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\( \cos\left(\sin^{-1}\frac{3}{5}+\sin^{-1}\frac{5}{13}+\sin^{-1}\frac{33}{65}\right) \) is equal to:

Let \( f: R \rightarrow R \) be a function defined by \( f(x)=(2+3a)x^{2}+\left(\frac{a+2}{a-1}\right)x+b, \, a\neq 1 \). If \( f(x+y)=f(x)+f(y)+1-\frac{2}{7}xy \), then the value of \( 28\sum_{i=1}^{5} |f(i)| \) is:

Let for some function \( y=f(x) \), \( \int_{0}^{x} t f(t)\,dt = x^{2} f(x), \, x>0 \) and \( f(2)=3 \). Then \( f(6) \) is equal to:

The sum, of the squares of all the roots of the equation \( x^{2}+|2x-3|-4=0 \), is:

The number of different \( 5 \)-digit numbers greater than \( 50000 \) that can be formed using the digits \( 0, 1, 2, 3, 4, 5, 6, 7 \), such that the sum of their first and last digit should not be more than \( 8 \), is:

Let the equation of the circle, which touches \( x \)-axis at the point \( (a,0) \), \( a>0 \) and cuts off an intercept of length \( b \) on \( y \)-axis be \( x^{2}+y^{2}-\alpha x+\beta y+\gamma=0 \). If the circle lies below \( x \)-axis, the ordered pair \( (2a, b^{2}) \) is equal to:

Let ⟨ \( a_n \) ⟩ be a sequence such that \( a_0 = 0 \), \( a_1 = \frac{1}{2} \) and \( 2a_{n+2} = 5a_{n+1} - 3a_n \), \( n = 0, 1, 2, 3, \ldots \).
Then \( \displaystyle \sum_{k=1}^{100} a_k \) is equal to:

Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + i^{k_2} \) \( (i = \sqrt{-1}) \) is non-zero, equals:

The relation \( R = \{(x, y) : x, y \in \mathbb{Z} \ \text{and} \ x + y \ \text{is even}\} \) is:

Let \( A(x, y, z) \) be a point in \( xy \)-plane, which is equidistant from three points \( (0, 3, 2) \), \( (2, 0, 3) \) and \( (0, 0, 1) \). Let \( B = (1, 4, -1) \) and \( C = (2, 0, -2) \). Then among the statements: \( (S1) \): \( \triangle ABC \) is an isosceles right angled triangle, and \( (S2) \): the area of \( \triangle ABC \) is \( \dfrac{9\sqrt{2}}{2} \).

Let \( {}^nC_{r-1} = 28 \), \( {}^nC_r = 56 \) and \( {}^nC_{r+1} = 70 \).
Let \( A(4\cos t, 4\sin t) \), \( B(2\sin t, -2\cos t) \) and \( C(3r - n, r^2 - n - 1) \) be the vertices of a triangle \( ABC \), where \( r \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals:

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:

Let \( T_r \) be the \( r^{th} \) term A.P. If for some, \( m \), \( T_m = \frac{1}{25} \), \( T_{25} = \frac{1}{20} \) and \( 20 \displaystyle \sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle \sum_{r=m}^{2m} T_r \) is equal to:

Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If \( x \) denote the number of defective oranges, then the variance of \( x \) is:

Let \( ABCD \) be a trapezium whose vertices lie on the parabola \( y^2 = 4x \). Let the sides \( AD \) and \( BC \) of the trapezium be parallel to \( y \)-axis. If the diagonal \( AC \) is of length \( \frac{25}{4} \) and it passes through the point \( (1, 0) \), then the area of \( ABCD \) is:

If \( f(x) = \frac{2^x}{2^x + \sqrt{2}} , \ x \in \mathbb{R} \), then \( \displaystyle \sum_{k=1}^{81} f\!\left( \frac{k}{82} \right) \) is equal to:

If the image of the point \( (4, 4, 3) \) in the line \( \frac{x-1}{2} = \frac{y-2}{1} = \frac{z-1}{3} \) is \( (\alpha, \beta, \gamma) \), then \( \alpha + \beta + \gamma \) is equal to:

The area (in sq. units) of the region \( \{(x, y) : 0 \le y \le 2|x| + 1, \ 0 \le y \le x^2 + 1, \ |x| \le 3\} \) is:

The sum of all local minimum values of the function

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

where \([.]\) denotes greatest integer function. If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals ______.

If \( \alpha = 1 + \displaystyle \sum_{r=1}^{6}(-3)^{\,r-1} \, {}^{12}C_{2r-1}, \) then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}\, y + 1 = 0 \) is ______.

Let \( M \) denote the set of all real matrices of order \( 3 \times 3 \) and let \( S=\{-3,-2,-1,1,2\} \). Let \( S_{1}=\{A=[a_{ij}] \in M: A=A^{T} \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \), \( S_{2}=\{A=[a_{ij}] \in M: A-A^{T} \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \), \( S_{3}=\{A=[a_{ij}] \in M: a_{11}+a_{22}+a_{33}=0 \text{ and } a_{ij}\in S, \\ \forall \ i,j\} \). If \( n(S_{1}\cup S_{2}\cup S_{3}) = 125\,\alpha \), then \( \alpha \) equals ______.

Let \( \vec{a} = \hat{i} + \hat{j} + \hat{k} , \\ \vec{b} = 2\hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{d} = \vec{a} \times \vec{b} \). If \( \vec{c} \) is a vector such that \( \vec{a}\cdot\vec{c} = |\vec{c}| \), \( |\vec{c} - 2\vec{a}|^{2} = 8 \) and the angle between \( \vec{d} \) and \( \vec{c} \) is \( \frac{\pi}{4} \), then \( |10 - 3\vec{b}\cdot\vec{c}| + |\vec{d}\times\vec{c}|^{2} \) is equal to ______.

Let \( E_1 : \frac{x^2}{9} + \frac{y^2}{4} = 1 \) be an ellipse. Ellipses \( E_i \)'s are constructed such that their centres and eccentricities are same as that of \( E_1 \), and the length of minor axis of \( E_i \) is the length of major axis of \( E_{i+1} \) \( (i \geq 1) \). If \( A_i \) is the area of the ellipse \( E_i \), then \( \frac{5}{\pi}\left( \displaystyle \sum_{i=1}^{\infty} A_i \right) \) is equal to ______.

JEE Main 2025 28th Jan Morning Shift

Mathematics Physics Chemistry

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Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + i^{k_2} \) \( (i = \sqrt{-1}) \) is non-zero, equals: