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If 1, $\omega$ , ${\omega }^2$ are the roots of the $x^3$ + p$x^2$ + qx + r = 0 , find ( p , q, r)

Find the equation of the tangent and normals to the curve $y=x^3+2x+6$ which are parallel to the line x+14y+4=0.

If $y=(cosx{)}^{(cosx{)}^{{cosx}^{-\infty }}}$, then show that $\frac{dy}{dx}=\frac{y^{2}tanx}{ylog\left(cosx\right)-1}$

In a hand at Whist, what is the probability that four kings are held by a specified player?

The centre of circle $x^2+y^2+16x-22y-20=0,$ is

$ta{n}^{-1}\frac{1}{\sqrt{x^2-1}}=$

The Van’t Hoff factor for 0.1 M $Ba(N{O}_3{)}_2$ solution is 2.74. The degree of dissociation is:

Let $\alpha$ and $\beta$ be the roots of equation $p{x}^2+qx+r=0,p\ne 0.$ If $p,q,r$ are in A.P. and $\frac{1}{\alpha }+\frac{1}{\beta }=4,$ then the value of $|\alpha -\beta |$ is

If angle between line $\overrightarrow{r}=\hat{i}+2\widehat{+}2\hat{k}+\lambda (4\hat{j}-3\hat{k})$ and $XY$ plane is $\alpha$, and angle between the planes $x+2y=0$ and $2x+y=0$ is $\beta$, then $\frac{{\cos }^2\alpha }{{\sin }^2\beta }=$

If $f_n\left(\theta \right)=\sum _{r=0}^n\frac{1}{4^r}{\sin }^4\left(2^r\theta \right)$, then which of the following alternative(s) is/are correct?

Solve the inequalities in Exercieses 7 to 10 and represent the solution graphically on number line:

2(x-1)< x+5, 3(x+2)>2-x

Let the function, $f:[-7,0]\to R$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }\left(x\right)\le 2,$ for all $x\in (-7,0),$ then for all such functions $f,f(-1)+f\left(0\right)$ lies in the interval:

Table below shows the frequency f with which 'x' alpha particles were radiated from a diskette

$\begin{array}{cccccccccccccc}x:& 0& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ f:& 51& 203& 383& 525& 532& 408& 273& 139& 43& 27& 10& 4& 2\end{array}$

Find the local maxima and local minima, if any of the following function. Also, find the local maximum and the local minimum values, as the case may be as follows.

$h\left(x\right)=sinx+cosx,0<x<\frac{\pi }{2}$

Consider two functions defined on R as f(x) = 2 + |x – 1| and g(x) = min (f(t)) where $x\le t\le x^2+x+1$. If $n_1$ denotes number of points of discontinuity of g(x) and $n_2$ denotes number of points where g(x) is non-differentiable then $(n_1+n_2)$ is equal to ___.

1. 3

   
   

The quadrilateral formed by the lines $y=ax+c$, $y=ax+d$, $y=bx+c$ and $y=bx+d$ has area $18$. The quadrilateral formed by the lines $y=ax+c$, $y=ax-d$, $y=bx+c$ and $y=bx-d$ has area $72$. If $a,b,c,d$ are positive integers then the least possible value of the sum $a+b+c+d$ is

1) Sin(40+A)cos(10+A)-cos(40+A)sin(10+A)

If $x+\frac{1}{x}=4,$ then the value of $x^4+\frac{1}{x^4}$ is

If the sides a, b, c of a triangle ABC are the roots of the equation $x^3-13x^2+54x-72=0$, then the value of $\frac{cosA}{a}+\frac{cosB}{b}+\frac{cosC}{c}$ is equal to

If tan A +cot A =4, then the value of $ta{n}^{4}A+co{t}^{4}A.$

$3\times 1^2+5\times 2^2+7\times 3^2+...$

If the locus of the midpoint of contact of tangent drawn to the parabola $y^2=8x$ and foot of perpendicular drawn from its focus to the tangents is a conic then length of latus rectum of this conic is

