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The total number of times, the digit ${}^{\prime }{3}^{\prime }$ will be written, when the integers having less than $4$ digits are listed, is

$si{n}^{6}A+co{s}^{6}A+3si{n}^{2}Aco{s}^{2}A=$

Express the complex numbers in the form of a + ib:

${(-2-\frac{1}{3}i)}^3$

If $\sin x+\cos x=\sqrt{y^2+\frac{1}{y^2}}$ for $x\in [0,\pi ]$ has a solution, then which of the following is/are correct?

Let $\alpha$ and $\beta$ are complex numbers satisfying $|\alpha +1+i|=1$ and $|\beta -2-3i|=6$ such that $6|\alpha {|}_{max}-|\beta {|}_{max}=\sqrt{a}-\sqrt{b};a,b\in R^{+}$ then the value of $\sqrt{b^2-2a}$ is

The foot of perpendicular from $A(1,1,1)$ to the line joining $B(-8,5,6)$ and $C(12,1,0)$ lies on the plane(s)

If sin A, cos A and tan A are in geometric progression, then $co{t}^{6}A-co{t}^2$ A is equal to

One of the diameter of the circle circumscribing the rectangle ABCD is 4y = x + 7. If A and B are the points (-3, 4) and (5, 4) respectively, then the area of the rectangle is (in square units)

Let $a,b,x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$. If the complex number $z=x+iy$ satisfies $Im\left(\frac{az+b}{z+1}\right)=y$, then which of the following is(are) possible value(s) of $x$?

A random variable X has the following probability distribution.

$\begin{array}{ccccccccc}X& 0& 1& 2& 3& 4& 5& 6& 7\\ P\left(X\right)& 0& k& 2k& 2k& 3k& k^2& 2k^2& 7k^2+k\end{array}$

P(X > 6)

Integrate the function.
$\int xsinxdx.$

What is the locus of the midpoint of chords of ($x-1{)}^2+y^2=1$ that passes through the origin.

Two circles with equal radii are intersecting at the points $(0,1)$ and $(0,-1)$. The tangent at the point $(0,1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

A function f(x) is such that it is not differentiable at two points h, k in its domain and f’(x) becomes zero at 3 points a, b, c in the domain. The critical points and stationary points will be -

If $3\sin \theta +4\cos \theta =5$, then the value of $4\sin \theta -3\cos \theta$ is

What is an abelian group ? Could you please provide an example as well ?

If $\sin \theta =\frac{3}{5},\cos \varphi =\frac{12}{13}$ where $\theta ,\varphi \in (0,\pi /2)$, then the value of $\tan (\theta +\varphi )$ is

Which of the following is the identity function?

In an A.P., let $T_n$ denotes the $n^{th}$ term and $S_n$ denotes the sum of first $n$ terms. If $T_7=\frac{1}{9}$ and $T_9=\frac{1}{7},$ then the value of $S_{63}$ is

The equation of the ellipse, whose axes are coincident with the co-ordinates axis and which touches the straight lines $3x-2y-20=0$ and $x+6y-20=0,$ is

If the ordered pairs (x, -1) and (5, y) belong to the set (a, b) ; b = 2a - 3}, find the values of x and y.

If 2cosx < $\sqrt{3}$ and x $\in [-\pi ,\pi ]$ then the solution set for x is

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two vectors with $|\overrightarrow{a}|=13,$$|\overrightarrow{b}|=19$ and $|\overrightarrow{a}-\overrightarrow{b}|=22.$ The value of $|\overrightarrow{a}+\overrightarrow{b}|$ is

General values of $x$ for which $\sin 2x+\cos x=0$ is/are :

The polynomial $(x+y{)}^9$ is expanded in decreasing powers of $x.$ The second and third terms have equal values when evaluated at $x=p$ and $y=q,$ where $p$ and $q$ are positive numbers whose sum is one. The value of $p$ is

If $\overrightarrow{a}=2\hat{i}-\hat{j}+\hat{k}$, $\overrightarrow{b}=\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{c}=\hat{i}+3\hat{j}-({\lambda }^2+3\lambda )\hat{k}$ (where $\lambda$ is constant) and $\overrightarrow{a}$ is perpendicular to $\overrightarrow{c}-\lambda \overrightarrow{b}$, then sum of different values of $\lambda$ is -

If $x=2+2^{2/3}+2^{1/3}$, then the value of $x^3-6x^2+6x$ is

Find the integrals of the functions.
$\int \frac{cos2x}{(cosx+sinx{)}^2}dx.$

$tan\frac{7\pi }{6},tan\frac{9\pi }{4}andtan\frac{10\pi }{3}arein$

Classify the following as scalar and vector quantities:
(i) Force

The period of $f\left(x\right)=sin\frac{\pi }{4}\left[x\right]+cos\frac{\pi x}{2}$, where [.] denotes greatest integer function, is___.

Which of the following is/are the equation of a circle?

For the given circles $x^2+y^2-6x-2y+1=0$ and $x^2+y^2+2x-8y+13=0$, which of the following is true

Which of the following statements are correct?

1. The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equal is called an equilateral hyperbola.

2. Eccentricity of equilateral hyperbola = $\sqrt{2}$
3. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=0$

4. Equation of pair of asymptotes of rectangular hyperbola $x^2-y^2=a^2$ is $x^2-y^2=-a^2$

The solution set of the equation $si{n}^{-1}x=2ta{n}^{-1}x$ is

If the roots of $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)$ = 0 are equal then a,b,c are in:

If $f\left(x\right)=\begin{array}{c}1-cos3x,x\ne 0\\ k,x=0\end{array}$ is a continuous function, then the value of $\stackrel{Lim}{x\to 1}\frac{tan(1-x)f(x-1)}{x^2+3x+2}$is.

Given 3 points given by position vectors $\overline{a},\overline{b}$ and $\overline{c}$. The plane which passes through these 3 points can be given by

Find the slope of the normal to the curve $x=aco{s}^3\theta ,y=asi{n}^3\theta at\theta =\frac{\pi }{4}$.

If $Z_1,Z_2$ ∈ C, which of the following is true?

The equation of the plane passing through the points (1,-1,2) and (2, -2 2) and which is perpendicular to the plane 6x -2y +2z =9 is

A horizontal chord of a circle has midpoint (x, y) and midpoint of a vertical chord is (p,q). What is the center of the circle?