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The curve $y-e^{xy}+x=0$ has a vertical tangent at the point

The range of $f\left(x\right)=-x^2+7x+60$ in $x\in [-3,2]$ is

If |z - 3i| = 3, (where i = $\sqrt{-1}$) and argz $\in (0,\frac{\pi }{2})$, then cot(arg(z)) - $\frac{6}{z}$ is equal to

How many of the following statements are true about the identity function?

(a) It's graph is shown above.

(b) Domain is R

(c) Range is R

(d) Its an even function

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A rectangle of maximum area is inscribed in the circle $|z-3-4i|=1$. If one vertex of the rectangle is $4+4i,$ then another adjacent vertex of this rectangle can be

Which of the following is logically equivalent to $\sim (\sim p\Rightarrow q)$

Let $P=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \alpha \\ 3& -5& 0\end{array}\right]$, where $\alpha \in R$. Suppose $Q=\left[q_{ij}\right]$ is a matrix such that $PQ=kI$, where $k\in R,k\ne 0$ and $I$ the identity matrix of order $3$. If $q_{23}=-\frac{k}{8}$ and $det\left(Q\right)=\frac{k^2}{2}$, then:

If complex numbers $z_1$ and $z_2$ both satisfy $z+\overline{z}=2|z-1|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then find the value of $\text{Im}(z_1+z_2)$. (where $\text{Im}\left(z\right)$ denotes the imaginary part of $z$)

If $\cos \frac{A}{2}=\sqrt{\frac{b+c}{2c}}$, then :

If x, y, z are in G.P. and $a^x$ = $b^y$ = $c^z$, then

(i) Evaluate ${lim}_{x\to 1}\frac{x+x^2+x^3+...+x^n-n}{x-1}$

(ii) Find the derivative $\text{sqrt{sin x}}$ from first principle.

Consider the parabola whose focus at $(0,0)$ and tangent at vertex is $x-y+1=0$.

The length of chord of a parabola on the $x-$axis is

Let $\alpha ,\beta$ are the roots of the equation $2x^2-3x-7=0$, then the quadratic equation whose roots are $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ is

Perpendicular distance of the point (3, 4, 5) from the y-axis, is [MP PET 1994, Pb. CET 2002]

The domain of the function $f\left(x\right)={\log }_{10}{\log }_{10}{\log }_{10}{\log }_{10}x$ is

The square root of $-1+2\sqrt{2}i$ is

If the line, $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane, lx+my-z = 9, then $l^2+m^2$ is equal to

The probability that the ${13}^{th}$ day of a randomly chosen month is a Friday, is

If $f\left(x\right)$ is differentiable and ${\int }_0^{t^2}xf\left(x\right)dx=\frac{2}{5}{t}^5,\text{then}f\left(\frac{4}{25}\right)$ equals

How many of the following are matched correctly?

$\begin{array}{cc}\text{Degree measurement}& \text{Radian measurement}\\ \text{(A)}{180}^{\circ }& \text{(1)}\pi \\ \text{(B)}{60}^{\circ }& \text{(2)}\frac{\pi }{6}\\ \text{(C)}{0}^{\circ }& \text{(3)}0\\ \text{(D)}{120}^{\circ }& \text{(4)}2\frac{\pi }{6}\\ \text{(E)}{360}^{\circ }& \text{(5)}2\pi \\ \text{(F)}{30}^{\circ }& \text{(6)}\frac{\pi }{3}\\ \text{(G)}{90}^{\circ }& \text{(7)}\frac{\pi }{2}\\ \text{(H)}{45}^{\circ }& \text{(8)}\frac{\pi }{4}\\ \text{(I)}{270}^{\circ }& \text{(9)}3\pi \end{array}$
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Let ‘head’ means one and ‘tial’ means two and the coefficients of the equation $a{x}^2+bx+c=0$ are chosen by tossing a coin. The probability that the roots of the equation are non – real, is equal to:

Solve:$2co{s}^{2}x+3sinx=0$

Find the value of ${\sec }^{2}x$ - ${\text{cosec}}^{2}x$.

If $A=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$, then the matrix $A^{-50}$ when $\theta =\frac{\pi }{12}$, is equal to :

The area (in sq. units) of the region

$A=\text{{}(x,y):|x|+|y|\le 1,2y^2\ge |x|\text{}}$ is :

Real part of $(1-\cos \theta +2i\sin \theta {)}^{-1}$ is:

If the line $x+2y+4=0$ cutting the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points whose eccentric angles are ${30}^{\circ }$ and ${60}^{\circ }$ subtends a right angle at the origin then its equation is

If \omega = $\alpha$ + i$\beta$ where $\alpha$, $\beta$ are real, $\beta$ $\ne$ 0 and z $\ne$ 1 satisfies the condition that $\frac{\omega -\overline{\omega }z}{1-z}$ is purely real then the set of values of z is

The equation whose roots are the values of $r$ satisfying the equation ${}^{69}{C}_{3r-1}-{}^{69}{C}_{r^2}={}^{69}{C}_{r^2-1}-{}^{69}{C}_{3r}$ is

Insert 2 no.s between 1 & $\frac{1}{3}$ so that the sequence becomes an Harmonic progression --

If $A=\{x:x$ is a letter of the word 'RAMANA'$\}$,

$B=\{x:x$ is a letter of the word 'MISSISSIPPI'$\}$,

$C=\{x:x$ is a letter of the word 'NOOKBOOK'$\}$,

Then relation between cardinality of sets $A,B$ and $C$ is

If $z$ is a complex number, then the number of solution(s) for the equation $z^2=\overline{z}$ is

The maximum area (in sq. units) of a rectangle having its base on the $x$-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is:

A solid sphere of radius 20 cm is subjected to a uniform pressure of ${10}^{6}N{m}^{-2}$. If the bulk modulus of the solid is $1.7\times {10}^{11}N{m}^{-2}$, the decrease in the volume of the solid is approximately equal to

If $z_1,z_2,z_3$ are the solutions of $z^2+\overline{z}=z,$ then $z_1+z_2+z_3$ is equal to

($z$ is a complex number on the Argand plane and $i=\sqrt{-1}$)

Two statements p and q are given below

p: It is snowing
q: I am cold

The compound statement "It is snowing and it is not that I am cold" is given by

The circle $x^2+y^2-8x=0$ and hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ intersect at the points A and B.

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

2 tangents $P{T}_1$ and $P{T}_2$ are drawn as shown in the figure and PQ passes through the centre O of the circle. Which segment(s) is/are the chord of contact of P.

Mean of 100 observations is 45. It was later found that two observations 19 and 31 were incorrectly recorded as 91 and 13. The correct mean is

1. 5

The value of the expression ${\cos }^4\frac{\pi }{8}+{\cos }^4\frac{3\pi }{8}+{\cos }^4\frac{5\pi }{8}+{\cos }^4\frac{7\pi }{8}$ is

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{ax+b}{cx+1}$

The set of real values x satisfying ${\log }_{0.3}$ (x-3)>3.

Describe the sample space for the indicated experiment.
2 boys and 2 girls are in a Room X and 1 boy and 3 girls in Room Y. Specify the sample space for the experiment in which a room is selected and then a person.