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${lim}_{x\to 0}\frac{cosax-cosbx}{coscx-1}$

A line passes through the centre of a sphere whose radius is 5 and one of the intercept points is $(1,-2,2)$. If the equation of the line is
$\frac{x}{1}=\frac{y}{-2}=\frac{z}{2}$
,then the equation of the sphere can be

If the probability of A to fail in an examination is $\frac{1}{5}$ and that of B is $\frac{3}{10}$.

Then, the probability that either A or B fails is

If $(2x-1{)}^{20}-(ax+b{)}^{20}={(x^2+px+q)}^{10}$ holds true $\forall x\in R$ where $a,b,p$ and $q$ are real numbers, then which of the following is (are) CORRECT?

The function $f\left(x\right)=\underset{-1}{\overset{x}{\int }}t(e^t-1)(t-1)(t-2{)}^3(t-3{)}^{5}dt$ has a maximum value at $x=k$. If $k=\cos \theta +\sec \theta$, then ${\cos }^6\theta +{\sec }^6\theta$ is equal to

How many different selections of 4 books can be made from 10 different books, if

(i) There is no restriction ;

(ii) Two particular books are always selected ;

(iii) Two particular books are never selected ?

Volume of parallelopiped whose coterminous edges are given by $\overrightarrow{u}=\hat{i}+\hat{j}+\lambda \hat{k},\overrightarrow{v}=\hat{i}+\hat{j}+3\hat{k}$ and $\overrightarrow{w}=2\hat{i}+\hat{j}+\hat{k}$ is $1$ cu. unit. If $\theta$ be the angle between the edges $\overrightarrow{u}$ and $\overrightarrow{w}$, then $\cos \theta$ can be:

If the vertices A, B, C of a triangle ABC have position vectors (1, 2, 3), (-1, 0, 0), (0, 1, 2) respectively then find $\angle ABC$ ($\angle ABC$ is the angle between the vectors BA and BC).

If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.

The domain of definition of $f\left(x\right)=\sqrt{\frac{1-|x|}{2-|x|}}$ is

Pointing to a photograph Arun said, “She is the mother of my brother’s son’s wife’s daughter.” How is Arun related to the lady?

From among the 36 teachers in a school, one pricipal and one vice-principal are to be appointed. In how many ways can this be done ?

If the sum of two numbers $p$ and $q$ is $\sqrt{10}$ and their difference is $\sqrt{6}$, then the value of ${\log }_{q}p$ is

Which of the following sets can be the subset of the general solution of the equation $1+\cos 3x=2\cos 2x$ ?

If the pairs of lines $x^2+2xy+a{y}^2=0$ and $a{x}^2+2xy+y^2=0$ have exactly one line in common, then the combined equation of the other two lines is given by

Which of the following could be the $n^{th}$ term of any AP?

A box contains 24 identical balls, of which 12 are white and 12 are black. The balls are drawn at random from the box one at a time with replacement. The probability that a white ball is drawn for the $4^{th}$ time on the $7^{th}$ draw is

A natural number is chosen at random from amongst first 500. What is hte probability that the number so chosen is divisbile by 3 or 5 ?

A value of $\alpha$ such that $\underset{\alpha }{\overset{\alpha +1}{\int }}\frac{dx}{(x+\alpha )(x+\alpha +1)}={\log }_e\left(\frac{9}{8}\right)$ is :

$\text{If}f\left(x\right)=\{\begin{array}{c}\frac{sinx}{x}+cosx,x\ne 0\\ k,x=0\end{array}\text{is continuos at}x=0,thenk=$

The least value of k which makes the roots of the equation

$x^2+5x+k=0$imaginary is

If (1 + i) z = (1 - i) $\overline{z}$, then show that z = -i$\overline{z}$.

Let $z\in C$ be such that $|z|<1$. If $\omega =\frac{5+3z}{5(1-z)}$, then :

In the figure shown a thin parallel beam of light is incident on a plane mirror $M_1$ at small angle ‘$\theta$’. $M_2$ is a concave mirror of focal length ‘f ’. After three successive reflections of this beam the x and y coordinates of the image is:

The components of a vector along $x$ and $y$ directions are $(n+1)$ and $1$ respectively. If the coordinate system is rotated by an angle ${60}^{\circ }$, then the components change to $n$ and $3$. The value of $n$ is

If $A(-2,1),B(2,3)$ and $C(-2,-5)$ are the vertices of an acute angled $△ABC,$ then the value of $\tan \angle B$ is

The angle of intersection of the normals at the point $(-\frac{5}{\sqrt{2}},\frac{3}{\sqrt{2}})$ of the curves $x^2-y^2=8$ and $9x^2+25y^2=225$ is

Let $f\left(x\right)=x^2-2|x|$ and $g\left(x\right)=\left\{\begin{array}{cc}\text{min}\left\{f\right(t\left)\right\},& -2\le t\le x,-2\le x<0\\ \text{max}\left\{f\right(t\left)\right\},& 0\le t\le x,0\le x\le 2,\\ f\left(x\right),& x>2\end{array}\right\}$
then $g\left(x\right)$ is not differentiable at

What is the difference between a collection and a set ? GIve reasona to support your answer ?

If $|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|$,then the vectors $\overrightarrow{a}and\overrightarrow{b}$are orthogonal

Determine $P\left(\frac{E}{F}\right)$
A die is thrown three times
E: 4 appears on the third toss
F: 6 and 5 appears, respectively on first two tosses.

The mean and standard deviation of marks of $10$ students in a class test of $10$ marks is $7$ and $1$ respectively. The marks of $A$ is not known and the standard deviation of the remaining $9$ students is $0$. If $A$ does not score full marks, then the marks of $A$ is

Evaluate the integrals using substitution.
${\int }_0^{2}x\sqrt{x+2}dx.$

Using elementary transformations, find the inverse of the followng matrix.

$\left[\begin{array}{cc}6& -3\\ -2& 1\end{array}\right]$

in a vernier calliper N vernier scale division are equal to ------------------ main scale divisions.

cos{π/4 - x}cos{π/4 - y} - sin{π/4 - x}sin{π/4 - y} = sin(x+y)