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Let $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ be two vectors such that $|\vec{a}| = 1$, $\vec{a} \cdot \vec{b} = 2$ and $|\vec{b}| = 4$. If $\vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to:

If \( 2 \sin^{3} x + \sin 2x \cos x + 4 \sin x - 4 = 0 \) has exactly \( 3 \) solutions in the interval \( \left[ 0, \dfrac{n\pi}{2} \right], \; n \in \mathbb{N} \), then the roots of the equation \( x^{2} + nx + (n - 3) = 0 \) belong to:

Consider the system of linear equations
\( x + y + z = 4\mu, \quad x + 2y + 2\lambda z = 10\mu, \quad x + 3y + 4\lambda^{2} z = \mu^{2} + 15 \), where, \( \lambda, \mu \in \mathbb{R} \). Which one of the following statements is NOT correct?

The area (in square units) of the region bounded by the parabola \( y^{2} = 4(x - 2) \) and the line \( y = 2x - 8 \), is:

Let \( M \) denote the median of the following frequency distribution

Then \( 20\,M \) is equal to:

A line passing through the point \( A(9, 0) \), makes an angle of \( 30^{\circ} \) with the positive direction of \( x \)-axis. If this line is rotated about \( A \) through an angle of \( 15^{\circ} \) in the clockwise direction, then its equation in the new position is:

If the domain of the function \( f(x) = \cos^{-1}\!\left(\dfrac{2 - |x|}{4}\right) + \{\log_{e}(3 - x)\}^{-1} \) is \( [ -\alpha, \beta ) - \{\gamma\} \), then \( \alpha + \beta + \gamma \) is equal to:

Let \( S_n \) denote the sum of first \( n \) terms of an arithmetic progression. If \( S_{20} = 790 \) and \( S_{10} = 145 \), then \( S_{15} - S_{5} \) is:

Let \( f : \left[ -\dfrac{\pi}{2}, \dfrac{\pi}{2} \right] \to \mathbb{R} \) be a differentiable function such that \( f(0) = \dfrac{1}{2} \). If the \( \displaystyle \lim_{x \to 0} \dfrac{x \int_{0}^{x} f(t)\,dt}{e^{x^{2}} - 1} = \alpha \), then \( 8\alpha^{2} \) is equal to:

Let \( g : \mathbb{R} \to \mathbb{R} \) be a non constant twice differentiable function such that \( g'\!\left(\dfrac{1}{2}\right) = g'\!\left(\dfrac{3}{2}\right) \). If a real valued function \( f \) is defined as \( f(x) = \dfrac{1}{2}\,[g(x) + g(2 - x)] \), then:

Let \( y = y(x) \) be the solution of the differential equation \( \sec x \, dy + \{2(1 - x)\tan x + x(2 - x)\}dx = 0 \) such that \( y(0) = 2 \). Then \( y(2) \) is equal to:

then \( \dfrac{1}{5} f'(0) \) is equal to:

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

Two integers \( x \) and \( y \) are chosen with replacement from the set \( \{ 0, 1, 2, 3, \ldots, 10 \} \). Then the probability that \( |x - y| > 5 \), is:

Let \( A(2,3,5) \) and \( C(-3,4,-2) \) be opposite vertices of a parallelogram \( ABCD \). If the diagonal \( \overrightarrow{BD} = \hat{i} + 2\hat{j} + 3\hat{k} \), then the area of the parallelogram is equal to:

If \( z = x + iy, \ xy \neq 0 \), satisfies the equation \( z^{2} + i\overline{z} = 0 \), then \( |z|^{2} \) is equal to:

Let \( (\alpha, \beta, \gamma) \) be the foot of perpendicular from the point \( (1, 2, 3) \) on the line \( \frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3} \). Then \( 19(\alpha + \beta + \gamma) \) is equal to:

If the circles \( (x + 1)^{2} + (y + 2)^{2} = r^{2} \) and \( x^{2} + y^{2} - 4x - 4y + 4 = 0 \) intersect at exactly two distinct points, then:

