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Find the lengths of the axes; the coordinates of the vertices and the foci; the eccentricity and length of the latus rectum of the hyperbola $9x^2-16y^2=144.$

If a, b, c and d are in G.P show that $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd{)}^2.$

The equation(s) of normal(s) to the curve $3x^2-y^2=8$ which is (are) parallel to the line x + 3y = 4 is (are)

What is well defined collection of an object with an example

If $-5<p<-2$ and $7<q<9$, then what is the range of p + q?

There are $2$ brothers among a group of $20$ persons. The number of ways the group can be arranged around a circle so that there is exactly one person between the two brothers is

Range of g(x) is

AZ, FU,KP, ?, UF

The co-ordinates of the point in which the line joining the points (3, 5, -7) and (-2, 1, 8) is intersected by the plane yz are given by
[MP PET 1993]

In the expansion of ${}^{20}{C}_r$.$4^{\frac{20-r}{3}}{.6}^{-\frac{r}{4}}$

If S = $co{s}^2\frac{\pi }{n}+co{s}^2\frac{2\pi }{n}+........+co{s}^2\frac{(n-1)\pi }{n}$ then S is equal to

In a "keep-fit” gymnasium class, there are fifteen females enrolled in a weight-loss program. They all have been grouped in any one of the five weight-groups $W_1,W_2,W_3,W_4$ or $W_5$One instructor is assigned to one weight-group only. A, B, C, and D belong to the same weight-group. A and E are in one weight-group, F and G are also in one weight-group. E, H, G, I, and J belong to different weight-groups. K cannot be with J, and L cannot be with H. M cannot be with H, K, or J. D is in $W_1$ and K is in $W_4$with I. N and O cannot be with F, but are in a weight-group with total membership of four. No weight-group can have more than five or less that one member. P, Q, R, S, and T are instructors of weight-groups with membership sizes 5, 4, 3, 2 and 1, respectively. Who is the instructor of H? /p>

If A is a square matrix of order 3 and |A| = 5 then |3A| = ?

If f(x) = (a – xⁿ) ^1/n, find f{f(x)}

If $Z_r=\sin \frac{2\pi r}{11}-i\cos \frac{2\pi r}{11}$ then $\sum _{r=0}^{10}{Z}_r=$

$x+y=k$ will be a tangent to $16x^2+9y^2=144$ if $k=$

The number of distinct solutions of the equation
$\frac{5}{4}co{s}^{2}2x+co{s}^{4}x+si{n}^{4}x+co{s}^{6}x+si{n}^{6}x=2$
In the interval $[0,2\pi ]$ is ___

The locus of mid point of that chord of parabola which subtends right angle on the vertex will be

Given that $\alpha ,\beta ,a,b$ are in A.P., $\alpha ,\beta ,c,d$ are in G.P. and $\alpha ,\beta ,e,f$ are in H.P. If b, d, f are in G.P., then $\frac{{\beta }^{-}{\alpha }^6}{\alpha \beta ({\beta }^4-{\alpha }^4)}$ equals to

A bag contains an assortment of blue and red balls. Two balls are drawn at random. The probability of drawing two red balls is six times the probability of drawing two blue balls. Furthermore, the probability of drawing one ball of each colour is six times the probability of drawing two blue balls. The number of red and blue balls in the bag is

If the function $f:[1,\infty )\to [1,\infty )$ is defined by $f\left(x\right)=2^{x(x-1)}$, then $f^{-1}\left(x\right)$ is

Let f : R → R and f(x) = x² and g: R → R such that g(x) = sinx. Then, g o f = ?

Differentiate the following functions with respect to x :

$\frac{1+logx}{1-logx}$

The distance between the point (2, 3, 1) and (–1, 2, – 3) is:

The statement $p\to (q\to p)$ is equivalent to

If $x_1,x_2,...x_n$ are n-positive real numbers then $\frac{x_1+x_2+...+x_n}{n}\ge \sqrt[n]{n{x}_1{x}_2....x_n}$, which is known as AM - GM inequality and the equality holds only when $x_1=x_2=x_3=...=x_n$.

$\Delta$ABC is acute angled triangle then the least value of tan A tan B tan C is

Let $S_1$ and $S_2$ be circles of radii 1 and r (r > 1) respectively touching the coordinate axes.
Column-1: Conditions between circles $S_1$ and $S_2$
Column-2: Values of r for conditions in Column-1.
Column-3: Number of common tangents between S1 and S2 for conditions in column-1.
$\begin{array}{cccccc}& \text{Column 1}& & \text{Column 2}& & \text{Column 3}\\ \text{(I)}& S_2\text{passes through the centre}& \text{(i)}& 3& \text{(P)}& 1\\ & \text{of}{S}_1.& & & & \\ \text{(II)}& S_1\text{and}{S}_2\text{touch each other}& \text{(ii)}& 2+\sqrt{2}& \text{(Q)}& 2\\ \text{(III)}& S_1\text{and}{S}_2\text{are orthogonal}& \text{(iii)}& 2+\sqrt{3}& \text{(R)}& 3\\ \text{(IV)}& S_1\text{and}{S}_2\text{have longest}& \text{(iv)}& 3+2\sqrt{2}& \text{(S)}& 4\\ & \text{common chord}& & & & \end{array}$

Which of the following options is the only CORRECT combination?

Given $a_1=\frac{1}{2}(a_0+\frac{A}{a_0})$,

$a_2=\frac{1}{2}(a_1+\frac{A}{a_1})and{a}_{n+1}=\frac{1}{2}(a_n+\frac{A}{a_n})$ for n $\ge$ 2, where a > 0, A > 0. Prove that $\frac{a_n-\sqrt{A}}{a_n+\sqrt{A}}={\left(\frac{a_1-\sqrt{A}}{a_1+\sqrt{A}}\right)}^{2n-1}$

Does there exist a function which is continuous every where but not differentiable at exactly two points? Justify your answer.

${lim}_{x\to -\infty }(\sqrt{4x^2-7x}+2x)$

In a cylindrical vessel containing liquid of density $\rho$, there are two holes in the side wall at heights of $h_1$ and $h_2$ respectively such that the range of efflux at the bottom of the vessel is same. The height of a hole, for which the range of efflux would be maximum will be

If $cosec\theta +cot\theta =\frac{11}{2}$, then $tan\theta =$

If A and B are two sets such that $A\subset B$, then write B' - A' in terms of A and B.

A unit vector perpendicular to the plane determined by the points P(1,-1,2), Q(2,0,-1) and R(0,2,1) is

If $\int \frac{x+1}{\sqrt{2x-1}}dx=f\left(x\right)\sqrt{2x-1}+C$, where $C$ is a constant of integration, then $f\left(x\right)$ is equal to :

${lim}_{x\to -1}\frac{x^2-x-2}{(x^2+x)+sin(x+1)}$

A spiral is made up of successive semicircles, with centres alternately at $A$ and $B,$ starting with centre at $A$. If the radii are in sequence, $0.5\text{cm},1.0\text{cm},1.5\text{cm},2.0\text{cm},\dots$ as shown in figure below.


Then the total length of the spiral made by thirteen consecutive semicircles is
$(\text{Take}\pi =\frac{22}{7})$