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The determinant $\left|\begin{array}{ccc}1& ab& \frac{1}{a}+\frac{1}{b}\\ 1& bc& \frac{1}{b}+\frac{1}{c}\\ 1& ca& \frac{1}{c}+\frac{1}{a}\end{array}\right|$ is equal to

Tangents are drawn from the points on the parabola $y^2=-8(x+4)$ to the parabola $y^2=4x.$ Then the locus of mid-point of chord of contact of $y^2=4x$ is

Let $S_1,S_2,S_3,\dots ,S_n$ be squares such that for each $n\ge 1$, the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10$ cm, then for which of the following values of $n$ is the area of $S_n$ less than $1$ sq. cm?

The eccentricity of the ellipse which meets the straight line $\frac{x}{7}+\frac{y}{2}=1$ on the axis of x and the straight line $\frac{x}{3}-\frac{y}{5}=1$ on the axis of y and whose axes lie along the axes of coordinates, is

The length of intercepts made by the circle $x^2+y^2-4x+6y+4=0$ on $X$ and $Y$ axis respectively, are

Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is $4$ and the second term is $\frac{3}{4}$, then

The value of $\underset{n\to \infty }{lim}\frac{1}{n^3}(\sqrt{n^2+1}+2\sqrt{n^2+2^2}+\cdots +n\sqrt{n^2+n^2})$ is

The area enclosed by 2|x| + 3|y| $\le$ 6 is

An equilatral triangle inscribed in parabola $y^2=4ax$ whose one vertex is at the vertex of parabola. Then the length of the side of the triangle is

Find the integral $\underset{0}{\overset{\infty }{\int }}{e}^{-x}dx$.

If a variable tangent to the curve $x^{2}y=c^3$ makes intercepts $a,b$ on $x$ and $y-$axis respectively, then the value of $a^{2}b$ is

Two straight lines are perpendicular to each other one of them touches the parabola $y^2=4a(x+a)$ and the other touches $y^2=4b(x+b).$ The locus of the point of intersection of these two lines is

If a circle passes through the point (a, b) and cuts the circle $x^2+y^2=4$ orthogonally, then the locus of its Centre of the circle is _____

If $(m_i,\frac{1}{m_i})$, i = 1, 2, 3, 4 are con - cyclic points, then the value of $m_1{m}_2{m}_3{m}_4$ is

If $z_1$ and $z_2$ are two complex numbers, then the inequality $|z_1+z_2{|}^2\le (1+c)|z_1{|}^2+(1+c^{-1})|z_2{|}^2$ is true if

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

If the roots of the equation $6x^2-7x+k=0,k>-3$ are rational, then possible integral value(s) of $k$ is/are

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are vectors such that $|\overrightarrow{a}+\overrightarrow{b}|=\sqrt{29}$ and $\overrightarrow{a}\times (2\hat{i}+3\hat{j}+4\hat{k})=(2\hat{i}+3\hat{j}+4\hat{k})\times \overrightarrow{b},$ then a possible value of $(\overrightarrow{a}+\overrightarrow{b}).(-7\hat{i}+2\hat{j}+3\hat{k})$ is

Let a,b,c be the sides of a triangle where a$\ne b\ne c$ and $\lambda ϵ$R.If the roots of the equation $x^2+2(a+b+c)x+3\lambda (ab+bc+ca)=0$ are real. then

If the ratio of sum of first $n$ terms of two A.P.s is $2n+8:5n-3,$ then the ratio of $n^{\text{th}}$ terms of those two A.P.s is

If $A$ is a $3\times 3$ matrix and $detA=5,$ then $det\left(adjA\right)$ is equal to

If $5x+9=0$ is the directrix of the hyperbola $16x^2-9y^2=144,$ then its corresponding focus is :

Three fair and unbiased dice and rolled at a time. The probability that the numbers shown are totally different is.

