List of top Mathematics Questions asked in JEE Main

$y=tan^{-1}(sec \;x^3 - tan\; x^3), \frac{π}{2} < x^3< \frac{3π}{2},$ then

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

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $-\hat{ i }+2 \hat{ j }-2 \hat{ k }$ is $30$, then $\alpha$ is equal to

For the hyperbola $H: x^2 – y^2 = 1$ and the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b>0$, let the
(1) eccentricity of E be reciprocal of the eccentricity of H, and
(2) the line $y= \sqrt\frac{5}{2} x+k$ be a common tangent of E and H.
Then $4(a^2 + b^2)$ is equal to _______.

Let a triangle be bounded by the lines L₁ : 2x + 5y = 10; L₂ : –4x + 3y = 12 and the line L₃, which passes through the point P(2, 3), intersects L₂ at A and L₁ at B. If the point P divides the line-segment AB, internally in the ratio 1 : 3, then the area of the triangle is equal to

If α, β, γ, δ are the roots of the equation x⁴ + x³ + x² + x + 1 = 0, then α²⁰²¹ + β²⁰²¹ + γ²⁰²¹ + δ²⁰²¹ is equal to

If [t] denotes the greatest integer ≤ t, then the value of $\int\limits_{0}^{1}$[2x−|$3x^2$−5x+2|+1]dx is

Let a₁, a₂, a₃,…. be an A.P. If
$\begin{array}{l} \displaystyle\sum\limits_{r=1}^\infty\frac{a_r}{2^r}=4,\end{array}$
then 4a₂ is equal to ________.

Let
$f(x) = \begin{cases} x^3 - x^2 + 10x - 7, & x \leq 1 \\ -2x + \log_2(b^2 - 4), & x > 1 \end{cases}$
Then the set of all values of b, for which f(x) has maximum value at x = 1, is

The plane passing through the line L :lx – y + 3(1 – l) z = 1, x + 2y – z = 2 and perpendicular to the plane 3x + 2y + z = 6 is 3x – 8y + 7z = 4. If θ is the acute angle between the line L and the y-axis, then 415 cos²θ is equal to ________.

Let $y = y(x)$ be the solution of the differential equation $(1 - x^2) \, dy = (xy + (x^3 + 2) \sqrt{1 - x^2}) \, dx \quad -1 < x < 1$, and $y(0) = 0.$ If $\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1 - x^2} \, y(x) \, dx = k$ then $k^{−1}$ is equal to_______.

Let f : [0, 1] → R be a twice differentiable function in (0, 1) such that f(0) = 3 and f(1) = 5.
If the line y = 2x + 3 intersects the graph of f at only two distinct points in (0, 1) then the least number of points x ∈ (0, 1) at which f”(x) = 0, is ___________.

The system of equations

–kx + 3y – 14z = 25

–15x + 4y – kz = 3

–4x + y + 3z = 4

is consistent for all k in the set

For real number a, b (a > b > 0), let
$\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi$
and
$\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi$
Then the value of (a – b)² is equal to _____.

If the solution curve of the differential equation $\frac{dy}{dx} = \frac{x + y - 2}{x - y}$ passes through the points $(2, 1)$ and (k + 1, 2), k > 0, then

A point P moves so that the sum of squares of its distances from the points (1, 2) and (–2, 1) is 14. Let f(x, y) = 0 be the locus of P, which intersects the x-axis at the points A, B and the y-axis at the points C, D. Then the area of the quadrilateral ACBD is equal to

If $a_1, a_2, a_3$ ….. and $b_1, b_2, b_3$ ….. are A.P., and $a_1 = 2, a_{10} = 3, a_1b_1 = 1 = a_{10}b_{10}$, then $a_4b_4$ is equal to

If $\begin{array}{l} \frac{6}{3^{12}}+\frac{10}{3^{11}}+\frac{20}{3^{10}}+\frac{40}{3^9}+\cdots +\frac{10240}{3}=2^n\cdot m, \end{array}$where m is odd, then m⋅n is equal to ________.

Let
$\begin{array}{l}f\left(x\right) = min \{\left[x – 1\right], \left[x – 2\right], …., \left[x – 10\right]\}\end{array}$
where [t] denotes the greatest integer ≤t. Then
$\begin{array}{l} \displaystyle\int\limits_{0}^{10}f\left(x\right)dx+\displaystyle\int\limits_{0}^{10}\left(f\left(x\right)\right)^2dx+\displaystyle\int\limits_{0}^{10}\left|f\left(x\right)\right|dx\end{array}$
is equal to ________.

Let y = y(x) be the solution curve of the differential equation
$\sin(2x^2) \log_e(\tan(x^2)) \,dy + (4xy - 4\sqrt{2}x\sin(x^2 - \frac{\pi}{4})) \,dx = 0, \quad 0 < x < \sqrt{\frac{\pi}{2}}$
which passes through the point $(\sqrt{\frac{π}{6}},1)$. Then $|y(\sqrt{\frac{π}{3}})|$
is equal to _______.

The value of $\cot\left(\sum_{n=1}^{50} \tan^{-1}\left(\frac{1}{1+n+n^2}\right)\right)$ is

The locus of the mid-point of the line segment joining the point (4, 3) and the points on the ellipse x² + 2y² = 4 is an ellipse with eccentricity:

The remainder when (2021)²⁰²³ is divided by 7 is

The value of the integral $∫^2_{-2}\frac{ |x^3+x|}{(e^x|x|+1)}dx$ is equal to:

Let f : R → R be a continuous function satisfying f(x) + f(x + k) = n, for all x ∈ R where k > 0 and n is a positive integer. If
$l_1 = \int_{0}^{4nk} f(x) \, dx$ and $l_2 = \int_{-k}^{3k} f(x) \, dx$, then