Let $A$ and $B$ be two events such that $P(\overline{A\cup B})=\dfrac{1}{6}$, $P(A\cap B)=\dfrac{1}{4}$ and $P(\overline{A})=\dfrac{1}{4}$, where $\overline{A}$ stands for the complement of the event $A$. Then the events $A$ and $B$ are :

1

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha\log|x|+\beta x^{2}+x$ then

Topic: JEE Main 2014 (Offline)

2

If $g$ is the inverse of a function $f$ and $f'(x)=\dfrac{1}{1+x^{5}}$, then $g'(x)$ is equal to:

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3

If $\alpha,\beta \ne 0$, and $f(n) = \alpha^{n} + \beta^{n}$ and $\begin{vmatrix} 3 & 1 + f(1) & 1 + f(2) \\ 1+…

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4

If $A$ is a $3\times 3$ non-singular matrix such that $AA' = A'A$ and $B = A^{-1}A'$, then $BB'$ equals :

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5

The integral $\displaystyle \int (1+x-\frac{1}{x})e^{x+\frac{1}{x}}\,dx$ is equal to

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6

The area of the region described by $A=\{(x,y):x^{2}+y^{2}\le 1 \text{ and } y^{2}\le 1-x\}$ is :

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7

The integral $\displaystyle \int_{0}^{\pi}\sqrt{1+4\sin^{2}\frac{x}{2}-4\sin\frac{x}{2}}\,dx$ equals:

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8

Let the population of rabbits surviving at time $t$ be governed by the differential equation $\dfrac{dp(t)}{dt}=\dfrac{…

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9

$\displaystyle \lim_{x\to 0}\frac{\sin(\pi \cos^{2}x)}{x^{2}}$ is equal to :

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10

The variance of first $50$ even natural numbers is :

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1

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2}+3y^{2}=6$ on any tangent to it is :

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2

Let $a,b,c$ and $d$ be non-zero numbers. If the point of intersection of the lines $4ax+2ay+c=0$ and $5bx+2by+d=0$ lies…

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3

Let $PS$ be the median of the triangle with vertices $P(2,2)$, $Q(6,-1)$ and $R(7,3)$. The equation of the line passing…

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4

Three positive numbers form an increasing G.P. If the middle term in this G.P. is doubled, the new numbers are in A.P. …

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5

Let $\alpha$ and $\beta$ be the roots of equation $px^{2}+qx+r=0$, $p\ne 0$. If $p,q,r$ are in A.P. and $\dfrac{1}{\alp…

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6

If $a\in\mathbb{R}$ and the equation $-3(x-[x])^{2}+2(x-[x])+a^{2}=0$ (where $[x]$ denotes the greatest integer $\le x$…

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7

If $z$ is a complex number such that $|z|\ge 2$, then the minimum value of $\left|z+\dfrac{1}{2}\right|$ :

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8

Let $f_{k}(x)=\dfrac{1}{k}(\sin^{k}x+\cos^{k}x)$ where $x\in\mathbb{R}$ and $k\ge 1$. Then $f_{4}(x)-f_{6}(x)$ equals :

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9

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^{2}=m^{2}+n^{2}$ is :

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