{"id":5100,"date":"2025-06-04T08:26:26","date_gmt":"2025-06-04T08:26:26","guid":{"rendered":"https:\/\/diznr.com\/?p=5100"},"modified":"2025-06-04T08:26:26","modified_gmt":"2025-06-04T08:26:26","slug":"permutation-and-combination-trigonometry-triangle-circle-mathematical-induction","status":"publish","type":"post","link":"https:\/\/www.reilsolar.com\/pdf\/permutation-and-combination-trigonometry-triangle-circle-mathematical-induction\/","title":{"rendered":"Permutation and Combination Trigonometry Triangle Circle Mathematical Induction"},"content":{"rendered":"
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\u092f\u0939\u093e\u0901 \u0906\u092a\u0915\u0947 \u0926\u094d\u0935\u093e\u0930\u093e \u092a\u0942\u091b\u0947 \u0917\u090f \u0938\u092d\u0940 \u091f\u0949\u092a\u093f\u0915 \u2014 Permutation and Combination, Trigonometry, Triangle, Circle, \u0914\u0930 Mathematical Induction<\/strong> \u2014 \u0915\u0947 \u0932\u093f\u090f \u090f\u0915\u0926\u092e \u0906\u0938\u093e\u0928, \u0936\u0949\u0930\u094d\u091f \u091f\u094d\u0930\u093f\u0915 \u0915\u0947 \u0938\u093e\u0925 \u0938\u0902\u0915\u094d\u0937\u093f\u092a\u094d\u0924 \u0928\u094b\u091f\u094d\u0938 \u0939\u093f\u0902\u0926\u0940 \u092e\u0947\u0902 \u0926\u093f\u090f \u091c\u093e \u0930\u0939\u0947 \u0939\u0948\u0902\u0964 \u092f\u0947 \u0916\u093e\u0938\u0924\u094c\u0930 \u092a\u0930 \u0917\u0947\u091f (GATE), JEE, NDA, SSC, UPSC, \u092c\u094b\u0930\u094d\u0921 \u092a\u0930\u0940\u0915\u094d\u0937\u093e<\/strong> \u0906\u0926\u093f \u0915\u0940 \u0924\u0948\u092f\u093e\u0930\u0940 \u0915\u0930\u0928\u0947 \u0935\u093e\u0932\u0947 \u091b\u093e\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0932\u093f\u090f \u0909\u092a\u092f\u094b\u0917\u0940 \u0939\u0948\u0902\u0964<\/p>\n

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\ud83e\uddee 1. Permutation and Combination (\u0938\u0902\u091a\u092f \u0914\u0930 \u0938\u0902\u091a\u092f\u0928)<\/strong><\/h2>\n

\ud83d\udd39 Permutation (\u0915\u094d\u0930\u092e\u091a\u092f)<\/strong> \u2013 \u091c\u092c \u0915\u094d\u0930\u092e (order)<\/strong> \u092e\u0939\u0924\u094d\u0935\u092a\u0942\u0930\u094d\u0923 \u0939\u094b\u0924\u093e \u0939\u0948\u0964<\/h3>\n

nPr=n!(n\u2212r)!^nP_r = \\frac{n!}{(n – r)!}<\/span>\ud83d\udd38 Trick:<\/strong> \u092f\u0926\u093f \u0938\u0940\u091f\u094b\u0902 \u092a\u0930 \u0932\u094b\u0917 \u092c\u0948\u0920 \u0930\u0939\u0947 \u0939\u094b\u0902, \u0930\u0902\u0917\u094b\u0902 \u0915\u094b \u0905\u0932\u0917-\u0905\u0932\u0917 \u0915\u094d\u0930\u092e \u092e\u0947\u0902 \u0932\u0917\u093e\u092f\u093e \u091c\u093e \u0930\u0939\u093e \u0939\u094b \u2014 \u0924\u094b permutation \u0932\u0917\u0947\u0917\u093e\u0964<\/p>\n

\ud83e\udde0 Example:<\/h4>\n

5 \u0932\u094b\u0917\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 3 \u0915\u094b \u090f\u0915 \u0932\u093e\u0907\u0928 \u092e\u0947\u0902 \u092c\u0948\u0920\u093e\u0928\u093e \u0939\u0948 \u2014<\/p>\n

