Odd Extension of a Function

Q. Solve the following equations:
3 – 2 cos x – 4 sin x – cos 2x + sin 2x = 0

Q. If f(x) = cos (log x), then the value of f(x²) f(y²) −$\frac{1}{2}\left\{f\left(\frac{x^2}{y^2}\right)+f\left(x^2{y}^2\right)\right\}$is
(a) −2
(b) −1
(c) 1/2
(d) None of these

Q. The solution of the equation ${\cos }^{2}x+\sin x+1=0$ lies in the interval
(a) $\left(-\pi /4,\pi /4\right)$
(b) $\left(\pi /4,3\pi /4\right)$
(c) $\left(3\pi /4,5\pi /4\right)$
(d) $\left(5\pi /4,7\pi /4\right)$​

Q. The number of values of x in the interval [0, 5 π] satisfying the equation $3{\sin }^{2}x-7\sin x+2=0$ is
(a) 0
(b) 5
(c) 6
(d) 10

Q.

$\mathrm{If}f\left(x\right)=(\begin{array}{c}{x}^3+x^2,for0\le x\le 2\\ x+2,for2\le x\le 4\end{array}$
then the odd extension of f(x) would be -

1. $f\left(x\right)=(\begin{array}{c}{x}^3+x^2,for-2\le x\le 0\\ x+2,for-4\le x\le -2\end{array}$

2. $f\left(x\right)=(\begin{array}{c}-x^3+x^2,for-2\le x\le 0\\ -x+2,for-4\le x\le -2\end{array}$

3. 
   $f\left(x\right)=(\begin{array}{c}{x}^3-x^2,for-2\le x\le 0\\ x-2,for-4\le x\le -2\end{array}$

4. $\text{None of these}$
   

Q. If f(x) = cos (log x), then value of$f\left(x\right)f\left(4\right)-\frac{1}{2}\left\{f\left(\frac{x}{4}\right)+f\left(4x\right)\right\}$is
(a) 1
(b) −1
(c) 0
(d) ±1

Q.

Odd extension is obtained by replacing x by (-x) in the equation of f(x).

1. True

2. False

Q.

The number of solution(s) of the equationsin(-1)2x - cos(-1)x + tan(-1)2x = pi/2is

(1) Zero (2) One(3) Two (4) Infinitely many