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The interval(s) which satisfies the solution set of the inequality $4x-2>x-\frac{3-x}{4}>3$ is/are

The vectors $\overrightarrow{a}=3\hat{i}-2\hat{j}+2\hat{k}$ and $\overrightarrow{b}=-\hat{i}-2\hat{k}$ are the adjacent sides of a parallelogram.Then,angle between its diagonals is

If $\alpha$ and $\beta$ are the roots of the equation $x^2+3x+1=0$, then the value of ${\left(\frac{\alpha }{1+\beta }\right)}^2+{\left(\frac{\beta }{\alpha +1}\right)}^2$ is equal to

The equation(s) of the angle bisectors of the lines $3x-4y+7=0$ and $12x-5y-8=0$ is/are

Which of the following is/are singleton sets?

A coin whose faces marked 2 and 3 is thrown 5 times, then chance of obtaining a total of 12 is

Let $\overrightarrow{a}=a_1\hat{i}+{\alpha }_2\hat{j}+a_3\hat{k},\overrightarrow{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}and\overrightarrow{c}=c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three non-zero vectors such that $\overrightarrow{c}$ is a unit vector perpendicular to both the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi }{6},$

then ${\left|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}\right|}^2$ is equal to

1. 96

Let $O$ be the origin, and $\overrightarrow{OX},\overrightarrow{OY},\overrightarrow{OZ}$ be three unit vectors in the directions of the sides $\overrightarrow{QR},\overrightarrow{RP},\overrightarrow{PQ},$ respectively, of a triangle $PQR.$



$|\overrightarrow{OX}\times \overrightarrow{OY}|=$

The equations of the directrices of the hyperbola $16x^2-9y^2=-144$ are:

The velocity $v($in $\text{m/s})$ of a train in time $t($in $\text{sec})$ is given by $v=52+7t$. The minimum time $t($in $\text{sec})$ when its velocity is atleast $73\text{m/s}$ is

We can’t apply rolle’s theorem on f(x) = |x| on the interval [-2, 2] because -

Let two non-collinear unit vectors $\hat{a}\text{and}\hat{b}$ form an acute angle. A point P moves so that at any time t the position vector

$\overrightarrow{OP}$ (where, O is the origin) is given by

$\hat{a}\cos t+\hat{b}\sin t$. When P is farthest from origin O, let M be the length of $\overrightarrow{OP}$ and $\hat{u}$ be the unit vector along $\overrightarrow{OP}$. Then,

If the sum of the first $15$ terms of the series

${\left(\frac{3}{4}\right)}^3+{\left(1\frac{1}{2}\right)}^3+{\left(2\frac{1}{4}\right)}^3+3^3+{\left(3\frac{3}{4}\right)}^3+....$



is equal to $225k$, then $k$ is equal to :

In triangle $ABC,a^2+c^2=2002b^2$, then $\frac{cotA+cotC}{cotB}$ is equal to

If $\int \frac{se{c}^{2}x-2010}{si{n}^{2010}x}dx=\frac{P\left(x\right)}{si{n}^{2010}x}+C,$ then value of $P\left(\frac{\pi }{3}\right)$ is

If the line $\frac{x}{a}+\frac{y}{b}=\sqrt{2}$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then the eccentric angle of point of contact is

The coefficient of $x^{10}$ in the expansion of $[1+x^2(1-x){]}^8$ is

The average value of $\sin 2°,\sin 4°,\sin 6°,...,\sin 180°\mathrm{is}$

The locus of the point of intersection of perpendicular tangent drawn to each one of the parabola $y^2=4x+4$ and $y^2=8x+16$ is

For a>0, b>0, c>0, which of the following hold good?

The number of integral solutions for the equation $x+y+z+t=20,$ where $x,y,z,t\ge -1,$ is

If the tangents to the parabola $x=y^2+c$ from origin are perpendicular, then $c$ is equal to

Select the correct graph of the function $f\left(x\right)=\tan |x+\frac{\pi }{3}|.$

If a and b are two different positive real numbers, then which of the following relations is true

The smallest set $A$ such that $A\cup \{1,2\}=\{1,2,3,5,9\}$ is

Let $f\left(n\right)=\left(\sin 1\right)\left(\sin 2\right)\left(\sin 3\right)\cdots \left(\sin \right(n\left)\right)\forall n\in N$ where $n$ is in radians. Then the number of elements in the set $A=\left\{f\right(1),f(2),\dots ,f(6\left)\right\}$ that are positive, is

Each of the circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2+2x-2y+1=0$ touches internally a circle of radius $2$. The equation of circles touching all the three circles, is

The first term of an AP is $5$, last term is $45$ and the sum is $400,$ then the number of terms and common difference of the series are respectively :

Positive integers from $1$ to $45,$ are placed in $5$ groups of $9$ each. Then highest possible average of the medians of these $5$ groups is

The value of ${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$ is equal to



${\cot }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$ का मान है

The number of integeral values of $m,$ for which the $x$ -coordinate of the point of intersection of the lines $3x+4y=9$ and $y=mx+1$ is an integer is

The equation of the circle passing through points of intersection of the circle $x^2+y^2-2x-4y+4=0$ and the line $x+2y=4$ and touches the line $x+2y=0$, is

If the mean of the data : $7,8,9,7,8,7,\lambda ,8$ is $8$, then the variance of this data is :

If $a=cis2\alpha ,b=cis2\beta$, then $\cos (\alpha -\beta )$ is

where $cis\theta =\cos \theta +i\sin \theta$

If the sides of a right-angled triangle form an A.P. Then the sines of the acute angle are

If $f:[0,2]\to A,f\left(x\right)=\left[x^2\right]-[x{]}^2$ is a real valued function, then minimum elements required in set $A$ is

(where [.] denotes greatest integer function)

Number of positive integral solutions of $abc=30$ is

If $\underset{x\to 0}{lim}\frac{a\tan 3x+(1-\cos 2x)}{x+\sin x+\tan x}=1$, then the value of $a$ is

The normal at a point P on the ellipse $x^2+4y^2=16$ meets the X - axis at Q. If M is the mid-point of the line segment PQ, then the locus of M intersects the latusrectum of the given ellipse at the points

If $y_1\left(x\right)$ is a solution of the differential equation $\frac{dy}{dx}-f\left(x\right)y=0$, then a solution of the differential equation $\frac{dy}{dx}-f\left(x\right)y=r\left(x\right)$ is