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If a < b < c < d, then the roots of the equation (x-a)(x-c) + 2(x-b)(x-d) = 0 are

${lim}_{x\to 0}\frac{a^x+b^x-2}{x}$

If $|x^2+3x-4|=|x+4|$, then $x=$

Reduce the equation $\sqrt{3}x+y+2=0$ to:

(i) slope-intercept form and find slope and y-intercept;

(ii) intercept form and find intercept on the axes;

(iii) the normal form and find p and $\alpha .$

The number of solutions of $x^2+|x-1|=1is$

Consider the curve $x=1-3t^2,y=t-3t^3$, Let P(-2, 2) be a point on the curve. Let the tangent at P
cuts the curve again at Q. Then answer the following
The angle between the tangents at P and Q will be

The area (in sq. units) of the region consisting of the points $x,y$ on $X-Y$ plane which satisfy
$|x|\le 1+|y|$ and $|y|\le 1$ is

Find the equation of the side BC of the triangle ABC whose vertices are A (-1, -2), B (0, 1) and C (2, 0) respectively. Also, find the equation of the median through A (-1, -2).

If -3x+17<-13,then

By using properties of definite integrals, evaluate the integrals
${\int }_0^{2\pi }co{s}^{5}xdx.$

Mark the correct alternative in each of the following :

$Inany\Delta ABC,a(bcosC-ccosB)=$

Mark the correct alternative in each of the following :

$Inany\Delta ABC,2(bccosA+cacosB+abcosC)=$

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

A coin is tossed and then a die is thrown. Describe the sample space for this experiment.

Number of integral values of x satisfying $(\frac{3}{4}{)}^{6x+10-x^2}<\frac{27}{64}$ is ___

Write 1 - i in polar form.

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

Let $E_1$ and $E_2$ be two ellipse whose centres are at the origin. The major axes of $E_1$ and $E_2$ lie along x-axis and y-axis respectively. Let S be the circle $x^2+(y-1{)}^2=2$ the straight line $x+y=3$ touches the curves S, $E_1$ and $E_2$ at P, q and R, respectively.
Suppose that $PQ=PR=\frac{2\sqrt{2}}{3}.$ If $e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2$ respectively, then the correct expression(s) is/are

If the circles $x^2+y^2-2x-4y=0$ and $x^2+y^2-8y-k=0$ touch each other internally, then the value of $k$ is

The equation of the bisector of the acute angle between $4x+3y-6=0$ and $5x+12y+9=0$ is

${}^n{C}_{r-1}=330,{}^n{C}_r=462,{}^n{C}_{r+1}=462\Rightarrow r=$

If $x-iy=\sqrt{\frac{a-id}{c-id}},$ prove that $(x^2+y^2{)}^2=\frac{a^2+b^2}{c^2+d^2}$.

If $e_1$ and $e_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}{4}=1$ and the hyperbola, $\frac{x^2}{9}-\frac{y^2}{4}=1$ respectively and $(e_1,e_2)$ is a point on the ellipse, $15x^2+3y^2=k$. Then $k$ is equal to

If ${\sin }^{2}x+{\cos }^{2}y=2{\sec }^{2}z$, then
(where $m,n,t\in Z$)

Prove the following:

$\frac{\text{tan}(\frac{\pi }{4}+x)}{\text{tan}(\frac{\pi }{4}-x)}$ = ${\left[\frac{1+\text{tan x}}{1-\text{tan x}}\right]}^2$

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.

$g\left(x\right)=\frac{1}{x^2+2}$

A line a drawn through A (4, -1) parallel to the line 3 x - 4 y + 1 = 0. Find the coordinates of the two points on this line which are at a distance of 5 units from A.

If $ABCD$ (in order) is a quadrilateral inscribed in a circle, then which of the following is/are always true?

There are $10$ white and $10$ black balls marked $1,2,3,...,10.$ The number of ways in which we can arrange these balls in a row in such a way that neighbouring balls are of different colour, is

If ${\log }_{\sin \frac{\pi }{6}}\left\{\frac{|z-2|+3}{3|z-2|-1}\right\}>1$, then

The equation of a line inclined at an angle $\frac{\pi }{4}$ to the $x-$axis, such that the two circles $x^2+y^2=4,$ $x^2+y^2-10x-14y+65=0$ intercept equal lengths on it, is

The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is

Integrate the following functions.
$\int \frac{1}{\sqrt{(x-a)(x-b)}}dx$

Let $E_1$ and $E_2$ be two independent events such that $P\left(E_1\right)=P_1$ and $P\left(E_2\right)=P_2$. Describe in words of the events whose probabilities are

$P_1+P_2-2P_1{P}_2$

Assume that each child born is equaly likely to be boy or a girl. IF a family has two children, what is the conditional probability that both are girls given that
the youngest is a girl?

atleast one is a girl?

Integrate the following functions.
$\int \frac{1}{\sqrt{7-6x-x^2}}dx.$

How can I save time in solving vector problems where just Ā B and such sort of variables are given and ask for angles between them and their properties what's way?

what is the value of c in the equation y = 4 x square - 3 x ?

Prove that the relation R defined on set A of all lines as : R={(L1,L2): L1and L2 are parallel lines} is an equivalence relation

If A is a square matrix of order 3 and determinant of A=3 find (AA^T)

The point on $X-$ axis at a distance of $10$ units from $(6,10)$ is

The point $Q$ is the image of the point $P(1,5)$ about the line $y=x$ and $R$ is the image of the point $Q$ about the line $y=–x$. The circumcenter of the $\Delta PQR$ is

Value of $\left[\right({\log }_{b}a\left)\right({\log }_{c}b\left)\right({\log }_{a}c\left)\right]$

In a quadrilateral ABCD,
Let $△=\left|\begin{array}{ccc}cosA& sinA& cos(A+D)\\ cosB& sinB& cos(B+D)\\ cosC& sinC& cos(C+D)\end{array}\right|,$ then $△$ is

It is given that at x=1, the function $x^4-62x^2+ax+9$ attains its maximum value, on the interval [0,2]. Find the value of a.

Integrate the rational functions.
$\int \frac{2x-3}{(x^2-1)(2x+3)}dx.$