List of top Mathematics Questions asked in JEE Main

If the value of \(\frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ}\) is \(\frac{\alpha + \beta\sqrt{5}}{\gamma}\) then value of \((\alpha + \beta + \gamma)\) (where \(\alpha, \beta, \gamma \in \mathbb{N}\) and are in lowest form) :

If \[ \frac{\cos^2 48^\circ - \sin^2 12^\circ}{\sin^2 24^\circ - \sin^2 6^\circ} = \frac{\alpha + \sqrt{5}\,\beta}{2}, \] then the value of \((\alpha + \beta)\) is

The number of values of \(x\) satisfying \[ \tan^{-1}(4x)+\tan^{-1}(6x)=\frac{\pi}{6}, \quad x\in\left[-\frac{1}{2\sqrt{6}},\,\frac{1}{2\sqrt{6}}\right] \] is

Let \( A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix} \) and \( B = \begin{bmatrix} 29 & 49 \\1 & 2 \end{bmatrix} \). If \( (A^5 + B)\begin{bmatrix} x\\ y \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \), then find \( (x, y) \).

The value of \[ \int_{\frac{\pi}{2}}^{\pi} \frac{dx}{[x]+4} \] where \([\,\cdot\,]\) denotes the greatest integer function, is

If the differential equation for the family of circles passing through the points \((19, 0)\) and \((-19, 0)\) is \(2xy\,\mathrm{d}x + (\alpha x^{2} + \beta y^{2} + \gamma)\,\mathrm{d}y = 0\), then \(\alpha + \beta + \gamma\) is equal to ________.

If \( \sum_{x=1}^{n^{2}} \sqrt{\left(x \cos \left(\cot ^{-1} x\right)+\sin \left(\cot ^{-1} x\right)\right)^{2}-1}=A n^{4}+B n^{2} \), then \( 2(A+B) \) equals ________.

Let \( \vec{a}=2 \hat{i}+\hat{j}-\hat{k} \) and \( \vec{b}=3 \hat{i}+2 \hat{j}+4 \hat{k} \). Let for some \( \lambda, \mu \in \mathbf{R} \), a vector \( \vec{c}=\lambda \vec{a}+\mu \vec{b} \). If \( \vec{a} \cdot \vec{c}=16 \) and \( \vec{b} \times \vec{c}=-12 \hat{i}+22 \hat{j}-2 \hat{k} \), then \( |\vec{c}|^{2} \) equals ________.

If \( \left({ }^{10} C_{1}\right)^{2}+2\left({ }^{10} C_{2}\right)^{2}+\ldots+10\left({ }^{10} C_{10}\right)^{2}=m\left({ }^{n} C_{m}\right) \), then \( m+n \) equals ________.

If \( \int_{4}^{9} \dfrac{x-\sqrt{x}}{x(x-1)}\, \mathrm{d}x=\alpha \), then \( \mathrm{e}^{\alpha} \) is equal to :

The area of the region, enclosed between the curves \( y=\cos x \) and \( y=|\cos x| \) for \( 0 \leq x \leq 3 \pi \), is :

Let \( A=\begin{bmatrix}1 & 2 & 3\\ 0 & 2 & 3\\ 0 & 0 & 1\end{bmatrix} \) and let \( P \) be a \( 3 \times 3 \) matrix. If \( AP=\begin{bmatrix}1 & 0 & 1\\ 1 & 1 & 0\\ 0 & 1 & 1\end{bmatrix} \) and \( \begin{bmatrix}1 & 0 & 2\end{bmatrix} P\begin{bmatrix}1\\ 0\\ 2\end{bmatrix}=[k]_{1 \times 1} \), then \( k \) is equal to ________.

If the function \( f(x) \) defined by \( f(x)=\begin{cases}\dfrac{\log_{e}(1+5x)-\log_{e}(1-2x)}{x}, & x \neq 0 \\ l, & x=0\end{cases} \) is continuous at \( x=0 \), then the value of \( l \) is :

Let \( f \) be a differentiable non-zero function and \( 3\int_0^x f(t)\,dt = 2(f(x))^3 \). Then \( \int_0^1 \frac{3(f(x))^4}{\sqrt{1 + (f(x))^{12}}}\,dx \) is equal to :

The length of the perpendicular from the point \( (1, 2, 3) \) on the line \( \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{-2} \) is :

Let \( f(x)=\sum_{r=1}^{10} \cot^{-1}\left(1-(x+r)+(x+r)^{2}\right) \), \( x \geq 0 \). Then \( \lim_{x \rightarrow \infty}\left(\tan (f(x))+\sec^{2}(f(x))\right) \) is :

Let \( L_1 \) be the line that passes through the point \( (1, 2, -4) \) and be perpendicular to the two lines \( \frac{x - 8}{3} = \frac{y + 19}{-16} = \frac{z - 10}{7} \) and \( \frac{x - 15}{3} = \frac{y - 29}{8} = \frac{z - 5}{-5} \). If \( L_2 \) is the line \( \vec{r} = 7\hat{i} - 6\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k}) \), \( \lambda \in \mathbf{R} \), then an angle between \( L_1 \) and \( L_2 \) is equal to :

Let \( A\left(3, 3\sqrt{3}\right) \), \( B(6, 0) \) and \( C(\alpha, \beta) \) be the vertices of an equilateral triangle. Let its incentre be \( D \), and \( P \) and \( Q \) be the feet of perpendiculars drawn from the point \( D \) on \( AC \) and \( BC \), respectively. Then the area of quadrilateral \( CPDQ \) is :

Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), \( a > b \) be an ellipse of eccentricity \( \frac{1}{\sqrt{3}} \). If the distance between the foci of \( E \) is 4, then the length of its latus rectum is :

Let a circle of radius 3 pass through the origin \( O \) and meet the coordinate axes at the points \( A \) and \( B \). Then the locus of the centroid of the triangle \( AOB \) is :

Let the line \( y = x + 3 \) intersect the parabola \( x^2 = 4y \) at the points \( A \) and \( B \). If \( F \) is the focus of the parabola, then the area of triangle \( AFB \) is :

In a question paper containing 10 questions, A can solve 70% of the questions and B can solve 50% of the questions. The probability, that none of A and B will be able to solve the question, selected at random from the paper, is :

If the mean of the frequency distribution

Variable (\(x_i\))	1	3	4	7	9	Total
Frequency (\(f_i\))	1	\(a\)	\(b\)	3	1	10

is 5, then its variance is :