The maximum area of a triangle whose one vertex is at \( (0, 0) \) and the other two vertices lie on the curve \( y = -2x^2 + 54 \) at points \( (x, y) \) and \( (-x, y) \), where \( y > 0 \), is:

The value of \( \displaystyle \lim_{n \to \infty} \sum_{k=1}^{n} \dfrac{n^3}{(n^2 + k^2)(n^2 + 3k^2)} \) is:

Let the latus rectum of the hyperbola \( \dfrac{x^2}{9} - \dfrac{y^2}{b^2} = 1 \) subtend an angle of \( \dfrac{\pi}{3} \) at the centre of the hyperbola. If \( b^2 \) is equal to \( \dfrac{l}{m}(1 + \sqrt{n}) \), where \( l \) and \( m \) are co-prime numbers, then \( l^2 + m^2 + n^2 \) is equal to \( \_\_\_\_\_\_\_\_ \).

If \( d_1 \) is the shortest distance between the lines \( x + 1 = 2y = -12z \) and \( x = y + 2 = 6z - 6 \) and \( d_2 \) is the shortest distance between the lines \( \dfrac{x - 1}{2} = \dfrac{y + 8}{-7} = \dfrac{z - 4}{5} \) and \( \dfrac{x - 1}{2} = \dfrac{y - 2}{1} = \dfrac{z - 6}{-3} \), then the value of \( \dfrac{32\sqrt{3}\, d_1}{d_2} \) is:

A group of \( 40 \) students appeared in an examination of \( 3 \) subjects – Mathematics, Physics and Chemistry. It was found that all students passed in at least one of the subjects, \( 20 \) students passed in Mathematics, \( 25 \) students passed in Physics, \( 16 \) students passed in Chemistry, at most \( 11 \) students passed in both Mathematics and Physics, at most \( 15 \) students passed in both Physics and Chemistry, at most \( 15 \) students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is \( \_\_\_\_\_ \).

is differentiable on \( \mathbb{R} \), then \( 48(a + b) \) is equal to \( \_\_\_\_ \).

Let $y = y(x)$ be the solution of the differential equation $\left(1 - x^2\right)dy = \left[ xy + \left(x^3 + 2\right)\sqrt{3\left(1 - x^2\right)} \right] dx$, $-1 < x < 1, y(0) = 0$. If $y\left(\frac{1}{2}\right) = \frac{m}{n}$, $m$ and $n$ are co-prime numbers, then $m + n$ is equal to _____.

Let \( \alpha, \beta \in \mathbb{N} \) be roots of the equation \( x^2 - 70x + \lambda = 0 \), where \( \frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N} \). If \( \lambda \) assumes the minimum possible value, then \( \dfrac{\left(\sqrt{\alpha - 1} + \sqrt{\beta - 1}\right)(\lambda
+ 35)}{|\alpha - \beta|} \) is equal to: ______.

Let \( \alpha = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + \ldots \) up to \( 10 \) terms and \( \beta = \displaystyle \sum_{n=1}^{10} n^4 \). If \( 4\alpha - \beta = 55k + 40 \), then \( k \) is equal to ________.

Number of integral terms in the expansion of $\left\{ 7^{\left(\frac{1}{2}\right)} + 11^{\left(\frac{1}{6}\right)} \right\}^{824}$ is equal to _________.

The value of $9 \int_{0}^{9} \left[ \sqrt{\frac{10x}{x+1}} \right] dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is

Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \to P(A) \) such that \( a \in f(a),\ \forall\, a \in A \) is \( m^n \), \( m \) and \( n \in \mathbb{N} \) and \( m \) is least, then \( m + n \) is equal to ______.

JEE Main 2024 30th Jan Morning Shift

Mathematics Physics Chemistry

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is differentiable on \( \mathbb{R} \), then \( 48(a + b) \) is equal to \( \_\_\_\_ \).