For a given parabola $y^2=4ax$, two variable chords PQ and RS at right angles are drawn through the fixed point $A(x_1,y_1)$ inside the parabola, making variable angles $\theta$ and $\alpha$ with x-axis. If $r_1,r_2,r_3,r_4$ are distances of P, Q, R and S from A, then the value of $\frac{1}{r_1{r}_2}+\frac{1}{r_3{r}_4}$

1.If A and B are non-singular matrices, then

[MP PET 1991; Kurukshetra CEE 1998]

Given the parametric equations $x=f\left(t\right),y=g\left(t\right)$, then $\frac{d^{2}y}{d{x}^2}$ equals

Coordinates of the centre of a circle whose radius is $2$ units and touches the pair of lines $x^2-y^2-2x+1=0$ are

If $y=ta{n}^{-1}\left(\frac{1}{1+x+x^2}\right)+ta{n}^{-1}\left(\frac{1}{x^2+3x+3}\right)+ta{n}^{-1}\left(\frac{1}{x^2+5x+7}\right)+....+$ up to n terms. Then y’(0) is equal to

One of the solutions of the equation $8{\sin }^3\theta -7\sin \theta +\sqrt{3}\cos \theta =0$ lies in the interval

Let $I_1={\int }_{\frac{\pi }{3}}^{\frac{\pi }{6}}\frac{\sin x}{x}dx,I_2={\int }_{\frac{\pi }{3}}^{\frac{\pi }{6}}\frac{\sin \left(\sin x\right)}{\sin x}dx,I_3={\int }_{\frac{\pi }{3}}^{\frac{\pi }{6}}\frac{\sin \left(\tan x\right)}{\tan x}dx$, then

$\int secx(secx+tanx)dx=$

Integrate the following functions.
$\int \frac{si{n}^{-1}x}{\sqrt{1-x^2}}dx.$

If $\overrightarrow{b}$ is a vector whose initial point divides the join of $5\hat{i}$ and $5\hat{j}$ in the ratio $k:1$ and whose terminal point is the origin and $|\overrightarrow{b}|\le \sqrt{37}$, then $k$ lies in the interval

If $S=4-9x+16x^2-25x^3+36x^4-49x^5+......\infty$. Find the value of S for x=$\frac{-1}{2}$. ___

The equation of the straight line passing through the origin and the middle point of the portion of the line ax + by + c = 0 between the axes, is

There are two bags, one of which contains 5 black and 4 white balls while the other contains 3 black and 6 white balls. A die is thrown. If it shows up 1 or 3, a ball is taken from the first bag. But if it shows up any other number, a ball is chosen from the second bag. Find the probability of choosing a white ball.

If sum of n, 2n, 3n terms of an A.P. are $S_1,S_2,S_3$ respectively, then the value of $\frac{S_3}{S_2-S_1}$ is

Consider the plane $\pi :x+y=z$, point A(1, 2, -3) and a line $L:\frac{x-1}{3}=\frac{y-2}{-1}=\frac{z-3}{4}$

The coordinates of a point B on L such that AB is parallel to the plane is ___

$\frac{d}{dx}\left[(x+1)(x^2+1)(x^4+1)(x^8+1)\right]=(15x^p-16x^q+1){(x-1)}^{-2}$
$\Rightarrow (p,q)=$

The ratio of the coefficient of $x^2$ to the coefficient of $x^{10}$ in the expansion of ${(x^5+4\cdot 3^{-{\log }_{\sqrt{3}}\sqrt{x^3}})}^{10}$ is

If $C_0,C_1,\cdots C_n$ are the coefficient of $x$ in expansion of $(1+x{)}^n$, then $C_0-C_2+C_4-C_6+\cdots +(-1{)}^n{C}_n=$

If the following functions have both domain and co-domain as $[-1,1]$, then select those which are not bijective?