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If $sin\frac{x}{2}=sin\beta$
where $-\frac{\pi }{2}\le \beta \le \frac{\pi }{2}$ then,

The general solution of the inequality $-2\le \frac{1-x}{4}<3$ is

If $\alpha$ and $\beta$ are the roots of the equation, $x^2+x\sin \theta -2\sin \theta =0,\theta \in (0,\frac{\pi }{2})$ , then $\frac{{\alpha }^{12}+{\beta }^{12}}{({\alpha }^{-12}+{\beta }^{-12})(\alpha -\beta {)}^{24}}$ is equal to

Let a square with side length ${}^{\prime }{p}^{\prime }$ and making an angle of $\theta$ with $x-$ axis, has one vertex at origin. If $0<\theta <\frac{\pi }{2}$, then the equation of the diagonals of the square is

Let $f:[0,\infty )\to$ [0,2] be defined by $f\left(x\right)=\frac{2x}{1+x}$ , then f is

Perpendicular are drawn from points on the line $\frac{x+2}{2}=\frac{y+1}{-1}=\frac{z}{3}$ to the plane x + y + z = 3. The feet of perpendiculars lie on the line

If n > 1, the value of $\frac{1}{lo{g}_{2}n}+\frac{1}{lo{g}_{3}n}+....+\frac{1}{lo{g}_{53}n}$ is

What is the principal solution of $sinx+\sqrt{sinx}=0?$

$tan{20}^{\circ }+tan{40}^{\circ }+\sqrt{3}tan{20}^{\circ }tan{40}^{\circ }$ =

If $(A\times A)$ has 9 elements two of which are (-1,0) and (0,1), find the set A and the remaining elements of $(A\times A)$

${}^n{P}_r÷{}^n{C}_r$ =

Ammonium Hydrogen sulphide dissociates according to the equation,

$N{H}_{4}HS$(s) $\leftrightharpoons$ $N{H}_3$(g) + $H_{2}S$(g) if the observed pressure of the mixture is 1.12 atm at ${106}^{\circ }$C. What is the KP of the reaction -

If $(1+x{)}^{15}$ = $C_0+C_{1}x+C_2{x}^2+.........+C_{15}{x}^{15}$, then

$C_2+2C_3+3C_4+........+14C_{15}$=

Let $y=y\left(x\right)$ be the solution of the differential equation, $x\frac{dy}{dx}+y=x{\log }_{e}x,(x>1).$ If $2y\left(2\right)={\log }_{e}4-1,$ then $y\left(e\right)$ is equal to:

Find the set of values of $\alpha$ for which point the $P(\alpha ,-\alpha )$ is inside

$\frac{x^2}{16}+\frac{y^2}{9}=1$

The number of subsets of the power set of a singleton set is

Coefficient of $x^{25}$ in $(1+x+x^2+x^3+...+x^{10}{)}^7$ is:

The number of ways in which $5$ balls can be selected from a bag containing $5$ identical and $5$ different balls is

Which of the following options holds true for the system of equations,

x + y + z =6

x + 2y + 3z = 12

x + 4y + 7z =30

If $a_1,a_2,a_3,\cdots ,a_n$ are in A.P. and $a_1+a_4+a_7+\cdots +a_{16}=114$ , then $a_1+a_6+a_{11}+a_{16}$ is equal to :

What is the polar of the point (2, 3) with respect to the circle $x^2+y^2-2x-4y-4=0$

The value of $\underset{x\to 0}{lim}\frac{(1+x{)}^{\frac{1}{x}}-e+\frac{1}{2}ex}{x^2}$ is ---------

$C_0-C_1+C_2-C_3+........+(-1{)}^n{C}_n$ is equal to

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

Which of the following intervals are subsets of the interval $(3,11]$

The equation of second degree $x^2+2\sqrt{2}xy+2y^2+4x+4\sqrt{2}y+1=0$ represents a pair of straight lines.The distance between them is

If the truth value of the Boolean expression $\left(\right(p\vee q)\wedge (q\to r)\wedge (\sim r\left)\right)\to (p\wedge q)$ is false, then the truth values of the statements $p,q,r$ respectively can be

Let $P\left(z\right)$ be a point in complex plane satisfying $z\overline{z}+(4-5i)\overline{z}+(4+5i)z=40.$ If $a=\text{max}|z+2-3i|$ and $b=\text{min}|z+2-3i|,$ then

One of the two events must occur. If the chance of one is $\frac{2}{3}$ of the other, then odds in favour of the other are

The mean of the series $x_1,x_2,x_3,.......,x_n$ is $\overline{x}$. If $x_2$ is replaced by $\lambda$, then the new mean is

An aeroplane can carry a maximum of 200 passengers. A profit of Rs 1000 is made on each executive class ticket and a profit of Rs 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class and 80 for economy class, then the number of tickets of each class must be sold in order to maximise the profit for the airline is

$[$ where $n\left(E\right)=$ number of executive class tickets and $n\left(E^{\prime }\right)=$ number of economy class tickets $]$

If $\left|\frac{|x|-2}{7-2|x|}\right|=1$, then $x$ can be

Suppose $a,b,c$ are in A.P. and $a^2,b^2,c^2$ are in G.P.. If $a<b<c$ and $a+b+c=\frac{3}{2}$ , then value of a is

How many values of $\theta ϵ[0,\frac{\pi }{2}]$, satisfy the relation cos $\theta +cos3\theta +cos5\theta +cos7\theta =0$ ?

Locus of the point whose sum of distances from the origin and the $x-$ axis is $4$ units is

$If\alpha ,\beta \text{are the roots of the equation}{x}^2-x-1=0\text{and}{A}_n={\alpha }^n+{\beta }^n\text{then}{A}_{n+2}+A_{n-2}=--$

If x < 0, then find the value of

$2\left({\tan }^{-1}\frac{1}{x}+{\tan }^{-1}x\right)$

The eccentricity of the ellipse, whose end points of major axis and minor axis are $(\pm \sqrt{5},0)$ and $(0,\pm 1)$ respectively, is

A committee of $5$ men and $3$ women is to be formed out of $7$ men and $6$ women. If two particular women are not to be included together in the committee, then the number of committees that can be formed is

How many of the following are matched correctly?

Degree measurement Radian measurement

(A) ${180}^{\circ }$ (1) $\pi$
(B) ${60}^{\circ }$ (2) $\frac{\pi }{6}$
(C) $0^{\circ }$ (3) 0
(D) ${120}^{\circ }$ (4) $\frac{2\pi }{6}$
(E) ${360}^{\circ }$ (5) $2\pi$
(F) ${30}^{\circ }$ (6) $\frac{\pi }{3}$
(G) ${90}^{\circ }$ (7) $\frac{\pi }{2}$
(H) ${45}^{\circ }$ (8) $\frac{\pi }{4}$
(I) ${270}^{\circ }$ (9) $3\pi$

Total number of values in $(-2\pi ,2\pi )$ and satisfying
$lo{g}_{|cosx|}|sinx|+lo{g}_{|sinx|}|cosx|=2$ is

Find the sum of the series s = 1 + $\frac{1}{2}$(1 + 2) + $\frac{1}{3}$(1 + 2 + 3) + $\frac{1}{4}$(1 + 2 + 3 + 4) + .........upto 40 terms.

Which of the following is the graph of the function $y=e^x-1$