Exercise 3.3

Find the transpose of each of the following matrices: (i) 5 1/2 /   /        −  (ii) 1 1 / 2 3 / −     (iii) 1 5 6 / 3 5 6 / 2 3 1 / −  /       − 

Solve the following pair of linear equations by the elimination method and the substitution method :

(i) $x + y = 5$ and $2x - 3y = 4$

(ii) $3x + 4y = 10$ and $2x - 2y = 2$

(iii) $3x - 5y - 4 = 0$ and $9x = 2y + 7$

(iv) $\dfrac{2}{x} + \dfrac{3}{y} = 2$ and $\dfrac{3}{x} - \dfrac{2}{y} = 3$

sin² π / 6 + cos² 3 π - tan² 1-4 2 π = 2. 2sin 2 π + cosec² 27 3cos6 3 2 π π = 3. 2 25cot cosec 3tan 66 6 6 π π π+ + = 4. 2 2 232sin 2cos 2sec 104 4 3 π π π+ + =

Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) 3 5 / 1 1 /     −  (ii) 6 2 2 / 2 3 1 / 2 1 3 / −   − −   −  (iii) 3 3 1 / 2 2 1 / 4 5 2 / −   − −   − −  (iv) 1 5 / 1 2 /    −  Choose the correct answer in the Exercises 1 1 and 12.

sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x 11. 3 3cos cos 2 sin4 4x x xπ π   + − − = −      

If A, B are symmetric matrices of same order , then AB - BA is a (A) Skew symmetric matrix (B) Symmetric matrix (C) Zero matrix (D) Identity matrix

If cos sinA ,sin cos α − α =  α α  and A + A′ = I, then the value of α is (A) 6 / π (B) 3 / π (C) π (D) 3 / π

sin² 6x - sin² 4x = sin 2x sin 10x 13. cos² 2x - cos² 6x = sin 4 x sin 8x

sin² x + 2 sin 4x + sin 6x = 4 cos² x sin 4x

cot 4x (sin 5x + sin 3x) = cot x (sin 5x - sin 3 x) 16. cos cos sin sin sin cos 9 5 17 3 2/10 x x x x x x − − = − 17. sin sin cos cos tan5 3 5 3 4x x x x x+ + = 18. sin sin cos cos tanx y x y x y− + = − 2 19. sin sin cos cos tanx x x x x+ + =3 3 2 20. sin sin sin cos sinx x x x x− − =3 22 2 21. cos cos cos sin sin sin cot4 3 2 4 3 2 3x x x x x x x+ + + + = 22. cot x cot 2x - cot 2x cot 3x - cot 3x cot x = 1 23. 2 4 4tan (1 tan )tan 4 1 6 tan tan x xx x x −= − + 24. cos 4x = 1 - 8sin 2 x cos² x 25. cos 6x = 32 cos 6 x - 48cos 4 x + 18 cos 2 x - 1

Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method :

(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes $\dfrac{1}{2}$ if we only add 1 to the denominator. What is the fraction?

(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

(iv) Meena went to a bank to withdraw $\text{₹ } 2000$. She asked the cashier to give her $\text{₹ } 50$ and $\text{₹ } 100$ notes only. Meena got 25 notes in all. Find how many notes of $\text{₹ } 50$ and $\text{₹ } 100$ she received.

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid $\text{₹ } 27$ for a book kept for seven days, while Susy paid $\text{₹ } 21$ for the book she kept for five days. Find the fixed charge and the charge for each extra day.

If / 1 2 3 4 1 5 A 5 7 9 and B 1 2 0 2 1 1 1 3 1 − − −       = =       −    , then verify that (i) (A + B)′ = A′ + B′, (ii) (A - B)′ = A′ - B′

If 3 4 1 2 1A 1 2 and B 1 2 30 1   −  ′ = − =        , then verify that (i) (A + B)′ = A′ + B′ (ii) (A - B)′ = A′ - B′

If 2 3 1 0A and B1 2 1 2 − −   ′ = =       , then find (A + 2B)′

Find the value of: (i) sin 75° (ii) tan 15° Prove the following:

For the matrices A and B, verify that (AB) ′ = B′A′, where (i) [ ] A 4 , B 1 2 1    = − = −     (ii) [ ] A 1 , B 1 5 7    = =    

cos cos sin sin sin ( )4 4 4 4x y x y x yπ π π π       − − − − − = +               7. πtan 1 tan4 π 1 tantan 4 x x xx   +   +  =  −   −   8. 2cos ( ) cos ( ) cot sin ( ) cos 2

x x π + − =π π − +   

If (i) cos sinA sin cos α α =  − α α  , then verify that A′ A = I (ii) If sin cosA cos sin α α =  − α α  , then verify that A′ A = I

(i) Show that the matrix 1 1 5

5 1 3 / −   = −     is a symmetric matrix. (ii) Show that the matrix 0 1 1

1 1 0 / −   = −    −  is a skew symmetric matrix.

For the matrix 1 5A 6 7  =    , verify that (i) (A + A′) is a symmetric matrix (ii) (A - A′) is a skew symmetric matrix

Find ( )1 A A2 ′+ and ( )1 A A2 ′− , when a b a c b c    = −   − − 

3π 3πcos cos (2 π ) cot cot (2π ) 12 2x x x x    + + − + + =        