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If E and F are events such that $P\left(E\right)=\frac{1}{4},$
$P\left(F\right)=\frac{1}{2}$ and$P\left(\text{E and F}\right)$ $=\frac{1}{8}$ Find
$\text{(i) P (E or F) (ii) P (not E and not F)}$

A point on the ellipse $x^2+3y^2=37$ where the normal is parallel to the line $6x-5y=2$ is

If the ellipse $\frac{x^2}{4}+y^2=1$ meets the ellipse $x^2+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-5b+7,$ then $b$ does not lie in

Prove that $co{s}^{2}X+co{s}^2(X+\frac{\pi }{3})+co{s}^2(X-\frac{\pi }{3})+co{s}^2=\frac{3}{2}.$

Consider a parabola $y^2=4x$, Let $A$ be the vertex of parabola, $P$ be any point on the parabola and $B$ is a point on the axis of parabola, if $PA\perp PB$, then the locus of centroid of $△PAB$ is

Which among the following can be called as identity matrix/matrices.

A circle of radius $2\sqrt{10}$ cm is divided by a chord of length $12$ cm into two segments. Then maximum area of a rectangle that can be inscribed into the smaller circular segment is

Sum of first n terms of an A.P is $2n^2$. Find its nth term.

The number of numbers that can be formed by the digits 1, 2, 3, 4, 3, 2, 1 with the odd digits at odd places is

The value of $x,$ satisfying the inequality $x^{{\log }_{2}x}>2$ lies in

If $x^2+2ax+10-3a>0$ for all x ∈ R, then

If the 7th term of a H.P is $\frac{1}{10}$ and the 12th term is $\frac{1}{25}$, then the 20th term is

If the sets A and B are defined as

A = {(x, y) : y = $\frac{1}{x}$, 0 ≠ x ∈ R}

B = {(x, y) : y = -x, x ∈ R}, then

Let $A=\{x:x=2n+1,n\in W\}$ and $B=\{y:y=2n,n\in N\}.$ If a relation $R$ is defined from $A$ to $B,$ then which of the following is void relation?

The angle made by a double ordinate of length 8a at the vertex of the parabola $y^2=4ax$ is

A number of lock on a suitcase has three wheels each labelled with ten digits $0$ to $9$. The opening of the lock is a particular sequence of three digits with no repeats. If a person tries to open the lock, then the number of unsuccessful attempts made by him is

If a plane passes through the point (1,1,1) and is perpendicular to the line $\frac{x-1}{3}=\frac{y-1}{0}=\frac{z-1}{4}$ then its perpendicular distance from the origin is

A set S has 7 elements. How many subsets having at most 5 elements does it have?

How many of the following statements are true about the identity function.

(a) It's graph is shown above.
(b) Domain is R
(c) Range is R
(d) Its an even function

Which of the following is a fallacy?

If B $\times$ A = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c), (z, a), (z, b), (z, c)} Find set A and set B.

If all integers satisfying the inequality $\frac{(x-1{)}^2(x-2{)}^3(x-4{)}^5(x-5{)}^5}{(x-5{)}^2}\ge 0$ are arranged in increasing order then the quadratic equation with the first and fifth integers in the list as roots is -

If $si{n}^2(\theta -\alpha )cos\alpha =co{s}^2(\theta -\alpha )sin\alpha =msin\alpha cos\alpha$ then

The normal at a point $P$ on the ellipse $x^2+4y^2=16$ meets the $x-$ axis at $Q$. If $M$ is the midpoint of the line segment $PQ$, then the locus of $M$ intersects the latus rectum of the given ellipse at the points

$co{s}^2{48}^{\circ }-si{n}^2{12}^{\circ }=$

The centre of the conic section $2x^2+4y^2+2xy+4x-6y+17=0$ is

If a tangent to the parabola $y^2=8x$ meets the $x$-axis at $T$ and intersect the tangent at vertex $A$ at $P$, and the rectangle $TAPQ$ is completed, then the locus of the point $Q$ is

The distance of the point (2,0,-6) from the x-z plane is ___ units.

Given that $P(3,2,-4),Q(5,4,-6)$ and $R(9,8,-10)$ are collinear. Find the ratio in which Q divides PR.

The solution set of $\frac{1}{x-1}+\frac{1}{x+1}\le \frac{1}{x}$ is

Find the value of x if ${\left(x^{{\log }_{10}3}\right)}^2-\left(3^{{\log }_{10}x}\right)-6=0$

Find the angle between the vectors 1 + i and -1 + i

Given the parabola $y^2=4ax,$ find the pair of tangents from the point (2, 6) where a =2.

Let $A=\{y:y={\log }_{2}x,x<16,x,y\in N\}$ and $B=\{x:x^2-7x+12=0\}.$ Then $(A\cup B)\times (A\cap B)$ is

If X = {$4^n$ - 3n - 1 : n ∈ N} and Y = { 9(n-1) : n ∈ N}, then

X ∪ Y is equal to

The range of values of $\theta$ in the interval $(0,\pi )$ such that the points $(3,5)$ and $(\sin \theta ,\cos \theta )$ lie on the same side of the line $x+y-1=0$, is

Two numbers are selected simultaneously from the set {6, 7, 8, 9, ………. 39}. If the sum of selected numbers is even then the probability that both the selected numbers are odd, is equal to:

If the reduction formula for $I_n=\int {\tan }^n\mathrm{xdx}$ is given by

$I_n=\frac{1}{n-1}{\tan }^{n-1}x-I_{n-2}$, then $\int {\tan }^{3}x\mathrm{dx}\mathrm{is}$

If $\int \sqrt{1+\sin x}f\left(x\right)dx=\frac{2}{3}(1+\sin x{)}^{3/2}+c,$ then $f\left(x\right)$ equals

More than One Answer Type

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Given $\cos \theta =\frac{1}{3},(\theta \in R)$, then sinθ can be



दिया है $\cos \theta =\frac{1}{3},(\theta \in R)$, तब sinθ हो सकता है

The slope of the line joining the points (3,5) and (4,8) is

A ladder20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

The value of the determinant

$\left|\begin{array}{ccc}a-b-c& 2a& 2a\\ 2b& b-c-a& 2b\\ 2c& 2c& c-a-b\end{array}\right|$ will be