List of top Mathematics Questions asked in JEE Main

Which one of the following is a tautology ?

Let P be the point of intersection of the plane \(4x-6y+14z=57\) with the line joining the points A\((2,3,5)\) and B\((1,-1,4)\). If \(BP=\lambda\,AP\), then the value of \(\lambda\) is :

Let \(\vec{a}, \vec{b}, \vec{c}\) be any three non-coplanar vectors. If the points \(\mathrm{A}(-2 \vec{a}+3 \vec{b}+5 \vec{c})\), \(\mathrm{B}(\vec{a}+2 \vec{b}+3 \vec{c})\) and \(\mathrm{C}(7 \vec{a}-\lambda \vec{c})\) are collinear, then \(\lambda\) is equal to :

If \(2 \sin x-\cos 2 x=1\), then :

If a hyperbola has length of its transverse axis \(=\sqrt{2}\) and it has the same foci as that of the ellipse \(4x^{2}+3y^{2}=24\), then this hyperbola also passes through the point whose coordinates are :

Let ABC be a triangle such that the vertex A has coordinates \((-3,1)\). If the equation of the median through B be \(2x+y-3=0\) and the equation of the bisector of angle C be \(7x-4y-1=0\), then the equation of the line BC is :

If the ellipse centered at the origin has a focus at \((0,3)\) and the corresponding directrix is \(y=9\), then which one of the following is not true for this ellipse ?

The area (in sq. units) of the region \(\{(x,y)\in\mathbb{R}^{2}:2|x|+|y|\le 2\text{ and }x+y\ge 1\}\) is :

If \(y(x)=\int_{\pi^{2}/16}^{x^{2}}\frac{\sin x\,\sin\sqrt{\theta}}{1+\cos^{2}\sqrt{\theta}}\,d\theta\), then \(y'\left(\frac{\pi}{4}\right)\) is equal to :

If for \(x>0\), \(\int \frac{e^{\sqrt{x}}\sin\left(\frac{\pi}{4}+\sqrt{x}\right)}{\sqrt{x}}\,dx=e^{\sqrt{x}}\cdot f(x)+C\), where \(C\) is a constant of integration, then \(f(x)\) is equal to :

Let \(f(x)=\left(2x^{2}-9x+11\right)\cdot e^{x},\ -\infty<x<\infty\). Then \(f\) has :

Let \(f\) be a twice differentiable function on \(\mathbb{R}\) such that \(f(0)=4\) and the function, \(g(x)=\frac{1}{2}f^{2}(x)+3x^{3}-5\) is monotonically increasing in \(\mathbb{R}\). Then \(f(1)\) cannot be :

Let the functions \(f\) and \(g\) be defined as \(f(x)=2\cos|x-2|+3|x|\sin|x|\) and \(g(x)=x^{5}+x\). If S be the set of points where the function \((f\circ g)(x)\) is not differentiable, then S is equal to :

For \(\theta\in(0,\pi/2)\), \(\theta\in(\pi,2\pi)\) and \(\beta\in\mathbb{R}\), the value of the determinant
\(\begin{vmatrix}\cos 2\alpha & \cos(\alpha+\theta) & -\sin(\alpha+\theta)\\ \sin\alpha & \sin\theta & \cos\theta\\ \beta\cos\alpha & -\cos\theta & \sin\theta\end{vmatrix}\) :

If \(z=-1+i\sqrt{2}\), then which one of the following is not true ?

Let \(A=\begin{pmatrix}\alpha & 1 & 2\\ 1 & 2 & 3\\ 3 & 1 & 1\end{pmatrix}\), \(\alpha\in\mathbb{R}\) and \(B=[b_{ij}]_{3\times3}\) such that \(AB=rI\), \(r\neq0\) in \(\mathbb{R}\). If \(b_{32}=-\frac{3}{2}r\) and \(|\operatorname{adj}B|=r^{5}\), then \(|\operatorname{adj}(AB)|\) equals :

Let \(a_1, a_2, a_3, a_4\) and \(a_5\) be in A.P. with the common difference greater than 2. If \(\sum_{i=1}^{5}a_i=30\) and \(a_1\cdot a_2\cdot a_3\cdot a_4\cdot a_5=3840\), then \(a_1\cdot a_5\) is equal to :

Let \(\alpha\) be a common root of the equations, \(x^{2}-x-12=0\) and \(\lambda x^{2}+10x+3=0\) (where \(\lambda\) is an integer). If \(\beta\) is the other root of the second equation, then the ordered pair \((\lambda,\beta)\) is equal to :

Let a function, \(f:\mathbb{R}\to\mathbb{R}\) be defined by \(f(x)=1+x|x|\) and let \(\operatorname{sgn}(x)=\begin{cases}1, & \text{if } x\ge 0\\ -1, & \text{if } x<0\end{cases}\)
Then :

If \(\int \mathrm{e}^{2x}(\cos x+7\sin x)\,\mathrm{d}x=\mathrm{e}^{2x}g(x)+\mathrm{c}\), where c is a constant of integration, then \(g(0)+g\left(\frac{\pi}{2}\right)\) is equal to ________.

If the total number of ways in which 8-digit numbers can be formed by using all the digits \(0,1,2,3,4,5,7,9\) such that no two even digits appear together is \((5!)\,k\), then \(k\) is equal to ________.

If the line joining the points \((-1,2,5)\) and \((3,4,-10)\) intersects the \(xy\)-plane at the point \((x, y, z)\), then \(\dfrac{y}{x}\) is equal to ________.

Let \(f(x)=\begin{cases}|x-3|, & \text{if } x<-1\\ 3x+4, & \text{if } x\geq -1\end{cases}\), \(g(x)=x^{2}-\mathrm{b}x-2\ (x\in\mathbf{R})\) and \(\mathrm{b}\) is a real constant. If \(g\circ f\) is continuous at \(x=-1\), then \(\mathrm{b}\) is equal to ________.

Let a, \(\mathrm{b}\in\mathbf{R}\) and \(\mathrm{a}>0\). If the tangent at the point \((2,2)\) to the circle \(x^{2}+y^{2}=8\) touches the parabola \(y^{2}=4\mathrm{a}(x-\mathrm{b})\), then \(\mathrm{b}-\mathrm{a}\) is equal to ________.

Let \(\mathrm{A}=\left[\begin{array}{cc}x & 2 y \\ -1 & y\end{array}\right]\); \(x, y \in \mathbf{R}\).
If \(\mathrm{AA}^{\mathrm{T}}=\left[\begin{array}{ll}1 & 0 \\ 0 & \alpha\end{array}\right]\) \((\alpha \in \mathbf{R})\), then \(\alpha+y^{2}\) is equal to :