List of top Mathematics Questions asked in JEE Main

The sum of all local minimum values of the function \( f(x) \) as defined below is:
\[ f(x) = \begin{cases} 1 - 2x & \text{if } x < -1, \\[10pt] \frac{1}{3}(7 + 2|x|) & \text{if } -1 \leq x \leq 2, \\[10pt] \frac{11}{18}(x-4)(x-5) & \text{if } x > 2. \end{cases} \]

Let \[ \int x^3 \sin x \, dx = g(x) + C, \quad \text{where \( C \) is the constant of integration.} \] If \[ g\left( \frac{\pi}{2} \right) + g\left( \frac{\pi}{2} \right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \quad \alpha, \beta, \gamma \in {Z}, \] then \[ \alpha + \beta - \gamma \text{ equals:} \]

The value of \( (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) \) is:

Let the curve \(z(1 + i) +\) \(\overline{z(1 - i) = 4}\), z \(\in\)\( \mathbb{C}\), divide the region \(|z - 3| \leq 1\) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals:}

For a \( 3 \times 3 \) matrix \( M \), let trace(M) denote the sum of all the diagonal elements of \( M \). Let \( A \) be a \( 3 \times 3 \) matrix such that \( |A| = \frac{1}{2} \) and \( \text{trace}(A) = 3 \). If \( B = \text{adj}(\text{adj}(2A)) \), then the value of \( |B| + \text{trace}(B) \) equals:

If the system of linear equations: \[ x + y + 2z = 6, \] \[ 2x + 3y + az = a + 1, \] \[ -x - 3y + bz = 2b, \] where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to:

The area of the region enclosed by the curves \( y = x^2 - 4x + 4 \) and \( y^2 = 16 - 8x \) is:

The perpendicular distance of the line \( \frac{x - 1}{2} = \frac{y + 2}{-1} = \frac{z + 3}{2} \) from the point \( P(2, -10, 1) \) is:

The variance of the numbers 8, 21, 34, 47, \dots, 320, is:

The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is.

Let \( \frac{\overline{z} - i}{z - i} = \frac{1}{3}, \, z \in \mathbb{C} \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0), C \) and \( (\alpha, 0) \), is 11 square units, then \( \alpha^2 \) equals:

If the function \[ f(x) = \begin{cases} \frac{2}{x} \left( \sin(k_1 + 1)x + \sin(k_2 -1)x \right), & x<0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e \left( \frac{2 + k_1 x}{2 + k_2 x} \right), & x>0 \end{cases} \] is continuous at \( x = 0 \), then \( k_1^2 + k_2^2 \) is equal to:

The square of the distance of the point \(\left( \frac{15}{7}, \frac{32}{7}, 7 \right)\) from the line \(\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}\) in the direction of the vector \(\mathbf{i} + 4\mathbf{j} + 7\mathbf{k}\) is:

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4\pi} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is:

Let \[ L_1 : \frac{x-1}{1} = \frac{y-2}{-1} = \frac{z-1}{-1} \quad {and} \quad L_2 : \frac{x+1}{1} = \frac{y-2}{2} = \frac{z-2}{2} \] be two lines. Let \( L_3 \) be a line passing through the point \( (\alpha, \beta, \gamma) \) and be perpendicular to both \( L_1 \) and \( L_2 \). If \( L_3 \) intersects \( L_1 \) where \( 5x - 11y - 8z = 1 \), then \( 5x - 11y - 8z \) equals:

The integral is given by:

\[ 80 \int_{0}^{\frac{\pi}{4}} \frac{\sin\theta + \cos\theta}{9 + 16 \sin 2\theta} d\theta \]

is equals to?

If \[ \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{96x^2 \cos^2 x}{1 + e^x} dx = \pi(a\pi^2 + \beta), \quad a, \beta \in \mathbb{Z}, \] then \( (a + \beta)^2 \) equals:

Find the value of \[ \sin 70^\circ \left( \cot 10^\circ \cot 70^\circ - 1 \right). \]

Let the line \(x + y = 1\) meet the axes of x and y at A and B, respectively. A right-angled triangle AMN is inscribed in the triangle OAB, where O is the origin and the points M and N lie on the lines OB and AB, respectively. If the area of the triangle AMN is \(\frac{4}{9}\) of the area of the triangle OAB and \(AN : NB = \lambda : 1\), then the sum of all possible values of \(\lambda\) is:

Let the position vectors of three vertices of a triangle be \( \overrightarrow{p} = 4\hat{i} + \hat{j} - 3\hat{k} \), \( \overrightarrow{q} = -5\hat{i} + 2\hat{j} + 3\hat{k} \), and \( \overrightarrow{r} = -5\hat{i} + 3\hat{j} + 2\hat{k} \). Then \( \alpha + 2\beta + 5\gamma \) is equal to:

Let \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}, \, \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) and \( \vec{c} \) be three vectors such that \( \vec{c} \) is coplanar with \( \vec{a} \) and \( \vec{b} \). If the vector \( \vec{c} \) is perpendicular to \( \vec{b} \) and \( \vec{a} \cdot \vec{c} = 5 \), then \( |\vec{c}| \) is equal to:

Let O be the origin, the point A be \( z_1 = \sqrt{3} + 2\sqrt{2}i \), the point B \( z_2 \) be such that \( \sqrt{3}|z_2| = |z_1| \) and \( \arg(z_2) = \arg(z_1) + \frac{\pi}{6} \). Then: