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Let $Q(acos\theta ,bsin\theta )$ be a point on the auxiliary circle. Then the corresponding point with respect to Q on the ellipse when a line drawn perpendicular to major axis AA' will be.

If matrix $A=\left[\begin{array}{ccc}1& 0& -1\\ 3& 4& 5\\ 0& 6& 7\end{array}\right]$ and its inverse is denoted by $A^{-1}=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$, then the value of $a_{23}$=

Which of the following statments are correct?

1. The coordinates of the point which divides the line segment joining the points

$(1,-2,3)$ and $(3,4,-5)$ internally in the ratio $2:3$ is $(-3,-14,19)$.

2. The coordinates of the point which divides the line segment joining the points

$(1,-2,3)$ and $(3,4,-5)$ externally in the ratio $2:3$ is $(\frac{9}{5},\frac{2}{5},\frac{-1}{5})$.

If $|z-\frac{4}{z}|$ =2, then the maximum value of $\left|z\right|$ is

${lim}_{x\to 3}\frac{x^4-81}{x^2-9}$

A wooden cube with edge length ‘n’ (>2) units is painted red all over. By cutting parallel to faces, the cube is cut into $n^3$ smaller cubes each of unit edge length. If the number of smaller cubes with just one face painted Red is equal to the number of smaller cubes completely un painted, then n=

Solve y² +1/2y-1=0 by Factorisation

If coefficients of $5^{\text{th}},6^{\text{th}}$ and $7^{\text{th}}$ terms in the expansion of $(1+x{)}^n$ are in A.P., then the value(s) of $n$ is/are

How will you deal with the following items while preparing for the Bombay Women Cricket Club its income and expenditure account for the year ending 31.3.2007 and its Balance Sheet as on 31.3.2007

$\begin{array}{ccc}& & \\ & & \left(Rs\right)\\ \text{(a)}& \text{Donation received during the year for the construction}& \\ & \text{of a permanent pavilion}& 12,25,000\\ & \text{Expenditure incurred up to 31.3.2007 on its construction}& 10,80,000\\ & \text{The Total estimated expenditure on construction}& \\ & \text{of Pavilion being}& 25,00,000\\ \text{(b)}& \text{Tournament Fund}& \\ & \text{Balance as on 1.4.2006}& 10,700\\ & \text{Subscriptions for tournament received during the year}& 65,800\\ & \text{Expenditure incurred during the year on}& \\ & \text{conducting tournaments}& 72,400\\ \left(c\right)& \text{Life Membership fee received during the year}& 28,000\end{array}$

Give reasons for your answers.

In a village, there are 87 families of which 52 families have at most 2 children. In a rural development programme, 20 families are to be helped chosen for assistance, of which at least 18 families must have at most 2 children. In how many ways can the choice be made ?

The equation of the circle concentric with the circle $x^2+y^2+8x+10y-7=0$ and passing through the centre of the circle $x^2+y^2-4x-6y=0$ is

If x-y =5,xy=84,findx+y

A rectangle ABCD of dimensions r and 2r is folded along diagonal BD such that planes ABD and CBD are perpendicular to each other. Let the position of the vertex A remains unchanged and $C_1$ is the new position of C.
If $\angle AB{C}_1=\theta$, then cos $\theta$ is equal to

Solution of the equation $co{s}^{2}x\frac{dy}{dx}-\left(tan2x\right)y=co{s}^{4}x,|x|<\frac{\pi }{4},$ where $\left(\frac{\pi }{6}\right)=\frac{3\sqrt{3}}{8}$ is

A tangent is drawn to the parabola $y^2=6x$ which is perpendicular to the line $2x+y=1$. Which of the following points does $\text{NOT}$ lie on it ?

If $f\left(x\right)=x^2+1$ and $g\left(x\right)=2x$, then find the domain of the function
$h\left(x\right)=\frac{(f+g)x}{(f-g)x}$

The domain of $\sqrt{x+2}+\frac{1}{{\log }_{10}(x+1)}$ is

A chord $AB$ is drawn from the point $A(0,3)$ on the circle $x^2+4x+(y-3{)}^2=0,$ and is extended to $M$ such that $AM=2AB$. The locus of $M$ is

If A, B are, respectively m × n, k × l matrices, then both AB and BA are defined if and only if

If ${}^{\prime }{z}^{\prime }$ lies on the circle $|z-2i|=2\sqrt{2}$, then the value of $\arg \left(\frac{z-2}{z+2}\right)$ is equal to

If $z=\left(\frac{1+i}{1-i}\right)$, then $z^4$ equals

If a + b + c = 0 the roots of the quadratic equation $a{x}^2+bx+c=0$ is/are

Consider set of observations $x_1,x_2,x_3,\cdots ,x_{101}$. It is given that $x_1<x_2<x_3\cdots <x_{100}<x_{101}$ then the mean deviation of the set of observations about a point $k$ is minimum when $k$ equals

${lim}_{x\to 0}\frac{ta{n}^{2}3x}{x^2}$

A real function f is said to be continuous if it is continuous at every point in ...... .

If $A=\{1,2,3,4,5\}$ then which of the following is correct?

The variance of 20 observation is 5. If each observation is multiplied by 2, find the variance of the resulting observations.

Show that the sequence $<a_n>$, defined by $a_n=\frac{2}{3^n},nϵN$ is a G.P.

Find the critical points of the function $f\left(x\right)=2x^3+15x^2+36x.$

If sum of the coefficients of first, second and third terms in the expansion of ${(x^2+\frac{1}{x})}^m$ is $46,$ then the coefficient of the term that is independent of $x,$ is

Let C be the circle with centre at (1, 1) and radius 1. If T is the circle centred at (0, y) passing through origin and touching the circle C externally, then the radius of T is equal to

If $p,q,r,s,t$ and $u$ are in A.P. then difference $(t-r)$ is equal

Let $\overrightarrow{p},\overrightarrow{q}$ and $\overrightarrow{r}$ be three non-coplanar unit vectors equally inclined to each other at an angle of $\frac{\pi }{3}$. Then the value of $|\overrightarrow{p}\times (\overrightarrow{q}\times \overrightarrow{r})|$ is

The speed of a projective when it is at its greatest height is $\sqrt{\frac{2}{5}}$ times its speed at half the maximum height, What is its angle of projection?

Probability that a student will succeed in IIT entrance test is 0.2 and that he will succeed in Roorkee entrance test is 0.5. If the probability that he will be successful at both the places is 0.3, then the probability that he does not succeed at both the places is

On March 31, 2006 after the close of accounts, the capitals of Mountain, Hill and Rock stood in the books of the firm at Rs 4,00,000, Rs 3,00,000 and Rs 2,00,000 respectively. Subsequently, it was discovered that the interest on capital 10% pa had been omitted. The profit for the year amounted to Rs 1,50,000 and the partner's drawings had been Mountain; Rs 20,000, Hill Rs 15,000 and Rock Rs 10,000.

Calculate interest on capital.

${\int }_0^{\frac{\pi }{2}}\text{log sin x dx =}$

A relation n$\varphi$ from C to R is defined by $x\varphi y\iff |x|=y.$ Which one is correct ?

In the expansion of ${(\frac{1}{2}{x}^{1/3}+x^{-1/5})}^8$, the term independent of x is