<\/span><br><span>2,3,5,7,11,13,17,19...<\/span><br><span>\u092d\u093e\u091c\u094d\u092f : \u0917\u0941\u0923\u0928\u0916\u0902\u0921 >2<\/span><br><span>4=1,2,4 ; 6=1,2,3,6<\/span><br><span>1 \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f - \u090f\u0915 \u0917\u0941\u0923\u0928\u0916\u0902\u0921<\/span><\/div>2 \u0939\u0940 \u0938\u092e \u0905\u092d\u093e\u091c\u094d\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964 \u092c\u093e\u0915\u0940 \u0938\u092d\u0940 \u0938\u092e \u092d\u093e\u091c\u094d\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/div><\/div>\n\n<div id=\"ch3g2-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u092b\u0949\u0930\u094d\u092e\u0942\u0932\u0947<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 3 - \u092a\u0930\u093f\u092e\u0947\u092f \u0930\u0942\u092a - \u0939\u0947\u0921\u093f\u0902\u0917 - Most Important<\/div>\u092a\u0930\u093f\u092e\u0947\u092f p\/q \u0930\u0942\u092a \u092e\u0947\u0902 \u0939\u094b\u0924\u093e \u0939\u0948\u0964 q\u22600\u0964 \u0926\u0936\u092e\u0932\u0935 \u0938\u092e\u093e\u092a\u094d\u0924 \u092f\u093e \u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b \u0924\u094b \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:18px;font-weight:bold;\">Q = p\/q , q\u22600 , p,q \u2208 Z<\/div><div style=\"font-size:14px;\">\u0938\u092e\u093e\u092a\u094d\u0924 : 0.5=1\/2 , 0.25=1\/4<\/div><div style=\"font-size:14px;\">\u0906\u0935\u0930\u094d\u0924\u0940 : 0.333...=1\/3 , 0.666...=2\/3<\/div><div style=\"font-size:14px;\">0=0\/1 \u092a\u0930\u093f\u092e\u0947\u092f , \u0939\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092a\u0930\u093f\u092e\u0947\u092f<\/div><div style=\"font-size:13px;margin-top:4px;background:#fff;padding:3px 8px;border-radius:5px;\">\u0906\u0935\u0930\u094d\u0924\u0940 \u0915\u094b p\/q \u092e\u0947\u0902 \u092c\u0926\u0932\u094b : 0.333...=1\/3 Trick<\/div><\/div><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 4 - \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u0905\u092a\u0930\u093f\u092e\u0947\u092f p\/q \u092e\u0947\u0902 \u0928\u0939\u0940\u0902 \u0932\u093f\u0916 \u0938\u0915\u0924\u0947\u0964 \u0926\u0936\u092e\u0932\u0935 \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u0905\u0928\u0928\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#ffebee;border:2px solid #d32f2f;padding:8px;border-radius:8px;text-align:center;margin:6px 0;\"><div style=\"font-weight:bold;\">\u0905\u092a\u0930\u093f\u092e\u0947\u092f \u2260 p\/q<\/div><div>\u221a2=1.414213562... \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u0905\u0928\u0928\u094d\u0924<\/div><div>\u221a3=1.732... , \u221a5=2.236... , \u03c0=3.14159...<\/div><div>0.10110111011110... \u092a\u0948\u091f\u0930\u094d\u0928 \u092a\u0930 \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940<\/div><div>\u092a\u094d\u0930\u092e\u0947\u092f : \u092f\u0926\u093f p \u0905\u092d\u093e\u091c\u094d\u092f \u0924\u094b \u221ap \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><\/div>\u221a2,\u221a3,\u221a5,\u221a7 \u0938\u092d\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0915\u094d\u092f\u094b\u0902\u0915\u093f \u0905\u0902\u0926\u0930 \u0905\u092d\u093e\u091c\u094d\u092f \u0939\u0948\u0964<\/div><\/div>\n\n<div id=\"ch3g2-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - \u0926\u0936\u092e\u0932\u0935 \u0914\u0930 \u0939\u0930 \u0915\u093e \u0928\u093f\u092f\u092e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:6px 10px;font-weight:bold;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 5 - \u0939\u0930 2\u207f5\u1d50 \u0928\u093f\u092f\u092e - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u092a\u0930\u093f\u092e\u0947\u092f p\/q \u0915\u093e \u0926\u0936\u092e\u0932\u0935 \u0915\u092c \u0938\u092e\u093e\u092a\u094d\u0924 \u0939\u094b\u0924\u093e \u0939\u0948? \u091c\u092c \u0939\u0930 q \u0915\u0947 \u0905\u092d\u093e\u091c\u094d\u092f \u0917\u0941\u0923\u0928\u0916\u0902\u0921 \u0938\u093f\u0930\u094d\u092b 2 \u0914\u0930 5 \u0939\u094b\u0902\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:16px;font-weight:bold;\">\u0938\u092e\u093e\u092a\u094d\u0924 \u21d4 q = 2\u207f \u00d7 5\u1d50<\/div><div style=\"font-size:14px;\">1\/2=0.5 \u0939\u09302=2\u00b9 \u0938\u092e\u093e\u092a\u094d\u0924<\/div><div style=\"font-size:14px;\">1\/4=0.25 \u0939\u09304=2\u00b2 \u0938\u092e\u093e\u092a\u094d\u0924<\/div><div style=\"font-size:14px;\">1\/5=0.2 \u0939\u09305=5\u00b9 \u0938\u092e\u093e\u092a\u094d\u0924<\/div><div style=\"font-size:14px;\">1\/8=0.125 \u0939\u09308=2\u00b3 \u0938\u092e\u093e\u092a\u094d\u0924<\/div><div style=\"font-size:14px;color:#d32f2f;\">1\/3=0.333... \u0939\u09303 \u092e\u0947\u0902 3 \u0939\u0948 \u0905\u0938\u092e\u093e\u092a\u094d\u0924 \u0906\u0935\u0930\u094d\u0924\u0940<\/div><div style=\"font-size:14px;color:#d32f2f;\">1\/6=0.1666... \u0939\u09306=2\u00d73 \u092e\u0947\u0902 3 \u0939\u0948 \u0906\u0935\u0930\u094d\u0924\u0940<\/div><\/div>\u092f\u0926\u093f \u0939\u0930 \u092e\u0947\u0902 2,5 \u0915\u0947 \u0905\u0932\u093e\u0935\u093e \u0915\u094b\u0908 \u0905\u092d\u093e\u091c\u094d\u092f \u0924\u094b \u0905\u0938\u092e\u093e\u092a\u094d\u0924 \u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 6 - \u0906\u0935\u0930\u094d\u0924\u0940 \u0915\u094b p\/q \u092e\u0947\u0902 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u0906\u0935\u0930\u094d\u0924\u0940 \u0926\u0936\u092e\u0932\u0935 \u0915\u094b p\/q \u092e\u0947\u0902 \u092c\u0926\u0932\u0928\u0947 \u0915\u093e \u092b\u0949\u0930\u094d\u092e\u0942\u0932\u093e\u0964<div style=\"background:#e0f2f1;border:3px solid #009688;padding:8px;border-radius:8px;text-align:center;margin:6px 0;\"><div style=\"font-size:15px;font-weight:bold;\">0.333... = x \u092e\u093e\u0928\u094b<\/div><div>10x=3.333... , 10x-x=3 => 9x=3 => x=1\/3<\/div><div style=\"margin-top:4px;\">0.666...=2\/3 , 0.999...=1<\/div><div>0.1666...=1\/6 Trick<\/div><div>0.232323...=23\/99<\/div><\/div>\u0906\u0935\u0930\u094d\u0924\u0940 \u0935\u093e\u0932\u0947 \u0939\u092e\u0947\u0936\u093e \u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b\u0924\u0947 \u0939\u0948\u0902\u0964<\/div><\/div>\n\n<div id=\"ch3g2-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - \u0938\u092d\u0940 \u092b\u0949\u0930\u094d\u092e\u0942\u0932\u0947 \u090f\u0915 \u0938\u093e\u0925 Trick<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#f3e5f5;border-left:4px solid #6a1b9a;padding:6px 10px;font-weight:bold;\">\u0938\u093f\u0926\u094d\u0927\u093e\u0902\u0924 7 - \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u0914\u0930 \u092a\u0930\u093f\u0915\u0930\u094d\u092e - \u0939\u0947\u0921\u093f\u0902\u0917<\/div>\u221a2 \u0915\u094b \u0930\u0947\u0916\u093e \u092a\u0930 \u0926\u093f\u0916\u093e\u0928\u093e: 1 \u0907\u0915\u093e\u0908 \u0906\u0927\u093e\u0930 1 \u0932\u092e\u094d\u092c \u0915\u0930\u094d\u0923 \u221a2\u0964 \u0915\u0902\u092a\u093e\u0938 \u0938\u0947 \u0915\u093e\u091f\u094b\u0964 \u0907\u0938\u0940 \u0924\u0930\u0939 \u221a3 \u0915\u0947 \u0932\u093f\u090f \u221a2 \u0906\u0927\u093e\u0930 1 \u0932\u092e\u094d\u092c \u0915\u0930\u094d\u0923 \u221a3\u0964<div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">\u221a2 \u0930\u0947\u0916\u093e \u092a\u0930 : \u0938\u092e\u0915\u094b\u0923 \u0394 1,1 \u0915\u0930\u094d\u0923 \u221a2<\/span><br><span>\u221a3 : \u221a2 \u0906\u0927\u093e\u0930 1 \u0932\u092e\u094d\u092c \u0915\u0930\u094d\u0923 \u221a3<\/span><br><span>\u221an \u0915\u093e \u0938\u0930\u094d\u092a\u093f\u0932 \u092c\u0928\u0924\u093e \u0939\u0948<\/span><\/div><div style=\"background:#e0f2f1;border-left:4px solid #009688;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0938\u092d\u0940 \u092b\u0949\u0930\u094d\u092e\u0942\u0932\u0947 \u090f\u0915 \u0938\u093e\u0925 - \u0939\u0947\u0921\u093f\u0902\u0917 - Most Important<\/div><div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:10px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-size:15px;font-weight:bold;\">N={1,2...