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The mid - points of class intervals $x_i$ along with respective frequencies $f_i$ for recorded observations are given. Calculate the mean deviations about mean $x$

The transformed coordinates of the point $(4,3)$ when the axes are translated to point $(3,1)$ and then rotated through ${30}^{\circ }$ in anticlockwise direction is

Let $A\equiv (a,b)$ and $B\equiv (c,d)$ where $c>a>0$ and $d>b>0$. Then point $C$ on the $x-$axis such that $AC+BC$ is minimum,is

Let z be a complex number such that the imaginary part of z is non - zero and $a=z^2+z+1$ is real. Then, $a$ cannot take the value

Find the equation of normal to the circle $x^2+y^2-5x+2y-48=0$ which is parallel to the line 14x - 5y - 30 = 0

The sum of $\frac{3}{1\cdot 2}\left(\frac{1}{2}\right)+\frac{4}{2\cdot 3}{\left(\frac{1}{2}\right)}^2+\frac{5}{3\cdot 4}{\left(\frac{1}{2}\right)}^3+\cdots$ upto $20$ terms is

The locus of the center of a variable circle which cuts-off the intercept of 2 units and 4 units respectively on the X-axis and the Y-axis is .

The angle (in degrees) subtended at the centre of a circle of diameter $50$ cm by an arc of length $11$ cm is

The equation of a tangent to the hyperbola $4x^2-5y^2=20$ parallel to the line $x-y=2$ is :

Given $\sin x+\cos x$ = $\frac{5}{4}$. If $1+2\sin x\cos x=a$, find the value of $32a$.

Let L be a normal to the parabola $y^2=4x$. If L passes through the point (9, 6), then L is given by

Let S be the set of all non -zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_{1}and{x}_2$satisfying the inequality $|x_1-x_2|<1.$ Which of the folowing intervals is (are) a subset (s) of S?

How many of the following are not properties of equivalence classes?

1. All elements of an equivalence class will be related to each other


2. No element of an equivalence class will be related to an element of another equivalence class


3. All the equivalence classes, which are sets, are disjoint


4. Union of all the equivalence classes of particular relation will give the set A on which we defined the relation

Tangents drawn from the point P(1, 8) to the circle $x^2+y^2-6x-4y-11=0$ touch the circle at points A and B. The equation of the circumcircle of triangle PAB is

In a group of $115$ people, each has at least one among passport and voter ID. If $65$ had passport and $30$ had both, how many had only voter ID but not passport?

let $S$,$S^1$ be the foci of an ellipse. If $\angle BS{S}^1$ = $\theta$,where B is any point on the ellipse. Then its eccentricity is

If f is an even function defined on the interval (-5,5), then the real values of x, satisfying the equation $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are ___.

The length of the latus rectum of the ellipse $5x^2+9y^2=45$ is

For two complex numbers $z_1$ and $z_2;(a{z}_1+b\overline{z_1})(c{z}_2+d\overline{z_2})=(c{z}_1+d\overline{z_1})(a{z}_2+b\overline{z_2})$ $b\ne 0,d\ne 0$ if

$\underset{x\to -\infty }{lim}\frac{x^{5}tan\left(\frac{1}{\pi x^2}\right)+3|x{|}^2+7}{|x{|}^3+7|x|+8}\text{is equal to}$

Let $f\left(x\right)$ be a non-negative function. If $f^{\prime }\left(x\right)\cos x\le f\left(x\right)\sin x,\forall x\ge 0,$ then value of $f\left(\frac{5\pi }{3}\right)$ is

Let $p,q$ be integers and let $\alpha ,\beta$ be the roots of the equation, $x^2-x-1=0,$ where $\alpha \ne \beta .$ For $n=0,1,2,....,$ let $a_n=p{\alpha }^n+q{\beta }^n.$

FACT: If $a$ and $b$ are rational numbers and $a+b\sqrt{5}=0.$ then $a=0=b.$

If $a_4=28,$ then $p+2q=$

In how many ways 6 letters can be put in 6 addressed envelopes so that only 5 of them will go in the correct envelopes.

