List of top Mathematics Questions asked in JEE Main

A person has three different bags and four different books. The number of ways, in which he can put these books in the bags so that no bag is empty, is:

If \(26\left(\frac{2^3}{3}\binom{12}{2} + \frac{2^5}{5}\binom{12}{4} + \frac{2^7}{7}\binom{12}{6} + \dots + \frac{2^{13}}{13}\binom{12}{12}\right) = 3^{13} - \alpha\), then \(\alpha\) is equal to:

Let a line \(L_1\) pass through the origin and be perpendicular to the lines \(L_2: \vec{r} = (3 + t)\hat{i} + (2t - 1)\hat{j} + (2t + 4)\hat{k}\) and \(L_3: \vec{r} = (3 + 2s)\hat{i} + (3 + 2s)\hat{j} + (2 + s)\hat{k}\). If \((a, b, c)\), \(a \in \mathbb{Z}\), is the point on \(L_3\) at a distance of \(\sqrt{17}\) from the point of intersection of \(L_1\) and \(L_2\), then \((a + b + c)^2\) is equal to ________.

A set of four observations has mean 1 and variance 13. Another set of six observations has mean 2 and variance 1. Then, the variance of all these 10 observations is equal to:

Let \(f\) be a polynomial function such that \(\log_2(f(x)) = \left(\log_2\left(2 + \frac{2}{3} + \frac{2}{9} + \dots \infty\right)\right) \cdot \log_3\left(1 + \frac{f(x)}{f(1/x)}\right)\), \(x>0\) and \(f(6) = 37\). Then \(\sum_{n=1}^{10} f(n)\) is equal to ________.

If \( \sin\!\left(\tan^{-1}(x\sqrt2)\right)=\cot\!\left(\sin^{-1}\!\sqrt{1-x^2}\right),\; x\in(0,1) \), then the value of \(x\) is :

If \[ (1-x^3)^{10}=\sum_{r=0}^{10}a_r x^r(1-x)^{30-2r}, \] then \( \dfrac{9a_9}{a_{10}} \) is equal to ________.

The area of the region \( \{(x,y): x^2-8x \le y \le -x\} \) is :

Let \[ A= \begin{bmatrix} 1 & 3 & -1\\ 2 & 1 & \alpha\\ 0 & 1 & -1 \end{bmatrix} \] be a singular matrix. Let \[ f(x)=\int_{0}^{x}(t^2+2t+3)\,dt,\quad x\in[1,\alpha]. \] If \(M\) and \(m\) are respectively the maximum and the minimum values of \(f\) in \([1,\alpha]\), then \(3(M-m)\) is equal to :

The eccentricity of an ellipse \(E\) with centre at the origin \(O\) is \( \frac{\sqrt3}{2} \) and its directrices are \( x=\pm \frac{4\sqrt6}{3} \). Let \( H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \) be a hyperbola whose eccentricity is equal to the length of semi-major axis of \(E\), and whose length of latus rectum is equal to the length of minor axis of \(E\). Then the distance between the foci of \(H\) is :

Let \( f : \mathbb{R} \to \mathbb{R} \) be defined as \( f(x) = \dfrac{2x^2 - 3x + 2}{3x^2 + x + 3} \). Then \( f \) is:

Let \(C\) be a circle having centre in the first quadrant and touching the \(x\)-axis at a distance of \(3\) units from the origin. If the circle \(C\) has an intercept of length \(6\sqrt{3}\) on \(y\)-axis, then the length of the chord of the circle \(C\) on the line \(x-y=3\) is:

Let the image of the point \(P(0,-5,0)\) in the line \[ \frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2} \] be the point \(R\) and the image of the point \(Q(0,-\frac12,0)\) in the line \[ \frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1} \] be the point \(S\). Then the square of the area of the parallelogram \(PQRS\) is ______.

Let \(x=-9\) be a directrix of an ellipse \(E\), whose centre is at the origin and eccentricity is \( \frac13 \). Let \(P(\alpha,0), \alpha>0\), be a focus of \(E\) and \(AB\) be a chord passing through \(P\). Then the locus of the mid point of \(AB\) is :

Let \( \vec a = 2\hat i + 3\hat j + 3\hat k \) and \( \vec b = 6\hat i + 3\hat j + 3\hat k \). Then the square of the area of the triangle with adjacent sides determined by the vectors \( (2\vec a + 3\vec b) \) and \( (\vec a - \vec b) \) is :

Let \(f:\mathbb{R}\to\mathbb{R}\) be such that \(f(x+y)=f(x)f(y)\) for all \(x,y\in\mathbb{R}\) and \(f(0)\neq0\). Let \(g:[1,\infty)\to\mathbb{R}\) be a differentiable function such that \[ x^2g(x)=\int_1^x\big(t^2f(t)-tg(t)\big)\,dt \] Then \(g(2)\) is equal to :

The sum \( 1 + \frac{1}{2}(1^2+2^2) + \frac{1}{3}(1^2+2^2+3^2) + \ldots \) upto \(10\) terms is equal to:

Let the mean and the variance of seven observations \(2,4,\alpha,8,\beta,12,14\), \( \alpha < \beta \), be \(8\) and \(16\) respectively. Then the quadratic equation whose roots are \(3\alpha+2\) and \(2\beta+1\) is :

Let \( S = \{z \in \mathbb{C} : z^2 + \sqrt{6}\,iz - 3 = 0 \}. \) Then \( \displaystyle \sum_{z \in S} z^8 \) is equal to:

The value of the integral \[ \int_{-1}^{1}\left(\frac{x^3+|x|+1}{x^2+2|x|+1}\right)dx \] is equal to :

Let \( R=\{(x,y)\in \mathbb{N}\times\mathbb{N}:\log_e(x+y)\le2\} \). Then the minimum number of elements required to be added in \(R\) to make it a transitive relation is ________.

A building has ground floor and 10 more floors. Nine persons enter in a lift at the ground floor. The lift goes up to the 10th floor. The number of ways, in which any 4 persons exit at a floor and the remaining 5 persons exit at a different floor, if the lift does not stop at the first and the second floors, is equal to :

Consider the quadratic equation \( (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0, \; n \in \mathbb{R}. \) Let \( \alpha \) be the minimum value of the product of its roots and \( \beta \) be the maximum value of the sum of its roots. Then the sum of the first six terms of the G.P., whose first term is \( \alpha \) and the common ratio is \( \dfrac{\alpha}{\beta} \), is:

Let \[ \lim_{x\to2}\frac{\tan(x-2)\,[x^2+(p-2)x-2p]}{(x-2)^2}=5 \] for some \(p,r\in\mathbb{R}\). If the set of all possible values of \(q\), such that the roots of the equation \(rx^2-px+q=0\) lie in \( (0,2) \), be the interval \( (\alpha,\beta) \), then \(4(\alpha+\beta)\) equals :

The shortest distance between the lines \[ \frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3} \] and \[ \frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5} \] is :