Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions

Prove that:

`tan^(-1)  63/16 = sin^(-1)  5/13 + cos^(-1)  3/5`

Prove `tan^(-1)   1/5 + tan^(-1)  (1/7) + tan^(-1)  1/3 + tan^(-1)  1/8 = pi/4`

Prove that:

`tan^(-1) sqrtx = 1/2 cos^(-1)  (1-x)/(1+x)`, x ∈ [0, 1]

Prove that:

`cot^(-1)  ((sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1- sinx))) = x/2, x in (0, pi/4)`

Prove `(9pi)/8 - 9/4  sin^(-1)  1/3 = 9/4 sin^(-1)  (2sqrt2)/3`

Solve the following equation:

2 tan⁻¹ (cos x) = tan⁻¹ (2 cosec x)

sin (tan^–1 x), |x| < 1 is equal to ______.

sin^–1 (1 – x) – 2 sin^–1 x = `pi/2`, then x is equal to ______.

Solve  `tan^(-1) -  tan^(-1)  (x - y)/(x+y)` is equal to

(A) `pi/2`

(B). `pi/3` 

(C) `pi/4` 

(D) `(-3pi)/4`

Find the value of x, y, and z from the following equation:

`[(4,3),(x,5)] = [(y,z),(1,5)]`

Find the value of x, y, and z from the following equation:

`[(x+y, 2),(5+z, xy)] = [(6,2), (5,8)]`

Find the value of x, y, and z from the following equation:

`[(x+y+z), (x+z), (y+z)] = [(9),(5),(7)]`

Find the value of a, b, c, and d from the equation:

`[(a-b, 2a+c),(2a-b, 3x+d)] = [(-1,5),(0,13)]`

`A = [a_(ij)]_(mxxn)` is a square matrix, if ______.

if `A = [(0, -tan  alpha/2), (tan  alpha/2, 0)]` and I is the identity matrix of order 2, show that I + A = `(I -A)[(cos alpha, -sin alpha),(sin alpha, cos alpha)]`

Let A = `[(0,1),(0,0)]`show that (aI+bA)ⁿ  = aⁿI + na^n-1 bA , where I is the identity matrix of order 2 and n ∈ N

if A = [(1,1,1),(1,1,1),(1,1,1)], Prove that A" = `[(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1)),(3^(n-1),3^(n-1),3^(n-1))]` `n in N`

if `A = [(3,-4),(1,-1)]` then prove A"=` [(1+2n, -4n),(n, 1-2n)]` where n is any positive integer

Find the matrix X so that  X`[(1,2,3),(4,5,6)]= [(-7,-8,-9),(2,4,6)]`

If A and B are square matrices of the same order such that AB = BA, then prove by induction that AB" = B"A. Further, prove that (AB)" = A"B" for all n ∈ N