Probability a blade is defective $=0.002$, packet of $10$ blades. Find packets with no defective blades in $10000$ packets.

Regression lines: $3x+2y=26$, $6x+y=31$ Correlation between $x,y$ is

Car travels half distance with $v_1$, half with $v_2$. Average speed is

Find least integer $k$ such that $(k-2)x^2 + k + 8x + 4 > 0$ for all $x\in\mathbb{R}$.

If $\displaystyle \sum_{K=0}^{2n}(-1)^K\binom{2n}{K}^2 = A$, find $\displaystyle \sum_{K=0}^{2n}(-1)^K(K-2n)\binom{2n}{K}^2$.

Solve inequality $\log_3\big((x+2)(x+4)\big)+\log_{1/3}(x+2)<\dfrac12\log_{\sqrt{3}}7$

$a,b,c$ are positive and $c>a$ and in H.P. Compute $\log(a+c)+\log(a-2b+c)$.

Area enclosed by $|x|+|y|=1$

$A$ polygon has $44$ diagonals, the number of its sides is

Let $X$ be the universal set for sets $A$ and $B$. If $n(A)=200,;n(B)=300,;n(A\cap B)=100$, then $n(A'\cap B')=300$ provided $n(X)$ is equal to

In a college of $300$ students, every student reads $5$ newspapers and every newspaper is read by $60$ students. The number of newspapers is

The number of ways of forming different $9$-digit numbers from $223355588$ by rearranging digits so that odd digits occupy even positions is

An anti-aircraft gun fires at a plane. Probabilities of hitting at slots 1,2,3,4 are $0.4,;0.3,;0.2,;0.1$. Probability that the gun hits the plane is

The minimum value of $px + qy$ when $xy=r^2$ and $p,q,x,y$ are positive numbers is

If $a$ is a positive integer, then the number of values satisfying $ \displaystyle \int_{0}^{\pi/2} \left[ a^{2}\left(\frac{\cos 3x}{4}+\frac{3}{4}\cos x\right)+a\sin x - 20\cos x \right] dx \le -\frac{a^{2}}{3} $ is

Find $ \displaystyle \frac{d}{dx}\left( \sqrt{x} - \frac{5}{\sqrt{x}} \right) $

$ \displaystyle \lim_{x\to 0} \frac{x+\sin x}{\sqrt{x}-\cos x} $

If $ f(x)=\displaystyle \int_{0}^{x} t\sin t, dt $, then $f'(x)$ is

The value of $ \sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ $

In $\triangle ABC$, $B = 45^\circ, C = 105^\circ, c=\sqrt{2}$. Find side $a$ and $b$.

If $ \displaystyle \tan \theta = \frac{b}{a} $, then the value of $ a\cos 2\theta + b\sin 2\theta $ is

The general solution of $ \sqrt{3}\cos x + \sin x = 3 $ is:

$ \displaystyle \text{The value of } \frac{1 - \tan^{2} 15^\circ}{1 + \tan^{2} 15^\circ} \text{ is:} $

$ \displaystyle \int_{0}^{1/2} \frac{dx}{\sqrt{x - x^{2}}} $

If area between $ y=x^{2} $ and $ y=x $ is $ A $, then area between $ y=x^{2} $ and $ y=1 $ is:

If $ a,b,c $ are coplanar, evaluate $ [,2a - b, 2b - c,2c - a,] $

$ \vec{a} = x\hat{i} - 3\hat{j} - \hat{k},\quad \vec{b} = 2x\hat{i} + x\hat{j} - \hat{k} $ Angle between $ \vec{a} $ and $ \vec{b} $ is acute and angle between $ \vec{b} $ and $ +y $ axis lies in $ \left(\dfrac{\pi}{2}, \pi\right) $ Find $x$.

Lines $2x + 3y - 6 = 0$ and $9x + 6y - 18 = 0$ cut coordinate axes in concyclic points. Center of circle is:

Number of distinct solutions of $ x^{2} = y^{2} $ and $ (x - a)^{2} + y^{2} = 1 $ where $a$ is any real number:

Vertex of parabola $ y^{2} - 8y + 19 = 0 $

Eccentricity of ellipse $ 9x^{2} + 5y^{2} - 30y = 0 $

$ \vec{v} = 2\hat{i} + \hat{j} - \hat{k},\quad \vec{w} = \hat{i} + 3\hat{k} $ If $ \vec{u} $ is a unit vector, maximum value of $ [\vec{u}\ \vec{v}\ \vec{w}] $ is:

If the function $f:[1,\infty)\to[1,\infty)$ is defined by $f(x)=2^{x(x-1)}$, then $f^{-1}(x)$ is:

A random variable $X$ has the probability distribution: \[\begin{array}{c|ccccccccc} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(X=x) & a & 3a & 5a & 7a & 9a & 11a & 13a & 15a & 17a \end{array} \]The value of $a$ is:

$ \text{The sum of } 11^{2} + 12^{2} + \cdots + 30^{2} \text{ is} $

$ \text{If } B = -A^{-1}BA,\ \text{then } (A+B)^{2} = $

Roots of $x^{2} - 2x + 4 = 0$ are $\alpha, \beta$. Compute $ \alpha^{6} + \beta^{6} $.

If $ |\vec{a}\times \vec{b}| = |\vec{a}\cdot \vec{b}| $, then angle $\theta$ between $\vec{a},\vec{b}$ is:

$ABCD$ is a parallelogram with diagonals $AC$ and $BD$. Compute $ \overrightarrow{AC} - \overrightarrow{BD} $.

If $\sin x,\ \cos x,\ \tan x$ are in GP, find $\cot 6x - \cot 2x$.

Triangle sides: $x^{2}+x+1,\ 2x+1,\ x^{2}-1$. Largest angle?

Solve: $ 2\sin^{2}\theta - 3\sin\theta - 2 = 0$

Let: $ X = 2^{100},\quad Y = 3^{100},\quad Z = 4^{100} $ Which statement is true?