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Let X = {1,2,3,4,5} and Y = {1,3,5,7,9}. Which of the following is/are relations from X to Y

What is the area of the triangle formed by the vertices $(0,0),(2,1)\text{and}(5,3).$

The principal amplitude of $(2-i)(1-2i{)}^2$ is in the interval :

Let origin is one vertex of an equilateral triangle of side length $a$ units. If other vertex lies on the line $x-\sqrt{3}y=0$ in the first quadrant, then the co-ordinates of third vertex is/are

Six-digit odd numbers, greater than $6,00,000$ that can be formed using the digits $5,6,7,8,9$ and $0$ if repetition of digits is not allowed is :

If $x^2+y^2=25,xy=12$,then complete set of $x=$

If |a+b| = |a-b| then (a,b) =

If $(\cos p-1)x^2+\left(\cos p\right)x+\sin p=0,x\in R$ has real roots for $x$, then the range of $p$ is

If a variable plane forms a tetrahedron of constant volume $64k^3$ with the co-ordinate planes, then the locus of the centroid of the tetrahedron is

Which among the following is the correct graphical representation of $y=-x^2+4x+1$ ?

A, B have position vectors $\overrightarrow{a},\overrightarrow{b}$ relative to the origin O and X, Y divide $\overrightarrow{AB}$ internally and externally respectively in the ratio 2 : 1. Then,$\overrightarrow{XY}$ is equal to

The coordinates of the point of contact of the tangent to the parabola $y^2=16x$, which is perpendicular to the line $2x-y+5=0$ are

The equation to the locus of the midpoints of chords of the circle $x^2+y^2=r^2$ having a constant length 2l is

Total number of solution of ${\cos }^{2}x+\frac{\sqrt{3}+1}{2}\sin x-\frac{\sqrt{3}}{4}-1=0$ in $x\in [-\pi ,\pi ]$ is :

If the term independent of $x$ in the expansion of ${(\sqrt{x}-\frac{k}{x^2})}^{10}$ is $405$, then the value(s) of $k$ can be

Number of words that can be formed by using the $4$ letters of the word $MISSISSIPPI$ is

In the figure, length of subnormal is the length P1N (tangent and normal is drawn at the point P)

1. T

If $A$ and $B$ are two positive acute angles satisfying the equations $4-3{\cos }^{2}A=2{\cos }^{2}B$ and $\cos (A+2B)=0$, then the value of $\frac{3\sin A}{2\cos B}$ is

The coefficient of the middle term in the binomial expansion in powers of $x$ of $(1+\alpha x{)}^4$ and of $(1-\alpha x{)}^6$ is the same, if $\alpha$ equals

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=4\hat{i}+3\hat{j}+4\hat{k}$ and $\overrightarrow{c}=\hat{i}+\alpha \hat{j}+\beta \hat{k}$ are linearly dependent vectors and $|\overrightarrow{c}|=\sqrt{3},$ then

If $\alpha ,\beta$ be the roots of the equation $u^2-2u+2=0$ and if $\cot \theta =x+1$, then $\frac{(x+\alpha {)}^n-(x+\beta {)}^n}{(\alpha -\beta )}$ is equal to

Find the equation of the chord of contact of tangents to the parabola

$y^2=4x$ from the point $P(3,4)$.

If $|z|=1$ and $|\omega -1|=1$ where $z,\omega \in C,$ then the largest set of values of $|2z-1{|}^2+|2\omega -1{|}^2$ equals

The value of sum $\sum _{n=1}^{\infty }\frac{n}{7^n}$ is

The value of the determinant $\left(\begin{array}{ccc}x& a& x+a\\ y& b& y+b\\ z& c& z+c\end{array}\right|$ is

Let $\overrightarrow{a}=\hat{i}-\hat{j},\overrightarrow{b}=\overrightarrow{j}-\hat{k},\overrightarrow{c}=\hat{k}-\hat{i}.If\overrightarrow{d}$ is a unit vector such that $\overrightarrow{a}.\overrightarrow{d}=0=\left[\overrightarrow{b}\overrightarrow{c}\overrightarrow{d}\right],$ then $\overrightarrow{d}$ equals

If $f\left(x_1\right)-f\left(x_2\right)=f\left(\frac{x_1-x_2}{1-x_1{x}_2}\right)$ for $x_1,x_2ϵ$ (-1, 1), then f(x) is

abc $\ne$ 0 & a, b, c $ϵ$ R. If $x_1$ is a root of $a^2{x}^2+bx+c=0,x_2$ is a root of $a^2{x}^2-bx-c=0$ and $x_1>x_2>0$, then the equation $a^2{x}^2+2bx+2c=0$ has a root $x_3$ such that

The chord $x+y=1$ of the curve $y^2=12x$ cuts it at the points $A$ and $B.$ The normals at $A$ and $B$ intersect at $C.$ If a third line from $C$ cuts the curve normally at $D,$ then the co-ordinates of $D$ are

$IfmatrixA=[a_{ij}{]}_{3\times 3},matrixB=[b_{ij}{]}_{3\times 3}where{a}_{ij}+a_{ji}=0and{b}_{ij}-b_{ji}=0,then|A^4.B^3|is$

The distance between the two lines represented by the equation $9x^2-24xy+16y^2-12x+16y-12=0$ is __ units

The maximum value of $cos{\alpha }_1.cos{\alpha }_2......cos{\alpha }_n,$

under the restrictions $0\le {\alpha }_1{\alpha }_2,.....,{\alpha }_n\le \frac{\pi }{2}andcot{\alpha }_1.cot{\alpha }_2......cot{\alpha }_n=1$ is

Given that $|z-1|=1$, where $z$ is a non zero point on the complex plane, then $\frac{z-2}{z}$ is equal to :

If both the roots of $x^2+2ax+a=0$ are less than $2$, then the set values of ${}^{\prime }{a}^{\prime }$ is

If $\log (x+z)+\log (x-2y+z)=2\log (x-z)$, then

(1) The sum of all the values of $r$ satisfying ${}^{39}{C}_{3r-1}{-}^{39}{C}_{r^2}{=}^{39}{C}_{r^2-1}{-}^{39}{C}_{3r}\text{is}{\alpha }_1.$



(2) If ${}^{2n+3}{C}_{2n}{-}^{2n+2}{C}_{2n-1}=15.(2n+1)$ then the value of $n$ is ${\alpha }_2.$



(3) If ${}^{56}{P}_{r+6}{:}^{54}{P}_{r+3}=30800:1$ then value of $r$ is ${\alpha }_3.$



(4) ${}^{n+2}{C}_8{:}^{n-2}{P}_4=57:16$ then the value of $n$ is ${\alpha }_4.$



$\begin{array}{cccc}& \text{List-I}& & \text{List-II}\\ \left(I\right)& \text{The value of}{\alpha }_1\text{is}& \left(P\right)& 41\\ \left(II\right)& \text{The value of}{\alpha }_2\text{is}& \left(Q\right)& 8\\ \left(III\right)& \text{The value of}{\alpha }_3\text{is}& \left(R\right)& 14\\ \left(IV\right)& \text{The value of}{\alpha }_4\text{is}& \left(S\right)& 19\end{array}$



$\text{Which of the following is only CORRECT Combination?}$

An equation of a plane parallel to the plane x - 2y + 2z - 5 = 0 and at a unit distance from the origin is

$\underset{n\to \infty }{\overset{}{lim}}$ $\sum _{x=1}^{20}$ $cos{}^{2n}(x-10)$ is equal to

A rod of length $l$ moves such that its ends $A$ and $B$ always lie on the lines $3x-y+5=0$ and $y+5=0$ respectively. Then the locus of the point $P$ which divides $AB$ internally in the ratio of $2:1,$ is

The condition that the straight line $lx+my+n=0$ touches the parabola $x^2=4ay$ is

The range in meters of a projectile launched over a flat ground from the origin with positive velocity $V$ in $m/s$ at an angle $\theta$ given in radian is given by $R=\frac{V^2\sin \left(2\theta \right)}{g}$ where $g$ is a positive constant, assume $V=2m/s,g=10m/s^2$ and $\theta$ was measured to be $\frac{\pi }{12}$ radians. If there was a possible error in the measurement of $\theta$ of $\frac{1}{10\sqrt{3}}$ radians, estimate the corrosponding error in the computation of the range.

Match the entries of col. I with those of col. II.

$\begin{array}{cc}Column-I& Column-II\\ \left(a\right)f\left(x\right)=\frac{1-x+x^2}{1+x-x^2}\text{on}[0,1]& \left(p\right)Greatestvalueoff=1\\ \left(b\right)f\left(x\right)=2tanx-ta{n}^{2}x\text{on}[0,\frac{\pi }{2}]& \left(q\right)Leastvalueoff=\frac{3}{5}\\ \left(c\right)f\left(x\right)=\frac{2}{\pi }(sin2x-x)\text{on}[-\frac{\pi }{2},\frac{\pi }{2}]& \left(r\right)Leastvalueoff=-1\\ \left(d\right)f\left(x\right)=\frac{1}{2},(x^3-3x^2+6x-2)\text{on}(-1,1)& \left(s\right)Leastvalueoff=-6\end{array}$

The product of the perpendicular from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes, is equal to

In a battle 70% of the combatants lost eye, 80% an ear, 75% an arm, 85% a leg, x% lost all of four body parts. The minimum value of x is

If words are formed by taking only $4$ at a time out of the letters of the word "PHYSICAL", then the number of words in which 'Y' occur is