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If $\sqrt{2x-5}<3$, then $x\in$

If $n$ is even and the middle term in the expansion of ${(x^2+\frac{1}{x})}^n$ is $924x^6$, then $n$ is equal to

Let $A(1,2),B(\text{cosec}\alpha ,-2)$ and $C(2,\sec \beta )$ are $3$ points such that $(OA{)}^2=OB\cdot OC$,($O$ is the origin) then the value of $2{\sin }^2\alpha -{\tan }^2\beta$ is

Let $PQR$ be a right angled isosceles triangle right angled at $P(2,1)$. If the equation of the line $QR$ is $2x+y=3$, then the equation representing the pair of lines $PQ$ and $PR$ is

The two equations $x^3+1=0$ and $a{x}^2+bx+c=0,a,b,c\in R$ have two roots in common. Then $a+b$ is equal to

If $tan{\theta }_1,tan{\theta }_2,tan{\theta }_3$ are the real roots of $x^3-(a+1)x^2+(b-a)x-b=0where{\theta }_1,{\theta }_2,{\theta }_3$ are acute then ${\theta }_1+{\theta }_2+{\theta }_3=$

The result of $21$ football matches (win, lose, draw) are to be predicted. The number of different forecasts that can contain $19$ wins is

In triangle $ABC,\frac{\sin A+\sin B+\sin C}{\sin A+\sin B-\sin C}$ is equal to

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the top faces will be at least 10 is

Parametric coordinates of a point on ellipse, whose foci are $(-1,0)$ and $(7,0)$ and eccentricity is $\frac{1}{2},$ is

Let $f$ be a differentiable function such that

$f\left(1\right)=2$ and $f^{\prime }\left(x\right)=f\left(x\right)$ for all $x\in R$. If $h\left(x\right)=f\left(f\right(x\left)\right)$, then $h^{\prime }\left(1\right)$ is equal to :

Tangent drawn at any point on $y^2=4ax$ meets the axis of parabola at $T$ and tangent at vertex at $S$. If $TASG$ is a rectangle, where $A$ is the vertex, then locus of $G$ is

Complete solution set of $|x^2-5x+7|+|x^2-5x-14|=21$ is -

If $\frac{1}{a+\omega }+\frac{1}{b+\omega }+\frac{1}{c+\omega }+\frac{1}{d+\omega }=\frac{2}{\omega },$ where $a,b,c$ are real and $\omega$ is non real cube root of unity, then:

Which of the following is/are not the solution of ${\log }_3\left(x^2-2\right)<{\log }_3\left(\frac{3}{2}\left|x\right|-1\right)$

A class contains three girls and four boys. Every Saturday, a group of $5$ students go on a picnic (a different group of students is sent every week). During the picnic, each girl in the group is given a doll by the accompanying teacher. If all possible groups of five have gone for picnic once, the total number of dolls that the girls have got is

The shortest distance of the point (a, b, c) from the x-axis is

[MP PET 1999; DCE 1999]

Pair of tangents are drawn from a point $P$ on $x^2+y^2=4$ to $x^2+y^2-20x-20y+199=0$. If $A$ and $B$ are points of contact of these tangents, then the area bounded by the locus of circumcentre of $△PAB$ is (in sq. units)

The number of arrangements of the letters of the word ‘NAVA NAVA LAVANYAM’ which begin with N and end with M is :

The common tangents to the circles $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$ is/are

If $a{\sin }^{-1}x-b{\cos }^{-1}x=c,$ then $a{\sin }^{-1}x+b{\cos }^{-1}x$ is equal to

The vertex of a parabola is the point (a,b) and latus rectum is of length /. If the axis of the parabola is along the positive direction of y-axis, then its equation is

The point of intersection of the line joining the points $(2,4,5)$ and $(-4,3,-2)$ and the xy plane is

The integral $\underset{\pi /6}{\overset{\pi /4}{\int }}\frac{dx}{\sin 2x({\tan }^{5}x+{\cot }^{5}x)}$ equals :

If ratio of the sum of $n$ terms of $2$ different A.P. is $\frac{3n+2}{4n+23}$, then the ratio of ${10}^{\text{th}}$ term is

The number of different ways in which a committee of $4$ members formed out of $6$ Asians, $3$ Europeans and $4$ Americans if the committee is to have at least one member from each of the regional groups, is

If $\int \frac{\mathrm{dx}}{4{\sin }^{2}x+6{\cos }^{2}x}$$=a{\tan }^{-1}\left(\frac{2\mathrm{tanx}}{\sqrt{6}}\right)+C$, then $a$ is

In a triangle ABC,

$\angle A=60°andb:c=\sqrt{3}+1:2,thenthevalueof\angle B-\angle C=$ .

