List of top Mathematics Questions asked in JEE Advanced

If $ lim_{ x \to \infty} \bigg( \frac{x^2 + x + 1}{x + 1} - ax - b \bigg) $ = 4, then

Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$, parallel to the straight line $2x-y=1.$ The points of contacts of the tangents on the hyperbola are

If $X$ and $Y$ are two events such that $P (X | Y )=\frac{1}{2}, P(X | Y)=\frac{1}{3.} \, and \, P(X \cap Y)\frac{1}{6}$ Then, which of the following is/are correct?

Four fair dice $D_1, D_2, D_3$ and $D_4$ each having six faces numbered $1, 2, 3, 4, 5$ and $6$ are rolled simultaneously. The probability that D$_4$ shows a number appearing on one of $D_1, D_2$ and $D_3$, is

The point $P$ is the intersection of the straight line joining the points $Q(2, 3, 5)$ and $R (1, - 1, 4)$ with the plane $5x - 4y - z = 1$. If $S$ is the foot of the perpendicular drawn from the point $T (2,1, 4)$ to $QR$, then the length of the line segment $PS$ is

The equation of a plane passing through the line of intersection of the planes $x + 2y + 3z = 2$ and $x - y + z = 3$ and at a distance $\frac{2}{\sqrt 3}$ from the point $(3,1, -1)$ is

If $\overrightarrow{a} $ and $ \overrightarrow{b}$ are vectors such that $| \overrightarrow{a}+\overrightarrow{b} |=\sqrt{29}$ and $\overrightarrow{a}\times(2\widehat{i}+3\widehat{j}+4\widehat{k})=(2\widehat{i}+3\widehat{j}+4\widehat{k})\times\overrightarrow{b},$ then a possible value of $ (\overrightarrow{a}+\overrightarrow{b})(-7\widehat{i}+2\widehat{j}+3\widehat{k} is $

A ship is fitted with three engines $ E_1, E_2$ and $E_3 $ The engines function independently of each other with respective probabilities $\frac {1}{2}, \frac {1}{4}$ and $\frac{1}{4}$. For the ship to be operational at least two of its engines must function. Let $X$ denotes the event that the ship is operational and let $ X_1 , X _2 $ and $ X_ 3 $ denote, respectively the events that the engines $ E_1 , E_2$ and $E_3 $ are functioning. Which of the following is/are true?

The probabilities that a student passes in Mathematics, Physics and Chemistry are m, p and c, respectively. Of these subjects, the students has a 75% chance of passing in at least one, a 50% chance of passing in at least two and a 40% chance of passing in exactly two. Which of the following relations are true

A value of $b$ for which the equations $x^2+bx-1=0,x^2+x+b=0$ have one root in common is

Let $E$ and $F$ be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability of none of them occurring is $\frac{2}{25}$ . If $P(T)$ denotes the probability of occurrence of the event $T$, then

The circle passing through the point (-1,0) and touching the Y-axis at (0, 2) also passes through the point

Let $\overrightarrow{a}=\overrightarrow{i}+\overrightarrow{j}+\hat{k}, \overrightarrow{b}=\hat{i}-\hat{j}+\hat{k} $ and $\overrightarrow{c}=\overrightarrow{i}-\overrightarrow{j}-\overrightarrow{k} $ be three vectors. A vector $\hat{v}$ in the plane of $\overrightarrow{a}$ and $ \overrightarrow{b},$ whose projection on $\overrightarrow{c} $ is $\frac{1}{\sqrt{3}},$ is given by

The vector(s) which is/are coplanar with vectors $\widehat{i} + \widehat{j}+2\widehat{k}$ and $\widehat{i} + 2\widehat{j}+\widehat{k},$ are perpendicular to the vector $\widehat{i} + \widehat{j}+\widehat{k}$ is / are

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be reciprocal to that of the ellipse $x^2+4y^2=4.$ If the hyperbola passes through a focus of the ellipse, then

Let $f : R \rightarrow R$ be a function such that $f (x + y) = f (x) + f (y), \forall x, y \in R$. If $f (x )$ is differentiable at $x = 0$, then

Let P(6,3) be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$ If the norm al at the point P intersects the X -axis at (9, 0), then the eccentricity of the hyperbola is

The number of $3 \times 3$ matrices $A$ whose entries are either $0$ or $1$ and for which the system A $\begin {bmatrix} x \\ y \\ z \end {bmatrix}-\begin {bmatrix} 1 \\ 0 \\ 0 \end {bmatrix}$ has exactly two distinct solutions, is

The value(s ) of $ \int^1_0 \frac { x^4 ( 1 - x )^4 }{ ( 1 + x^2 ) } \, dx $ is are

Let $\omega$ be a complex cube root of unity with $\omega \, \ne$ 1. A fair die is thrown three times. If r$_1$, r$_2$ and r$_3$ are the numbers obtained on the die, then the probability that $\omega ^{r_1} +\omega ^{r_2} +\omega ^{r_3}=0 , $ is

Equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}= \frac{z}{4}$ and perpendicular to the plane containing the staight lines $\frac{x}{2}=\frac{y}{4}= \frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}= \frac{z}{3}$ is

If the distance of the point $P (1, - 2,1)$ from the plane $x + 2y - 2z = a$, where $a > 0$, is $5$, then the foot of the perpendicular form $ P$ to the plane is

If the angles $A, B$ and $C$ of a triangle are in an arithmetic progression and if $a, b$ and $c$ denote the lengths of the sides opposite to $A, B$ and $C$ respectively, then the value of the expression $ \frac{a}{c} sin \, 2 C + \frac{c}{a} sin \, 2 A $ is

If $\frac{sin^4 x}{2}+\frac{cos^4 x}{3}=\frac{1}{5}$ then

For $0 < \theta < \frac{\pi}{2}$, the solution(s) of $\displaystyle\sum _{m=1}^{6} cosec \Bigg(\theta+\frac{(m-1)\pi}{4}\Bigg)cosec\Bigg(\theta+\frac{m\pi}{4}\Bigg)=4\sqrt{2}$ is/are