List of practice Questions

If sin $\left(\theta-\phi\right) = n \, sin (\theta - \phi),n \ne1,$ then the value of $\frac{\tan\theta}{\tan\phi}$ is equal to

If the projection of the vector $\vec {a}$ on $\vec{b}$ is $ \overrightarrow{a} $ on $ \overrightarrow{b} $ is $ |\overrightarrow{a}\times \overrightarrow{b}| $ and if $ 3\overrightarrow{b}=\vec{i}+\vec{j}+\vec{k}, $ then the angle between $ \vec{a} $ and $ \vec{b} $ is

If $\begin{vmatrix}3i&-9i&1\\ 2&9i&-1\\ 10&9&i\end{vmatrix} = x + iy $, then

If $xy\, = \,A \,sinx \,+ \,B \,cos \,x$ is the solution of the differential equation $x\frac{d^{2}y}{dx^{2}}-5a\frac{dy}{dx}+xy=0$ then the value of $a$ is equal to

If $ y={{\sin }^{2}}{{\cot }^{-1}}\sqrt{\frac{1+x}{1-x}}, $ then $ \frac{dy}{dx} $ is equal to

In a certain town $25\%$ families own a cell phone, $15\%$ families own a scooter and $65\%$ families own neither a cell phone nor a scooter. If $1500$ families own both a cell phone and a scooter, then the total number of families in the town is

In the expansion of $ {{(1+x+{{x}^{2}}+{{x}^{3}})}^{6}}, $ the coefficient of $ {{x}^{14}} $ is

Let $A (6, -1), B (1, 3)$ and $C (x, 8)$ be three points such that $AB = BC$. The values of $x$ are

The domain of the function $f\left(x\right) = sin^{-1}\left(\frac{x+5}{2}\right)$ is

The general solution of the differential equation $(x + y + 3) \,\frac{dy}{dx}\, =\,1$ is

The principal argument of the complex numb $Z=\frac{1+\sin \frac{\pi}{3}+i \cos\frac{\pi}{3} }{1+\sin \frac{\pi}{3} - i \cos\frac{\pi}{3} }$ is

The set $\{(x, y) : x + y =1\}$ in the $xy$ plane represents

The value of $\frac{\sqrt{3}}{\sin15^{\circ}} - \frac{\sqrt{1}}{\cos15^{\circ}}$ is equal to

If the mean of the numbers $a, b, 8,5,10$ is $6$ and their variance is $6.8$, then $ab$ is equal to

If the position vectors of three consecutive vertices, of a parallelogram are $ \vec{i}+\vec{j}+\vec{k}, $ $ \vec{i}+3\vec{j}+5\vec{k} $ and $ 7\vec{i}+9\vec{j}+11\vec{k}, $ then the coordinates of the fourth vertex are

If the standard deviation of $3$, $8$, $6$, $10$, $12$, $9$, $11$, $10$, $12$, $7$ is $2.71$, then the standard deviation of $30$, $80$, $60$, $100$, $120$, $90$, $110$, $100$, $120$, $70$ is

If $ \sqrt{x+iy}=\pm (a+ib), $ then $ \sqrt{-x-iy} $ is equal to

If $ x $ satisfies the in equations $ 2x-7<11 $, $ 3x+4

If $ y={{\sin }^{-1}}(3x-4{{x}^{3}})+{{\cos }^{-1}}(4{{x}^{3}}-3x) $ $ +{{\tan }^{-1}}(e), $ then $ \frac{dy}{dx} $ is equal to

Let $ \alpha $ and $ \beta $ be the roots of $ a{{x}^{2}}+bx+c=0 $ . Then, $ \underset{x\to \alpha }{\mathop{\lim }}\,\frac{1-\cos (a{{x}^{2}}+bx+c)}{{{(x-\alpha )}^{2}}} $ is equal to

Standard deviation of first $n$ odd natural numbers is

The $A$.$M$. of $9$ terms is $15$. If one more term is added to this series, then the $A$.$M$. becomes $16$. The value of the added term is

The angle between the straight lines $x-1=\frac{2y+3}{3}=\frac{z+5}{2}$ and $x-3r+2; y=-2r-1; z=2,$ where $r$ is a parameter, is

The argument of the complex number $ \left( \frac{i}{2}-\frac{2}{i} \right) $ is equal to

The locus of a point which is equidistant from the points $(1,1)$ and $(3, 3)$ is