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In a class of 100 students there are 70 boys whose average marks in a subject are 75. If the average marks of the complete class is 72, then what is the average marks of the girls?

Urn A contains 6 red, 4 white balls and urn B contains 4 red and 6 white balls. One ball is drawn at random from the urn A and placed in the urn B. Then one ball is drawn at random from the urn B and placed in the urn A. If one ball is now drawn from the urn A, the probability that it is found to be red is:

Two perpendicular unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are such that $\left[\overrightarrow{r}\overrightarrow{a}\overrightarrow{b}\right]=\frac{5}{4},$ $\overrightarrow{r}\cdot (3\overrightarrow{a}+2\overrightarrow{b})=0$ and $\underset{-2\overrightarrow{r}.\overrightarrow{a}}{\overset{\frac{-4}{3}\overrightarrow{r}.\overrightarrow{b}}{\int }}\frac{x+1}{x^2+1}dx=\frac{\pi }{2}$. Then which of the following is(are) correct ?

The probability that length of a randomly selected chord of a circle lies between $\frac{1}{2}$ and $\frac{3}{2}$ of its radius is

(correct answer + 1, wrong answer - 0.25)

Let $f:R\to R$ be defined as

$f\left(x\right)=\{\begin{array}{cc}-55x,& \text{if}x<-5\\ 2x^3-3x^2-120x,& \text{if}-5\le x\le 4\\ 2x^3-3x^2-36x-336,& \text{if}x>4\end{array}$



Let $A=\{x\in R:f$ is increasing$\}.$ Then $A$ is equal to

Find the derivative of $\frac{x^n-a^n}{x-a}$ for some constant a.

If $(h,k)$ is a point on the axis of the parabola $2(x-1{)}^2+2(y-1{)}^2=(x+y+2{)}^2$ from where three distinct normals may be drawn, then

If the normal at the point $P\left(\theta \right)$ to the ellipse $\frac{x^2}{14}+\frac{y^2}{5}=1$ intersects it again at the point $Q\left(2\theta \right)$, then

A student appears for tests I, II and III. The student is successful if he passes either in tests I and II or tests I and III. The probabilities of the student passing in tests I, II and III are, respectively, $p$, $q$ and $1/2$. if the probability that the student is successful is $1/2$, then $p(1+q)=$

If $\cot \alpha +\tan \alpha =m$ and $\frac{1}{\cos \alpha }-\cos \alpha =n$, then

In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?

The vertices of $\Delta PQR$ are P(2, 1), Q(-2, 3) and R(4, 5). Find equation of the median through the vertex R.

1. 51

Equation of the hyperbola with length of the latusrectum 4 and e = 3 is

The function $f\left(x\right)=\frac{9x}{5}+32$ is the formula to convert $x^{\circ }C$ to Fahrenheit units. Then at what ${}^{\circ }C$ both Celsius and Fahrenheit is same?

If the probability of hitting a target by a shooter, in any shot, is $\frac{1}{3}$, then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than $\frac{5}{6}$, is :

If (p+q)th term of G.P. is m and (p-q)th term is n, then pth term will be

In a factory 70% of the workers like oranges and 64% likes apples. If each worker likes at least one fruit, What is the minimum percentage of workers who like both the fruits?

The equation of the pair of straight lines, each of which makes as angle $\alpha$ with the line y = x, is

In a team of 11, seven are batsmen, six are bowlers and one of them is a wicket keeper-batsman. How many of them are all-rounders (both batting and balling)?

If the coefficients of $a^{r-1},a^r$ and $a^{r+1}$ in the binomial expansion $(1+a{)}^n$ are in the arithmetic progression, prove that $n^2-n(4r+1)+4r^2-2=0$

Or

The 2nd, 3rd and 4th terms in the expansion of $(x+y{)}^n$ are 240, 720 and 1080, respectively. Find the values of x, y and n.

If n(A) = m and n(B)= n, find the total number of non-empty relations that can be defined from A to B.

(a) The mode and mean are 26.6 and 28.1 respectively in an asymmetrical distribution. Find out the value of median.

(b) Explain the comparative features of mean, mode and median.

If $I=\int \frac{dx}{x^3\sqrt{x^2-1}}$, then $I$ equals

The line $y=mx+c$ becomes a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,then the value of c is

If $(5,12)$ and $(24,7)$ are the foci of a conic passing through the origin, then the eccentricity of conic can be:

Prove that tan $\text{tan}4\theta =\frac{4\text{tan}\theta (1-{\text{tan}}^2\theta )}{1-6{\text{tan}}^2\theta +{\text{tan}}^4\theta }$. Find the angle $\theta ϵ(0,\frac{\pi }{2})$, in which the given proof does not hold.

Or

Prove that $\frac{\text{cos}4x+\text{cos}3x+\text{cos}2x}{\text{sin}4x+\text{sin}3x+\text{sin}2x}$ = cot 3x . Do you think at $x=\frac{\pi }{3}$, the given proof holds true?

Number of ordered pairs (x,y) satisfying |y|=cosx, y=$si{n}^{-1}\left(sinx\right),x\in [-2\pi ,3\pi ]$ is

If $\frac{m}{n}=\frac{tanA}{tanB}$ , find the value of $\frac{m+n}{m-n}$

If tangents $OQ$ and $OR$ are drawn to variable circles having radius $r$ and the centre lying on the rectangular hyperbola $xy=1,$ then locus of circumcentre of triangle $OQR$ is $(O$ being the origin$).$

What is the mode for the set of numbers "1, 3, 5, 3, 7, 3, 5"?

Find the equation of the line which passes through the point (-4,3) and the portion of the line intercepted between the axes is divided internally in the ratio 5:3 by this point.

Let $A$ be a $3\times 3$ square matrix. If $B=adj\left(A\right),$ $C=adj\left(adj\right(A\left)\right)$ and $D=adj\left(adj\right(adj\left(A\right)\left)\right),$ then $|adj\left(adj\right(adj\left(adj\right(ABCD\left)\right)\left)\right)|,$ in terms of $|A|$ is

An equilateral triangle is inscribed in the parabola $y^2=4ax,$ such that one vertex of this triangle coincides with the vertex of the parabola. Side length of this triangle is

The ratio in which $x-$ axis divides the line segment joining $(3,-4)$ and $(-5,6)$ is

The sums of n terms of two arithmetic progressions are on the ratio 5n + 4 : 9n + 6. Find the ratio of their ${18}^{th}$ terms.

If $\left|z\right|=1$ prove that $\frac{1+z}{1+z}=z.$

The largest number common to both the sequences $1,11,21,31,\cdots \text{upto}100\text{terms}$ and $31,36,41,46,\cdots \text{upto}100\text{terms}$ is

The sum of the digits in the unit place of all numbers formed with the help of 3,4,5,6 taken all at a time is

If $\frac{3}{5}+\frac{5}{15}+\frac{7}{45}+\frac{9}{135}+\dots =\frac{a}{b},$ then

(where $a$ and $b$ are coprime)

The graph of $|x|+|y|=1$ is

Solution set of $lo{g}_{0.25}(x^2+2x-8{)}^2-lo{g}_{0.5}(10+3x+x^2)=1$

The function : $R\to [-\frac{1}{2},\frac{1}{2}]$ defined as $f\left(x\right)=\frac{x}{1+x^2}$ is

Let (h, k) be a fixed point, where h > 0, k > 0. A straight line passing through this point cuts the positive direction of the coordinate axes at the points P and Q. The minimum area of the $\Delta$ OPQ, O being the origin, is