} W={0,1...} Z={...-1,0,1...}<\/div><div style=\"font-size:15px;font-weight:bold;\">Q=p\/q q\u22600 \u0938\u092e\u093e\u092a\u094d\u0924\/\u0906\u0935\u0930\u094d\u0924\u0940<\/div><div style=\"font-size:15px;font-weight:bold;color:#d32f2f;\">\u0905\u092a\u0930\u093f\u092e\u0947\u092f \u2260p\/q \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u221ap p \u0905\u092d\u093e\u091c\u094d\u092f<\/div><div style=\"font-size:15px;font-weight:bold;\">R=Q+\u0905\u092a\u0930\u093f\u092e\u0947\u092f N\u2282W\u2282Z\u2282Q\u2282R<\/div><div style=\"font-size:14px;margin-top:4px;\">\u0938\u092e\u093e\u092a\u094d\u0924 \u21d4 q=2\u207f5\u1d50 \u0935\u0930\u0928\u093e \u0906\u0935\u0930\u094d\u0924\u0940<\/div><div style=\"font-size:14px;\">\u0906\u0935\u0930\u094d\u0924\u0940 p\/q : 0.333..=1\/3 0.666..=2\/3 0.999..=1<\/div><div style=\"font-size:13px;margin-top:4px;background:#fff;padding:3px;border-radius:5px;\">\u0905\u092d\u093e\u091c\u094d\u092f 2 \u0939\u0940 \u0938\u092e 1 \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f<\/div><\/div>\u092f\u0939\u0940 \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930 \u092e\u0947\u0902 \u092a\u0915\u094d\u0915\u093e \u092a\u0942\u091b\u0924\u0947 \u0939\u0948\u0902\u0964 \u0939\u0930 2\u207f5\u1d50 \u0938\u092c\u0938\u0947 \u091c\u094d\u092f\u093e\u0926\u093e\u0964<\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"ch3g2p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"ch3g2i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-size:15px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"ch3g2n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('ch3g2-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('ch3g2i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('ch3g2n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('ch3g2p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-941-2\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-941-2-0\"  class=\"panel-grid-cell\" ><div id=\"panel-941-2-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"2\" ><div class=\"textwidget custom-html-widget\"><div id=\"FILE3-CH3-GANIT-FINAL\" style=\"max-width:860px;margin:12px auto;border:3.5px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:8px 10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:6px 18px;border-radius:20px;font-weight:bold;font-size:11px;\">Ch3 - \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 - \u092d\u093e\u0917 3\/4 - \u0905\u092d\u094d\u092f\u093e\u0938 3.2 \u0938\u0947 \u0905\u0902\u0924\u093f\u092e - REVISED \u0915\u094b\u0908 \u0928\u0939\u0940\u0902 \u091b\u0942\u091f\u093e<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"ch3g3f-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - 3.2 \u0938\u092d\u0940 \u092a\u094d\u0930\u0936\u094d\u0928 - \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0938\u093f\u0926\u094d\u0927 - Detail<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;\">3.2 Q1 - \u221a2 \u0905\u092a\u0930\u093f\u092e\u0947\u092f - \u092a\u0942\u0930\u093e Step - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q1. \u0938\u093f\u0926\u094d\u0927 \u221a2 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0939\u0948\u0964<\/b><br>\u092e\u093e\u0928\u094b \u221a2 = p\/q \u0938\u0939\u0905\u092d\u093e\u091c\u094d\u092f q\u22600\u0964<br>2 = p\u00b2\/q\u00b2 => p\u00b2=2q\u00b2 => p\u00b2 \u0938\u092e => p \u0938\u092e =2k<br>4k\u00b2=2q\u00b2 =>2k\u00b2=q\u00b2 => q\u00b2 \u0938\u092e => q \u0938\u092e => p,q \u0926\u094b\u0928\u094b\u0902 \u0938\u092e => \u0938\u0939\u0905\u092d\u093e\u091c\u094d\u092f \u0928\u0939\u0940\u0902 \u0935\u093f\u0930\u094b\u0927\u0964 \u0907\u0938\u0932\u093f\u090f \u221a2 \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<br><br><b>Q2. \u221a3 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u094b\u0964<\/b><br>\u092e\u093e\u0928\u094b \u221a3=p\/q p\u00b2=3q\u00b2 => p\u00b2 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f => p 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f p=3k =>9k\u00b2=3q\u00b2 =>3k\u00b2=q\u00b2 => q 3 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f => \u0935\u093f\u0930\u094b\u0927\u0964 \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<br><br><b>Q3. \u221a5 \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<\/b> \u0935\u0939\u0940 \u0935\u093f\u0927\u093f 5 \u0938\u0947 \u0935\u093f\u092d\u093e\u091c\u094d\u092f\u0964<br><br><b>Q4. \u092a\u094d\u0930\u092e\u0947\u092f \u0932\u093f\u0916\u094b: p \u0905\u092d\u093e\u091c\u094d\u092f => \u221ap \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<\/b><br>\u0909\u092a\u0930\u094b\u0915\u094d\u0924 \u0924\u0940\u0928\u094b\u0902 \u0938\u0947 \u0938\u093f\u0926\u094d\u0927\u0964 \u221a2,\u221a3,\u221a5,\u221a7,\u221a11,\u221a13 \u0938\u092d\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">Q1-Q4 3.2 \u0916\u0924\u094d\u092e - \u0915\u094b\u0908 \u0928\u0939\u0940\u0902 \u091b\u0942\u091f\u093e<\/span><br><span>\u221ap p \u0905\u092d\u093e\u091c\u094d\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/span><\/div><div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:6px 10px;font-weight:bold;margin-top:8px;\">3.2 Q5 - \u092a\u0930\u093f\u092e\u0947\u092f+\u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0914\u0930 \u0917\u0941\u0923\u093e - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q5. \u0926\u093f\u0916\u093e\u0913 \u092a\u0930\u093f\u092e\u0947\u092f+\u0905\u092a\u0930\u093f\u092e\u0947\u092f=\u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<\/b><br>\u092e\u093e\u0928\u094b 3+\u221a2 \u092a\u0930\u093f\u092e\u0947\u092f =r \u0924\u094b \u221a2 = r-3 \u092a\u0930\u093f\u092e\u0947\u092f-\u092a\u0930\u093f\u092e\u0947\u092f=\u092a\u0930\u093f\u092e\u0947\u092f \u0935\u093f\u0930\u094b\u0927\u0964 \u0907\u0938\u0932\u093f\u090f \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<br><br><b>Q6. \u092a\u0930\u093f\u092e\u0947\u092f\u00d7\u0905\u092a\u0930\u093f\u092e\u0947\u092f=\u0905\u092a\u0930\u093f\u092e\u0947\u092f (0 \u091b\u094b\u0921\u093c)\u0964<\/b><br>\u092e\u093e\u0928\u094b 2\u221a2 \u092a\u0930\u093f\u092e\u0947\u092f =r \u0924\u094b \u221a2=r\/2 \u092a\u0930\u093f\u092e\u0947\u092f \u0935\u093f\u0930\u094b\u0927\u0964 \u0907\u0938\u0932\u093f\u090f \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964 0\u00d7\u221a2=0 \u092a\u0930\u093f\u092e\u0947\u092f \u0907\u0938\u0932\u093f\u090f 0 \u091b\u094b\u0921\u093c\u0964<br><br><b>Q7. \u221a2+\u221a3 \u0905\u092a\u0930\u093f\u092e\u0947\u092f? \u0939\u093e\u0901\u0964 \u0926\u094b \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0915\u093e \u092f\u094b\u0917 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0939\u094b \u0938\u0915\u0924\u093e \u0939\u0948 \u092a\u0930 \u0939\u092e\u0947\u0936\u093e \u0928\u0939\u0940\u0902\u0964 \u221a2+(-\u221a2)=0 \u092a\u0930\u093f\u092e\u0947\u092f \u092c\u0928 \u091c\u093e\u0924\u093e \u0939\u0948\u0964 \u0907\u0938\u0932\u093f\u090f \u0926\u094b \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0915\u093e \u092f\u094b\u0917 \u0915\u092d\u0940 \u092a\u0930\u093f\u092e\u0947\u092f \u0915\u092d\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<\/b><\/div><\/div>\n\n<div id=\"ch3g3f-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - 3.3 \u0926\u0936\u092e\u0932\u0935 - \u0938\u092d\u0940 \u092a\u094d\u0930\u0936\u094d\u0928 - Detail<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;\">3.3 Q1 - 1\/4 2\/3 3\/7 4\/11 \u0938\u092d\u0940 \u0926\u0936\u092e\u0932\u0935 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q1. \u0926\u0936\u092e\u0932\u0935 \u0928\u093f\u0915\u093e\u0932\u094b \u0914\u0930 \u092c\u0924\u093e\u0913 \u0938\u092e\u093e\u092a\u094d\u0924\/\u0906\u0935\u0930\u094d\u0924\u0940\u0964<\/b><br>(i) 1\/4=0.25 \u0938\u092e\u093e\u092a\u094d\u0924 \u0939\u0930 4=2\u00b2 2\u207f5\u1d50<br>(ii) 2\/3=0.666...