Extrema

Q. Prove that: $\sin 5x=5\sin x-20{\sin }^{3}x+16{\sin }^{5}x$

Q. The smallest positive angle which satisfies the equation ​$2{\sin }^{2}x+\sqrt{3}\cos x+1=0$ is
(a) $\frac{5\pi }{6}$

(b) $\frac{2\pi }{3}$

(c) $\frac{\pi }{3}$

(d) $\frac{\pi }{6}$

Q.

The function $f\left(x\right)={\int }_{-1}^{x}t\left(e^t-1\right)\left(t-1\right){\left(t-2\right)}^3{\left(t-3\right)}^{5}dt$ for $t\in \left[-1,x\right]$ has a local minimum at,

1. $0$, $4$

2. $1$, $3$

3. $0$, $2$

4. $2$, $4$

Q. Prove that:

${\cos }^2\frac{\pi }{8}+{\cos }^2\frac{3\pi }{8}+{\cos }^2\frac{5\pi }{8}+{\cos }^2\frac{7\pi }{8}=2$

Q. find the derivative of sin^3(3x+5) using first principles

Q.

The points of extremum of the function $F\left(x\right)={\int }_1^x{e}^{-t^2/2}(1-t^2)$ dt are

1. $\pm 1$
2. 0
3. $\pm \frac{1}{2}$
4. $\pm 2$

Q.

Now, as you know the product of roots is 30, let those roots be a, b, c $ϵR$. The minimum value of 10a + 9b + 10c is __

Q. $\underset{x\to 0}{\lim }\frac{\sin x^0}{x}$ is equal to

(a) 1
(b) π
(c) x
(d) π/180

Q. Let $f\left(x\right)=\frac{x^2+3}{[x+1]},0\le x\le 2$, where $[.]$ is the greatest integer function. Then the sum of the least value and the greatest value of $f\left(x\right)$ is

Q. If $\omega$ is a complex cube root of unity, then the determinant $\left|\begin{array}{ccc}2& 2\omega & -{\omega }^2\\ 1& 1& 1\\ 1& -1& 0\end{array}\right|=$

1. 0
2. 1
3. -1
4. None of these

Q. Let $A=a_{ij}$ be a matrix of order $3,$ where
$a_{ij}=\{\begin{array}{cc}x;& \text{if}i=j,x\in R\\ 1;& \text{if}|i-j|=1\\ 0;& \text{otherwise}\end{array},$
then which of the following hold(s) good:

1. for $x=2,A$ is a diagonal matrix
2. $A$ is a symmetric matrix
3. Let $f\left(x\right)=detA$, then the function$f\left(x\right)$ has only maxima
4. Let $f\left(x\right)=detA$, then the function$f\left(x\right)$ has both the maxima and minima

Q. Write the value of sin $\frac{\mathrm{\pi }}{12}$ sin $\frac{5\mathrm{\pi }}{12}$.

Q. If $x=-1$ and $x=2$ are extreme points of $f\left(x\right)=\alpha \log |x|+\beta x^2+x$ then :

1. $\alpha =-6,\beta =\frac{1}{2}$
2. $\alpha =-6,\beta =-\frac{1}{2}$
3. $\alpha =2,\beta =-\frac{1}{2}$
4. $\alpha =2,\beta =\frac{1}{2}$

Q.

In h(x) = f(x) + f(-x), then h (x) has got an extreme value at a point where f'(x) is

1. even function.

2. odd function.

3. zero

4. None of these

Q. ${\sin }^{3}A+{\sin }^3\left(\frac{2\pi }{3}+A\right){\sin }^3\left(\frac{4\pi }{3}+A\right)=-\frac{3}{4}\sin 3A$

Q.

If $ax+\frac{b}{x}\le$ c for all positive x, where a, b > 0, then

1. $ab<\frac{c^2}{4}$

2. $ab\ge \frac{c^2}{4}$

3. $ab\ge \frac{c}{4}$

4. $ab=c/4$