For all the sets A, B and C,

(A – B) ∩ (C – B) =

If $3\text{and}4$ lies between the roots of the equation $x^2+2kx+9=0$ then $k$ lies in the interval

The point which divides the line segment joining the points $(6,3)$ and $(-4,5)$ in the ratio $3:2$ externally is

If $PSQ$ is the focal chord of a parabola such that $SP=2$ and $SQ=4$ then the length of the latus rectum is

In a beauty contest, half the number of experts voted for Miss $A$ and two-third voted for Miss $B$, $10$ voted for both and $6$ did not vote for either. Then how many experts were there?

Find the length of the tangent from a point (6, 1) to the circle $x^2+y^2-4x=0.$

If $f\left(x\right)={\int }_1^x\frac{dt}{2+t^4}$, then

Each entry of $\text{List I}$ is to be matched with one entry of $\text{List II}.$



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& 100(\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\cdots +\frac{1}{99\cdot 100})\text{equals}& \text{(P)}& 7\\ \text{(B)}& \text{If}x\text{is the arithmetic mean between two real numbers}a\text{and}b\text{,}& \text{(Q)}& 9\\ & y=a^{2/3}\cdot b^{1/3}\text{and}z=a^{1/3}\cdot b^{2/3},\text{then}\frac{y^3+z^3}{xyz}\text{equals}& & \\ \text{(C)}& \text{If}198\text{arithmetic means are inserted between}\frac{1}{4}\text{and}\frac{3}{4}\text{, then}& \text{(R)}& 99\\ & \text{the sum of these arithmetic means is}& & \\ \text{(D)}& \text{If}n\text{is a positive integer such that}n\text{,}\frac{n(n-1)}{2}\text{and}& \text{(S)}& 100\\ & \frac{n(n-1)(n-2)}{6}\text{are in A.P., then the value of}n\text{is}& & \\ & & \text{(T)}& 2\end{array}$



Which of the following is the only CORRECT combination?

In a survey of $200$ students of a higher secondary school, it was found that $120$ studied mathematics; $90$ studied physics and $70$ studied chemistry; $40$ studied mathematics and physics; $30$ studied physics and chemistry; $50$ studied chemistry and mathematics, and $20$ studied none of these subjects. Then the number of students who studied all the three subjects, is

In how many ways one can post three letters in four letter boxes?

The length of the line segments joining focus to the point of intersection of angular bisector of co-ordinate axes (in the first quadrant) and the parabola $y^2=lx$ is

If a variable chord $PQ$ of the parabola $y^2=4ax$ is drawn parallel to $y=x,$ then the locus of point of intersection of normals at $P$ and $Q$ is

Let $a=p+2$ and $b=3-2p$. If $a$ and $b$ have same absolute value, then the value(s) of $p$ is/are

$\underset{x\to 1}{lim}\frac{\log x}{x-1}$ is equal to

The equation of directrix of a conic is

$L:x+y-1=0$ and the focus is the point $(0,0)$. Find the equation of the conic if its eccentricity is

$\frac{1}{\sqrt{2}}$

The general solution(s) of $\theta$ which satisfy $3-2\cos \theta –4\sin \theta -\cos 2\theta +\sin 2\theta =0$ is/are (where $n\in Z$)

Which of the following statements is/are correct?

1. With respect to a tangent both the circles lie on the same side, this tangent is called direct common tangent.

2. With respect to a tangent both the circles lie on the opposite side, this tangent is called transverse (indirect) common tangent.

Let $S$ be the solution set of the inequality $4x+5\le 2x+17$ (where $x$ is a whole number), then $n\left(S\right)$ is equal to

For $0<\theta <\frac{\pi }{2},$ the solution (s) of

${\sum }_{m=1}^{6}cosec(\theta +\frac{(m-1)\pi }{4})cosec(\theta +\frac{m\pi }{4})=4\sqrt{2}$ is/are

The complex number $z$ satisfying the equation $|z|=z+1+2i$ is

The equation $x^2-5xy+p{y}^2+3x-8y+2=0$ represents a pair of straight lines. If $\theta$ is the acute angle between them, then $\sin \theta$ equals