=0.(6) \u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u0930 3 \u092e\u0947\u0902 3<br>(iii) 3\/7=0.428571428571...=0.(428571) \u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u0930 7<br>(iv) 4\/11=0.363636...=0.(36) \u0906\u0935\u0930\u094d\u0924\u0940 \u0939\u0930 11<br>(v) 13\/25=0.52 \u0938\u092e\u093e\u092a\u094d\u0924 25=5\u00b2<br>(vi) 7\/8=0.875 \u0938\u092e\u093e\u092a\u094d\u0924 8=2\u00b3<br>(vii) 15\/16=0.9375 \u0938\u092e\u093e\u092a\u094d\u0924 16=2\u2074<br>(viii) 17\/12=1.41666...=1.41(6) \u0906\u0935\u0930\u094d\u0924\u0940 12=2\u00b2\u00d73<br><br><div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:6px 10px;font-weight:bold;\">3.3 Q2 - \u0906\u0935\u0930\u094d\u0924\u0940 \u0915\u094b p\/q - \u0938\u092d\u0940 \u092a\u094d\u0930\u0936\u094d\u0928 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q2. 0.666... p\/q?<\/b> x=0.666... 10x=6.666... 9x=6 x=2\/3<br><b>Q3. 0.999... p\/q?<\/b> x=0.999... 10x=9.999... 9x=9 x=1 => 0.999...=1<br><b>Q4. 0.353535... p\/q?<\/b> x=0.3535... 100x=35.3535... 99x=35 x=35\/99<br><b>Q5. 0.1636363... =0.16(36)?<\/b> x=0.163636... 10x=1.63636... 1000x=163.636... 990x=162 =>x=162\/990=9\/55 Detail Step<br><b>Q6. 0.1666... =0.1(6)?<\/b> x=0.1666... 10x=1.666... 100x=16.666... 90x=15 x=1\/6<br><b>Q7. 0.142857142857... =?<\/b> 142857\/999999=1\/7<br><b>Q8. 2.357357... =2.(357) =?<\/b> x=2.357357... 1000x=2357.357... 999x=2355 x=2355\/999=785\/333<br><div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">3.3 \u0916\u0924\u094d\u092e - \u0915\u094b\u0908 \u0928\u0939\u0940\u0902 \u091b\u0942\u091f\u093e - Rule q=2\u207f5\u1d50 \u0938\u092e\u093e\u092a\u094d\u0924<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3g3f-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - 3.4 \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e - \u0938\u092d\u0940 \u092a\u094d\u0930\u0936\u094d\u0928 - Detail<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:6px 10px;font-weight:bold;\">3.4 Q1 - \u221a2 \u221a3 \u221a5 \u0930\u0947\u0916\u093e \u092a\u0930 - \u0938\u092d\u0940 Step - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q1. \u221a2 \u0930\u0947\u0916\u093e \u092a\u0930\u0964<\/b> Step 0-1 1 \u092a\u0930 \u0932\u092e\u094d\u092c1 \u0915\u0930\u094d\u0923\u221a2 \u0915\u0902\u092a\u093e\u0938 \u0938\u0947 \u0915\u093e\u091f\u094b \u092c\u093f\u0902\u0926\u0941 \u221a2\u0964<br><b>Q2. \u221a3 \u0930\u0947\u0916\u093e \u092a\u0930\u0964<\/b> \u221a2 \u092e\u093f\u0932 \u0917\u092f\u093e \u221a2 \u092a\u0930 \u0932\u092e\u094d\u092c1 \u0915\u0930\u094d\u0923 \u221a(\u221a2\u00b2+1)=\u221a3 \u0915\u093e\u091f\u094b\u0964<br><b>Q3. \u221a5 \u0930\u0947\u0916\u093e \u092a\u0930 \u0938\u0930\u094d\u092a\u093f\u0932\u0964<\/b> \u221a4=2 \u092a\u0930 \u0932\u092e\u094d\u092c1 \u0915\u0930\u094d\u0923 \u221a5\u0964 \u0938\u0930\u094d\u092a\u093f\u0932: 1,1=>\u221a2, \u221a2,1=>\u221a3, \u221a3,1=>\u221a4=2, 2,1=>\u221a5, \u221a5,1=>\u221a6...<br><br><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;margin-top:8px;\">3.4 Q4 - \u092c\u0940\u091a \u092e\u0947\u0902 \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f - \u0938\u092d\u0940 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q4. 1 \u0914\u0930 2 \u092c\u0940\u091a 5 \u092a\u0930\u093f\u092e\u0947\u092f\u0964<\/b> 1.1,1.2,1.3,1.4,1.5 \u0938\u092d\u0940 \u092a\u0930\u093f\u092e\u0947\u092f\u0964 \u092b\u0949\u0930\u094d\u092e\u0942\u0932\u093e (1+2)\/2=1.5 \u092b\u093f\u0930 1.25 etc \u0905\u0928\u0902\u0924\u0964<br><b>Q5. 2 \u0914\u0930 3 \u092c\u0940\u091a 3 \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<\/b> \u221a5=2.236...,\u221a6=2.449...,\u221a7=2.645...,\u221a8=2.828... \u0938\u092d\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f 2 \u0914\u0930 3 \u092c\u0940\u091a\u0964<br><b>Q6. 0.2 \u0914\u0930 0.3 \u092c\u0940\u091a \u0905\u092a\u0930\u093f\u092e\u0947\u092f?<\/b> 0.20110111011110... 0.221221222... 0.252252225... \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u092c\u0928\u093e\u0913 \u092c\u0940\u091a \u092e\u0947\u0902\u0964<br><b>Q7. \u221a2 \u0914\u0930 \u221a3 \u092c\u0940\u091a \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f?<\/b> \u221a2=1.414 \u221a3=1.732 \u092c\u0940\u091a \u092a\u0930\u093f\u092e\u0947\u092f 1.5=3\/2 \u0905\u092a\u0930\u093f\u092e\u0947\u092f 1.505505550...<br><b>Q8. \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u092a\u0930 2.26(47) \u0924\u0915 \u0926\u0947\u0916\u094b\u0964<\/b> 2.26 \u092b\u093f\u0930 2.2647... \u0906\u0935\u0930\u094d\u0924\u0940 \u092c\u093f\u0902\u0926\u0941 \u0915\u0947 \u092a\u093e\u0938\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">3.4 \u0916\u0924\u094d\u092e - \u0915\u094b\u0908 \u0928\u0939\u0940\u0902 \u091b\u0942\u091f\u093e - \u092c\u0940\u091a \u092e\u0947\u0902 \u0905\u0928\u0902\u0924 \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u0928\u0902\u0924 \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3g3f-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - 3.5 \u0905\u0902\u0924\u093f\u092e \u0924\u0915 - \u0938\u092d\u0940 \u092a\u094d\u0930\u0936\u094d\u0928 - Detail<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#f3e5f5;border-left:4px solid #6a1b9a;padding:6px 10px;font-weight:bold;\">3.5 Q1 - \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u092a\u0939\u091a\u093e\u0928 \u0905\u0902\u0924\u093f\u092e - \u0938\u092d\u0940 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q1. \u092c\u0924\u093e\u0913 \u092a\u0930\u093f\u092e\u0947\u092f \u092f\u093e \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0905\u0902\u0924\u093f\u092e \u0924\u0915\u0964<\/b><br>(i) \u221a4=2 \u092a\u0930\u093f\u092e\u0947\u092f 2=2\/1<br>(ii) 3\u221a18=3\u00d73\u221a2=9\u221a2 \u0905\u092a\u0930\u093f\u092e\u0947\u092f<br>(iii) \u221a1.44=1.2=12\/10 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u092e\u093e\u092a\u094d\u0924<br>(iv) \u221a9\/\u221a27=3\/3\u221a3=1\/\u221a3 \u0905\u092a\u0930\u093f\u092e\u0947\u092f<br>(v) 0.3796 \u0938\u092e\u093e\u092a\u094d\u0924 3796\/10000 \u092a\u0930\u093f\u092e\u0947\u092f<br>(vi) 7.478478...=7.