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If the line $\frac{x}{a}+\frac{y}{b}=1$ intersects the curve $5x^2+5y^2+5bx+5ay-9ab=0$ at $P$ and $Q$ such that $\angle POQ={90}^{\circ },$ where $O$ is the origin, then the value of $\frac{a}{b}$ is

Let the equations of two sides of a triangle be $3x-2y+6=0$ and $4x+5y-20=0$. If the orthocentre of this triangle is at $(1,1)$, then the equation of its third side is :

A bag contains 4 white, 3 red and 2 blue balls. A ball is drawn at random. Find the probability of the event 'the ball drawn is white or red'.

Let $A=\left[\begin{array}{ccc}2& 0& 7\\ 0& 1& 0\\ 1& -2& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-x& 14x& 7x\\ 0& 1& 0\\ x& -4x& -2x\end{array}\right]$ be two matrices such that $AB=(AB{)}^{-1}$ and $AB\ne I,$ where $I$ is an identity matrix of order $3\times 3.$ Then the value of $tr(AB+(AB{)}^2+(AB{)}^3+\cdots +(AB{)}^{100})$ is

$($ Here, $tr\left(A\right)$ denotes the trace of matrix $A,$ i.e., sum of diagonal elements of $A.)$

Range of $f\left(x\right)=\frac{tan\left(\pi [x^2-x]\right)}{1+sin\left(cosx\right)}$ is where $\left[x\right]$ denotes the greatest integer function

The values of $m$ such that exactly one root of $x^2+2(m-3)x+9=0$ lies between $1$ and $3$, is

If two distinct tangents can be drawn from the point $(\alpha ,2)$ on different branches of the hyperbola

$\frac{x^2}{9}-\frac{y^2}{16}=1$, then

The value of integral ${\int }_1^{e^6}\left[\frac{logx}{3}\right]dx$, where [.] denotes the greatest integer function, is

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

The parametric form the curve

$\frac{(x+1{)}^2}{16}-\frac{(y-2{)}^2}{4}=1$ is

The number of points of intersection of the curve $y=\sin 3x$ with $x$-axis in the interval $(0,\pi )$ is

If $z_1$ and $z_2$ are any two complex numbers, then $|z_1+\sqrt{z_1^2-z_2^2}|+|z_1-\sqrt{z_1^2-z_2^2}|$ is equal to

A factory has $80$ workers and $3$ machines. Each worker knows to operate at least two machines. If there are $65$ persons who know to operate machine $\text{I}$, $60$ for machine $\text{II}$ and $55$ for machine $\text{III}$, what can be the minimum number of persons who know to operate all the three machines ?

The center of the circle given by $\overrightarrow{r}\cdot (\hat{i}+2\hat{j}+2\hat{k})=15$ and $|\overrightarrow{r}-(\hat{j}+2\hat{k})|=4$ is

If $si{n}^{-1}x=\frac{\pi }{5}$ for some $xϵ(-1,1),$ then the value of $co{s}^{-1}x$ is

[IIT 1992]

The length of the chord of the parabola $x^2=4y$ passing through the vertex and having slope $cot\alpha$ is

The sum of values of $x$ satisfying the equation $\sqrt{\frac{x}{1-x}}+\sqrt{\frac{1-x}{x}}=\frac{13}{6}$ is

In $\Delta ABC$, right angled at $B$, if one angle is ${45}^{\circ }$, find the value of $\sin A,\cos C,\cot A$ and $\tan C$ respectively.

A ladder $12$ units long slides in a vertical plane with its ends in contact with a vertical wall and a horizontal floor along $x-$axis. Then the locus of a point on the ladder $4$ units from its foot, is

The least positive integer n which will reduce ${\left(\frac{i-1}{i+1}\right)}^n$to a real number , is

If ${\tan }^{2}x=1-{\alpha }^2$, then the possible values of $\alpha$ satisfying the equation $\sec x+{\tan }^{3}x\text{cosec}x=(2-{\alpha }^2{)}^{3/2}$ is

The value of $\frac{\cos {58}^{\circ }}{\sin {32}^{\circ }}+\frac{\sin {22}^{\circ }}{\cos {68}^{\circ }}-\frac{\cos {38}^{\circ }\text{cosec}{52}^{\circ }}{\tan {18}^{\circ }\tan {35}^{\circ }\tan {72}^{\circ }\tan {55}^{\circ }}$

is

Let $f(x,y)=0$ be the equation of a circle. If $f(0,\lambda )=0$ has equal roots $\lambda =2$ and $f(\lambda ,0)=0$ has root $\lambda =\frac{4}{5},5$, then the centre of the circle is

If $3k,$ $k,$ $[k^2-34]$ are the first three terms of a geometric progression, where $k\in R^{+}$ and $[.]$ is the greatest integer function, then the value of $\sum _{r=1}^{10}(r{)}^{k/2}$ is

Which of the following expression(s) is/are not representing a polynomial for $n\in N$.

