Q. Give the answer in one word:
If the limits of an integral are mutually changed, then what is the effect of sign in integral?

Q. Prove that coefficient of correlation lies between $-1$ and $1$.

Q. The data are received from a firm for advertisement and sale as given below:

	Sales (in crores)	Advertisement (in crores)
Mean	$40$	$6$
Standard Deviation	$10$	$1.5$
Coefficient of correlation $r=0.9$

If the firm fixed the target for sale as $60$ crores rupees, then calculate the amount for advertisement.

Q.

Q. Give the answer in one word:
What is the differential coefficient of $\log \log \sin x$?

Q. Find the line of regression of $y$ on $x$ for the following data:

$x$	$y$
$10$	$8$
$9$	$12$
$8$	$7$
$7$	$10$
$6$	$9$
$5$	$6$

Q. Give the answer in one word
What type of quantity work is?

Q. Find the differential coefficient of ${\tan }^{-1}x$

Q. Show that $\overline{a}\times (\overline{b}\times \overline{c})+\overline{b}\times (\overline{c}\times \overline{a})+\overline{c}\times (\overline{a}\times \overline{b})=0$

Q. Find the equation of planes which passes through the intersection of the lines $x+3y+6=0$ and $3x-y-4z=0$ and their perpendicular distance from the origin is $1$.

Q. If $y={\left({\sin }^{-1}x\right)}^2$, then prove that $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}-2=0$

Q. Value of ${\cos }^{-1}\frac{3}{5}-{\sin }^{-1}\frac{4}{5}$ is

1. $1$
2. $0$
3. $-1$
4. $\frac{4}{5}$

Q. If $x^y=e^{x-y}$ then prove that:
$\frac{dy}{dx}=\frac{\log x}{{(1+\log x)}^2}$

Q. Differentiate w.r.t $x$, the following function:
$\log \sqrt{\frac{1+\cos x}{1-\cos x}}$

Q. Verify Rolle's theorem for the function $f\left(x\right)=\sin x+\cos x-1$ in the interval $[0,\frac{\pi }{2}]$

Q. If ${\sin }^{-1}x+{\tan }^{-1}x=\frac{\pi }{2}$ then prove that: $2x^2=\sqrt{5}-1$

Q. Give answer in one word:
Displacement is which quantity in vectors algebra?

Q. Prove that ${\cos }^{-1}\frac{3}{11}-{\sin }^{-1}\frac{3}{4}={\sin }^{-1}\frac{19}{44}$

Q. Find the coefficient of correlation by step deviation method from the given data:

$x$	$y$
$10$	$18$
$14$	$12$
$18$	$24$
$22$	$6$
$26$	$30$
$30$	$36$

Q. Prove by the vector method, the law of sine in trignometry:
$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$

Q. The value of $\int \frac{1}{1-\sin x}dx$ is _____.

Q. Partial fractions of $\frac{1}{x(x+a)}$ are:

1. $[\frac{1}{x}-\frac{1}{x+a}]$
2. $\frac{1}{a}[\frac{1}{x}+\frac{1}{x+a}]$
3. $[\frac{1}{x}+\frac{1}{x+a}]$
4. $\frac{1}{a}[\frac{1}{x}-\frac{1}{x+a}]$

Q. Find the value of
$\underset{x\to 0}{lim}\frac{x(e^x-1)}{1-\cos x}$

Q. Find the equation of the straight line passing through the points $(2,3,4)$ and $(1,-2,3)$

1. $\frac{x-2}{1}=\frac{y-3}{-5}=\frac{z-4}{-1}$
2. $\frac{x-2}{-1}=\frac{y-3}{-5}=\frac{z-4}{-1}$
3. $\frac{x-2}{-1}=\frac{y-3}{5}=\frac{z-4}{-1}$
4. $\frac{x-2}{-1}=\frac{y-3}{-5}=\frac{z-4}{1}$

Q. Break $\frac{2x+1}{(x-1)(x^2+1)}$ into partial fractions.

Q. Differential coefficient of $\sin x^3$ will be _________.

Q. Break $\frac{x^4}{x^3+1}$ into partial fractions.

Q. Prove that the maximum value of $\sin x+\cos x$ is $\sqrt{2}$.

Q. The perpendicular distance of the point $(5,12,13)$ from x-axis is:

1. $\sqrt{131}$
2. $\sqrt{313}$
3. $\sqrt{12}$
4. $\sqrt{5}$

Q. A variable plane keeping at a constant distance $P$ from the origin $O$ cuts the intercept on the axis at the points $A,B,C$. Show that the locus of centroid of tetrahedron $OABC$ is $x^{-2}+y^{-2}+z^{-2}=16P^{-2}$