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Tangents $PA$ and $PB$ are drawn to $x^2+y^2=4$ from the point $P(3,0).$ Then the area (in sq. units) of $△PAB$ is

Given $f\left(x\right)=|x-1|+|x+1|$. Then f(x) is

What are the eigen values of the followign $2\times 2$ matrix \($$\left[\begin{array}{cc}2& -1\\ -4& 5\end{array}\right]$$

$2ta{n}^{-1}\left(cosx\right)=ta{n}^{-1}\left(cose{c}^{2}x\right)$,then x=

In acute angled triangle $ABC,r=r_2+r_3-r_1$ and $\angle B>\frac{\pi }{3}$ then exhaustive range of $\frac{a-c}{b}$ is

A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of the perpendicular $Q$ also lies on the plane $x-y+z=3$. Then the co-ordinates of $Q$ are :

What are the conditions for a polynomial $f\left(x\right)=a{x}^3+b{x}^2+cx+d$ to be a cubic equation?

The graph of $y=(x-1{)}^2+2$ is

If $|x|=1,|y|=2,$ then the least value of $|x-y|$ is

How many odd numbers less than 1000 can be formed by using the digits 0, 2, 5, 7 when the repetition of digits is allowed ?

Numerically greatest term in the expansion of $(x-\frac{1}{x}{)}^{11}$ when x=2 is given by

Let axes of ellipse be coordinate axes, $S$ and $S^{\prime }$ be foci, $B$ and $B^{\prime }$ are the endpoints of the minor axis. If $\sin \left(\angle SB{S}^{\prime }\right)=\frac{4}{5}$ and area of $SB{S}^{\prime }{B}^{\prime }$ is $20$ sq. unit, then the equation of ellipse is

The sum ${\sum }_{r=1}^{n}r.{}^{2n}{C}_r$ is equal to

The value of the integral is $\left(\frac{2z+1}{z^2+1}\right)dz$ where $c$ is $|z|=\frac{1}{2}$

If $\sqrt{1-x^{2n}}+\sqrt{1-y^{2n}}=a(x^n-y^n)$ then $\sqrt{\frac{1-x^{2n}}{1-y^{2n}}}.\frac{dy}{dx}=$

There were 80 persons on a trip organised by a school. Out of which 60 were students, 14 were teachers and 6 (all males) were peons. Out of total persons, 16 were females including one lady teacher. Present the above information in a table.

The differential equation for all the straight lines which are at a unit distance from the origin is

If $9x=5(y–32)$ and $x\in (30,35)$, then $y$ lies in the interval

Find the principal solution of $ta{n}^{2}x+(\sqrt{3}-1)tanx-\sqrt{3}=0$

List all the elements of the following sets:

(i) A = {x : x is an odd natural number}

(ii) B = {x : x is an integer, 1/2 < x < 9/2}

(iii) C = {x : x is an integer, $x^2\le 4$}

(iv) D = {x : x is a letter in the word "LOYAL"}

(v) E = {x : x is a month of a year not having 31 days}

(vi) F = {x : x is a consonant in the English alphabet which precedes K}

If f(x+y) = f(x) . f(y) , then

For all natural numbers n, $2^{3n}-7n-1$ is divisible by

What is the geometry of $H_{2}S{O}_4$?

The solution set of the inequality $\frac{(3^x-4^x)\cdot \ln (x+2)}{x^2-3x-4}\le 0$ is

Let $A$ be a $3\times 3$ matrix such that $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right],$ $P=adj\left(A\right)$ and $|A|=4,$ then the value of $\alpha$ is

If a, b,and c are in G.P. and x, y, respectively, be arithmetic means between a, b and b, c, then the value of $\frac{a}{x}+\frac{c}{y}$is

If the abscissas and ordinates of two points $P$ and $Q$ are the roots of the equations $x^2-7x+10=0$ and $x^2+7x+12=0$ respectively, then the equation of the circle with $PQ$ as diameter is

If the pair of straight lines $a{x}^2+2hxy+b{y}^2=0$ is rotated about the origin through ${90}^{\circ },$ then the equations in the new position is

Between 1 and 31 m arithmetic means are inserted so that the ratio of the $7^{th}$ and ${(m-1)}^{th}$ means is 5 : 9. Then the value of m is

The equation(s) of the circle(s) having radius $5$, centre on the line $y=x$ and touching both the coordinate axes is(are)

The equation to the pair of tangents drawn from (–1, –2) to parabola $x^2=2y$ is

If $3+{\log }_{5}x=2{\log }_{25}y$, then the value of $x$ is

Which of the following are the conditions to be satisfied in the axiomatic approach to probability?

Does there exists a function which is continuous everywhere but not differentiable at exactly two points? Justify your answer.

Which of the following inequalities represent the statement - ' k cannot exceed 5' ?

Match the following:

Given sin x = $\frac{2}{5}$ and x $ϵ$ $(0,\frac{\Pi }{2})$

(p) cosx (1) $\frac{5}{2}$

(q) tanx (2) $\frac{\sqrt{21}}{2}$

(r) cosecx (3) $\frac{\sqrt{21}}{5}$

(4) $\frac{2}{\sqrt{21}}$

Find the sum of the series $\frac{1}{1\times 3}+\frac{1}{3\times 5}+\frac{1}{5\times 7}+......$ up to 10 terms.

The set $(A\cap B^{\prime }{)}^{\prime }\cup (B\cap C)$ is equal to

$\frac{{}^{11}{C}_0}{1}$ + $\frac{{}^{11}{C}_1}{2}$ + $\frac{{}^{11}{C}_2}{3}$+.............. $\frac{{}^{11}{C}_{10}}{11}$ =

The value of $\sum _{r=0}^4\frac{{}^4{C}_r}{{}^4{C}_r+{}^4{C}_{r+1}}$ is

Find the interval(s) in which the expression (4-x)(x+2) is positive.

If $P$ is the number of natural numbers whose logarithms to the base $10$ have the characteristic $p$ and $Q$ is the number of natural numbers, logarithms of whose reciprocals to the base $10$ have the characteristic $-q$, then the value of ${\log }_{10}P-{\log }_{10}Q$ is

Ravi obtained 70 and 75 marks in first two unit tests. Find the minimum marks he should get in the third test to have on average of at least 60 marks.

If $x_1$ and $x_2$ are the values of x satisfying the equation |x| = 6, then $x_1$ + $x_2$ = ,then

An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is