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The equation of circle touching the line $2x+3y+1=0$ at $(1,-1)$ and orthogonally cutting the cirlce whose endpoints of diameter are $(0,3)$ and $(-2,-1)$ is

Find the value of k for which the quadratic equation $(3k+1)x^2+2(k+1)x+1=$has equal roots. Also, find the roots.

If $co{s}^{-1}\left(\frac{x^2-y^2}{x^2+y^2}\right)=loga$ then $\frac{dy}{dx}$ is equal to

$a+ar+a{r}^2+......+a{r}^{n-1}=a\left(\frac{r^n-1}{r-1}\right)$ $r\ne 1$.

Solve:
(i) $\frac{16x^2-20x+9}{8x^2+12x+21}=\frac{4x-5}{2x+3}$
(ii) $\frac{5y^2+40y-12}{5y+10y^2-4}=\frac{y+8}{1+2y}$

Let $\hat{a}$ and $\hat{b}$ be two unit vectors. If the vectors $\overrightarrow{c}=\hat{a}+2\hat{b}$ and $\overrightarrow{d}=5\hat{a}-4\hat{b}$ are perpendicular to each other, the angle between $\hat{a}$ and $\hat{b}$ is:

If one geometric mean $G$ and two arithmetic means $A_1$ and $A_2$ are inserted between two distinct positive numbers, then $\left(\frac{2A_1-A_2}{G}\right)\left(\frac{2A_2-A_1}{G}\right)$ is equal to

Solve the following system of inequalities graphically: 3x + 2y ≤ 150, x + 4y ≤ 80, x ≤ 15, y ≥ 0, x ≥ 0

If $2x^2+2y^2-12x+8y+k=0$ is a point circle, then the value of $k$ is

Pair together the addition and multiplication statements which give the same answers.

$cos\frac{\pi }{7}$. $cos\frac{3\pi }{7}$. $cos\frac{5\pi }{7}$ are the roots of the equation $8x^3-4x^2-4x+1=0$. Then the value of $sec\frac{\pi }{7}+sec\frac{3\pi }{7}+sec\frac{5\pi }{7}$ is

$FourpointsA(6,3),B(-3,5),C(4,-2)andD(x,3x)aregiveninsuchawaythat\frac{\Delta DBC}{\Delta ABC}=\frac{1}{2},findx.$

The value of $\underset{x\to 0}{lim}\frac{2x\sin 2x+3\tan x+x^2\sin x}{\sin 3x}$ is

If $\alpha ,\beta$ are the roots of $2x^2-3x+4=0$, then the equation whose roots are ${\alpha }^2$ and ${\beta }^2$ is

A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale, otherwise, it is rejected. Find the probability that a box containing $15$ oranges out of which $10$ are good and $5$ are bad ones will be approved for sale.

If $P=2^4\cdot 6^3\cdot 5^2\cdot {15}^2$, then the number of proper even divisors of $P$ will be

Two continuous random variables $X$ and $Y$ are related as $Y=4X+2.$ The correct relation between the differential entropies of the two random variable is equal to

The lines $(lx+my{)}^2-3(mx-ly{)}^2=0$ and lx + my + n = 0 form

Let $P=\left[a_{ij}\right]$ be a 3 $\times$ 3 matrix and let $Q=\left[b_{ij}\right]$, where $b_{ij}=2^{i+j}{a}_{ij}for1\le i,j\le 3$. If the determinant of P is 2, then the determinant of the matrix Q is

Match the colums:



$\begin{array}{cc}Column1& Column2\\ \text{a. The set of points}z\text{satisfying}|z-i|z||=|z+i|z||& \text{p. an ellipse with eccentricity}\frac{4}{5}\\ \text{is contained in or equal to}& \\ \text{b. The set of points}z\text{satisfying}|z+4|+|z-4|=10& \text{q. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& \text{Img}z=0\\ \text{c. If}|w|=2,\text{then the set of points}z=w-\frac{1}{w}& \text{r. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Img}z|\le 10\\ \text{d. If}|w|=1,\text{then the set of points}z=w+\frac{1}{w}& \text{s. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Re}z|<2\\ & \text{t. the set of points}z\text{satisfying}\\ & |z|\le 3\end{array}$

The equation of the parabola whose axis is parallel to y – axis and passing through (4, 5), (–2, 11), (–4, 21) is

The sum of the coefficients of even power of $x$ in the expansion of ${(1+x+x^2+x^3)}^5$ is

The sum of two numbers is $30$. Then find one of the number, given that the product of those two numbers are maximum ?

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are three non coplanar, non zero vectors and $\overrightarrow{r}$ is any vector in space, then $(\overrightarrow{a}\times \overrightarrow{b})\times (\overrightarrow{r}\times \overrightarrow{c})+(\overrightarrow{b}\times \overrightarrow{c})\times (\overrightarrow{r}\times \overrightarrow{a})+(\overrightarrow{c}\times \overrightarrow{a})\times (\overrightarrow{r}\times \overrightarrow{b})$ is equal to $\lambda \left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$. Then the value of $\lambda$ is

Let $A=N\times N$ and $\ast$ be the binary operation on A defined by $(a,b)\ast (c,d)=(a+c,b+d)$. Show that $\ast$ is commutative and associative. Find the identity element for $\ast$ on A, if any.