5P3=5!(5\u22123)!=60^5P_3 = \\frac{5!}{(5 – 3)!} = 60<\/span><\/p>\n

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\ud83d\udd39 Combination (\u0938\u0902\u091a\u092f\u0928)<\/strong> \u2013 \u091c\u092c \u0915\u094d\u0930\u092e (order)<\/strong> \u092e\u093e\u092f\u0928\u0947 \u0928\u0939\u0940\u0902 \u0930\u0916\u0924\u093e\u0964<\/h3>\n

nCr=n!r!(n\u2212r)!^nC_r = \\frac{n!}{r!(n – r)!}<\/span>\ud83d\udd38 Trick:<\/strong> \u091f\u0940\u092e, \u0938\u092e\u0942\u0939, \u0938\u092e\u093f\u0924\u093f \u092c\u0928\u093e\u0928\u0940 \u0939\u094b \u2014 \u0924\u094b combination \u0915\u093e \u092a\u094d\u0930\u092f\u094b\u0917 \u0915\u0930\u094b\u0964<\/p>\n

\ud83e\udde0 Example:<\/h4>\n

5 \u0916\u093f\u0932\u093e\u0921\u093c\u093f\u092f\u094b\u0902 \u092e\u0947\u0902 \u0938\u0947 3 \u0915\u093e \u091a\u092f\u0928 \u0915\u0930\u0928\u093e \u0939\u0948 \u2014<\/p>\n

5C3=10^5C_3 = 10<\/span><\/p>\n

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\ud83d\udcd0 2. Trigonometry (\u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u093f)<\/strong><\/h2>\n

\ud83d\udd38 Basic Identities:<\/h3>\n

sin\u20612\u03b8+cos\u20612\u03b8=1\\sin^2\\theta + \\cos^2\\theta = 1 <\/span> 1+tan\u20612\u03b8=sec\u20612\u03b81 + \\tan^2\\theta = \\sec^2\\theta <\/span> 1+cot\u20612\u03b8=csc\u20612\u03b81 + \\cot^2\\theta = \\csc^2\\theta<\/span><\/p>\n

\ud83d\udd38 Important Values (0\u00b0, 30\u00b0, 45\u00b0, 60\u00b0, 90\u00b0):<\/h3>\n\n\n\n\n\n\n\n\n\n
\u03b8<\/th>\nsin\u03b8<\/th>\ncos\u03b8<\/th>\ntan\u03b8<\/th>\n<\/tr>\n<\/thead>\n
0\u00b0<\/td>\n0<\/td>\n1<\/td>\n0<\/td>\n<\/tr>\n
30\u00b0<\/td>\n1\/2<\/td>\n\u221a3\/2<\/td>\n1\/\u221a3<\/td>\n<\/tr>\n
45\u00b0<\/td>\n1\/\u221a2<\/td>\n1\/\u221a2<\/td>\n1<\/td>\n<\/tr>\n
60\u00b0<\/td>\n\u221a3\/2<\/td>\n1\/2<\/td>\n\u221a3<\/td>\n<\/tr>\n
90\u00b0<\/td>\n1<\/td>\n0<\/td>\n\u221e<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\ud83d\udd38 Trick to remember<\/strong>:
\nSin:<\/strong> \u221a0\/2, \u221a1\/2, \u221a2\/2, \u221a3\/2, \u221a4\/2 \u2192 sin0 to sin90
\nCos:<\/strong> \u0909\u0932\u094d\u091f\u093e (reverse of sin)<\/p>\n

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\ud83d\udd3a 3. Triangle (\u0924\u094d\u0930\u093f\u092d\u0941\u091c)<\/strong><\/h2>\n

\ud83d\udd38 Triangle Angle Sum Property:<\/h3>\n

\u2220A+\u2220B+\u2220C=180\u2218\\angle A + \\angle B + \\angle C = 180^\\circ<\/span><\/p>\n

\ud83d\udd38 Heron\u2019s Formula (area):<\/h3>\n

Area=s(s\u2212a)(s\u2212b)(s\u2212c)where\u00a0s=a+b+c2\\text{Area} = \\sqrt{s(s-a)(s-b)(s-c)} \\text{where } s = \\frac{a + b + c}{2}<\/span><\/p>\n

\ud83d\udd38 Pythagoras Theorem:<\/h3>\n

a2+b2=c2(Right-angled\u00a0triangle)a^2 + b^2 = c^2 \\quad (\\text{Right-angled triangle})<\/span><\/p>\n

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\u26aa 4. Circle (\u0935\u0943\u0924\u094d\u0924)<\/strong><\/h2>\n

\ud83d\udd38 Formulas:<\/h3>\n