List of practice Questions

If the coordinates at one end of a diameter of the circle $x^2 + y^2 - 8x - 4y + c = 0$ are $(-3, 2)$, then the coordinates at the other end are

The system of linear equations : $x + y + z = 0, 2x + y - z = 0, 3x + 2y = 0$ has :

The position vector of $A$ and $B$ are $ 2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ The length of the internal bisector of $?BOA$ triangle $AOB$ is

If vector equation of the line$\frac{x-2}{2}=\frac{2y-5}{-3}=z+1,$ is $\vec{r}=\left(2\hat{i}+\frac{5}{2} \hat{j}-\hat{k}\right)+\lambda\left(2\hat{i}-\frac{3}{2} \hat{j}+p\hat{k}\right)$ then $p$ is equal to

Let $a, b, c, be$ in $A.P.$ with a common difference $d.$ Then $e^{1/e}, e^{b/bc}, e^{1/a}$ are in :

Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of heads is

If sin y = x sin (a + y), then  $\frac{dy}{dx}$is

A value of $\theta \ \ \in \ (0, \pi /3)$ for which $\begin {vmatrix} 1 + cos^2 \theta & sin^2\theta & 4 cos 6\theta \\ cos^2 \theta & 1+sin^2 \theta & 4 cos6 \theta \\ cos^2 \theta & sin^2 \theta & 1+4 cos6 \theta \end {vmatrix} $ =0 , is :

If $\sum^\limits{25}_{r=0} \left\{^{50}C_{r} . ^{50-r}C_{25-r}\right\}=K\left(^{50}C_{25}\right) $ , then $K$ is equal to :

If [$x$] denotes the greatest integer $\le x$, then the system of linear equations [$sin\,\theta$] x + [$-cos\,\theta$]y=0 [$cot\,\theta$] $x + y = 0$

If the sum of the deviations of $50$ observations from $30$ is $50$, then the mean of these observation is :

If three distinct numbers a,b,c are in G.P. and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then which one of the following statements is correct?

Let $N$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g : N \to N$ such that : $f(n) = \begin{cases} \frac{n+1}{2} & \quad \text{if } n \text{ is odd}\\ \frac{n}{2} & \quad \text{if } n \text{ is even} \end{cases}$ and $g(n) = n-(-1)^n$. The $fog$ is :

Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4 |z_2|$. If $z = \frac{3z_{1}}{2z_{2}} + \frac{2z_{2}}{3z_{1}}$ then :

$\displaystyle\lim_{n\to\infty} \left(\frac{\left(n+1\right)^{\frac{1}{3}} }{n^{\frac{4}{3}}} + \frac{\left(n+2\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}} + ..... + \frac{\left(2n\right)^{\frac{1}{3}}}{n^{\frac{4}{3}}}\right) $ equal to :

A helicopter is flying along the curve given by $y - x^{3/2} = 7, (x \ge 0)$. A soldier positioned at the point $\left(\frac{1}{2}, 7\right)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

All possible numbers are formed using the digits \(1,1,2,2,2,2,3,4,4\) taken all at a time. The number of such numbers in which the odd digits occupy even places is :

Consider a triangular plot $ABC$ with sides $AB = 7m, BC = 5m$ and $CA = 6m$. A vertical lamp-post at the mid point $D$ of $AC$ subtends an angle $30^{\circ}$ at $B$. The height (in $m$) of the lamp-post is:

If $\begin {bmatrix} 1 & 1 \\ 0 & 1 \end {bmatrix}$ . $\begin {bmatrix} 1 & 2 \\ 0 & 1 \end {bmatrix} $ . $\begin {bmatrix} 1 & 3 \\ 0 & 1 \end {bmatrix}$ ..... $\begin {bmatrix} 1 & n-1 \\ 1 & 1 \end {bmatrix}$ = $\begin {bmatrix} 1 & 78 \\ 0 & 1 \end {bmatrix}$, then the inverse of $\begin {bmatrix} 1 & n \\ 0 & 1 \end {bmatrix}$ is

Let $a_1, a_2, ....., a_{10}$ be a G.P. If $\frac{a_3}{a_1} = 25 $, then $\frac{a_9}{a_5}$ equals :

The term independent of x in the expansion of $\bigg(\frac{1}{60} - \frac{x^8}{81}\bigg). \bigg(2x^2 - \frac{3}{x^2}\bigg)^6$ is equal to:

If $A = \begin{bmatrix}e^{t}&e^{t} \cos t&e^{-t}\sin t\\ e^{t}&-e^{t} \cos t -e^{-t}\sin t&-e^{-t} \sin t+ e^{-t} \cos t\\ e^{t}&2e^{-t} \sin t&-2e^{-t} \cos t\end{bmatrix} $ Then $A$ is -

If $f\left(x\right) = \frac{2- x\cos x}{2+x \cos x}$ and $ g\left(x\right) =\log_{e}x ., \left(x>0\right) $ then the value of integral $\int\limits^{\frac{\pi}{4}}_{-\frac{\pi}{4}} g\left(f\left(x\right)\right)dx $ is :

Let $f : (-1,1) \to R$ be a function defined by $f(x) = max \{- | x |,- \sqrt{ 1- x^2} \}$. If $K$ be the set of all points at which $f$ is not differentiable, then $K$ has exactly :

The maximum volume (in $cu. m$) of the right circular cone having slant height $3\,m$ is :