List of top Questions asked in JEE Main

The value of \(S = \sum_{r=1}^{20} \sqrt{\pi \int_{0}^{r} x |\sin \pi x| dx}\) is :
Let \(\int \frac{1 - 5 \cos^2 x}{\sin^5 x \cos^2 x} dx = f(x) + c\) then find \(f\left(\frac{\pi}{4}\right) - f\left(\frac{\pi}{6}\right)\) :
Given conic \(x^2 - y^2 \sec^2 \theta = 8\) whose eccentricity is '\(e_1\)' & length of latus rectum '\(l_1\)' and for conic \(x^2 + y^2 \sec^2 \theta = 6\), eccentricity is '\(e_2\)' & length of latus rectum '\(l_2\)'. If \(e_1^2 = e_2^2 (1 + \sec^2 \theta)\) then value of \(\frac{e_1 l_1}{e_2 l_2} \tan \theta\)
The sum of coefficients of \(x^{499}\) and \(x^{500}\) in the expression: \[ (1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \cdots + x^{1000} \] is:
If \( y = \operatorname{sgn}(\sin x) + \operatorname{sgn}(\cos x) + \operatorname{sgn}(\tan x) + \operatorname{sgn}(\cot x) \), where \(\operatorname{sgn}(p)\) denotes the signum function of \(p\), then the sum of elements in the range of \(y\) is:
Let matrix \[ A=\begin{pmatrix} 3 & -4\\ 1 & -1 \end{pmatrix} \] and \(A^{100} = 100B + I\). Find the sum of all the elements in \(B^{100}\).
Statement 1 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x}{1+|x|} \] is one–one.
Statement 2 : The function \(f:\mathbb{R}\to\mathbb{R}\) defined by \[ f(x)=\frac{x^2+4x-30}{x^2-8x+18} \] is many–one.
Which of the following is correct?
If \[ \sum_{r=1}^{25}\frac{r}{r^4+r^2+1}=\frac{p}{q}, \] where \(p\) and \(q\) are coprime positive integers, then \(p+q\) is equal to:
If the image of a point \(P(3,2,1)\) in the line \[ \frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} \] is \(Q\), then the distance of \(Q\) from the line \[ \frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2} \] is:
If the arithmetic mean of \(\dfrac{1}{a}\) and \(\dfrac{1}{b}\) is \(\dfrac{5}{16}\) and \(a,\,4,\,\alpha,\,b\) are in increasing A.P., then both the roots of the equation \[ \alpha x^2-ax+2(\alpha-2b)=0 \] lie between:
Let \(L\) be the distance of point \(P(-1,2,5)\) from the line \[ \frac{x-1}{2}=\frac{y-3}{2}=\frac{z+1}{1} \] measured {parallel to a line having direction ratios \(4,\,3,\,-5\). Then \(L^2\) is equal to: }
Let \(f:[-2a,2a]\to\mathbb{R}\) be a thrice differentiable function and define \[ g(x)=f(a+x)+f(a-x). \] If \(m\) is the minimum number of roots of \(g'(x)=0\) in the interval \((-a,a)\) and \(n\) is the minimum number of roots of \(g''(x)=0\) in the interval \((-a,a)\), then \(m+n\) is equal to:
The function \[ f(x)=\sin 2x+2\cos x,\qquad x\in\left(-\frac{3\pi}{4},\frac{3\pi}{4}\right) \] has:
Let the solution curve of the differential equation \[ x\,dy - y\,dx = \sqrt{x^2+y^2}\,dx,\quad x>0, \] with $y(1)=0$, be $y=y(x)$. Then $y(3)$ is equal to
Let \( S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\} \) be a set of polynomials. Then the number of polynomials in \( S \), which are divisible by \( x^2 + 2 \), is:
Let the line \( y - x = 1 \) intersect the ellipse \( \frac{x^2}{2} + \frac{y^2}{1} = 1 \) at the points A and B. Then the angle made by the line segment AB at the center of the ellipse is:
The value of the integral \( \int_{\frac{\pi}{24}}^{\frac{5\pi}{24}} \frac{dx}{1 + \sqrt[3]{\tan 2x}} \) is :
A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :
Let \( S = \{z : 3 \le |2z - 3(1+i)| \le 7\ \) be a set of complex numbers. Then \( \min_{z \in S} \left| z + \frac{1}{2}(5+3i) \right| \) is equal to :}
The value of \( \frac{100 C_{50}}{51} + \frac{100 C_{51}}{52} + \dots + \frac{100 C_{100}}{101} \) is :
The number of real solutions of the equation: $x|x+3| + |x-1| - 2 = 0$ is

Let each of the two ellipses $E_1:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1\;(a>b)$ and $E_2:\dfrac{x^2}{A^2}+\dfrac{y^2}{B^2}=1A$

The mean and variance of a data of 10 observations are 10 and 2, respectively. If an observation $\alpha$ in this data is replaced by $\beta$, then the mean and variance become $10.1$ and $1.99$, respectively. Then $\alpha+\beta$ equals
From a lot containing $10$ defective and $90$ non-defective bulbs, $8$ bulbs are selected one by one with replacement. Then the probability of getting at least $7$ defective bulbs is

If the domain of the function \[ f(x)=\log\left(10x^2-17x+7\right)\left(18x^2-11x+1\right) \] is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then $90(a+b+c+d+e)$ equals