Exercise miscellaneous

If A and B are symmetric matrices, prove that AB - BA is a skew symmetric matrix.

sin x = 4 1 , x in quadrant II

If the matrix A is both symmetric and skew symmetric, then (A) A is a diagonal matrix (B) A is a zero matrix (C) A is a square matrix (D) None of these

If A is square matrix such that A2 = A, then (I + A)³ - 7 A is equal to (A) A (B) I - A (C) I (D) 3A

(sin 3x + sin x) sin x + (cos 3x - cos x) cos x = 0

Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

A / y z

   = −    −  satisfy the equation A′A = I.

(cos x + cos y)2 + (sin x - sin y)2 = 4 cos 2 2 x y+

For what values of x : [ ] 1 2 0 0 1 2 1 2 0 1 2 1 0 2 x                    = O?

(cos x - cos y)2 + (sin x - sin y)2 = 4 sin 2 x y−

sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x 6. 6.6. 6.6. (sin 7 sin 5 ) (sin 9 sin 3 ) tan 6(cos 7 cos 5 ) (cos 9 cos 3 )

• • • =+ + +

If 3 1A 1 2  =  −  , show that A2 - 5A + 7I = 0.

Find x, if [ ] 1 0 2 5 1 0 2 1 4 O 2 0 3 1 x x        − − =          

sin 3x + sin 2x - sin x = 4sin x cos x 2 cos 3/2 x Find sin x 2 , cos x 2 and tan x 2 in each of the following :

A manufacturer produces three products x, y, z which he sells in two markets. Annual sales are indicated below: Market Products I 10,000 2,000 18,000 II 6,000 20,000 8,000 (a) If unit sale prices of x, y and z are 2.50, 1.50 and 1.00, respectively, find the total revenue in each market with the help of matrix algebra. (b) If the unit costs of the above three commodities are 2.00, ` 1.00 and 50 paise respectively. Find the gross profit.

Find the matrix X so that 1 2 3 7 8 9X 4 5 6 2 4 6 − − −   =       Choose the correct answer in the following questions:

tan x = − 4 3 , x in quadrant II 9. cos x = − 1 3, x in quadrant III

If A = is such that A² = I, then (A) 1 + α² + βγ = 0 (B) 1 - α² + βγ = 0 (C) 1 - α² - βγ = 0 (D) 1 + α² - βγ = 0