(478) \u0906\u0935\u0930\u094d\u0924\u0940 \u092a\u0930\u093f\u092e\u0947\u092f<br>(vii) 1.101001000100001... \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f<br>(viii) \u221a2+21 \u092a\u0930\u093f\u092e\u0947\u092f+\u0905\u092a\u0930\u093f\u092e\u0947\u092f=\u0905\u092a\u0930\u093f\u092e\u0947\u092f<br>(ix) \u03c0=3.14159... \u0905\u092a\u0930\u093f\u092e\u0947\u092f<br>(x) 0.10110111011110... \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940<br>(xi) \u221a2\u00d7\u221a3=\u221a6 \u0905\u092a\u0930\u093f\u092e\u0947\u092f<br>(xii) (\u221a2+\u221a3)(\u221a2-\u221a3)= -1 \u092a\u0930\u093f\u092e\u0947\u092f<br><br><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;\">3.5 Q2 - \u0915\u0930\u0923\u0940 \u091c\u094b\u0921\u093c \u0918\u091f\u093e\u0935 \u0917\u0941\u0923\u093e \u0905\u0902\u0924\u093f\u092e - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q2. \u0938\u0930\u0932 \u0915\u0930\u094b \u0905\u0902\u0924\u093f\u092e \u0924\u0915\u0964<\/b><br>(i) \u221a48=\u221a16\u00d73=4\u221a3<br>(ii) \u221a32+\u221a8=4\u221a2+2\u221a2=6\u221a2<br>(iii) \u221a45-3\u221a20+4\u221a5=3\u221a5-6\u221a5+4\u221a5=\u221a5<br>(iv) 2\u221a3\u00d75\u221a6=10\u221a18=10\u00d73\u221a2=30\u221a2<br>(v) 5\u221a3\u00d73\u221a2=15\u221a6<br>(vi) (3+\u221a3)(2+\u221a2)=6+3\u221a2+2\u221a3+\u221a6<br>(vii) (\u221a5+\u221a2)\u00b2=5+2+2\u221a10=7+2\u221a10<br>(viii) (\u221a5-\u221a2)(\u221a5+\u221a2)=5-2=3 \u092a\u0930\u093f\u092e\u0947\u092f<br><br><div style=\"background:#e0f2f1;border-left:4px solid #009688;padding:6px 10px;font-weight:bold;\">3.5 Q3 - \u0939\u0930 \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0905\u0902\u0924\u093f\u092e \u092a\u094d\u0930\u0936\u094d\u0928 - \u0939\u0947\u0921\u093f\u0902\u0917 - \u092c\u093f\u0932\u0915\u0941\u0932 \u0905\u0902\u0924\u093f\u092e<\/div><b>Q3. \u0939\u0930 \u0915\u093e \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u094b \u092c\u093f\u0932\u0915\u0941\u0932 \u0905\u0902\u0924\u093f\u092e \u0924\u0915\u0964<\/b><br>(i) 1\/\u221a2=\u221a2\/2<br>(ii) 1\/(2+\u221a3)=2-\u221a3 \u090a\u092a\u0930 \u0928\u0940\u091a\u0947 2-\u221a3<br>(iii) 1\/(\u221a5-\u221a3)=(\u221a5+\u221a3)\/2<br>(iv) 3\/(\u221a5+\u221a2)= (\u221a5-\u221a2) \u090a\u092a\u0930 \u0928\u0940\u091a\u0947 \u0917\u0941\u0923\u093e =3(\u221a5-\u221a2)\/3=\u221a5-\u221a2<br>(v) 5\/(3\u221a2)=5\u221a2\/6<br>(vi) (\u221a3+1)\/(\u221a3-1)= (\u221a3+1)\u00b2\/2=(4+2\u221a3)\/2=2+\u221a3<br><br><b>Q4. 1\/(\u221a7-\u221a6) - 1\/(\u221a6-\u221a5) +... \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u094b =5\u0964<\/b><br>\u0939\u0930 \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923: \u221a7+\u221a6 - (\u221a6+\u221a5)+ (\u221a5+\u221a4) -... =\u221a7+\u221a6-\u221a6-\u221a5+\u221a5+\u221a4... =5<br>Detail: \u0938\u092d\u0940 \u0915\u091f \u091c\u093e\u0924\u0947 \u0939\u0948\u0902 \u0905\u0902\u0924 \u092e\u0947\u0902 5 \u092c\u091a\u0924\u093e \u0939\u0948\u0964 \u092f\u0939\u0940 \u0905\u0902\u0924\u093f\u092e \u092a\u094d\u0930\u0936\u094d\u0928 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-weight:bold;\">3.5 \u0905\u0902\u0924\u093f\u092e \u0924\u0915 \u0916\u0924\u094d\u092e - \u0915\u094b\u0908 \u092a\u094d\u0930\u0936\u094d\u0928 \u0928\u0939\u0940\u0902 \u091b\u0942\u091f\u093e \u0905\u092c!<\/div><div>\u092a\u0939\u091a\u093e\u0928 \u221a4 \u092a\u0930\u093f\u092e\u0947\u092f \u221ap \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0906\u0935\u0930\u094d\u0924\u0940 \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><div>\u0915\u0930\u0923\u0940 \u221a48=4\u221a3 \u091c\u094b\u0921\u093c \u0918\u091f\u093e\u0935 \u0917\u0941\u0923\u093e (a+b)(a-b)=a\u00b2-b\u00b2<\/div><div>\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 1\/\u221a2=\u221a2\/2 conjugate \u0917\u0941\u0923\u093e<\/div><div>\u0905\u0902\u0924\u093f\u092e Q 1\/(\u221a7-\u221a6)...=5 Trick<\/div><\/div><div style=\"background:#ffebee;border:2px solid #d32f2f;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">REVISED - 3.2 \u0938\u0947 \u092c\u093f\u0932\u0915\u0941\u0932 \u0905\u0902\u0924\u093f\u092e \u0924\u0915 - \u0930\u094b\u091f\u0940 \u092a\u0915\u094d\u0915\u0940 \u0906\u092a\u0915\u0947 \u0938\u093e\u0925!<\/span><\/div><\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"ch3g3fp\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"ch3g3fi\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-size:15px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"ch3g3fn\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('ch3g3f-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('ch3g3fi').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('ch3g3fn').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('ch3g3fp').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-941-3\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-941-3-0\"  class=\"panel-grid-cell\" ><div id=\"panel-941-3-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"3\" ><div class=\"textwidget custom-html-widget\"><div id=\"FILE5-CH3-ANTIM-16Q\" style=\"max-width:860px;margin:12px auto;border:3.5px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:8px 10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:6px 18px;border-radius:20px;font-weight:bold;font-size:11px;\">Ch3 - \u0905\u0902\u0924\u093f\u092e \u092a\u094d\u0930\u0936\u094d\u0928\u093e\u0935\u0932\u0940 - 16 Q \u0905\u0932\u0917 \u0938\u0947<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"ch3a-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - Q1-Q4 - \u0905\u0902\u0924\u093f\u092e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;\">\u0905\u0902\u0924\u093f\u092e Q1-Q4 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q1. \u092a\u0930\u093f\u092e\u0947\u092f \u092f\u093e \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u092c\u0924\u093e\u0913\u0964<\/b><br>(i) \u221a23 = \u0905\u092a\u0930\u093f\u092e\u0947\u092f 23 \u0905\u092d\u093e\u091c\u094d\u092f<br>(ii) \u221a225 =15 \u092a\u0930\u093f\u092e\u0947\u092f<br>(iii) 0.3796 = \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u092e\u093e\u092a\u094d\u0924 3796\/10000<br>(iv) 7.478478... = 7.(478) \u0906\u0935\u0930\u094d\u0924\u0940 \u092a\u0930\u093f\u092e\u0947\u092f<br><br><b>Q2. \u0926\u094b \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0915\u093e \u092f\u094b\u0917 \u092a\u0930\u093f\u092e\u0947\u092f \u092c\u0928 \u0938\u0915\u0924\u093e? \u0909\u0926\u093e\u0939\u0930\u0923\u0964<\/b><br>\u0939\u093e\u0901\u0964 \u221a2 + (-\u221a2)=0 \u092a\u0930\u093f\u092e\u0947\u092f\u0964 \u221a3+ (2-\u221a3)=2 \u092a\u0930\u093f\u092e\u0947\u092f\u0964 \u0907\u0938\u0932\u093f\u090f \u0926\u094b \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0915\u093e \u092f\u094b\u0917 \u0915\u092d\u0940 \u092a\u0930\u093f\u092e\u0947\u092f \u0915\u092d\u0940 \u0905\u092a\u0930\u093f\u092e\u0947\u092f\u0964<br><br><b>Q3. \u0926\u094b \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0915\u093e \u0917\u0941\u0923\u093e \u092a\u0930\u093f\u092e\u0947\u092f \u092c\u0928 \u0938\u0915\u0924\u093e? \u0909\u0926\u093e\u0939\u0930\u0923\u0964<\/b><br>\u0939\u093e\u0901\u0964 \u221a2\u00d7\u221a2=2 \u092a\u0930\u093f\u092e\u0947\u092f\u0964 \u221a3\u00d7\u221a12=\u221a36=6 \u092a\u0930\u093f\u092e\u0947\u092f\u0964<br><br><b>Q4. 3 \u092a\u0930\u093f\u092e\u0947\u092f 3 \u0905\u092a\u0930\u093f\u092e\u0947\u092f 0.2 \u0914\u0930 0.3 \u092c\u0940\u091a\u0964<\/b><br>\u092a\u0930\u093f\u092e\u0947\u092f: 0.21, 0.22, 0.25 = 1\/4<br>\u0905\u092a\u0930\u093f\u092e\u0947\u092f: 0.201101110..., 0.212112111..., 0.241141114... \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940 \u092c\u0928\u093e\u0913\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">Q1-Q4 - \u221ap \u0905\u092d\u093e\u091c\u094d\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u221a225=15 \u092a\u0930\u093f\u092e\u0947\u092f<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3a-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - Q5-Q8 - \u0905\u0902\u0924\u093f\u092e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;\">\u0905\u0902\u0924\u093f\u092e Q5-Q8 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q5. 5 \u092a\u0930\u093f\u092e\u0947\u092f \u0938\u0902\u0916\u094d\u092f\u093e 3 \u0914\u0930 4 \u092c\u0940\u091a\u0964<\/b><br>3=3.0 4=4.0 \u092c\u0940\u091a 3.1,3.2,3.3,3.4,3.5 \u0938\u092d\u0940 \u092a\u0930\u093f\u092e\u0947\u092f\u0964 \u092f\u093e (3+4)\/2=3.5 \u092b\u093f\u0930 3.25 etc\u0964 \u0905\u0928\u0902\u0924\u0964<br><br><b>Q6. 0.999... =1 \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u094b\u0964<\/b><br>\u092e\u093e\u0928\u094b x=0.999... 10x=9.999... 10x-x=9 =>9x=9 =>x=1\u0964 \u0907\u0938\u0932\u093f\u090f 0.999...=1\u0964<br><br><b>Q7. 0.235...? \u0928\u0939\u0940\u0902 0.2353535...=0.23(53) \u0915\u094b p\/q \u092e\u0947\u0902\u0964<\/b><br>x=0.2353535... 10x=2.3535... 1000x=235.3535... 990x=233 =>x=233\/990<br><br><b>Q8. \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 1\/\u221a7\u0964<\/b><br>1\/\u221a7 = \u221a7\/7 \u090a\u092a\u0930 \u0928\u0940\u091a\u0947 \u221a7 \u0917\u0941\u0923\u093e\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">Q5-Q8 - (a+b)\/2 \u092e\u0927\u094d\u092f 0.999=1 0.23(53)=233\/990 1\/\u221a7=\u221a7\/7<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3a-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - Q9-Q12 - \u0905\u0902\u0924\u093f\u092e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:6px 10px;font-weight:bold;\">\u0905\u0902\u0924\u093f\u092e Q9-Q12 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q9. \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 1\/(\u221a7-\u221a6)\u0964<\/b><br>1\/(\u221a7-\u221a6)= (\u221a7+\u221a6)\/(7-6)=\u221a7+\u221a6<br><br><b>Q10. \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 1\/(\u221a5+\u221a2)\u0964<\/b><br>1\/(\u221a5+\u221a2)= (\u221a5-\u221a2)\/(5-2)= (\u221a5-\u221a2)\/3<br><br><b>Q11. \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 1\/(\u221a7-2)\u0964<\/b><br>1\/(\u221a7-2)= (\u221a7+2)\/(7-4)= (\u221a7+2)\/3<br><br><b>Q12. (\u221a11+\u221a7)(\u221a11-\u221a7)=?<\/b><br>(a+b)(a-b)=a\u00b2-b\u00b2=11-7=4 \u092a\u0930\u093f\u092e\u0947\u092f \u092c\u0928 \u0917\u092f\u093e\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">Q9-Q12 - conjugate \u0917\u0941\u0923\u093e (a+b)(a-b)=a\u00b2-b\u00b2<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3a-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - Q13-Q16 - \u092c\u093f\u0932\u0915\u0941\u0932 \u0905\u0902\u0924\u093f\u092e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#f3e5f5;border-left:4px solid #6a1b9a;padding:6px 10px;font-weight:bold;\">\u0905\u0902\u0924\u093f\u092e Q13-Q16 - \u092c\u093f\u0932\u0915\u0941\u0932 \u0905\u0902\u0924\u093f\u092e - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q13. \u221a2=1.4142 \u0924\u092c 2+\u221a2 , 3-\u221a2 , \u221a2\/2 \u0915\u093e \u092e\u093e\u0928\u0964<\/b><br>2+1.4142=3.4142<br>3-1.4142=1.5858<br>1.4142\/2=0.7071<br><br><b>Q14. \u221a3=1.732 \u0924\u092c 1\/\u221a3 , \u221a3\/2\u0964<\/b><br>1\/\u221a3=\u221a3\/3=1.732\/3=0.5773<br>\u221a3\/2=0.866<br><br><b>Q15. \u0926\u093f\u0916\u093e\u0913 1\/(\u221a7-\u221a6)-1\/(\u221a6-\u221a5)+1\/(\u221a5-\u221a4)-1\/(\u221a4-\u221a3)+1\/(\u221a3-1)=5<\/b><br>\u0939\u0930 \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923:<br>\u221a7+\u221a6 - (\u221a6+\u221a5) + \u221a5+\u221a4 - (\u221a4+\u221a3) + (\u221a3+1)\/2? \u0938\u0939\u0940 \u0915\u094d\u0930\u092e:<br>\u092a\u0939\u0932\u093e \u221a7+\u221a6<br>\u0926\u0942\u0938\u0930\u093e \u221a6+\u221a5 \u0918\u091f\u093e<br>\u0924\u0940\u0938\u0930\u093e \u221a5+2 \u091c\u094b\u0921\u093c<br>\u091a\u094c\u0925\u093e 2+\u221a3 \u0918\u091f\u093e<br>\u092a\u093e\u0902\u091a\u0935\u093e (\u221a3+1)\/2 \u091c\u094b\u0921\u093c... \u0905\u0902\u0924 \u092e\u0947\u0902 \u0938\u092d\u0940 \u0915\u091f\u0915\u0930 5\u0964 NCERT \u0905\u0902\u0924\u093f\u092e \u092a\u094d\u0930\u0936\u094d\u0928\u0964<br><br><b>Q16. \u0938\u0902\u0916\u094d\u092f\u093e \u0930\u0947\u0916\u093e \u092a\u0930 4.2626... \u0914\u0930 4.262... \u0926\u093f\u0916\u093e\u0913\u0964<\/b><br>4.2626..=4.(26) \u0906\u0935\u0930\u094d\u0924\u0940 4.26 \u0914\u0930 4.27 \u092c\u0940\u091a\u0964 \u0906\u0935\u0930\u094d\u0927\u0928 \u0938\u0947 \u0926\u0947\u0916\u094b\u0964 \u0930\u0947\u0916\u093e \u092a\u0930 4 \u0938\u0947 5 \u092c\u0940\u091a 10 \u092d\u093e\u0917 \u092b\u093f\u0930 4.2 \u0938\u0947 4.3 \u092c\u0940\u091a 10 \u092d\u093e\u0917 \u092b\u093f\u0930 4.26 \u0938\u0947 4.27\u0964<div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-weight:bold;\">Q13-Q16 \u092c\u093f\u0932\u0915\u0941\u0932 \u0905\u0902\u0924\u093f\u092e - \u0915\u094b\u0908 \u0928\u0939\u0940\u0902 \u091b\u0942\u091f\u093e<\/div><div>\u221a2=1.4142 \u221a3=1.732 \u092e\u093e\u0928<\/div><div>\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 conjugate<\/div><div>\u0905\u0902\u0924\u093f\u092e \u0938\u093f\u0926\u094d\u0927 5 \u0906\u0935\u0930\u094d\u0927\u0928<\/div><\/div><\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"ch3a5p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"ch3a5i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-size:15px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"ch3a5n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('ch3a-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('ch3a5i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('ch3a5n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('ch3a5p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><div id=\"pg-941-4\"  class=\"panel-grid panel-no-style\" ><div id=\"pgc-941-4-0\"  class=\"panel-grid-cell\" ><div id=\"panel-941-4-0-0\" class=\"widget_text so-panel widget widget_custom_html panel-first-child panel-last-child\" data-index=\"4\" ><div class=\"textwidget custom-html-widget\"><div id=\"FILE4-CH3-GANIT\" style=\"max-width:860px;margin:12px auto;border:3.5px solid #0f5e4b;border-radius:14px;background:#e6f4f1;font-family:'Noto Sans Devanagari',Arial,sans-serif;overflow:hidden;\">\n<div style=\"background:#0a5c4b;padding:8px 10px;text-align:center;\"><span style=\"background:#fff;color:#0a3d33;padding:6px 18px;border-radius:20px;font-weight:bold;font-size:11px;\">Ch3 - \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 - \u092d\u093e\u0917 4\/4 - MCQ \u0905\u0917\u094d\u0928\u093f\u0935\u0940\u0930<\/span><\/div>\n\n<div style=\"padding:10px 12px;\">\n<div id=\"ch3g4-1\" style=\"display:block;\"><div style=\"background:#d32f2f;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd34 \u092d\u093e\u0917 1\/4 - MCQ 1-8 - \u0938\u0902\u0916\u094d\u092f\u093e \u092a\u0930\u093f\u0935\u093e\u0930<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;\">\u092a\u0930\u093f\u0935\u093e\u0930 \u092a\u0939\u091a\u093e\u0928 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q1. \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094c\u0928?<\/b><br>A) 0 B) 1 C) -1 D) 1\/2<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) 1 N={1,2,3...}<\/span><br><br><b>Q2. \u092a\u0942\u0930\u094d\u0923 \u0938\u0902\u0916\u094d\u092f\u093e \u0915\u094c\u0928?<\/b><br>A) -1 B) 0 C) 1\/2 D) \u221a2<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) 0 W={0,1,2...}<\/span><br><br><b>Q3. \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u0915\u094c\u0928?<\/b><br>A) 0.5 B) -3 C) \u221a2 D) 1\/2<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) -3 Z={...-1,0,1...}<\/span><br><br><b>Q4. N\u2282W\u2282Z\u2282Q\u2282R \u0938\u0939\u0940?<\/b><br>A) \u0939\u093e\u0901 B) \u0928\u0939\u0940\u0902 C) \u0909\u0932\u094d\u091f\u093e D) \u0915\u094b\u0908 \u0928\u0939\u0940\u0902<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) \u0939\u093e\u0901<\/span><br><br><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u0905\u092d\u093e\u091c\u094d\u092f \u092d\u093e\u091c\u094d\u092f - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q5. \u0905\u092d\u093e\u091c\u094d\u092f \u0915\u094c\u0928?<\/b><br>A) 4 B) 9 C) 7 D) 1<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans C) 7 \u0926\u094b \u0917\u0941\u0923\u0928\u0916\u0902\u0921<\/span><br><br><b>Q6. \u092d\u093e\u091c\u094d\u092f \u0915\u094c\u0928?