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The value of $‘a^{\prime }$ in the following equation is:

${\left[\frac{9}{2}\right]}^{-3}\times {\left[\frac{2}{9}\right]}^{-6}={\left[\frac{2}{9}\right]}^{2a-1}$

If f:
R → R be given by,
then fof(x) is

(A) (B) x³ (C) x (D) (3
− x³)

Show that set of all points such that the difference of their distance from (4,0), and (-4,0) is always equal to 2 represents a hyperbola.

The coordinates of two consecutive vertices $A$ and $B$ of a regular hexagon $ABCDEF$ are $(1,0)$ and $(2,0),$ respectively. Then the equation of the diagonal $CE$ is

The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :

The values of 'm' for which $(m-2)x^2+8x+(m+4)$ > 0 $\forall$ x $ϵ$ R, are

The domain of the function $f\left(x\right)=\frac{\sin \left(\log \right(x^2-4x+4\left)\right)}{\left(\right[x]-1)}$, where $[.]$ denotes the greatest integer function, is

The value of integral $I=\underset{5\pi }{\overset{\frac{21\pi }{2}}{\int }}\sin 2xdx$ is equal to

One bisector of the angle between the lines given by $a(x-1{)}^2+2h(x-1)y+b{y}^2=0$ is 2x + y - 2 = 0.The other bisector is

If $f:[1,\infty )\to [0,\infty )andf\left(x\right)=\frac{x}{1+x}$ then f is

Let $f\left(x\right)={\int }_0^x{e}^{t}f\left(t\right)dt+e^x$ be a differentiable function for all $x\in R$. Then $f\left(x\right)$ equals.

The equation of diameter which bisects the chord $3x+y+5=0$ of the circle $x^2+y^2=16$ is

The value of $\underset{n\to \infty }{lim}\frac{n}{(n!{)}^{1/n}}$ is equal to

Rotate the red point about the black point 90 degrees counter clockwise.

Let
U ={1,
2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C =
{3, 4, 5, 6}. Find

(i)

(ii)

(iii)

(iv)

(v)

(vi)

The value of tan ${130}^{\circ }tan{140}^{\circ }$ is equal to

If $a,b,c$ are the sides of a triangle $ABC$ opposite to angles $A,B,C$ respectively and angle $C$ is ${90}^{\circ },$ then $\tan A+\tan B$ is equal to

What is the fraction represented by the unshaded portion?

Let $y=f\left(x\right)$ be defined parametrically as $y=t^2+t|t|,x=2t-|t|,t\in R$ and $f\left(x\right)=k$ has at least one real solution, then

Choose the correct option for the underlined phrase.

Meenakshi, a talented carnatic singer, was offered a chance to sing in the movie.

The locus of point of intersection of two tangents to $y^2=4ax$ at $t$ and $2t$ on the parabola is

If normal at point $(6,2)$ to the ellipse passes through its nearest focus $(5,2),$ having centre at $(4,2),$ then its eccentricity is

Calculate the mean deviation about the median of the following observations :

(i) 3011, 2780, 3020, 2354, 3541, 4150, 5000

(ii) 38, 70, 48, 34, 42, 55, 63, 46, 54, 44

(iii) 34, 66, 30, 38, 44, 50, 40, 60, 42, 51

(iv) 22, 24, 30, 27, 29, 31, 25, 28, 41, 42

(v) 38, 70, 48, 34, 63, 42, 55, 44, 53, 47

If $\frac{1+\sin 2x}{1-\sin 2x}={\cot }^2(a+x)$$\forall x\in R-(n\pi +\frac{\pi }{4}),$ $n\in N$ then the possible value of $a$ is

If $\int \frac{d\theta }{\left({\cos }^2\theta \right(\tan 2\theta +\sec 2\theta )}=\lambda \tan \theta +2{\log }_e|f\left(\theta \right)|+C$ where $C$ is constant of integration, then the ordered pair $(\lambda ,f(\theta \left)\right)$ is equal to:

The mean deviation about the median for the following data $3,7,8,9,4,6,8,13,12,10$ is:

Which of the following relations are transitive?

The value of $0.7+0.77+0.777+\dots \dots \text{upto}20\text{terms}$ is

If $\underset{-2}{\overset{5}{\int }}f\left(x\right)dx=4$ and $\underset{3}{\overset{5}{\int }}(2-f(x\left)\right)dx=6,$

The solution set of the inequation $|x+2|\le 5$ is

The probability that a certain beginner at golf gets good shot if he uses correct club is $\frac{1}{3},$ and the probability of a good shot with an incorrect club is $\frac{1}{4}.$ In his bag there are 5 different clubs only one of which is correct for the good shot. If he chooses a club at random and take a stroke, the probability that he gets a good shot is

Observe the graph:





Where the output has the highest increase, the required domain is

If 3 is root of the quadratic equation $x^2-x+k=0$, find the value of p so that the roots of the equation $x^2+k(2x+k+2)+p=0$ are equal.