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A straight line $L$ with negative slope, passes through the point $(12,3)$ and cuts the positive coordinate axes at points $P$ and $Q$. As $L$ varies, the absolute minimum value of $OP+OQ$ is $(O$ is the origin$)$

The equation of the straight line which passes through the point $P(-4,3)$ such that the portion of it between the $x$-axis and $y$-axis is divided by the point $P$ in the ratio $1:2$ respectively, is

The equations of circles with radius $3$ units and touching the circle $x^2+y^2-2x-4y-20=0$ at $(5,5)$ is/are

The range of $k$ for which the equation $k\cos x-3\sin x=k+1$ has a solution is

If the straight line $x(a+2b)+y(a+3b)=a+b$ passes through a fixed point for different values of $a$ and $b$, then the fixed point is

The angle of elevation of the top of a vertical tower from a point A, due east of it is ${45}^{\circ }$. The angle of elevation of the top of the same tower from a point B, due south of A is ${30}^{\circ }$. If the distance between A and B is $54\sqrt{2}$ m, then the height of the tower (in metres), is

The sum of the tangents of the interior angles of a triangle formed by the lines $L_1:2x+3y+3=0;L_2:y-x-2=0$ and $L_3:2x-3y-3=0$, is

Which of the following is/are infinite set?

Tangents are drawn to the ellipse $x^2+2y^2=2$, then the locus of the mid-point of the intercept made by the tangents between the coordinate axis is

If $\tan x=n\tan y,n\in R^{+}$, then maximum value of ${\sec }^2(x-y)$ is

The number of permutations of n dissimilar things taken not more than ‘r’ at a time, when each thing may occur any number of times is

A hyperbola passes through the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities is one, then the equation of the hyperbola is:

Let R be the relation over the set of straight lines of a plane, such that $l_{1}R{l}_2\iff l_1\pm l_2$ . Then, R is

A boy is throwing stones at a target. The probability of hitting the target at any trial is $\frac{1}{2}$ .The probability of hitting the target $5^{th}$ time at the ${10}^{th}$ throw is

Which of the following is NOT the graph of a quadratic polynomial ?

The number of solutions of ${\sin }^{5}x-{\cos }^{5}x=\frac{1}{\cos x}-\frac{1}{\sin x}$ (where $\sin x\ne \cos x)$ is

What is the distance of the plane $2x-3y+4z=6$ from the origin?

Find the equation of plane if it passes through a point (2, 3, - 4) and is perpendicular to the line with direction ratios (2,3, -1) .

Total number of polynomials of the form $x^3+a{x}^2+bx+c,$ that are divisible by $x^2+1,$ where $a,b,c\in \{1,2,3,\dots ,9,10\}$ is

Let R be a relation on the set N of natural numbers defined by nRm

⇔ n is a factor of m (i.e. n(m). Then R is

If one of the diameters of the circle, given by the equation $x^2+y^2-4x+6y-12=0$, is a chord of a circle S, whose centre is at (-3,2), then the radius of S is

The sum of the slopes of the tangents to the parabola $y^2=8x$ from the point $(-2,3),$ is

Which one of the following hold good?

If m is the AM of two distinct real numbers l and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals

If $A=(5,0)$ and $B=(0,4)$, then the locus of moving point $P$ such that $|PA{|}^2-|PB{|}^2=9$ is

If ${\log }_{4}5=a$ and ${\log }_{5}6=b$, then ${\log }_{3}2$ equal to

The perpendicular distance from the origin to the plane containing the two lines,

$\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$ , is:

If $D=0$ and at least one of $D_1,D_2,D_3$ is not 0 then according to Cramer's rule the system of linear equations will have

In a certain test, $a_i$ students gave wrong answers to at least i questions, where i = 1, 2, ......, k. No student gave more than k wrong answers. The total number of wrong answers given is ___.

$3x^2-4x+\frac{20}{3}=0$

Which of the following integral represents the summation ${\sum }^{}\left(4x^3\right)\delta x$ for the limit $\delta x\to 0$?

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non-coplanar vectors and $\overrightarrow{d}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}+\nu \overrightarrow{c},$ then $\lambda$ equal to

Two distinct chords of the parabola $y^2=4ax$ passing through $P(a,2a)$ are bisected by the line $x-y+1=0$. The possible length of the latus rectum of the parabola when $a>0$ is

The value of $\sum _{k=1}^{13}\frac{1}{\sin (\frac{\pi }{4}+\frac{(k-1)\pi }{6})\sin (\frac{\pi }{4}+\frac{k\pi }{6})}$ is equal to

The maximum value of the function $f\left(x\right)=2x^3-18x^2+48x-11$ over the set $S=\{x\in R:x^2+42\le 13x\}$ is

$\int si{n}^{-\frac{11}{3}}\left(x\right).co{s}^{-\frac{1}{3}}\left(x\right)dx$

The graph of $f\left(x\right)=-x^2+x-2$ is

Range of the rational expression $y=\frac{x+3}{2x^2+3x+9},x\in R$ is

If $\text{cosec}\theta -\cot \theta =\frac{1}{2},$ then the value of ${\sec }^2\theta -{\cos }^2\theta$ is equal to

Let the sequence $a_1,a_2,a_3.....a_n$ form an A.P. Then

$a_1^2-a_2^2+a_3^2-a_4^2+.....+a_{2n-1}^2-a_{2n}^2$ is equal to

The ratio in which $y-$ axis divides the line segment joining $(-3,5)$ and $(7,2)$ is

The equation $\left|\begin{array}{ccc}5& x& 4y\\ 2& -5& -12\\ 1& 0& 3\end{array}\right|=21$ represents a –

The sum of the series ${31}^3+{32}^3+\dots \dots +{50}^3$ is

If $f\left(x\right)=\{\begin{array}{cc}{x}^k\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$ is differentiable at $x=0,$ then (where $k$ is an integer)