<\/b><br>A) 2 B) 3 C) 4 D) 5<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans C) 4 \u0917\u0941\u0923\u0928\u0916\u0902\u0921 1,2,4<\/span><br><br><b>Q7. 1 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>A) \u0905\u092d\u093e\u091c\u094d\u092f B) \u092d\u093e\u091c\u094d\u092f C) \u0926\u094b\u0928\u094b\u0902 D) \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans D) \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f<\/span><br><br><b>Q8. \u0938\u092e \u0905\u092d\u093e\u091c\u094d\u092f \u0915\u094c\u0928?<\/b><br>A) 2 B) 4 C) 6 D) 8<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) 2 \u0939\u0940 \u0938\u092e \u0905\u092d\u093e\u091c\u094d\u092f<\/span><div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">N={1,2..} W={0,1..} Z={..-1,0,1..} \u0905\u092d\u093e\u091c\u094d\u092f 2 \u0939\u0940 \u0938\u092e 1 \u0928 \u0905\u092d\u093e\u091c\u094d\u092f \u0928 \u092d\u093e\u091c\u094d\u092f<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3g4-2\" style=\"display:none;\"><div style=\"background:#1565c0;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udd35 \u092d\u093e\u0917 2\/4 - MCQ 9-16 - \u092a\u0930\u093f\u092e\u0947\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;\">\u092a\u0930\u093f\u092e\u0947\u092f \u092a\u0939\u091a\u093e\u0928 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q9. \u092a\u0930\u093f\u092e\u0947\u092f \u0915\u094c\u0928?<\/b><br>A) \u221a2 B) \u221a3 C) 1\/2 D) \u03c0<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans C) 1\/2 p\/q<\/span><br><br><b>Q10. \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0915\u094c\u0928?<\/b><br>A) 1\/2 B) \u221a2 C) 0 D) 3<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u221a2 \u2260p\/q<\/span><br><br><b>Q11. 0 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>A) \u092a\u0930\u093f\u092e\u0947\u092f B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f C) \u0926\u094b\u0928\u094b\u0902 D) \u0928\u0939\u0940\u0902<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) \u092a\u0930\u093f\u092e\u0947\u092f 0\/1<\/span><br><br><b>Q12. \u03c0 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>A) \u092a\u0930\u093f\u092e\u0947\u092f B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f C) \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 D) \u092a\u094d\u0930\u093e\u0915\u0943\u0924<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f 3.14159.. \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940<\/span><br><br><div style=\"background:#ffebee;border-left:4px solid #d32f2f;padding:6px 10px;font-weight:bold;margin-top:8px;\">\u221ap \u092a\u094d\u0930\u092e\u0947\u092f - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q13. \u221a4 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>A) \u0905\u092a\u0930\u093f\u092e\u0947\u092f B) \u092a\u0930\u093f\u092e\u0947\u092f C) \u0926\u094b\u0928\u094b\u0902 D) \u0928\u0939\u0940\u0902<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u092a\u0930\u093f\u092e\u0947\u092f \u221a4=2<\/span><br><br><b>Q14. \u221ap p \u0905\u092d\u093e\u091c\u094d\u092f \u0924\u094b?<\/b><br>A) \u092a\u0930\u093f\u092e\u0947\u092f B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f C) \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 D) \u092a\u094d\u0930\u093e\u0915\u0943\u0924<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/span><br><br><b>Q15. \u221a7 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>A) \u092a\u0930\u093f\u092e\u0947\u092f B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f C) \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 D) \u092a\u094d\u0930\u093e\u0915\u0943\u0924<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f 7 \u0905\u092d\u093e\u091c\u094d\u092f<\/span><br><br><b>Q16. \u221a9 \u0915\u094d\u092f\u093e \u0939\u0948?<\/b><br>A) \u0905\u092a\u0930\u093f\u092e\u0947\u092f B) \u092a\u0930\u093f\u092e\u0947\u092f C) \u0926\u094b\u0928\u094b\u0902 D) \u0928\u0939\u0940\u0902<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u092a\u0930\u093f\u092e\u0947\u092f \u221a9=3<\/span><div style=\"background:#fff9c4;border:2px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">Q=p\/q q\u22600 \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u2260p\/q \u221ap p \u0905\u092d\u093e\u091c\u094d\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3g4-3\" style=\"display:none;\"><div style=\"background:#2e7d32;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe2 \u092d\u093e\u0917 3\/4 - MCQ 17-24 - \u0926\u0936\u092e\u0932\u0935<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#e8f5e9;border-left:4px solid #2e7d32;padding:6px 10px;font-weight:bold;\">\u0926\u0936\u092e\u0932\u0935 \u092a\u094d\u0930\u0915\u093e\u0930 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q17. \u0938\u092e\u093e\u092a\u094d\u0924 \u0915\u094c\u0928?<\/b><br>A) 1\/3 B) 1\/2 C) 1\/7 D) 1\/11<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) 1\/2=0.5 \u0939\u0930 2\u00b9<\/span><br><br><b>Q18. \u0906\u0935\u0930\u094d\u0924\u0940 \u0915\u094c\u0928?<\/b><br>A) 1\/2 B) 1\/4 C) 1\/3 D) 1\/8<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans C) 1\/3=0.333.. \u0939\u0930 3<\/span><br><br><b>Q19. 1\/8 \u0926\u0936\u092e\u0932\u0935?<\/b><br>A) 0.125 B) 0.333 C) 0.142 D) 0.5<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) 0.125 \u0939\u0930 2\u00b3 \u0938\u092e\u093e\u092a\u094d\u0924<\/span><br><br><b>Q20. 7\/6 \u0926\u0936\u092e\u0932\u0935?<\/b><br>A) \u0938\u092e\u093e\u092a\u094d\u0924 B) \u0906\u0935\u0930\u094d\u0924\u0940 C) \u0905\u092a\u0930\u093f\u092e\u0947\u092f D) \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u0906\u0935\u0930\u094d\u0924\u0940 6=2\u00d73 \u092e\u0947\u0902 3<\/span><br><br><div style=\"background:#e3f2fd;border-left:4px solid #1565c0;padding:6px 10px;font-weight:bold;margin-top:8px;\">p\/q \u092c\u0926\u0932\u094b - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q21. 0.333.. =?<\/b><br>A) 1\/3 B) 2\/3 C) 1\/6 D) 1<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) 1\/3<\/span><br><br><b>Q22. 0.666.. =?<\/b><br>A) 1\/3 B) 2\/3 C) 1\/6 D) 1\/2<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) 2\/3<\/span><br><br><b>Q23. 0.999.. =?<\/b><br>A) 0 B) 1 C) 2 D) 0.9<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) 1<\/span><br><br><b>Q24. 0.1666.. =?<\/b><br>A) 1\/6 B) 1\/3 C) 2\/3 D) 1\/2<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) 1\/6<\/span><div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:6px;border-radius:8px;text-align:center;margin:6px 0;\"><span style=\"font-weight:bold;\">\u0938\u092e\u093e\u092a\u094d\u0924 \u21d4 q=2\u207f5\u1d50 \u0935\u0930\u0928\u093e \u0906\u0935\u0930\u094d\u0924\u0940 0.333=1\/3 0.666=2\/3 0.999=1 0.166=1\/6<\/span><\/div><\/div><\/div>\n\n<div id=\"ch3g4-4\" style=\"display:none;\"><div style=\"background:#6a1b9a;color:#fff;padding:6px 14px;border-radius:20px;font-size:13px;font-weight:bold;display:inline-block;margin-bottom:8px;\">\ud83d\udfe3 \u092d\u093e\u0917 4\/4 - MCQ 25-32 - \u0905\u0902\u0924\u093f\u092e<\/div><div style=\"background:#fff;border:1.5px solid #0f8a6d;border-radius:10px;padding:12px;font-size:13.5px;line-height:1.9;height:380px;overflow-y:auto;\"><div style=\"background:#f3e5f5;border-left:4px solid #6a1b9a;padding:6px 10px;font-weight:bold;\">\u092a\u0930\u093f\u0915\u0930\u094d\u092e \u0914\u0930 \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 - \u0939\u0947\u0921\u093f\u0902\u0917<\/div><b>Q25. \u092a\u0930\u093f\u092e\u0947\u092f+\u0905\u092a\u0930\u093f\u092e\u0947\u092f=?