Class XI Chapter 3 – Trigonometric Functions Maths ______________________________________________________________________________ 4 15 2 8 2 8 2 15 16 8 2 15 4 8 2 15 x sin x 4 2 8 2 15 tan 2 x cos 8 2 15 8 2 15 2 4 8 2 15 8 2 15 8 2 15 8 2 15 2 8 2 15 8 2 15 4 15 64 60 2 x x x 2 2 2 Thus, the respective values are sin ,cos and tan are 82 15 82 15 , and 4 15 4 4 Printed from Vedantu.com. Register now to book a Free LIVE Online trial session with a Top tutor.

Class XI Chapter 4 – Principle of Mathematical Induction Maths ______________________________________________________________________________ Exercise 4.1 Book Name: **NCERT** Solutions Question 1: Prove that following by using the principle of mathematical induction for all n N: n 3 1 2 n1 1 3 3 ..... 3 2 Solution 1: Let the given statement be , P n i.e., n 3 1 2 n1 P n :1 3 3 ..... 3 2 For n 1, we have 1 3 1 31 2 P 1 : 1 2 2 2 Let Pk , which is true. be true for some positive integer k, i.e., k 3 1 2 k1 1 3 3 .... 3 ....... i 2 We shall now prove that Consider 2 1 1 1 1 3 3 .... 3 k 3 k 2 k1 k 1 3 3 .... 3 3 P k 1 k 3 1 k 3 Using i 2 k 3 1 2.3 2 k 12 3 1 2 3.3 k 1 2 k 1 3 1 2 Thus, 1 k P k is true whenever is true. Pk is true. Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., N.

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