Odd Extension of a Function

Q. Find acute angles A and B, if $\sin \left(A+2B\right)=\frac{\sqrt{3}}{2}\mathrm{and}\cos \left(A+4B\right)=0,A>B$.

Q.

The odd extension of the function y = $x^2$ is y = $-x^2$

1. True

2. False

Q.

Let $$\begin{gathered} f\left(x\right)=\left\{\left|x-1\right|+aifx\le 1 \\ 2x+3ifx>\hspace{0.17em}1\right\} \end{gathered}$$. If $f\left(x\right)$ has local minimum at $x=1$, then:

1. $7$

2. $8$

3. $9$

4. $a\le 5$

Q. Consider the real-valued function f satisfying 2 f(sin x)+ f(cos x)=x. Then, the

1. domain of f(x) is R
2. domain of f(x) is[-1, 1]
3. range of f(x) is $\left(-\frac{2\pi }{3},\frac{\pi }{3}\right]$
4. range of f(x) is R

Q. If $\sin \left(A-B\right)=\frac{1}{2}\mathrm{and}\cos \left(A+B\right)=\frac{1}{2},0°<A+B\ge 90°,A<B$ find A and B.

Q. $\mathrm{If}f\left(x\right)=\cos \left(\log x\right),x>0;\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}f\left(x\right)f\left(4\right)-\frac{1}{2}\left(f\left(\frac{x}{4}\right)+f\left(4x\right)\right\}=$

1. 1
2. -1
3. \N
4. $\pm 1$

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. $\text{None of these}$
   

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

Q. If sin (A + B) = 1 and cos (A − B) = 1, 0° < A + B ≤ 90°, A ≥ B find A and B.

Q. f(x) = 2x^2 - 1 is one - one for the domain _____.

1. R
2. I
3. R - N
4. N

Q.

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

1. True

2. False

Q. Solve this:

Q44. If sin $\alpha$ and cos $\alpha$ are the roots of the equation ax² + bx + c = 0, then :

A) a² – 2ac = b² B) a² + 2ac = b² C) a² – ac = b²
​
D) a² + ac = b² E) a² + b² = 2ac