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If two distinct tangents can be drawn from the point $(\alpha ,2)$ on different branches of the hyperbola
$\frac{x^2}{9}-\frac{y^2}{16}=1$, then

The graph of $f\left(x\right)=-x^2+x-2$ is

In parametric representation of a standard hyperbola in $\theta$ terms, if $x=asec\theta$. What is y.

The mean deviation about the mean for the following data :

$\begin{array}{cccccccc}{x}_i& 2& 3& 4& 5& 6& 7& 8\\ f_i& 5& 2& 3& 4& 5& 4& 2\end{array}$

is

1. 1.333

Let f(x) = cos 2$\pi$ x+x-[x] ([.] denotes the greatest integer function). Then number of points in [0, 10] in which f (x) assume its local maximum value, is

Find the coordinates of the focus, axis of the parabola, the equation of directrix and the length of the latus rectum for y² = 10x

The number of complex numbers $z$ such that $|\frac{z}{\overline{z}}+\frac{\overline{z}}{z}|=1$ and $|z|=1$ is:

Locus of a point, whose chord of contact with respect to the circle $x^2+y^2=4$ is a tangent to hyperbola $xy=1$ is :

If $p$ and $q$ are the lengths of the perpendiculars from the origin on the lines, $x\text{cosec}\alpha -y\sec \alpha =k\cot 2\alpha$ and $x\sin \alpha +y\cos \alpha =k\sin 2\alpha$ respectively, then $k^2$ is equal to :

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:

The value of $\underset{1}{\overset{2}{\int }}\sqrt{(x-1)(2-x)}dx$ is

The value of 'a' for which the point (-a,a) lies inside the circle $x^2+y^2-4x+2y-8=0$ is .

Primitive of $f\left(x\right)=x{.2}^{In(x^2+1)}$ with respect to x is

Find the
mean deviation about the mean for the data

38, 70,
48, 40, 42, 55, 63, 46, 54, 44

A circle of radius 4 units touches the coordinate axes in the first quadrant. Find the equations of its images with respect to the line mirrors x = 0 and y 0.

In a triangle $ABC,AD$ is altitude from $A,b>c,\angle C={23}^0,AD=\frac{abc}{b^2-c^2},$ then $\angle B=$

Direction cosines of the line bisecting the obtuse angle between lines whose direction ratios are $(-1,0,1)$ and $(0,1,-1),$ is

The value of ${\int }_{-\pi }^{\pi }\frac{co{s}^{2}x}{1+a^x}dx,a>0is$

Which of the following is a unit vector in the direction of $\hat{i}+\hat{j}+\hat{k}$.

The solution of the differential equationis

[MP PET 1993; AISSE 1985]

Let

$p:2$ is a prime number

$q:\cos {30}^{\circ }=\frac{1}{2}$

$r:{\sec }^{2}x+{\tan }^{2}x=1$

$s:\sqrt{7}$ is an irrational number

$u:{\pi }^2$ is greater than $10$



Then which among the following statements are valid

The value of c, for which the function $f\left(x\right)=x^2+2x–8,xϵ[–4,2]$ satisfies Rolle’s theorem, is

Consider the following parametric equation of a curve :
$x\left(\theta \right)=|\cos 4\theta |\cos \theta y\left(\theta \right)=|\cos 4\theta |\sin \theta$
for $0\le \theta \le 2\pi$. Which one of the following graphs represents the curve?