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Let $f$ be a positive function

Let $I_1={\int }_{1-k}^{k}xf\left\{x\right(1-x\left)\right\}dxand{I}_2={\int }_{1-k}^{k}f\left\{x\right(1-x\left)\right\}dx$

where $2k-1>0,$ then $\frac{I_1}{I_2}$ is

If a, b, c are positive rational numbers such that a > b > c and the quadratic equation

$(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0$ has a root in the interval (-1, 0), then

If the two lines $(a+b)x-4y+5=0$ and $x+(a-b)y+10=0$ are perpendicular to each other then

$\underset{x\to 0}{lim}$ $\frac{x(e^x-1)}{1-cosx}$ =

A point moves so that its distance from the point (-1,0) is always three times its distance from the point (0,2). The locus of the point is

The graph for the linear function $y=4x+5$ is

If the area enclosed between the curves $y=k{x}^2$ and $x=k{y}^2,(k>0)$, is $1$ square unit. Then $k$ is

If $y=x\sqrt{2}-8\sqrt{2}$ is a normal chord to $y^2=8x$.Then its length (in units) is

The integral $\int x{\cos }^{-1}\left(\frac{1-x^2}{1+x^2}\right)dx$, where $x>0$, is equal to

(where $c$ is constant of integration)

If 9 AM's are inserted between 2 & 3, the general term of i^th AM is

There are ten numbers in A.P.. If the sum of first three terms is $321$ and the sum of last three numbers is $405$, then the sum of all ten numbers is

Let $z$ be a complex number such that $|2z+\frac{1}{z}|=1$ and $\arg \left(z\right)=\theta$, then minimum value of $8{\sin }^2\theta$ is

If the angles of a triangle are in the ratio 1:2:3, then the ratio of its corresponding sides is .

If $z_r$ = cos $\left(\frac{\pi }{2^r}\right)$ + i sin $\left(\frac{\pi }{2^r}\right)$ where i = $\sqrt{-1}$ . Find the value of $z_1$.$z_2$.$z_3$.$z_4\dots \dots \infty$

If A + C = B, then tan A tan B tan C =

From a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ lines are drawn to focus S

and directrix perpendicular to it as shown.Then,

Find the length of the perpendicular from the point $(x^1,y^1)$ to the straight line Ax + By + C = 0, the axes being inclined at an angle $\omega$, and the equation being written such that C is a negative quantity.

The length of the latus-rectum of the parabola $169\left\{\right(x-1{)}^2+(y-3{)}^2\}=(5x-12y+17{)}^2$ is

Prove that $4cos{12}^{\circ }cos{48}^{\circ }cos{72}^{\circ }=1-2si{n}^2{18}^{\circ }$.

The general solution of the differential equation $(y^2+e^{2x})dy-y^{3}dx=0$ (C being the constant of integration), is

Three athletes A, B and C participate in a race. Both A and B have the same probability of winning the race and each is twice as likely to win as C. The probability that B or C wins the race is

Three boys and two girls stand in a queue. The probability that the number of boys ahead of every girl is at least one more than the number of girls ahead of her, is?

Numerically greatest term in the expansion of $(3-5x{)}^{11}whenx=\frac{1}{5}$ is:

If the ratio of the coefficient of third and fourth term in the

expansion of $(x-\frac{1}{2x}{)}^n$ is 1:2, then the value of n will be

Find the number of integers satisfying the condition |x-3| $>$ 5 and |x + 1 | $\le$ 4

If the first term of a G.P., $a_1,a_2,a_3,\dots$ is unity, then the value of $4a_2+5a_3$ will be minimum when the common ratio is

If the line $y=x$ cuts the curve $y=2x^3+6x^2+x-4$ at three points $A,B$ and $C$. Then the value of $|OA.OB.OC|$ with $O$ being the origin is

A circle of radius $\sqrt{5}$ units has diameter along the angle bisector of the lines $x+y=2$ and $x-y=2$. If chord of contact from the origin makes an angle of ${45}^{\circ }$ with the positive direction of $x$-axis, then the equation of the circle is

Let $\alpha ,\beta$ be the roots of $a{x}^2+bx+c=0$. The roots of $a(x-2{)}^2-b(x-2\left)\right(x-3)+c(x-3{)}^2=0$, where $a\ne 0$ are

Two sets $M\&N$ are represented as shown below, then

If $p=(8+3\sqrt{7}{)}^n$ and $f=p-\left[p\right]$, then the value of $p(1-f)$ is

(where [.] denotes the greatest integer function)

If $co{s}^{3}x.sin2x={\sum }_{r=0}^n{a}_{r}sin\left(rx\right),\forall x\in R$, then choose the correct option(s).

Two numbers are selected simultaneously from the set {6, 7, 8, 9, ………. 39}. If the sum of selected numbers is even then the probability that both the selected numbers are odd, is equal to:

If $y=(a-2)x^2+(b-3)x,$ where $a,b\in R$ is a linear function and $|a-b|=4$, then the possible value(s) of $b$ is/are

In a G.P. (consisting of positive terms), if each term equals the sum of the next two terms, then the common ratio of the G.P. is

If $\tan \theta =-\frac{1}{\sqrt{3}},$ then the value of $\theta \in [0,2\pi ]$ for which $\frac{\cos \theta -{\cos }^3\theta }{\tan \theta +2}$ is always positive is

The number of integers greater than a million (Ten lakhs) that can be formed using the digits

$2,3,0,3,4,2,3$ is

A possible value of $x$, for which the ninth term in the expansion of ${\{3^{{\log }_3\sqrt{{25}^{x-1}+7}}+3^{(-\frac{1}{8}){\log }_3(5^{x-1}+1)}\}}^{10}$ in the increasing powers of $3^{(-\frac{1}{8}){\log }_3(5^{x-1}+1)}$ is equal to $180$, is

The range of values of 'a' such that the angle $\theta$ between the pair of tangents drawn from (a, 0) to the circle $x^2+y^2=1$ satisfies $\frac{\pi }{2}<\theta <\pi$, is

In the expansion of ( $\frac{a}{x}+bx{)}^{12}$, the coefficient of $x^{-10}$ will be

If $A=\{1,2,3,4,5,6\},B=\{3,6,9,12\},C=\{6,12,18,20\}$, then $n\left\{\right(A\times B)\cap (A\times C\left)\right\}=$

If $\sum _{r=0}^n\left(\frac{r+2}{r+1}\right)C_r=\frac{2^8-1}{6}$, where $C_r={}^n{C}_r$, then the value of $n$ is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $P(x_1,y_1),Q(x_2,y_2)R(x_3,y_3),S(x_4,y_4)$, then which of the following need not hold?

What is the equation of the normal which is perpendicular to 3x + 4y = 5 for the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

if z and $\omega$ be two non-zero compex numbers such that $|z|=|\omega |$ and arg(z) + arg$\left(\omega \right)=\pi$, then z equals