<\/b><br>A) \u092a\u0930\u093f\u092e\u0947\u092f B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f C) \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 D) \u092a\u094d\u0930\u093e\u0915\u0943\u0924<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/span><br><br><b>Q26. \u092a\u0930\u093f\u092e\u0947\u092f\u00d7\u0905\u092a\u0930\u093f\u092e\u0947\u092f (0 \u091b\u094b\u0921\u093c)=?<\/b><br>A) \u092a\u0930\u093f\u092e\u0947\u092f B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f C) 0 D) 1<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/span><br><br><b>Q27. \u221a2 \u0930\u0947\u0916\u093e \u092a\u0930 \u0915\u0948\u0938\u0947?<\/b><br>A) 1,1 \u0915\u0930\u094d\u0923 \u221a2 B) 2,2 C) 1,2 D) 2,1<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) 1,1 \u0938\u092e\u0915\u094b\u0923 \u0915\u0930\u094d\u0923 \u221a2<\/span><br><br><b>Q28. \u221a45=?<\/b><br>A) 3\u221a5 B) 5\u221a3 C) 9\u221a5 D) \u221a45<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) 3\u221a5 \u221a9\u00d75<\/span><br><br><b>Q29. (\u221a2+\u221a3)(\u221a2-\u221a3)=?<\/b><br>A) -1 B) 1 C) 5 D) 0<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) -1 2-3<\/span><br><br><b>Q30. 1\/\u221a2 =?<\/b><br>A) \u221a2\/2 B) \u221a2 C) 2\/\u221a2 D) 1<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans A) \u221a2\/2<\/span><br><br><b>Q31. 1\/(2+\u221a3)=?<\/b><br>A) 2+\u221a3 B) 2-\u221a3 C) \u221a3-2 D) 1<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) 2-\u221a3 conjugate<\/span><br><br><b>Q32. 0.10110111011110... \u0915\u094d\u092f\u093e?<\/b><br>A) \u092a\u0930\u093f\u092e\u0947\u092f B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f C) \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 D) \u092a\u094d\u0930\u093e\u0915\u0943\u0924<br><span style=\"background:#e8f5e9;padding:2px 8px;border-radius:5px;font-weight:bold;\">Ans B) \u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0905\u0928\u093e\u0935\u0930\u094d\u0924\u0940<\/span><div style=\"background:#fff9c4;border:3px solid #fbc02d;padding:8px;border-radius:10px;text-align:center;margin:6px 0;\"><div style=\"font-weight:bold;\">Trick \u0905\u0902\u0924\u093f\u092e<\/div><div>N\u2282W\u2282Z\u2282Q\u2282R Q=p\/q q\u22600 \u221ap p \u0905\u092d\u093e\u091c\u094d\u092f \u0905\u092a\u0930\u093f\u092e\u0947\u092f<\/div><div>\u0938\u092e\u093e\u092a\u094d\u0924 q=2\u207f5\u1d50 0.333=1\/3 0.999=1 1\/6<\/div><div>\u092a\u0930\u093f\u092e\u0947\u092f+\u0905\u092a\u0930\u093f\u092e\u0947\u092f=\u0905\u092a\u0930\u093f\u092e\u0947\u092f \u0917\u0941\u0923\u093e \u092d\u0940 0 \u091b\u094b\u0921\u093c<\/div><div>\u221a2 1,1 \u0915\u0930\u094d\u0923 \u221a45=3\u221a5 (a+b)(a-b)=a\u00b2-b\u00b2<\/div><div>\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 1\/\u221a2=\u221a2\/2<\/div><\/div><\/div><\/div>\n<\/div>\n\n<div style=\"background:#0a5c4b;padding:12px;display:flex;justify-content:space-between;align-items:center;\">\n<button type=\"button\" id=\"ch3g4p\" style=\"background:#ffcc00;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">&larr; \u092a\u0940\u091b\u0947<\/button>\n<span id=\"ch3g4i\" style=\"background:#fff;color:#0a5c4b;padding:8px 20px;border-radius:20px;font-size:15px;font-weight:bold;\">1 \/ 4<\/span>\n<button type=\"button\" id=\"ch3g4n\" style=\"background:#00e676;color:#000;border:3px solid #fff;border-radius:30px;padding:10px 22px;font-weight:bold;font-size:15px;cursor:pointer;min-width:100px;\">\u0906\u0917\u0947 &rarr;<\/button>\n<\/div>\n<\/div>\n\n<script>\n(function(){\nvar c=1,t=4;\nfunction s(n){\nfor(var i=1;i<=t;i++){\nvar el=document.getElementById('ch3g4-'+i);\nif(el)el.style.display=(i===n?'block':'none');\n}\ndocument.getElementById('ch3g4i').textContent=n+' \/ '+t;\nc=n;\n}\ndocument.getElementById('ch3g4n').onclick=function(){var n=c+1;if(n>t)n=1;s(n);};\ndocument.getElementById('ch3g4p').onclick=function(){var n=c-1;if(n<1)n=t;s(n);};\n})();\n<\/script><\/div><\/div><\/div><\/div><\/div>","protected":false},"excerpt":{"rendered":"<p>Ch3 &#8211; \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 &#8211; \u092d\u093e\u0917 1\/4 &#8211; \u092a\u0930\u093f\u091a\u092f \ud83d\udd34 \u092d\u093e\u0917 1\/4 &#8211; \u092a\u0930\u093f\u091a\u092f &#8211; \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915\u092a\u0930\u093f\u091a\u092f &#8211; \u0938\u0902\u0916\u094d\u092f\u093e\u090f\u0901 \u0915\u094d\u092f\u093e? &#8211; \u0939\u0947\u0921\u093f\u0902\u0917\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0938\u0947 \u0926\u0941\u0928\u093f\u092f\u093e \u091a\u0932\u0924\u0940 \u0939\u0948\u0964 \u0917\u093f\u0928\u0924\u0940 \u0938\u0947 \u0936\u0941\u0930\u0942 \u0939\u0941\u0908 \u0925\u0940\u0902\u0964 \u092b\u093f\u0930 \u0927\u0940\u0930\u0947 \u0927\u0940\u0930\u0947 \u0928\u090f \u092a\u094d\u0930\u0915\u093e\u0930 \u092c\u0928\u0947\u0964 \u092a\u094d\u0930\u093e\u0915\u0943\u0924 \u0938\u0947 \u092a\u0942\u0930\u094d\u0923 \u092b\u093f\u0930 \u092a\u0942\u0930\u094d\u0923\u093e\u0902\u0915 \u092b\u093f\u0930 \u092a\u0930\u093f\u092e\u0947\u092f\u0964\u092a\u094d\u0930\u093e\u0915\u0943\u0924 N : 1,2,3,4&#8230; \u0917\u093f\u0928\u0924\u0940 \u0935\u093e\u0932\u0940\u092a\u0942\u0930\u094d\u0923 W : 0,1,2,3&#8230; N + [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[10],"tags":[],"class_list":["post-941","post","type-post","status-publish","format-standard","hentry","category-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Mathematics &#187; *3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers &#187; Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.<\/title>\n<meta name=\"description\" content=\"Class 9 Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning, MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence &amp; Academic.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902-\u0915\u093e-\u0938\u0902\u0938\u093e\u0930-world-of-numbers\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Mathematics &#187; *3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers &#187; Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.\" \/>\n<meta property=\"og:description\" content=\"Class 9 Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning, MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence &amp; Academic.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902-\u0915\u093e-\u0938\u0902\u0938\u093e\u0930-world-of-numbers\/\" \/>\n<meta property=\"og:site_name\" content=\"Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.\" \/>\n<meta property=\"article:published_time\" content=\"2026-10-05T02:57:26+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2026-10-05T03:17:07+00:00\" \/>\n<meta name=\"author\" content=\"admin\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"admin\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/\"},\"author\":{\"name\":\"admin\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#\\\/schema\\\/person\\\/d4ecf71bc91964603da3f31c3b5d0715\"},\"headline\":\"*3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers\",\"datePublished\":\"2026-10-05T02:57:26+00:00\",\"dateModified\":\"2026-10-05T03:17:07+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/\"},\"wordCount\":48,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#organization\"},\"articleSection\":[\"Mathematics\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/\",\"url\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/\",\"name\":\"Mathematics &#187; *3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers &#187; Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#website\"},\"datePublished\":\"2026-10-05T02:57:26+00:00\",\"dateModified\":\"2026-10-05T03:17:07+00:00\",\"description\":\"Class 9 Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning, MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence & Academic.\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/2026\\\/10\\\/05\\\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"*3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#website\",\"url\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/\",\"name\":\"Class-6th - Agniveer Foundation | Pargai Defence & Academic.\",\"description\":\"Class 9th Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning,  MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence &amp; Academic.\",\"publisher\":{\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#organization\"},\"alternateName\":\"Agniveer Foundation Class-6th\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#organization\",\"name\":\"Pargai Defence & Academic.\",\"alternateName\":\"Pargai Institute of Defence Studies.\",\"url\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/wp-content\\\/uploads\\\/2026\\\/07\\\/cropped-channels4_profile.jpg\",\"contentUrl\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/wp-content\\\/uploads\\\/2026\\\/07\\\/cropped-channels4_profile.jpg\",\"width\":512,\"height\":512,\"caption\":\"Pargai Defence & Academic.\"},\"image\":{\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#\\\/schema\\\/logo\\\/image\\\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/#\\\/schema\\\/person\\\/d4ecf71bc91964603da3f31c3b5d0715\",\"name\":\"admin\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/d74d2ed3f17c48067bcb3273bb8f4de11563c441bc3d2855fbb8e5aea27f629f?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/d74d2ed3f17c48067bcb3273bb8f4de11563c441bc3d2855fbb8e5aea27f629f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/d74d2ed3f17c48067bcb3273bb8f4de11563c441bc3d2855fbb8e5aea27f629f?s=96&d=mm&r=g\",\"caption\":\"admin\"},\"sameAs\":[\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\"],\"url\":\"https:\\\/\\\/academic.pargaidefenceacademy.in\\\/Class-9th\\\/author\\\/admin\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Mathematics &#187; *3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers &#187; Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.","description":"Class 9 Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning, MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence & Academic.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902-\u0915\u093e-\u0938\u0902\u0938\u093e\u0930-world-of-numbers\/","og_locale":"en_US","og_type":"article","og_title":"Mathematics &#187; *3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers &#187; Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.","og_description":"Class 9 Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning, MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence & Academic.","og_url":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-\u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902-\u0915\u093e-\u0938\u0902\u0938\u093e\u0930-world-of-numbers\/","og_site_name":"Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.","article_published_time":"2026-10-05T02:57:26+00:00","article_modified_time":"2026-10-05T03:17:07+00:00","author":"admin","twitter_card":"summary_large_image","twitter_misc":{"Written by":"admin","Est. reading time":"7 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/#article","isPartOf":{"@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/"},"author":{"name":"admin","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#\/schema\/person\/d4ecf71bc91964603da3f31c3b5d0715"},"headline":"*3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers","datePublished":"2026-10-05T02:57:26+00:00","dateModified":"2026-10-05T03:17:07+00:00","mainEntityOfPage":{"@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/"},"wordCount":48,"commentCount":0,"publisher":{"@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#organization"},"articleSection":["Mathematics"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/","url":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/","name":"Mathematics &#187; *3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers &#187; Class-9th - Agniveer Foundation Hub | Pargai Defence &amp; Academic.","isPartOf":{"@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#website"},"datePublished":"2026-10-05T02:57:26+00:00","dateModified":"2026-10-05T03:17:07+00:00","description":"Class 9 Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning, MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence & Academic.","breadcrumb":{"@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/2026\/10\/05\/3-%e0%a4%b8%e0%a4%82%e0%a4%96%e0%a5%8d%e0%a4%af%e0%a4%be%e0%a4%93%e0%a4%82-%e0%a4%95%e0%a4%be-%e0%a4%b8%e0%a4%82%e0%a4%b8%e0%a4%be%e0%a4%b0-world-of-numbers\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/"},{"@type":"ListItem","position":2,"name":"*3* | \u0938\u0902\u0916\u094d\u092f\u093e\u0913\u0902 \u0915\u093e \u0938\u0902\u0938\u093e\u0930 | World of Numbers"}]},{"@type":"WebSite","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#website","url":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/","name":"Class-6th - Agniveer Foundation | Pargai Defence & Academic.","description":"Class 9th Online Classes in Bageshwar with NCERT Textbook based study material, chapter-wise learning,  MCQs, assignments and Agniveer Foundation. Complete online learning portal by Pargai Defence &amp; Academic.","publisher":{"@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#organization"},"alternateName":"Agniveer Foundation Class-6th","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#organization","name":"Pargai Defence & Academic.","alternateName":"Pargai Institute of Defence Studies.","url":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#\/schema\/logo\/image\/","url":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-content\/uploads\/2026\/07\/cropped-channels4_profile.jpg","contentUrl":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-content\/uploads\/2026\/07\/cropped-channels4_profile.jpg","width":512,"height":512,"caption":"Pargai Defence & Academic."},"image":{"@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/#\/schema\/person\/d4ecf71bc91964603da3f31c3b5d0715","name":"admin","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/d74d2ed3f17c48067bcb3273bb8f4de11563c441bc3d2855fbb8e5aea27f629f?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/d74d2ed3f17c48067bcb3273bb8f4de11563c441bc3d2855fbb8e5aea27f629f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d74d2ed3f17c48067bcb3273bb8f4de11563c441bc3d2855fbb8e5aea27f629f?s=96&d=mm&r=g","caption":"admin"},"sameAs":["https:\/\/academic.pargaidefenceacademy.in\/Class-9th"],"url":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/author\/admin\/"}]}},"_links":{"self":[{"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/posts\/941","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/comments?post=941"}],"version-history":[{"count":7,"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/posts\/941\/revisions"}],"predecessor-version":[{"id":948,"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/posts\/941\/revisions\/948"}],"wp:attachment":[{"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/media?parent=941"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/categories?post=941"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/academic.pargaidefenceacademy.in\/Class-9th\/wp-json\/wp\/v2\/tags?post=941"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}