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In a class there are $200$ students, at least $140$ of students like Maths, at least $150$ like Science and at least $160$ like English. All students like atleast $1$ subject. What is the minimum number of students who like all three subjects?

The $n^{\text{th}}$ term of a sequence of numbers is $a_n$ and given by the formula $a_n=a_{n-1}+2n$ for $n\ge 2$ and $a_1=1.$

The sum of first $20$ terms is

Read the information carefully and answer the following questions.
(i) ${}^{\prime }A+B^{\prime }$ means 'A is the father of B'.
(ii) ${}^{\prime }A\times B^{\prime }$ means 'A is the sister of B'.
(iii) ${}^{\prime }A\$B^{\prime }$ means 'A is the wife of B'.
(iv) ${}^{\prime }A\%B^{\prime }$ means 'A is the mother of B'.
(v) ${}^{\prime }A{B}^{\prime }$ means 'A is the son of B'.
Which of the following expressions is true, if 'Y is the son of X' is definitely false?

If ${\tan }^1\left(\frac{2y}{1y^2}\right)={\sin }^1\left(\frac{2a}{1+a^2}\right)+{\cos }^1\left(\frac{1b^2}{1+b^2}\right)$ then y equals

Let one root of $a{x}^2+bx+c=0$ where $a,b,c\in Z,a\ne 0$ be $3+\sqrt{5}$, then the other root of the equation will be

Which of the following is the principal value of ${\text{cosec}}^{-1}x$

A semi-circle of diameter 1 unit sits at the top of a semi-circle of diameter 2 units. The shaded region inside the smaller semi-circle but outside the larger semi-circle is called a lune. The area of the lune is.

If the transformed equation of $xy=c^2$ when the axis are rotated through an angle of $\frac{\pi }{4}$ (in the anti clockwise direction), is $p{X}^2+q{Y}^2=r{c}^2$, then

If $\tan \frac{\alpha }{2},\tan \frac{\beta }{2}$ are the roots of $8x^2-26x+15=0$, then the value of $\cos (\alpha +\beta )$ is

Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1)\) are in $A\times B,$ find A and B, where x,y,z are distinct elements.

Consider the family of all circles whose centers lie on the straight line $y=x$. If this family of circles is represented by the differential equation $P{y}^{\prime \prime }+Q{y}^{\prime }+1=0$, where $P,Q$ are functions of $x,y\text{and}{y}^{\prime }(\text{here}{y}^{\prime }=\frac{dy}{dx},y^{\prime \prime }=\frac{d^{2}y}{d{x}^2}),$ then which of the following statements is (are) true ?

The set of values of $x,$ satisfying the inequality $\left|\frac{1}{|x|-3}\right|>\frac{1}{2}$ is

If $f\left(x\right)=x^2+3x,$ then the differential coefficient of $f\left(x\right)$ at $x=1,$ is

Let $p,q$ and $r$ be the roots of the equation $y^3-3y^2+6y+1=0$. If the vertices of a triangle are $(pq,\frac{1}{pq}),$ $(qr,\frac{1}{qr})$ and $(rp,\frac{1}{rp}),$ then the coordinates of its centroid are

In parametric representation of a standard hyperbola in $\theta$ terms, if $x=asec\theta$. What is y.

A fair coin is tossed a fixed number of times. If the probability of getting $7$ heads is equal to probability of getting $9$ heads, then the probability of getting $2$ heads is :

Answer each of the following questions in one word or one sentence or as per exact requirement of for question:

$\text{If the sides of a triangle are proportional to}2,\sqrt{6}and\sqrt{3}-1,\text{find the measure of its greatest angle.}$

Which are of the following is standard form of a quadratic equation

If $\mathrm{cosec}x+\cot x=\frac{11}{2},$ then the value of tan x is ______________.

The value of integral $\underset{3}{\overset{5}{\int }}\frac{\ln x^2}{\ln x^2+\ln (64-16x+x^2)}dx,$ is

For water, coefficient of cubical expansion after $4{}^{o}C$ is ${\gamma }_1$ and before $4{}^{o}C$ is ${\gamma }_2$ then

The set of values of $a$ for which the function $f\left(x\right)=\frac{a{x}^3}{3}+(a+2)x^2+(a-1)x+2$ possesses a negative point of inflection is

The locus of centroid of a triangle whose vertices are $(a\cos t,a\sin t),(b\sin t,-b\cos t)$ and $(1,0),$ where $t$ is a parameter, is

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The value of $\sum _{r=1}^n\frac{r{}^n{C}_r}{{}^n{C}_{r-1}}$ is

The value of $\underset{x\to 0}{lim}\frac{\sin mx-\sin nx}{x-\sin nx}$ is