List of top Mathematics Questions asked in JEE Main

Match List I with List II

LIST I		LIST II
A.		I.	Neither surjective nor injective
B.		II.	Surjective but not injective
C.		III.	Injective but not surjective
D.		IV.	Injective and surjective

Choose the correct answer from the options given below:

There are two bags. The first bag contains 3 white and 4 red balls and the second bag contains 4 white and 3 red balls. Two balls are drawn at random from the first bag and are put into the second bag without noticing their colours. Then two balls are drawn from the second bag. If the probability that one ball is white and the other is red is \( \frac{x}{y} \), where \( \frac{x}{y} \) is in the simplest form, then value of \( (2x-y) \) is ________

The length of projection of the line segment joining the points \( (1,0,-1) \) and \( (-1,2,2) \) on the plane \( x+3y-5z=6 \) is \( d \), then find \( 70d^{2} \) ________

Let \( L \) denote the line in the \( xy \)-plane with \( x \) and \( y \) intercepts as 2 and 1 respectively. If the image of the point \( (-1,-3) \) in this line is \( \left(\frac{\alpha}{5}, \frac{\beta}{5}\right) \), then \( \alpha+\beta \) is equal to ________

In a \( \Delta ABC \), \( \angle A=\frac{\pi}{2} \). If \( \tan\frac{B}{2} \) and \( \tan\frac{C}{2} \) are the roots of \( 2x^{2}-bx-c=0 \), then what is the value of \( (b-c)^{3} \) ________ ?

Let \( y=y(x) \) be the solution of differential equation \( \sin x \frac{dy}{dx}+y\cos x=4x \). If \( y\left(\frac{\pi}{2}\right)=0 \) and \( y\left(-\frac{\pi}{6}\right)=\frac{\alpha}{\beta}\pi^{2} \), where \( \alpha \) and \( \beta \) are co-prime numbers, then value of \( (\alpha+\beta) \) is ________

The value of \( \int_{1}^{5} f(x)\, dx \), where \( f(x)=|x-1|+|x-2|+|x-3| \) is ________

If \( y=\tan^{-1}\left(\frac{3x}{1+54x^{2}}\right)+\tan^{-1}\left(\frac{3x}{1+18x^{2}}\right) \), then \( \frac{dy}{dx} \) at \( x=0 \) is ________

If \( r^{\text{th}} \) term in the expansion of \( \left(\sqrt[3]{\frac{a}{\sqrt{b}}}+\sqrt{\frac{b}{\sqrt[3]{a}}}\right)^{21} \) contains equal powers of \( a \) and \( b \), then value of \( r \) is ________

Let \( \alpha \) and \( \beta \) be the roots of the equation \( x^{2}-6x-2=0 \), with \( \alpha>\beta \). If \( a_{n}=\alpha^{n}-\beta^{n} \) for \( n\geq1 \), then the value of \( \frac{a_{10}-2a_{8}}{3a_{9}} \) is equal to ________

If \( \alpha,\beta,\gamma \) are the angles of a triangle, then the value of the determinant \( \begin{vmatrix}\cot\alpha&\cot\beta&\cot\gamma\\1&1&1\\\sin^{2}\alpha&\sin^{2}\beta&\sin^{2}\gamma\end{vmatrix} \) is equal to ________

The value of \( \sec 15^{\circ}-\sqrt{3}\,\operatorname{cosec} 15^{\circ} \) is

If \( A \) and \( B \) are two events such that \( P(A\cup B)=\frac{5}{8} \) and \( P(A'\cup B')=\frac{3}{4} \), then \( P(A')+P(B')= \)

The shortest distance between the lines \(\vec{r}=(\hat{i}+2 \hat{j}+3 \hat{k})+\lambda(\hat{i}-3 \hat{j}+2 \hat{k})\) and \(\vec{r}=(4 \hat{i}+5 \hat{j}+6 \hat{k})+\mu(2 \hat{i}+3 \hat{j}+\hat{k})\) is :

The projection vector of \(\vec{a}=\hat{i}-3 \hat{j}-3 \hat{k}\) on \(\vec{b}=3 \hat{i}-4 \hat{j}\) is

\( \cot^{-1}7+\cot^{-1}8+\cot^{-1}5= \)

Given \( |\vec{a}|=|\vec{b}|=1 \) and \( |\vec{a}+\vec{b}|=\sqrt{3} \). If \( \vec{c} \) is a vector such that \( \vec{c}-\vec{a}-2\vec{b}=3(\vec{a}\times\vec{b}) \), then \( \vec{c}\cdot\vec{b} \) is equal to

\(\int \frac{\mathrm{d} x}{\sin ^{2} x \cos ^{2} x}\) is equal to

The vertices of a triangle \(OAB\) are \((0,0)\), \((a, 0)\) and \((0, b)\) respectively. The distance between its orthocentre and circumcentre is

The solution of differential equation \(\left(x+2 y^{3}\right) \frac{d y}{d x}=y\) is

Let \(f:[0, \pi] \rightarrow \mathbb{R}\) be a function defined by \(f(x)=\begin{cases}x+\sqrt{2}\, a \sin x, & 0 \leq x<\frac{\pi}{4} \\ 2 x \cot x+b, & \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \\ a \cos 2 x-b \sin x, & \frac{\pi}{2}<x \leq \pi\end{cases}\)
If \(f\) is continuous for \(0 \leq x \leq \pi\), then \(a^{2}+140 b^{2}\) is equal to

If \(\lim_{x \to 1} \frac{x^{5}-1}{x-1}=\lim_{x \to k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}}\), then \(k\) is equal to

The point diametrically opposite to the point \((-1,1)\) on the circle \(x^{2}+y^{2}-x-4 y+1=0\) is

If the sum of first 10 terms and first 26 terms of an A.P. are 190 and 1326 respectively, then find the sum of its first 50 terms.

The coefficient of \(x^{24}\) in \(\left(1+x^{2}\right)^{12}\left(1+x^{12}\right)\left(1+x^{24}\right)\) is