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A straight line makes an angle $60°$ with the positive x- axis then its slope is

${\int }_0^{\frac{\pi }{2}}{e}^{x}sinxdx=$ [Roorkee 1978]

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball, the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle, then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is:

If $co{s}^{3}x.sin2x={\sum }_{x=0}^n{a}_{r}sin\left(\pi x\right),\forall x\in R$, then

In a lottery, a person chooses six different numbers at random from 1 to 20. If these six numbers match with the six numbers already fixed by the lottery committee he wins the prize. What is the probability of winning the prize in the game ?

If $a{x}^2+bx+c=0$ and $b{x}^2+cx+a=0$ have a common root

a ≠ 0, then $\frac{a^3+b^3+c^3}{abc}=$

If $2\times {}^n{C}_5$ = $9\times {}^{n-2}{C}_5$, then the value of n will be

Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$

for every real number $x$.

If $f\left(x\right)$ is strictly increasing function, then $h\left(x\right)$ is non-monotonic function given

Let $a={\log }_3{\log }_{3}2$. An integer $k$ satisfying $1<2^{(3^{-a}-k)}<2$ is

The equation of the line which is at a distance of $3$ units from the origin and the perpendicular from the origin makes an angle of ${30}^{\circ }$ with positive $x-$axis is

We have to choose $11$ players for cricket team from $8$ batsmen, $6$ bowlers, $4$ all rounders and $2$ wicket keepers. Number of selections, when two particular batsmen do not want to play when a particular bowler will play, is

$P\left(n\right):a^{2n}-b^{2n}\text{is divisible by}a+b,\forall nϵN$

To prove P(n) using mathematical induction, the base case is

If $a=cos2\alpha +isin2\alpha ,b=cos2\beta ,c=cos2\gamma +isin2\gamma$ and $d=cos2\delta +isin2\delta$, then $\sqrt{abcd}+\frac{1}{\sqrt{abcd}}$ is equal to

Match the following equation of parabola

$\begin{array}{cc}\text{Parabola}& \text{parametric equation}\\ \text{N)}{x}^2=4ay& \text{1)}(a{t}^2,2at)\\ \text{E)}{y}^2=4ax& \text{2)}(-at,2at)\\ \text{W)}{y}^2=-4ax& \text{3)}(2at,a{t}^2)\\ \text{S)}{x}^2=-4ay& \text{4)}(2at,a{t}^2)\end{array}$

If $|z-\frac{4}{z}|=2$, then the greatest value of $|z|$ is

Value(s) of $x$ for $\frac{|x^2-2|x|+1|}{|x|+1}=2$ is/are

If a circle and rectangular hyperbola $xy=c^2$ meet in the four points $t_1,t_2,t_3\&t_4$ . Then which of the following statements are correct?
1. The center of the mean position of the four points bisects the distance between the center of the two curve.
2. Center of the circle through the points $t_1,t_2\&t_3$ is:$\left[\right(\frac{c}{2}(t_1+t_2+t_3)+\frac{1}{t_1.t_2.t_3}),\frac{c}{2}(\frac{1}{t_1}+\frac{1}{t_2}+\frac{1}{t_3}+t_1{t}_2{t}_3\left)\right]$

Differentiate $sin2xcos3x.$

A wire elongates by $1.0\text{mm}$ when a load $W$ is hanged from it. If this wire goes over a pulley, and two weights W each are hung at the two ends, the elongation of the wire will be

Solution set of $(x^2-1)(x^3-1)(x^4-1)>0$ is

If a triangle has its orthocentre at $(1,1)$ and circumcentre at $(\frac{3}{2},\frac{3}{4}),$ then the coordinates of the centroid of the triangle are

f(x) = $\frac{(x-b)(x-c)}{(x-a)}$, where a,b,c are distinct real numbers, will assume all real values provided :

If $cos2B=\frac{cos(A+C)}{cos(A-C)}$ then

The value of ${\cos }^2(\frac{\pi }{8}+x)-{\sin }^2(\frac{3\pi }{8}-x)$ is

R is relation over the set of integers and it is given by (x, y) $ϵ$ R $\iff$ R |x - y| $\le$ 1. Then, R is

if $2co{s}^{-1}\sqrt{\frac{1+x}{2}}=\frac{\pi }{2}$, then x=

What is the point of contact between the hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and
the tangent $y=mx\pm \sqrt{a^2{m}^2-b^2}$.

If the $4^{th},{10}^{th}and{16}^{th}$ terms of a G.P, are x, y and z respectively. Prove that x, y, z are in G.P

Let y = $\frac{x^2+3x+1}{x^2+x+1}$ ∀ x $\in$ R , then

The orthocentre of the triangle formed by the lines xy = 0 and x + y = 1 is

Find the principal solution of $\sqrt{\frac{1+2sinx}{2}}=1$

In a class of $60$ students, $40$ opted for NCC, $30$ opted for NSS and $20$ opted for both NCC and NSS. If one of these students is selected at random, then the probability that the student selected has opted neither for NCC nor for NSS is :

If $2x+y=p$ is a chord to the parabola $y^2=16x$ whose midpoint is $(h,k)$, then which of the following is/are true?

The number of times the digit 3 will be written when listing the integers from 1 to 1000 is

The domain of the function $f\left(x\right)=e^{\left(\sqrt{5x-3-2x^2}\right)}$ is

The parabola $y^2=4ax$ passes through the centre of the circle $4x^2+4y^2-8x+12y-7=0$. The directrix of the parabola will be

Find the derivative of $f\left(x\right)=(x^3-2x{)}^2$

Calculate the mean and the variance of first n natural numbers.

Which of the following definite integrals reduces to $\frac{\pi }{2}$?

If the curve $x^2+2y^2=2$ intersects the line $x+y=1$ at two points $P$ and $Q,$ then the angle subtended by the line segment $PQ$ at the origin is:

If sum of the coefficients of first, second and third terms in the expansion of ${(x^2+\frac{1}{x})}^m$ is $46,$ then coefficient of the term that is independent of $x$ is

The negation of the statement $(p\wedge q)\to (\sim p\vee r)$ is

Find the sum to n terms

$3\times 8+6\times 11+9\times 14+\dots$

Two particles having position vectors ${\overrightarrow{r}}_1=(3\hat{i}+5\hat{j})m$ and ${\overrightarrow{r}}_2=(-5\hat{i}-3\hat{j})m$ are moving with velocities

$\overrightarrow{v_1}=(4\hat{i}+3\hat{j})m/s$ and ${\overrightarrow{v}}_2=(a\hat{i}+7\hat{j})m/s$.If they collide after 2s, find the value of a.