Tangents are drawn from a point $P$ to the parabola $y^2=4ax$. If the chord of contact of the parabola is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then the locus of $P$ is

If $t_1$ and $t_2$ are the roots of the equation $t^2+\lambda t+1=0$, where $\lambda$ is a parameter, then the line joining the points $(a{t}_1^2,2a{t}_1)$ and $(a{t}_2^2,2a{t}_2)$ always passes through

If $b{x}^2+acx+b^{2}c=0$ and $c{x}^2+abx+b^{2}c=0$ have a common root (where $a,b,c$ are non zero distinct real numbers), then which of the following is/are correct?

If for $x\in (0,\frac{1}{4})$, the derivative of ${\tan }^{-1}\left(\frac{6x\sqrt{x}}{1-9x^3}\right)$ is $\sqrt{x}\cdot g\left(x\right)$, then $g\left(x\right)$ equals:

If $\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})$ is perpendicular to $(\overrightarrow{a}\times \overrightarrow{b})\times \overrightarrow{c}$ then, we may have,

Complex number $z$ satisfying $|z+1|=z+2(1+i),$ is

For all permissible values of $A,2A$, following holds true.

$\left(i\right)\cot A+\tan A=\frac{1}{\sin A\cos A}=2\text{cosec}2A\left(ii\right)\cot A-\tan A=\frac{{\cos }^{2}A-{\sin }^{2}A}{\sin A\cos A}=2\cot 2A\left(iii\right)2\cot A=2(\text{cosec}2A+\cot 2A)\Rightarrow \text{cosec}2A+\cot 2A=\cot A$



Also to evaluate a series of form $f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)$ when $f\left(x\right)$ can be expressed as $g\left(x\right)-g\left(2x\right)$, we can use the following technique,

$f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)=\left(g\right(x)-g(2x\left)\right)+\left(g\right(2x)-g(4x\left)\right)+\cdots \left(g\right(2^{n}x)-g(2^{n+1}x\left)\right)=g\left(x\right)-g\left(2^{n+1}x\right)$



Based on the above information, solve the following questions for all permissible values of $x$.



The value of $\cot 37{\frac{1}{2}}^{\circ }$ is

A survey shows that $63\%$ of the people in a city read newspaper $A$ whereas $76\%$ read newspaper $B$. If $x\%$ of the people read both the newspapers, then a possible value of $x$ can be

Find the point(s) where the function f(x) = $x^3$ - 3x + 2 is increasing

The equation of the line passing through the point (1, 1, –1) and perpendicular to the plane x – 2y – 3z = 7 is

The equation(s) of the standard ellipse passing through the point (4,-1) and touching the line x + 4y - 10 = 0 is/are:

The coordinates of a point which is equidistant from the points $(0,0,0),(a,0,0),(0,b,0)and(0,0,c)$ are given by _____

Let $3600=x\cdot y$, where $x$ and $y$ are natural numbers, then the possible pairs $(x,y)$ is

The locus of centre of a circle which passes through the origin and cuts off a length of $4$ units on the line $x=3$ is

If $\frac{si{n}^{4}A}{a}+\frac{co{s}^{4}A}{b}=\frac{1}{a+b},$ then $\frac{si{n}^{8}A}{a^3}+\frac{co{s}^{8}A}{b^3}=$

The vertices of a hyperbola are at $(0,0)$ and $(10,0)$ and one of its foci is at $(18,0)$. The equation of the hyperbola is

The equation of circle passing through the origin and cutting off equal intercepts of $4$ units on the lines $xy=0$ can be

If $x\frac{dy}{dx}=y(logy-logx+1)$, then the solution of the equation is

If $A=\{a,b,2,3\},B=\{a,3,c\},$ $C=\{1,3,c\},$ then $n\left(\right(A\times B)\cap (A\times C\left)\right)$ is

Let $a,b,c$ be three distinct real numbers in geometric progression. If $x$ is real and $a+b+c=xb,$ then $x$ can be

Find the ratio of the length of the tangents from any point on the circle $15x^2+15y^2-48x+64y=0$ to the two circles $5x^2+5y^2-24x+32y+75=0,$ $5x^2+5y^2-48x+64y+300=0.$