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The radian measure of $-{37}^{\circ }{30}^{\prime }$ is

Three vertices of a parallelogram, taken in order ,are (-1,-6),(2,-5) and (7,2). Write the coordinates of its fouth vertex.

The coordinates of a point on the hyperbola, $\frac{x^2}{24}-\frac{y^2}{18}=1,$ which is nearest to the line $3x+2y+1=0$ are

Find the equation of the bisector of angle A of the triangle whose vertices are A (4, 3), B (0, 0) and C (2, 3).

If $-1+\sqrt{-3}=r{e}^{\theta }$ , then θ is equal to

Letters of the word "EDUCATION" is arranged. What is the probability that vowels and consonants are in alphabetical order?

If f, g, h are real functions defined by $f\left(x\right)=\sqrt{x+1},g\left(x\right)=\frac{1}{x}$ and $h\left(x\right)=2x^2-3$, then find the values of (2f + g - h) (1) and (2f + g - h) (0).

If $sin{\theta }_1+sin{\theta }_2+sin{\theta }_3=3$, then write the value of $cos{\theta }_1+cos{\theta }_2+cos{\theta }_3.$

$\alpha$ + $\beta$ - $\gamma$ = $\pi$ Then $si{n}^2\beta$ - $si{n}^2\gamma$ =

Thirty-two players ranked 1 to 32 are playing in a knockout tournament. Assume that in every match between any two players, the better-ranked player wins, the probability that ranked 1 and ranked 2 players are winner and runner up respectively, is

If $a\in R$ and the equation $-3(x-[x]{)}^2+2(x-\left[x\right])+a^2=0$, (where, $\left[x\right]$ denotes the greatest integer $\le x$) has no integral solution, then all the possible values of $a$ lie in the interval :

$\underset{x\to 1}{lim}sin|||x|-2|-3|is$

Let the first term $a$ of an infinite G.P. is the value of $x$, where the function $f\left(x\right)=7+2x{\log }_{e}25-5^{x-1}-5^{2-x}$ has the greatest value and the common ratio $r$ is equal to $\underset{x\to 0}{lim}\underset{0}{\overset{x}{\int }}\frac{t^2}{x^2\tan (\pi +x)}dt$. Also, let $S$ be the sum of infinite terms of G.P.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& a& \text{(P)}& 4\\ \text{(B)}& \frac{1}{r}& \text{(Q)}& 3\\ \text{(C)}& S& \text{(R)}& 2\\ \text{(D)}& a-rS& \text{(S)}& 1\\ & & \text{(T)}& 5\end{array}$

Which of the following is the only CORRECT combination?

The equation of the straight line passing through the point of intersection of $x+2y=5$ and $3x+7y=17$ and perpendicular to the straight line $3x+4y=10$ is

For what values of x, [1 2 1] $\left(\left[\begin{array}{ccc}1& 2& 0\\ 2& 0& 1\\ 1& 0& 2\end{array}\right]\left[\begin{array}{c}0\\ 2\\ x\end{array}\right]\right)=0$ ?

If the orthocenter of a triangle is $(6,3)$ and centroid is $(2,5)$, then find the circumcenter of the triangle.

The circle $C$ passing through the origin, has the line $3x+4y=0$ as its tangent at the origin. If the image of the centre of $C$ w.r.t the line $\frac{x}{\frac{25}{3}}+\frac{y}{\frac{25}{4}}=1$ lies on the line $3x+4y=0,$ then the equation of $C$ is

Compute the indicated product.
$\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]\left[234\right]$

Let A={-1,0,1,2}, B={-4,-2, 0,2} and $f,g:A\to B$ be the function defined by $f\left(x\right)=x^2-x,x\in A$ and $g\left(x\right)=2|x-\frac{1}{2}|-1,x\in A.$
Are f and g equal ? Justify your answer.

z_{​​​​​​1} ,z_{​​​​​​2} , z_{​​​​​​3} are complex numbers such that the modulus of each number is equal to the modulus of the sum of reciprocals of these numbers , which is equal to 1. Find the modulus of sum of these three numbers

The domain of the function $f\left(x\right)={\log }_2\left({\log }_3\right({\log }_4(x^2-3x+6)\left)\right)$ is

Find the value of $co{s}^{-1}\left(sin\frac{\pi }{9}\right)$ .

Choose the word which is MOST SIMILAR in MEANING to the word as used in the passage:

Embraced

If the line $\frac{x-x_1}{a_1}=\frac{y-y_1}{b_1}=\frac{z-z_1}{c_1}$ makes an angle $\theta$ with the plane $a_{2}x+b_{2}y+c_{2}z=d$, then = $\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$

If two vertices of a triangle are (1,3) & (4,-1) and the area of the triangle is 5sq. units, then the angle at the third vertex lies in

A. (0,2tan^-15/4). B. (0,tan^-15/4)

The value of is equal to

The equation $2^{2x}+(a-1)2^{x+1}+a=0$ has roots reciprocal to each, then set of values of 'a' is

If $a^{1/m}=b^{1/n}=c^{1/p}$ and abc = 1 , then m + n + p is equal to

If ${\alpha }_0,{\alpha }_1,{\alpha }_2\dots ,{\alpha }_{n-1}$ be the n, nth roots of the unity, then the value of ${\sum }_{i=0}^{n-1}\frac{{\alpha }_i}{(3-{\alpha }_i)}$ is equal to

Let $P(4,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at $P$ intersects the X-axis at $(16,0)$, then the eccentricity of the hyperbola is

If $\underset{x\to \infty }{lim}x\ln \left\{\left|\begin{array}{ccc}\frac{a}{x}& 1& c\\ 0& \frac{1}{x}& b\\ 1& 0& \frac{1}{x}\end{array}\right|\right\}=-5,$ where $a,b$ and $c$ are finite real numbers, then

The greatest positive integer $k$, for which ${49}^k+1$ is a factor of the sum ${49}^{125}+{49}^{124}+\cdots +{49}^2+49+1,$ is

There are five students $S_1,S_2,S_3,S_4$ and $S_5$ in a music class and for them there are five seats $R_1,R_2,R_3,R_4$ and $R_5$ arranged in a row, where initially the seat $R_i$ is allotted to the student $S_i,i=1,2,3,4,5$. But, on the examination day, the five students are randomly allotted the five seats.

The probability that, on the examination day, the student $S_1$ gets the previously allotted seat $R_1$, and $NONE$ of the remaining students gets the seat previously allotted to him/her is

If $f\left(x\right)=log\left(\frac{1+x}{1-x}\right)$ and $g\left(x\right)=\frac{3x+x^3}{1+3x^2}$, then f(g(x)) is equal to

Let from any point $P$ on the line $y=x,$ two tangents are drawn to the circle $(x-2{)}^2+y^2=1.$ Then the chord of contact of $P$ with respect to given circle always passes through a fixed point, whose coordinates are given by

X=(√5-2)÷(√5+2).find (I)x^2.

There are total ${}^{17}{C}_6-k{}^{11}{C}_5$ ways to get a sum of atmost $17$ by throwing six distinct dice ,then $k$ is equal to :

If $y=\log \left(\log x\right)$ then $\frac{d^{2}y}{d{x}^2}$ is equal to