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Find the equation of tangent at $(-\frac{3}{2},\frac{2}{2})$ to the ellipse

$7x^2+7y^2+2xy+10x-10y+7=0$

Let $A(6,4)$ and $B(2,12)$ be two given points, then the equation of perpendicular bisector of $AB$ is

Let $\overrightarrow{a}=2\overrightarrow{i}+\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}+2\hat{j}-\hat{k}$ and a unit vector $\overrightarrow{c}$ be coplanar. If $\overrightarrow{c}$ is perpendicular to $\overrightarrow{a},$ then $\overrightarrow{c}$ is equal to

f(x) = sin(x) defined on f: $[\frac{-\pi }{2},\frac{\pi }{2}]$ $\to$ $[-1,1]$ is -

Numbers are selected at random, one at a time, from the two-digit numbers 00, 01, 02,..., 99 with replacement. An event E occurs if and only if the product of the two digits of a selected number is 18. If four numbers are selected, find probability that the event E occurs at least 3 times.

Let $x_1,x_2(x_1\ne x_2)$ be the roots of the equation $x^2+2(m-3)x+9=0$. If $-6<x_1,x_2<1,$ then ${}^{\prime }{m}^{\prime }$ lies in the interval

Let $z$ and $\omega$ be complex numbers such that $\overline{z}+i\overline{\omega }=0$ and $argz\omega =\pi$. Then $argz$ equals

The least positive integer value of $a$ for which both roots of the quadratic equation $(a^2-6a+5)x^2+\left(\sqrt{a^2+2a}\right)x+(6a-a^2-8)=0$ lie on either side of origin, is -

An item is manufactured by three machines X, Y and Z. Out of the total number of items manufactured during a specified period, 40% are manufactured on X, 40% on Y and 20% on Z. 3% of the items produced on X and 2% of items produced on Y are defective, and 3% of these produced on Z is defective. All the items are stored at one godown. One item is drawn at random and is found to be defective. What is the probability that it was manufactured on machine Y?

If a chord which is normal to the parabola $y^2=4ax$ at one end subtends a right angle at the vertex, then its slope can be

The arithmetic mean of $5$ consecutive integers starting with ${}^{\prime }{s}^{\prime }$ is ${}^{\prime }{a}^{\prime }.$ Then the arithmetic mean of $9$ consecutive integers that start with $s+2$ is

If the radius of the sphere

$x^2+y^2+z^2-2x-4y-6z$ = 0

is r, then find the value of $r^2$

Let f:$[0,1]\to R$ (the set of all real numbers) be a function. Suppose the function f is twice differentiable,$f\left(0\right)=f\left(1\right)=0$ and satisfies $f^{\prime \prime }\left(x\right)-2f^{\prime }\left(x\right)+f\left(x\right)\ge e^x,x\in [0,1]$

If the function $e^{-x}f\left(x\right)$ assumes its minimum in the interval [0, 1] at $=\frac{1}{4}$, which of the following is true?

If '$a$' and '$b$' are the non-zero distinct zeros of $x^2+ax+b$, then the least value of quadratic polynomial $x^2+ax+b$ is

The value of the expression $(2+\sqrt{2}{)}^4$ lies between

In $△ABC,\frac{sin(A-B)}{sin(A+B)}=$

[MP PET 1986]

A tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ cuts the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ in points $P$ & $Q$. The locus of the mid- point $PQ$ is

The sum of an infinite number of terms of a G.P. is $20$, and the sum of their squares is $100$, then the first term of the G.P. is

If roots of the equation $a{x}^2+bx+c=0$ are $\alpha ,\beta$, then the equation whose roots are $\frac{1+\alpha }{1-\alpha },\frac{1+\beta }{1-\beta },$ where $\alpha \ne 1,\beta \ne 1$ is

Matrix $A=\left[\begin{array}{ccc}x& 3& 2\\ 1& y& 4\\ 2& 2& z\end{array}\right]$. If $xyz=60$ and $8x+4y+3z=20$, then $A\left(adjA\right)$ is equal to

If the graph of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ is


Then, the possible value(s) of $a$ can be

If the roots of the equation $a{x}^2+bx+a_1^2+b_1^2+c_1^2-a_1{b}_1-a_1{c}_1-b_1{c}_1=0$ are non real then

A group contains $10$ people. A rumour is spread from one person to another. The recipient of the rumour is chosen at random at each stage. However, the person who receives a rumour cannot transmit it back to the person from whom he/she received it. A rumour is passed by one person and spread to a total of five people . What is the number of ways in which it will not be repeated to the first recipient ?

