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The area (in sq. units) in the first quadrant bounded by the parabola, $y=x^2+1$, the tangent to it at the point $(2,5)$ and the coordinate axes is :

If ${}^n{\text{C}}_4,{}^n{\text{C}}_5$ and ${}^n{\text{C}}_6$ are in A.P., then $n$ can be :

An equation of the curve in which subnormal varies as the square of the ordinate is (k is constant of proportionality)

Let $a,b\in R,a\ne 0,$ such that the equation, $a{x}^2-2bx+5=0$ has a repeated root $\alpha ,$ which is also a root of the equation $x^2-2bx-10=0.$ If $\beta$ is the other root of this equation, then ${\alpha }^2+{\beta }^2$ is equal to:

The area bounded by the curves $y=(x-1{)}^2,$ $y=(x+1{)}^2$ and $y=\frac{1}{4}$ is

If $f\left(x\right)$ is defined on $(-1,1)$ , then the domain of $g\left(x\right)=f\left(e^x\right)+f\left({\log }_e|x|\right))$ is

The equation of pair of tangents drawn to the circle $x^2+y^2-2x+4y+3=0$ from point $(6,-5)$ is

If the double ordinate of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$ passes through the $(5,2),$ then the equation of double ordinate is

$\frac{1+sin{600}^{\circ }-cos{600}^{\circ }}{1+sin{600}^{\circ }+cos{600}^{\circ }}$ = ?

If $x+y=\pi +z$, then $si{n}^{2}x+si{n}^{2}y-si{n}^{2}z=$

If $\frac{1}{x-2}>0$, then $x$ lies in

If the tangent at $P$ on $y^2=4ax$ meets the tangent at the vertex in $Q$, and $S$ is the focus of the parabola, then $\angle SQP=$

If $\alpha and\beta$ are the roots of the equation $4x^2+3x+7=0$, then $\frac{1}{\alpha }+\frac{1}{\beta }=$

The number of terms in the sequence $3,7,11,\dots ,407$ is

Let $f:R\to R$ be such that for all $x\in R,(2^{1+x}+2^{1-x}),f\left(x\right)$ and $(3^x+3^{-x})$ are in A.P., then the minimum value of $f\left(x\right)$ is:

Let $\alpha$ and $\beta$ be the roots of $x^2-6x-2=0$. If $a_n={\alpha }^n-{\beta }^n$ for $n\ge 1$, then the value of $\frac{a_{10}-2a_8}{3a_9}$ is:

The value of $si{n}^{-1}\left\{cot(si{n}^{-1}\sqrt{\left(\frac{2-\sqrt{3}}{4}\right)}+co{s}^{-1}\frac{\sqrt{12}}{4}+se{c}^{-1}\sqrt{2})\right\}$ 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 :

If $f\left(x\right)=sinx+cosx,g\left(x\right)=x^2-1$ then $g\left(f\right(x\left)\right)$ in invertible in the Domain

A set $Y$ is set of all integers $k$ greater than $-10$ and less than $5$ can be represented as:

Find the value of

${\cos }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{5}{13}.$

The general solution of $4{\sin }^{2}x+{\tan }^{2}x+{\text{cosec}}^{2}x+{\cot }^{2}x-6=0$ is

(where $n\in Z$)

The equation of the ellipse having its centre at the point $(2,-3)$, one focus at $(3,-3)$ and one vertex at $(4,-3)$ is

Find the value of $\underset{n\to \infty }{lim}\frac{1+2+3+....n}{n^2}$

If all the roots of $z^3+a{z}^2+bz+c=0$ are of unit modulus, then

Equation of chord of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose mid point is $(x_1,y_1)$ is given by

If the line $3x+4y=12$ is a tangent to the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=2$ then find the point of contact.

If the length of the major axis of a vertical ellipse is three times length of the minor axis, then its eccentricity is equal to

The value of ${\log }_5{\log }_3\sqrt{\sqrt[5]{59}}$ is

$\frac{d^{2}x}{d{y}^2}$ equals:

The equation of the ellipse which passes through origin and has its foci at the points $(1,0)$ and $(3,0),$ is

If the mean deviation of the number $1,1+d,1+2d,....1+100d$ from their mean is $255$ then $d$ is equal to $(d>0)$

The equation of the line which is perpendicular to $x+4y-5=0$ at it's $y$ intercept

If $y=y\left(x\right)$ is the solution of the differential equation $\frac{dy}{dx}+2y\tan x=\sin x,y\left(\frac{\pi }{3}\right)=0$ then the maximum value of the function $y\left(x\right)$ over $R$ is equal to:

If $6^n-5n,n\in N$ is divided by $25$, then the remainder is

Three concentric circles of which biggest is $x^2+y^2=1$, have their radii in A.P. If the line $y=x+1$ cuts all the three circles in real and distinct points, then the interval in which the common difference of $AP$ will lie, is

Out of 18 points in a plane, no three are in the same straight line except 5 points which are collinear. How many straight lines can be formed by joining them?

If the reduction formula for $I_n=\int \frac{\sin \mathrm{nx}}{\cos x}\mathrm{dx}$ is given by

$I_n+I_{n-2}=-\frac{2\cos \left\{\right(n-1\left)x\right\}}{n-1}$, then $\int \frac{\sin 3x}{\cos x}\mathrm{dx}$ is.

The point (4, 1) undergoes the following transformation successively.

(i) reflection about the line y = x

(ii) translation through a distance 2 units along the positive direction of x - axis

(iii) rotation through an angle $\frac{\pi }{4}$ about the origin in the anticlockwise direction.

(iv) reflection aout x = 0

The final position of the given point is

If x is real, then the maximum and minimum values of expression $\frac{x^2+14x+9}{x^2+2x+3}$will be

If $n$ is the number of real solutions of the equation $min(e^{-|x|},1-e^{-|x|})=\frac{1}{4}$ and $L=\underset{x\to 0^{-}}{lim}(\frac{e^{2x}-1}{x}+\frac{e^{3x}-1}{x}+\frac{e^{4x}-1}{x}+\cdots \text{upto}n\text{terms}),$ then the value of $L$ is

If $\cos \alpha +\cos \beta +\cos \gamma =0=\sin \alpha +\sin \beta +\sin \gamma =0$ then $\cos (2\alpha -\beta -\gamma )+\cos (2\beta -\gamma -\alpha )+\cos (2\gamma -\alpha -\beta )=$

The values of $a$ for which the number $6$ lies in between the roots of the equation $x^2+2(a-3)x+9=0,$ belong to

Let $a-2b+c=1$, If $f\left(x\right)=\left|\begin{array}{ccc}x+a& x+2& x+1\\ x+b& x+3& x+2\\ x+c& x+4& x+3\end{array}\right|,$ then: