List of top Mathematics Questions asked in JEE Main

Let S be the set of all a∈ R for which the angle between the vectors$ \begin{array}{l} \overrightarrow{u}=a\left(\text{log}_e b\right)\hat{i}-6\hat{j}+3\hat{k}\end{array}$ and $\begin{array}{l} \overrightarrow{v}=\left(\text{log}_e b\right)\hat{i}+2\hat{j}+2a\left(\text{log}_e b\right)\hat{k},\left(b>1\right)\end{array}$ is acute. Then S is equal to

A vector $\vec{a}$
is parallel to the line of intersection of the plane determined by the vectors
$\hat{i},\hat{i}+\hat{j} $and the plane determined by the vectors
$\hat{i}−\hat{j},\hat{i}+\hat{k}$. The obtuse angle between
$\vec{a}$ and the vector $\vec{b}=\hat{i}−2\hat{j}+2\hat{k}$
is

$\begin{array}{l}\text{Let}~\vec{a} = \alpha \hat{i} + \hat{j} + \beta \hat{k}~\text{and}~ \vec{b} = 3 \hat{i} + 5\hat{j} + 4 \hat{k}~ \text{be two vectors, such that }\vec{a} \times \vec{b} = -\hat{i} + 9\hat{j} + 12 \hat{k}.\end{array}$
$\begin{array}{l}~\text{Then the projection of } \vec{b}-2\vec{a} ~\text{on}~ \vec{b}+ \vec{a}~\text {is equal to}\end{array}$

Numbers are to be formed between 1000 and 3000, which are divisible by 4, using the digits 1, 2, 3, 4, 5 and 6 without repetition of digits. Then the total number of such numbers is ______________.

The number of 7-digit numbers which are multiples of 11 and are formed using all the digits 1, 2, 3, 4, 5, 7 and 9 is _____.

The letters of the work ‘MANKIND’ are written in all possible orders and arranged in serial order as in an English dictionary. Then the serial number of the word ‘MANKIND’ is ______.

There are ten boys B₁, B₂, …, B₁₀ and five girls G₁, G₂,…, G₅ in a class. Then the number of ways of forming a group consisting of three boys and three girls, if both B₁ and B₂ together should not be the members of a group, is ________

If
$\sum\limits_{k=1}^{31}$ $(^{31}C_k) (^{31}C_{k-1})$ $-\sum\limits_{k=1}^{30}$ $(^{30}C_k) (^{30}C_{k-1})$ $= \frac{α (60!)} {(30!) (31!)}$
where $α ∈ R$, then the value of 16α is equal to

If
$(^{40}C_0) + (^{41}C_1) + (^{42}C_2) + ...... + (^{60}C_{20}) \frac{m}{n} ^{60}C_{20}$
m and n are coprime, then m + n is equal to _____.

The number of ways to distribute 30 identical candies among four children C₁, C₂, C₃ and C₄ so that C₂ receives atleast 4 and atmost 7 candies, C₃ receives atleast 2 and atmost 6 candies, is equal to:

The total number of four digit numbers such that each of first three digits is divisible by the last digit, is equal to _______.

A line, with the slope greater than one, passes through the point A (4,3) and intersects the line x-y-2=0 at the point B If the length of the line segment AB is $\frac{\sqrt{29}}{3},$ then B also lies on the line :

Let $P$ be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\frac{ x }{2}=\frac{ y }{3}=\frac{ z }{5}$ and $\frac{ x }{3}=\frac{ y }{7}=\frac{ z }{8}$ If $d$ is the distance of $P$ from the point $(2,-5,11)$, then $d ^2$ is equal to :

If $y = m_1x + c_1 $and $y = m_2x + c_2$, $m_1≠m_2$ are two common tangents of circle $x_2 + y_2 = 2$ and parabola $y^2 = x$, then the value of $8|m_1m_2|$ is equal to :

Let $Q$ be the mirror image of the point $P(1, 0, 1)$ with respect to the plane $S$: $x + y + z = 5$. If a line L passing through $(1, –1, –1)$, parallel to the line $PQ$ meets the plane $S$ at $R$, then $QR_2$ is equal to :

Let $x=2t,\ y=\frac {t^2}{3}$ be a conic. Let S be the focus and B be the point on the axis of the conic such that SA⊥BA, where A is any point on the conic. If k is the ordinate of the centroid of the ΔSAB, then $\lim\limits_{t \to 1} k$ is equal to

A ray of light passing through the point P(2, 3) reflects on the x-axis at point A and the reflected ray passes through the point Q(5, 4). Let R be the point that divides the line segment AQ internally into the ratio 2:1. Let the co-ordinates of the foot of the perpendicular M from R on the bisector of the angle PAQ be (α, β). Then, the value of $7α + 3β$ is equal to ________ .

Let l be a line which is normal to the curve $y = 2x^2 + x + 2$ at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then the area of the triangle OPQ is equal to _________ .

Let the lines
$y + 2x = \sqrt {11} + 7\sqrt 7$ and $2y + x = 2\sqrt {11} + 6\sqrt 7$
be normal to a circle $C : (x – h)^2 + (y – k)^2 = r^2$. If the line
$\sqrt {11}y - 3x =\frac { 5\sqrt {17}}{3} + 11$
is tangent to the circle C, then the value of $(5h – 8k)^2 + 5r^2$ is equal to _______.

A circle of radius 2 unit passes through the vertex and the focus of the parabola y² = 2x and touches the parabola $y=\left (x−\frac{1}{4}\right)^2+α,$ where $α > 0$. Then $(4α – 8)^2$ is equal to ___________.

Let the common tangents to the curves 4(x² + y²) = 9 and y² = 4x intersect at the point Q. Let an ellipse, centered at the origin O, has lengths of semi-minor and semi-major axes equal to OQ and 6, respectively. If e and I respectively denote the eccentricity and the length of the latus rectum of this ellipse, then
$\frac{1}{e^2}$ is equal to

Let the tangent drawn to the parabola y² = 24x at the point (α, β) is perpendicular to the line 2x + 2y = 5. Then the normal to the hyperbola
$\frac{x^2}{α^2}−\frac{y^2}{β^2}=1$
at the point (α + 4, β + 4) does NOT pass through the point

An ellipse
$E:\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
passes through the vertices of the hyperbola
$H:\frac{x^2}{49} - \frac{y^2}{64} = -1$
Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H, respectively. Let the product of the eccentricities of E and H be 1/2. If the length of the latus rectum of the ellipse E, then the value of 113l is equal to _____.

Let $O$ be the origin and $A$ be the point $z _1=1+2 i$. If $B$ is the point $z _2, \operatorname{Re}\left( z _2\right)<0$, such that $OAB$ is a right angled isosceles triangle with $OB$ as hypotenuse, then which of the following is NOT true ?

Let z₁ and z₂ be two complex numbers such that

$z_1=iz_2 \,and \,arg(\frac{z_1}{z_2})=π.$