Choose the correct representation for the sets $C=\{a,e,f\},D=\{d,e,f\}$ and $U=\{a,b,c,d,e,f\}.$

If ${\theta }_1$ and ${\theta }_2$ be the angles which the lines $(x^2+y^2)({\cos }^2\theta {\sin }^2\alpha +{\sin }^2\theta )=(x\tan \alpha -y\sin \theta {)}^2$ make with the axis of $x$, then if $\theta =\frac{\pi }{6},\tan {\theta }_1+\tan {\theta }_2$ is equal to

The set of all real values of $x$ which satisfy $(x-2)(x-3)\le 2\le (x-1)(x-2),$ is

If the cartesian coordinates of a point are $(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}),$ then the polar coordinates are

Find the equation of normal having slope 1 to the parabola

$y^2=4x$

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are perpendicular, then $\overrightarrow{a}\times (\overrightarrow{a}\times (\overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{b})))$ is equal to :

If $i=\sqrt{-1}$, then $4+5{(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{334}+3{(-\frac{1}{2}+\frac{\sqrt{3}}{2}i)}^{365}$ is equal to

Let $a_i,i=1,2,\dots ,n$ be an A.P. If $a_7=9,$ then the value of common difference of the A.P. such that $a_1\cdot a_2\cdot a_7$ is minimum, is

The graph of $f\left(x\right)=x^2-3|x|+2$ is

The line $x=y$ touches a circle at the point $(1,1).$ If the circle also passes through the point $(1,-3),$ then its radius (in units) is :

Let $S$ be the set of all $\alpha \in R$ such that the equation, $\cos 2x+\alpha \sin x=2\alpha -7$ has a solution. Then $S$ is equal to:

If $4a^2+9b^2+16c^2=2(3ab+6bc+4ca),$ where $a,b,c$ are non-zero numbers, then $a,b,c$ are in

If sin 3x + cos 2x = -2, then find the value of x

$Aforceof\overrightarrow{F}=3\hat{i}+4\hat{j}isactingonaboxatpoint\overrightarrow{A}whosepositionvectorwithrespecttooriginis<2,3>.Workdoneindisplacingtheparticlefrom\overrightarrow{A}to\overrightarrow{B}whosepositionvectorwithrespecttooriginis<5,6>willbe.......units$

Let $I_1={\int }_0^{\frac{\pi }{4}}{x}^{2008}(tanx{)}^{2008}dx,I_2={\int }_0^{\frac{\pi }{4}}{x}^{2009}(tanx{)}^{2009}dxand{I}_3={\int }_0^{\frac{\pi }{4}}{x}^{2010}(tanx{)}^{2010}dx$

The sum to $n$ terms of the series $(1\times 2\times 3)+(2\times 3\times 4)+(3\times 4\times 5)+\dots$ is

$\frac{1}{4}$$[\sqrt{3}cos{23}^{\circ }-sin{23}^{\circ }]$ =

Let $f:\to R\to R,g:R\to Randh:R\to R$ be differentiable functions such that $f\left(x\right)=x^3+3x+2,g\left(f\right(x\left)\right)=xandh\left(g\right(g\left(x\right)\left)\right)=xforallx\epsilon R.$ Then

Consider a family of circles which are passing through the point (-1, 1) and are tangent to the x-axis. If (h, k) are the coordinates of the centre of the circles, then the set of values of k is given by the interval

The locus of point of intersection of two normals drawn to the parabola $y^2=4ax$ which are at right angles is