List of top Mathematics Questions asked in JEE Main

Let \(f : R \to R\) be a continuous function such that \(f(x) + f(x+1) = 2\), for all \(x \in R\). If \(I_1 = \int\limits_0^8 f(x)dx\) and \(I_2 = \int\limits_{-1}^3 f(x)dx\), then the value of \(I_1 + 2I_2\) is equal to ____________.

If \(\lim\limits_{x \to 0} \frac{a e^x - b\cos x + c e^{-x}}{x \sin x} = 2\), then \(a + b + c\) is equal to ____________.

Let [ x ] denote greatest integer less than or equal to x. If for \(n \in N\), \((1 - x + x^3)^n = \sum\limits_{j=0}^{3n} a_j x^j\), then \(\sum\limits_{j=0}^{\left[\frac{3n}{2}\right]} a_{2j} + 4 \sum\limits_{j=0}^{\left[\frac{3n-1}{2}\right]} a_{2j+1}\) is equal to :

The number of roots of the equation, \((81)^{\sin^2 x} + (81)^{\cos^2 x} = 30\) in the interval \([\,0, \pi\,]\) is equal to :

If for \(a \gt 0\), the feet of perpendiculars from the points A(\(a, -2a, 3\)) and B(\(0, 4, 5\)) on the plane \(lx + my + nz = 0\) are points C(\(0, -a, -1\)) and D respectively, then the length of line segment CD is equal to :

If the three normals drawn to the parabola, \(y^2=2x\) pass through the point \((a,0)\) \(a\ne0\), then 'a' must be greater than :

Let the curve \(y = y(x)\) be the solution of the differential equation, \(\frac{dy}{dx} = 2(x+1)\). If the numerical value of area bounded by the curve \(y = y(x)\) and x-axis is \(\frac{4\sqrt8}{3}\), then the value of \(y(1)\) is equal to ____________.

Let P be a plane \(lx + my + nz = 0\) containing the line, \(\frac{1-x}{1} = \frac{y+4}{2} = \frac{z+2}{3}\). If plane P divides the line segment AB joining points A(\(-3, -6, 1\)) and B(\(2, 4, -3\)) in ratio k : 1 then the value of k is equal to :

A pack of cards has one card missing. Two cards are drawn randomly and are found to be spades. The probability that the missing card is not a spade, is :

If \(y=y(x)\) is the solution of the differential equation, \(\frac{dy}{dx}+2y\tan x=\sin x,\ y\left(\frac{\pi}{3}\right)=0\), then the maximum value of the function \(y(x)\) over \(\mathbb R\) is equal to:

Which of the following Boolean expression is a tautology?

If for \(x\in\left(0,\frac{\pi}{2}\right)\), \(\log_{10}\sin x+\log_{10}\cos x=-1\) and \(\log_{10}(\sin x+\cos x)=\frac12(\log_{10}n-1)\), \(n>0\), then the value of \(n\) is equal to :

Consider three observations a, b, and c such that \(b=a+c\). If the standard deviation of \(a+2,\ b+2,\ c+2\) is \(d\), then which of the following is true?

Let the functions \(f:\mathbb R\to\mathbb R\) and \(g:\mathbb R\to\mathbb R\) be defined as :
\(f(x)=\begin{cases}x+2, & x<0\\ x^2, & x\ge0\end{cases}\) and
\(g(x)=\begin{cases}x^3, & x<1\\ 3x-2, & x\ge1\end{cases}\)
Then, the number of points in \(\mathbb R\) where \((f\circ g)(x)\) is NOT differentiable is equal to :

The range of \(a\in\mathbb R\) for which the function \(f(x)=(4a-3)(x+\log_e5)+2(a-7)\cot\left(\frac{x}{2}\right)\sin^2\left(\frac{x}{2}\right)\), \(x\ne2n\pi,\ n\in N\) has critical points, is :

Let \(A=\begin{bmatrix} i & -i\\ -i & i\end{bmatrix},\ i=\sqrt{-1}\). Then, the system of linear equations \(A^8\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}8\\64\end{bmatrix}\) has :

If \(n\) is the number of irrational terms in the expansion of \(\left(3^{1/4}+5^{1/8}\right)^{60}\), then \((n-1)\) is divisible by :

Let a complex number \(z\), \(|z|\neq1\), satisfy \(\log_{\frac{1}{\sqrt2}}\left(\frac{|z|+11}{(|z|-1)^2}\right)\le2\). Then, the largest value of \(|z|\) is equal to ____________.

The locus of the midpoints of the chord of the circle, \(x^2+y^2=25\) which is tangent to the hyperbola, \(\frac{x^2}{9}-\frac{y^2}{16}=1\) is :

Let a vector \(\alpha\hat{i}+\beta\hat{j}\) be obtained by rotating the vector \(\sqrt{3}\,\hat{i}+\hat{j}\) by an angle \(45^\circ\) about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices \((\alpha, \beta)\), \((0, \beta)\) and \((0, 0)\) is equal to :

The number of elements in the set \(\{x\in\mathbb{R}:(|x|-3)\,|x+4|=6\}\) is equal to :

Let \(S_k=\sum_{r=1}^{k}\tan^{-1}\left(\frac{6^r}{2^{2r+1}+3^{2r+1}}\right)\). Then \(\lim_{k\to\infty}S_k\) is equal to :

Let \(X_1,X_2,\dots,X_{18}\) be eighteen observations such that
\(\sum_{i=1}^{18}(X_i-\alpha)=36\)
and
\(\sum_{i=1}^{18}(X_i-\beta)^2=90\)
, where \(\alpha\) and \(\beta\) are distinct real numbers. If the standard deviation of these observations is 1, then the value of \(|\alpha-\beta|\) is ____________.

Let \(a\) be an integer such that all the real roots of the polynomial \(2x^5+5x^4+10x^3+10x^2+10x+10\) lie in the interval \((a,a+1)\). Then, \(|a|\) is equal to ___________.

Let the normals at all the points on a given curve pass through a fixed point \((a,b)\). If the curve passes through \((3,-3)\) and \((4,-2\sqrt2)\), and given that \(a-2\sqrt2\,b=3\), then \((a^2+b^2+ab)\) is equal to __________.