The range of $y=\frac{3x-5}{5x-1},x\ne \frac{1}{5}$ is

The value of $\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{5\pi }{20}\right)\tan \left(\frac{7\pi }{20}\right)\tan \left(\frac{9\pi }{20}\right)$ is

Let $T_r$ be the $r^{th}$ term of an A.P. for $r=1,2,3,\dots$ If for some positive integers $m,n$, $T_m=\frac{1}{n}$ and $T_n=\frac{1}{m}$, then $T_{mn}$ equals

The line passing through the points $(5,1,a)\text{and}(3,b,1)$ crosses the $yz-$ plane at the point $(0,\frac{17}{2},\frac{-13}{2})$, then

If $l_1,m_1,n_1$ and $l_2,m_2,n_2$ are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be

If

A = {1, 2, 3, 4}

B = {3, 4, 5, 6, 7}

C = {6, 7, 8, 9}

D = {7, 8, 9, 10}

$$\begin{array}{cc}ColumnA& ColumnB\\ 1.A\cup B& a.\{3,4,6,7\}\\ 2.A\cap B& b.\varnothing \\ 3.(B\cap C)\cup D& c.\{3,4\}\\ 4.B\cup C\cup D& d.\{3,4,5,6,7,8,9,10\}\\ 5.(A\cap B)\cap (B\cup C)& e.\{1,2,3,4,5,6,7\}\\ & f.\{6,7,8,9,10\}\\ & g.\{7\}\end{array}$$

For a set $X=\{2,3,\{2,3\left\}\right\},P\left(X\right)=$

I) 10, 0, 101, 56, 72

II) 1021, 1011, 1058, 1100

(III) 256, 282, 320, 198, 200

(IV) 0, 93, 46, 10, 32, 46

Which of the given data sets have maximum range

If matrix $A=\left[\begin{array}{ccc}1& 0& -1\\ 3& 4& 5\\ 0& 6& 7\end{array}\right]$ and its inverse is denoted by $A^{-1}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, then the value of $a_{23}$=

Vector(s) perpendicular to both the vectors

$\overrightarrow{A}=3\hat{i}+5\hat{j}+2\hat{k}$ and

$\overrightarrow{B}=2\hat{i}+4\hat{j}+6\hat{k}$ is/are

In a class of 58 students, 20 follow cricket, 38 follow hockey and 15 follow basketball. Three students follow all the three games. How many students follow exactly two of these three games?

If f(x) = |x| + |x - 1| + |x - 2|, then

Let I $=\int \frac{e^x}{e^{4x}+e^{2x}+1}dx.J=\int \frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx,$Then, for an arbitrary constant c, the value of J-I equals

f(x) and f’(x) are differentiable at x = c. Which of the following is the condition for f(x) to have a local minimum at x = c, if f’(c) = 0

Let $f^{\prime }\left(x\right)=e^{x^2}$ and $f\left(0\right)=10.$ If $A<f\left(1\right)<B$ can be concluded from the mean value theorem, then the largest value of $(A-B)$ equals

If $x_1,x_2,x_3\cdots x_n$ are roots of $x^n+ax+b=0$, then the value of $(x_1-x_2)(x_1-x_3)(x_1-x_4)\cdots (x_1-x_n)$ =

If the two equations $x^3+3p{x}^2+3qx+r=0$ and $x^2+2px+q=0$ have a common root, then the value of $4(p^2-q)(q^2-pr)$ is

If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ at four points

$P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3),andS(x_4,y_4)$, then

If the ratio of sum of $n$ terms of $2$ different $A.P.$ is $\frac{2n-1}{5n+10}$, then the ratio of their ${15}^{th}$ term is

If $si{n}^{-1}\frac{3}{5}+co{s}^{-1}\left(\frac{12}{13}\right)=si{n}^{-1}C,thenC=$