List of top Mathematics Questions asked in JEE Main

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

Let for n = 1, 2, …, 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose common ratio is 
\(\frac{1}{(n+1)^2}\) . Then the value of 
\(\frac{1}{26} + \sum_{n=1}^{50} \left(S_n+\frac{2}{n+1}-n-1  \right)\)
 is equal to ________.

Let 
\(f(x)=max\left\{|x+1|,|x+2|,……,|x+5|\right\} \)
Then \(\int_{-6}^{0} f(x) \, dx\)
is equal to_______

Let S = {4, 6, 9} and T = {9, 10, 11, …,1000}. If A = {a1 + a2 + … +ak :k∈N, a1, a2, a3, …, ak∈S}, then the sum of all the elements in the set T – A is equal to ____________.
\(\alpha = \sin 36^\circ\)  is a root of which of the following equation?
Let the eccentricity of the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) be \(\frac{5}{4}\). If the equation of the normal at the point \((\frac{8}{√5}, \frac{12}{5})\) on the hyperbola is 8√5 x + β y = λ, then λ – β is equal to ____.
A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the side of the equilateral triangle, so that the combined area of the square and the equilateral triangle is minimum, is

If the lines
\(\stackrel{→}{r}= ( \hat{i} - \hat{j} + \hat{k} ) + λ (\hat{3j} - \hat{k} )= ( \hat{i} - \hat{j} + \hat{k} ) + λ (\hat{3j} - \hat{k} )\)
and
\(\stackrel{→}{r} = ( \alpha \hat{i} - \hat{j} ) + μ( \hat{2j} - \hat{3k} )\)
are co-planer , then the distance of the plane containing these two lines from the point \(( α , 0 , 0 )\) is :

The number of solutions of the equation \(2\theta - \cos^2(\theta) + \sqrt{2} = 0\) in R is equal to ___________.

Let f : R → R be a differentiable function such that
\(f(\frac{π}{4})=\sqrt2,f(\frac{π}{2})=0 \) and \(f′(\frac{π}{2})=1\)
and let
\(g(x) = \int_{x}^{\frac{\pi}{4}} \left(f'(t)\sec(t) + \tan(t)\sec(t)f(t)\right) \, dt\)
for\( x∈(\frac{π}{4},\frac{π}{2})\) Then \(\lim_{{x \to \frac{\pi}{2}^-}} g(x)\)is equal to

Let S be the set of all the natural numbers, for which the line 
\(\frac{x}{a}+\frac{y}{b}=2 \)
is a tangent to the curve
\((\frac{x}{a})^n+(\frac{y}{b})^n=2 \)
at the point (a, b), ab ≠ 0. Then :

The number of 5-digit natural numbers, such that the product of their digits is 36, is _____ .

Let \(y = y(x)\) be the solution of the differential equation \((x + 1)y' – y = e^{3x}(x + 1)^2\), with \(y(0)=\frac 13\). Then, the point \(x=−\frac 43\) for the curve \(y = y(x)\) is :
Let b1b2b3b4 be a 4-element permutation with bi{1, 2, 3,…..,100} for 1≤i≤4 and bi ≠bj for i≠ j, such that either b1, b2, b3 are consecutive integers or b2, b3, b4 are consecutive integers. Then the number of such permutations b1b2b3b4 is equal to _______ .