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Question: 1. Prove that the equation cos^{2}x + a sinx = 2a – 7 possesses a solution if
Ans: ⇒ cos^{2}x + a sinx = 2a – 7
⇒ 2sin^{2}x – asinx + (2a – 8) = 0
Since sin ∈IR
∴ Given equation has solution of
Question: 2. Find the values of x, (–S < x < S, x ≠ 0) satisfying the equation,
Ans:
Question: 3. Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is
Ans: Let centre be c(h, h + 1) , 0(0, 0)
Question: 4. Prove by induction that for all n∈ N, n^{2} + n is an even integer (n ≥ 1)
Ans: x = 1, x^{2} + x = 2 is an even integer
Let for n = k, k^{2} + k is even
Now for n = k + 1, (k + 1)^{2} + (k + 1) – (k^{2} + k)
= k^{2} + 2k + 1 + k + 1 – k^{2} – k = 2k + 2 which is even integer also k^{2} + k is even integer
Hence (k + 1)^{2} + (k + 1) is also an even integer
Hence n^{2} + n is even integer for all n∈ N.
Question: 5. If A, B are two square matrices such that AB = A and BA = B, then prove that B^{2} = B
Ans: B^{2} = B.B = (BA)B = B (AB) = B(A) = BA = B (Proved)
Question: 6.
Ans:
Question: 7. Use the formula
Ans:
Question: 8. prove that, where A is constant
Ans:
Question: 9. Evaluate the following integral
Ans:
Question: 10.
Ans:
= g'(a) f(a) – g(a) f'(a)
= (2)(2) – (–1) (1) = 4 + 1 = 5
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