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The eccentricity of an ellipse $9x^2+16y^2=144$ is

Let $f\left(x\right)$ be a continuous and not a constant function of all $x$ in its domain, such that
$\left(f\right(x){)}^2=\underset{0}{\overset{x}{\int }}f\left(t\right)\frac{4}{\sin 2t-4{\sin }^{2}t+4}dt$ and $f\left(0\right)=0,$ then

Form the differential equation of the family of circles having centre on Y-axis and radius 3 unit.

If sum of the squares of first n natural numbers exceeds their sum by 330, then find the value of n.

Find the values of x for which y=$\left[x\right(x-2){]}^2$ is an increasing function.

In a certain test, $a_i$ students gave wrong answers to at least i questions, where i = 1, 2, ......, k. No student gave more than k wrong answers. The total number of wrong answers given is ___.

The sum of coefficients of last 15 terms of the exapnsion $(1+x{)}^{29}$, when expanded in ascending powers of x is

If a matrix has 24 elements, what are the possible orders it can have? What, if it has 13 elements?

If the value of under root 6 and under root 5 are2.449 and 2.236 then the value of 1/root 6 + root 5 will be

${}^n{C}_{10}+4^n{C}_9+6^n{C}_8+4^n{C}_7{+}^n{C}_6=$

Statements: Z D, F c D, F $ G

Conclusion:

I. D c G

II. Z G

In a right angle triangle $PQR,$ right-angled at $P.$ Let $PQ=4,PR=3$ and $QR=5$. There is a point $O$ inside the triangle from which perpendicular are drawn on the sides $PQ,QR,RP$ with lengths $x,y,z$ respectively. The maximum integral value of $xyz$ is

Integrate the following functions.
$\int \frac{3x^2}{x^6+1}dx.$

What is the probability of getting at least one six in a single throw of three unbiased dice?

Find the common difference of the A.P created by inserting 20 A.Ms between 2 and 86.

If (n+3)! = 56 [(n-1)!] find n.

A person has to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{\text{th}}$ minute. If $a_1=a_2=\cdots =a_{10}=150$ and $a_{10},a_{11},a_{12},\dots$ are in A.P. with common difference as $-2$, then the time (in minutes) taken by him to count all notes, is

* is a binary operation on Z such that : a * b = - a + b + ab.Then the value of 2 *(-3)*5 is

If $A={\int }_0^{\pi }\frac{cosx}{(x+2{)}^2}dx,then{\int }_0^{\frac{\pi }{2}}\frac{sin2x}{(x+1)}$ dx is equal to

The radius of the circle $x^2+y^2+4x+6y+13=0$ is

Match the following
$\begin{array}{cccc}& Column-I& & Column-II\\ \left(A\right)& Theshortestdistancebetweenthelines& \left(P\right)& 2\\ & \frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}and\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}is& & \\ \left(B\right)& Thedistanceofthepointofintersectionoftheline& \left(Q\right)& 3\sqrt{30}\\ & \frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}andtheplane& & \\ & x-y+z=5fromthepoint(-1,-5,-10)is& & \\ \left(C\right)& Thelengthoftheperpendiculardrawnfromthe& \left(R\right)& 13\\ & point(5,4,-1)ontheline\frac{x-1}{2}=\frac{y}{9}=\frac{z}{5}is& & \\ \left(D\right)& ThedistancebetweenthepointsPandQisdand& \left(S\right)& \sqrt{\frac{2109}{110}}\\ & thelenghtoftheprojectionofPQontheco-ordinate& & \\ & planesare{d}_1,d_2,d_{3}then{d}_1^2+d_2^2+d_3^2=k{d}^{2}whenkis& & \end{array}$

If the $7^{th}$ term of a harmonic progression is 8 and the $8^{th}$ term is 7, then its ${15}^{th}$ term is

The value of the determinant
$\Delta =\left|\begin{array}{ccc}logx& logy& logz\\ log2x& log2y& log2z\\ log3x& log3y& log3z\end{array}\right|$ is

$\text{Range of the function}$ $f\left(x\right)=x^2+\frac{1}{x^2+1},is$

Angle between the line joining the origin to the points of intersection of the curves $2x^2+3y^2+10x=0$ and $3x^2+5y^2+16x=0$ is

Let $AB$ be a focal chord (not the latus rectum) of the parabola $y^2=4px$ where $p$ is a prime number$(p>0),$ such that the lengths, $SA$ and $SB$ are integers($S$ being the focus of the parabola). If the tangents to the parabola at $A$ and $B$ meet at point $P,$ then

The modulus of the complex number $\frac{1}{(2-i{)}^2}-\frac{1}{(2-i{)}^2}$ is

The circle passing through $(1,-2)$ and touching the axis of $x$ at $(3,0)$ also passes through the point

Input: zeal fan 52 31 heal 22 track 12

Which of the following will be the IIIrd step?

Which of the following should be the THIRD sentence after rearrangement?

Find the equation of the line through the points (3,4,-7) and (1,-1,6)

The converse of the proposition ‘If an integer is greater than 30, then it is greater than 10’.

The value of the integral
$\underset{-\pi /2}{\overset{\pi /2}{\int }}{\sin }^{4}x(1+\log \left(\frac{2+\sin x}{2-\sin x}\right))dx$ is :

The points (-a, -b), (a, b), ($a^2$, ab) are

The coefficient of $x^{-17}$ in the expansion of ${(x^4-\frac{1}{x^3})}^{15}$ is

The equation of the straight line through the origin making angle $\varphi$ with the line $y=mx+b$, is

Prove: $\cos \frac{2\pi }{15}\cos \frac{4\pi }{15}\cos \frac{8\pi }{15}\cos \frac{16\pi }{15}=\frac{1}{16}$

If the $x$-intercept of the focal chord of the parabola $y^2-2y-4x+5=0$ is $\frac{11}{4}$, then find the length of chord.

The number of integer values of m, for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer, is