If the roots of $a{x}^2-bx-c=0$ are increased by same quantity, then which of the following expressions does not change?

The value of $\underset{x\to \infty }{lim}\frac{5^{\sin x}+2x+1}{\cos x-\sqrt{1+x^2}}$ is

Let $p,q,r$ be the roots of $x^3+2x^2+3x+3=0$, then which of following is/are correct?

Tangents are drawn from the points on a tangent of the hyperbola $x^2-y^2=a^2$ to the parabola $y^2=4ax$. If all the chords of contact pass through a fixed point $Q$, then the locus of the point $Q$ for different tangents on the hyperbola is

If the centroid of triangle whose vertices are (a,1, 3), (– 2, b, –5) and (4, 7, c) be the origin, then the values of a, b, c are

A constant force of 5 N is acting on a body. If x(t) is the dispacement at time t, v(t) velocity at time t, m mass of the body, which of the following differential equations represent the situation?

If $a=i-j+k,a\cdot b=0,a\times b=c,$, where $c=-2i-j+k$, then $b$ is equal to

Let a,b,c,be any real numbers. Suppose that there are real numbers x, y, z not all zero such that x= cy +bz, y=az + cx and z = bx +ay. Then , a² + b²+ c²+ 2abc is equal to

If $z=\frac{(1+i)(1+2i)(1+3i)\dots \dots \dots (1+ni)}{(1-i)(2-i)(3-i)\dots \dots \dots (n-i)}$, where $i=\sqrt{-1},n\in N$, then principal argument of $z$ can be -

If the roots of the equation
$(1-q+\frac{p^2}{2})x^2+p(1+q)x+q(q-1)+\frac{p^2}{2}=0$ are equal, then

Consider the curve $\sin x+\sin y=1$, lying in the first quadrant , then

$\begin{array}{cccc}& \text{List- I}& & \text{List-II}\\ \text{(I)}& \underset{x\to \pi /2}{lim}\frac{d^{2}y}{d{x}^2}=& \text{(P)}& 0\\ \text{(II)}& \underset{x\to 0^{+}}{lim}{x}^{3/2}\frac{d^{2}y}{d{x}^2}=& \text{(Q)}& 1\\ \text{(III)}& \underset{x\to 0^{+}}{lim}{x}^2\frac{d^{2}y}{d{x}^2}=& \text{(R)}& \frac{1}{\sqrt{2}}\\ \text{(IV)}& \underset{x\to \pi /2}{lim}\frac{dy}{dx}=& \text{(S)}& \frac{1}{2\sqrt{2}}\\ & & \text{(T)}& \sqrt{2}\\ & & \text{(U)}& 3\end{array}$



Which of the following is the only INCORRECT combination?

From a bag containing 10 distinct balls, 6 balls are drawn simultaneously and replaced. Then 4 balls are drawn. The probability that exactly 3 balls are common to the drawings is

If $x+y-2=0,$ $2x-y+1=0$ and $px+qy-r=0$ are concurrent lines, then the slope of the member in the family of lines $2px+3qy+4r=0$ which is farthest from the origin is

The number of solutions of the equation $(|\sin x|-1)(5|\sin x|-1)=0$ in $[0,2\pi ]$ is

A line l passing through the origin is perpendicular to the lines

$l_1:(3+t)\hat{i}+(-1+2t)\hat{j}+(4+2t)\hat{k},-\infty <t<\infty l_2:(3+2s)\hat{i}+(3+2s)\hat{j}+(2+s)\hat{k},-\infty <s<\infty$

Then, the coordinate(s) of the point(s) on $l_2$ at a distance of $\sqrt{17}$ from the point of intersection of l and $l_1$ is (are)

A pair of perpendicular straight lines passing through the origin also passes through the points of intersection of the curve $x^2+y^2=4$ with the line $x+y=a$, then value(s) of $a$ can be

If $(2,0)$ is vertex and $y-$axis is the directrix of a parabola. Then the length of its latus rectum is