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4bb09cfb-2ab1-444e-bea8-964f5d829710 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 1 | The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is:
(1) \( \sqrt{17} \)
(2) \( \sqrt{15} \)
(3) \( \sqrt{14} \)
(4) \( \sqrt{13} \) | 3 |
e7f78510-f04f-4edc-ae36-a36fa1560105 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 2 | Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is:
(1) 15
(2) 24
(3) 18
(4) 12 | 4 |
99dc6eaa-8767-4188-ba6c-259aa4554208 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 3 | Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \) Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below:
(1) Both Statement I and Statement II are false
(2) Statement I is true but Statement II is false
(3) Both Statement I and Statement II are true
(4) Statement I is false but Statement II is true | 2 |
9d8b65c8-f334-4258-9396-87e8b05c040f | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 4 | Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals:
(1) 48
(2) 55
(3) 62
(4) 47 | 2 |
fd4b9955-e499-42a7-b307-dc01f26961e7 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 5 | A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to:
(1) 22
(2) 21
(3) 23
(4) 24 | 3 |
c1114f31-44a0-417b-bb0d-4021c582e91c | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 6 | If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to:
(1) 21
(2) 9
(3) 14
(4) 6 | 2 |
baf77833-80a8-4e67-be37-d8b8e6ea1c76 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 7 | \( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to:
(1) \( \frac{2}{3} \sqrt{e} \)
(2) \( \frac{3e}{5} \)
(3) \( \frac{2e}{3} \)
(4) \( \frac{3e}{5} \) | 3 |
0e467008-3d2e-4985-a888-c4c4c61ef89a | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 8 | Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is:
(1) \( \sqrt{7} \)
(2) 14
(3) 3
(4) 7 | 4 |
1e9fd5df-4e15-472f-8b30-eb508b1cd5d0 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 9 | The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is:
(1) \( \frac{5}{3} \sqrt{15} \)
(2) \( \frac{1}{3} \sqrt{15} \)
(3) \( \frac{2}{3} \sqrt{15} \)
(4) \( \sqrt{15} \) | 3 |
f72094ad-9378-463a-8afe-101e4c1feb7c | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 10 | The system of equations \( x + 2y + 5z = 9 \), has no solution if:
(x) \( x + 5y + \lambda z = \mu \),
(1) \( \lambda = 15, \mu \neq 17 \)
(2) \( \lambda \neq 17, \mu = 18 \)
(3) \( \lambda = 17, \mu \neq 18 \)
(4) \( \lambda = 17, \mu = 18 \) | 3 |
8767ea4d-8340-48c8-855d-b38999732170 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 11 | Let the range of the function
\[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \]
be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is:
(1) 11
(2) 8
(3) 10
(4) 9 | 1 |
1c0b8f27-c697-4c95-8608-edc4eac9c44f | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 12 | Let \( x = x(y) \) be the solution of the differential equation
\[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then
\[ \cos(x(2)) \] is equal to:
(1) \( 1 - 2(\log_2 2)^2 \)
(2) \( 1 - 2(\log_2 2) \)
(3) \( 2(\log_2 2)^2 - 1 \)
(4) \( 2(\log_2 2)^2 - 1 \) | 4 |
b99eecd0-86cd-4151-9d7a-05d422815525 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 13 | 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 \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is:
(1) 196π
(2) 256π
(3) 225π
(4) 128π | 2 |
fc5d4114-d3a7-440c-ab06-28e802c7e40f | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 14 | The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is:
(1) 4
(2) 8
(3) 10
(4) 6 | 2 |
0c203c20-5dc4-4021-aaa0-252c32704f27 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 15 | Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that
\[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals:
(1) -1
(2) 2
(3) 1
(4) 0 | 1 |
2760158c-5cbb-4b11-b37f-f567861c194e | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 16 | If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then
\[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals:
(1) \( \frac{x^2}{12} \)
(2) \( \frac{x^2}{4} \)
(3) \( \frac{x^2}{16} \)
(4) \( \frac{x^2}{8} \) | 3 |
c2ee8f45-0547-4c65-a11b-0105f122e588 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 17 | A board has 16 squares as shown in the figure:
Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:
(1) \(7/10\)
(2) \(4/5\)
(3) \(23/30\)
(4) \(3/5\) | 2 |
d128d6b6-a0f1-4ff4-b83f-13c828d9b444 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 18 | Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is:
(1) \(x^2 + y^2 - 10x + 9 = 0\)
(2) \(x^2 + y^2 - 6x + 5 = 0\)
(3) \(x^2 + y^2 - 4x + 3 = 0\)
(4) \(x^2 + y^2 - 8x + 7 = 0\) | 2 |
4a48d53e-0787-43ac-aa0c-e9efba9739d7 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 19 | If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to:
(1) 18
(2) 13
(3) 8
(4) 20 | 2 |
31fb90ca-bd84-4ca9-9b71-f32add59351d | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 20 | If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is:
(1) 8
(2) 7
(3) 5
(4) 6 | 3 |
f3c8fe19-bd8c-4f37-9e11-2b5930e3c8c6 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 21 | The variance of the numbers 8, 21, 34, 47, \ldots, 320 is | 3 |
0fb44c34-5e78-4474-a9c9-44ee4d1a0246 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 22 | The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to | 474 |
c355123b-d38e-40f8-942f-38ad395686a8 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 23 | The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is | 17,280 |
9c014fa2-7f21-45b7-9d92-468f1b26fbdc | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 24 | The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to | 15 |
25c19494-d9a2-408c-ade3-38e0a55aa8f2 | JEE Main 2025 (23 Jan Shift 2) | Mathematics | 25 | Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to | 31 |
716dfedd-e03b-472b-b096-06aa266e154c | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 1 | Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to
(1) $3 + \sqrt{3}$
(2) $4$
(3) $4 - \sqrt{3}$
(4) $3$ | 2 |
8ea3fe25-f9be-4a1f-bda3-27f56bb14b73 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 2 | Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is:
(1) $17$
(2) $21$
(3) $56$
(4) $42$ | 2 |
2a299a1a-0c06-4a60-ac2f-6bdaa40b0fc8 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 3 | Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is
(1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$
(2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$
(3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$
(4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$ | 2 |
ba619dfa-a9be-4aef-8a40-71019abd7612 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 4 | If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to
(1) $57$
(2) $59$
(3) $55$
(4) $56$ | 1 |
d46a273c-4f2e-406e-8349-7ebd51134569 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 5 | For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is:
(1) $20$
(2) $10$
(3) $35$
(4) $70$ | 3 |
1df5bc52-fb7d-4f2b-85ce-28e0f668fd1e | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 6 | The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to
(1) $14$
(2) $21$
(3) $28$
(4) $7$ | 1 |
36122fa4-0d5f-4c76-bd83-d0725f7934a9 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 7 | Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is
(1) $5$
(2) $5\sqrt{5}$
(3) $5\sqrt{6}$
(4) $10$ | 2 |
9fb4b7b3-0406-4aa9-b10b-3911e5b9f686 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 8 | Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to
(1) $84$
(2) $113$
(3) $91$
(4) $101$ | 3 |
fe6d35b5-ea23-494d-b95f-7df08f41b8a3 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 9 | If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to : (1) 441
(2) 398
(3) 312
(4) 409 | 1 |
70403eb0-a5a1-4b96-aa65-89a15344632f | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 10 | For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is
(1) 9
(2) 5
(3) 7
(4) 4 | 3 |
c826c0a1-6d4c-48c3-8879-78bd54e24c1d | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 11 | The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to
(1) 7
(2) 5
(3) 24/5
(4) 20/3 | 4 |
0b48edf8-5c2a-437c-9ccd-a97230a90e3d | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 12 | Let \(S_n = \frac{1}{2} + \frac{1}{9} + \frac{1}{12} + \frac{1}{21} + \ldots \) upto \(n\) terms. If the sum of the first six terms of an A.P. with first term \(-p\) and common difference \(p\) is \(\sqrt{2025} S_{2025}\), then the absolute difference between 20th and 15th terms of the A.P. is
(1) 20
(2) 90
(3) 45
(4) 25 | 4 |
0eb94d01-6c3e-47e4-84b4-ea1767364fb9 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 13 | Let \(f : R \to \{0\} \to R\) be a function such that \(f(x) = 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}\). If the limit as \(x \to 0\) \(\left(x^{\frac{1}{x}} + f(x)\right) = \beta; \alpha, \beta \in R\), then \(\alpha + 2\beta\) is equal to
(1) 5
(2) 3
(3) 4
(4) 6 | 3 |
1eced83e-3ea5-474f-9a8a-033ba116a923 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 14 | If \(I(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} \, dx, m, n > 0\), then \(I(9, 14) + I(10, 13)\) is
(1) \(I(19, 27)\)
(2) \(I(9, 1)\)
(3) \(I(1, 13)\)
(4) \(I(9, 13)\) | 4 |
4a14b483-58b6-4bca-9bf7-426433b14c59 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 15 | \(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8, and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5. The probability, that \(A\) wins if \(A\) makes the first throw, is
(1) \(\frac{9}{17}\)
(2) \(\frac{9}{17}\)
(3) \(\frac{9}{17}\)
(4) \(\frac{8}{17}\) | 2 |
0c6f607d-54d7-466d-be1c-db64fc917a2e | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 16 | Let \(f(x) = \frac{2x^2 + 16}{2x^3 + 2x^2 + 4 + 32}\). Then the value of \(8 \left(f\left(\frac{1}{15}\right) + f\left(\frac{2}{15}\right) + \ldots + f\left(\frac{59}{15}\right)\right)\) is equal to
(1) 92
(2) 118
(3) 102
(4) 108 | 2 |
a857fdf6-b070-4f2d-a94c-7f592eeb3779 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 17 | Let \(y = y(x)\) be the solution of the differential equation \((xy - 5x^2 \sqrt{1 + x^2}) \, dx + (1 + x^2) \, dy = 0, y(0) = 0\). Then \(y(\sqrt{3})\) is equal to
(1) \(\sqrt{15} \div 2\)
(2) \(\frac{1}{2} \sqrt{\frac{3}{2}}\)
(3) \(2\sqrt{2}\)
(4) \(\sqrt{\frac{14}{3}}\) | 2 |
74a5b313-5de1-4b42-afac-e4b11c1d75e0 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 18 | \(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is: (1) 0
(2) \( \frac{1}{\sqrt{15}} \)
(3) \( \frac{1}{2\sqrt{5}} \)
(4) \( -\frac{1}{2\sqrt{5}} \) | 4 |
4eabea93-8e5f-433c-914f-3757e0201d82 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 19 | Consider the region \( R = \{ (x, y) : x \leq y \leq 9 - \frac{1}{11}x^2, x \geq 0 \} \). The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in \( R \), is:
(1) \( \frac{90}{11} \)
(2) \( \frac{85}{11} \)
(3) \( \frac{61}{12} \)
(4) \( \frac{567}{121} \) | 4 |
a093afec-18f3-4da4-ae18-f21d8f60edb8 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 20 | 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
(1) \( \sqrt{\frac{11}{6}} \)
(2) \( \frac{1}{3\sqrt{2}} \)
(3) \( 16 \)
(4) \( 18 \) | 1 |
aa607395-f954-4395-99bd-bf683bb6e0f6 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 21 | Let \( S = \{ p_1, p_2, \ldots, p_{10} \} \) be the set of first ten prime numbers. Let \( A = S \cup P \), where \( P \) is the set of all possible products of distinct elements of \( S \). Then the number of all ordered pairs \( (x, y), x \in S, y \in A \), such that \( x \) divides \( y \), is ______. | 5,120 |
f5866438-d51e-4568-989c-3d999deac59e | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 22 | If for some \( \alpha, \beta, \alpha \leq \beta, \alpha + \beta = 8 \) and \( \sec^{-2} (\tan^{-1} \alpha) + \cosec^{-2} (\cot^{-1} \beta) = 36 \), then \( \alpha^2 + \beta^2 \) is ______. | 14 |
d29e5e0c-48f5-4303-b50d-3a8ff8eb8710 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 23 | Let \( A \) be a \( 3 \times 3 \) matrix such that \( X^TAX = O \) for all nonzero \( 3 \times 1 \) matrices \( X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \). If
\[
A = \begin{bmatrix} 1 & 4 \\ 1 & -5 \\ 4 & 1 \end{bmatrix}, \quad A = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}
\]
and \( \det(\text{adj}(2(A + 1))) = 2^\alpha 3^\beta 5^\gamma \), \( \alpha, \beta, \gamma \in N \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is______. | 44 |
7fca30a7-2f14-4248-8bf0-1b687fba8e51 | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 24 | Let \( f \) be a differentiable function such that \( 2(x + 2)^2 f(x) - 3(x + 2)^2 = 10 \int_0^x (t + 2) f(t) dt, x \geq 0. \) Then \( f(2) \) is equal to ______. | 19 |
599bb4f3-4909-4595-9cd4-3822f2ceae1c | JEE Main 2025 (24 Jan Shift 1) | Mathematics | 25 | The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______. | 125 |
ef0e84d8-2585-4467-b206-30704260a218 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 1 | Group A consists of 7 boys and 3 girls, while group B consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group B, is equal to:
(1) 8750
(2) 9100
(3) 8925
(4) 8575 | 3 |
42221b8f-55a0-4bd7-a037-6e5173757fe9 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 2 | If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to:
$14x + 3y + \mu z = 33$
(1) 13
(2) 10
(3) 12
(4) 11 | 3 |
6d3d3618-c7da-4b3a-8383-66b5b182ab6b | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 3 | Let $A = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{2/\pi} |\sin x| + \log_{2/\pi} |\cos x| = 2 \right\}$ and $B = \{ x \geq 0 : \sqrt{\sqrt{x} - 4} - 3\sqrt{\sqrt{x} - 2} + 6 = 0 \}$. Then $n(A \cup B)$ is equal to:
(1) 4
(2) 8
(3) 6
(4) 2 | 2 |
c8897713-8999-444b-84d6-3a54ba0b823d | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 4 | The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is:
(1) $1 - \log_e 2$
(2) $\log_e 2$
(3) $1 + \log_e 2$
(4) $2 \log_e 2 - 1$ | 1 |
35b0bdd6-4e7e-4ed5-86e3-484752574845 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 5 | The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is:
$25x + 101y = 176$
(1) $48x + 25y = 169$
(2) $5x + 16y = 31$
(3) $4x + 122y = 134$
(4) $4x + 122y = 134$ | 1 |
f602ed0f-1b06-458b-8ca3-2bf6c12b4f42 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 6 | Let the points $(\frac{11}{2}, \alpha)$ lie on or inside the triangle with sides $x + y = 11$, $x + 2y = 16$ and $2x + 3y = 29$. Then the product of the smallest and the largest values of $\alpha$ is equal to:
(1) 44
(2) 22
(3) 33
(4) 55 | 3 |
9e392e8e-9769-4730-8c3c-be055a34abcb | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 7 | Let $f : (0, \infty) \rightarrow \mathbb{R}$ be a function which is differentiable at all points of its domain and satisfies the condition $x^2 f'(x) = 2x f(x) + 3$, with $f(1) = 4$. Then $2f(2)$ is equal to:
(1) 39
(2) 19
(3) 29
(4) 23 | 1 |
9ea00ba8-51c4-4d92-b9fa-4f615e9937b2 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 8 | If $7 = 5 \times 1 + \frac{1}{7} (5 + \alpha) + \frac{1}{7^2} (5 + 2\alpha) + \frac{1}{7^3} (5 + 3\alpha) + \cdots \infty$, then the value of $\alpha$ is:
(1) $\frac{6}{7}$
(2) 6
(3) $\frac{1}{7}$
(4) 1 | 2 |
9f7ee42f-a422-42c0-a0cd-ff4f14da835a | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 9 | Let $[x]$ denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function $f(x) = [x] + [x - 2]$, $-2 < x < 3$, is not continuous and not differentiable. Then $m + n$ is equal to:
(1) 6
(2) 8
(3) 9
(4) 7 | 2 |
ff241288-2eaf-4e9a-8c75-5e9c5bbf48ec | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 10 | Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is: (1) \( \frac{3}{16} \)
(2) \( \frac{5}{8} \)
(3) \( \frac{3}{8} \)
(4) \( \frac{1}{8} \) | 3 |
01a34c0e-d55e-41c9-881a-c5007a131d39 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 11 | Let the position vectors of three vertices of a triangle be \(4\mathbf{p} + \mathbf{q} - 3\mathbf{r}, -5\mathbf{p} + \mathbf{q} + 2\mathbf{r}\) and \(2\mathbf{p} - \mathbf{q} + 2\mathbf{r}\). If the position vectors of the orthocenter and the circumcenter of the triangle are \(\frac{5\mathbf{p} + 2\mathbf{q} + 3\mathbf{r}}{14}\) and \(\alpha\mathbf{p} + \beta\mathbf{q} + \gamma\mathbf{r}\) respectively, then \(\alpha + 2\beta + 5\gamma\) is equal to:
(1) 3
(2) 4
(3) 1
(4) 6 | 1 |
b8677182-5ae1-4aed-ba9a-f5b13c00f01a | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 12 | Let \(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}, \mathbf{b} = \mathbf{a} \times (\mathbf{i} - 2\mathbf{k})\) and \(\mathbf{c} = \mathbf{b} \times \mathbf{k}\). Then the projection of \(\mathbf{c} - 2\mathbf{j}\) on \(\mathbf{a}\) is:
(1) \(2\sqrt{14}\)
(2) \(\mathbf{v}\)
(3) \(\sqrt{7}\)
(4) \(2\sqrt{7}\) | 1 |
92212052-fd2a-4fdb-9567-d15d7e04b3e2 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 13 | The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is:
(1) 1
(2) 0
(3) 2
(4) 3 | 3 |
66422792-577f-48e3-8577-6ce01f4feeb0 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 14 | The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is:
(1) Neither one-one nor onto
(2) Onto but not one-one
(3) Both one-one and onto
(4) One-one but not onto | 4 |
8b3060d6-97c3-427e-802a-831bef7af864 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 15 | In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to:
(1) 525
(2) 510
(3) 515
(4) 505 | 3 |
37be8789-ed18-4591-910c-57c0774c29c8 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 16 | Suppose \(A\) and \(B\) are the coefficients of 30th and 12th terms respectively in the binomial expansion of \((1 + x)^{2n-1}\). If \(2A = 5B\), then \(n\) is equal to:
(1) 22
(2) 20
(3) 21
(4) 19 | 3 |
8999ad58-5192-4896-bbd5-131e2f36c4a7 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 17 | Let \((2, 3)\) be the largest open interval in which the function \(f(x) = 2\log_e(x - 2) - x^2 + ax + 1\) is strictly increasing and \((b, c)\) be the largest open interval, in which the function \(g(x) = (x - 1)^3(x + 2 - a)^2\) is strictly decreasing. Then \(100(a + b - c)\) is equal to:
(1) 420
(2) 360
(3) 160
(4) 280 | 2 |
5dc239df-f2e9-4f64-956d-7cf33138be50 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 18 | For some \(a, b\), let \(f(x) = \frac{a + \sin x}{x} \begin{vmatrix} 1 & 1 & b \\ a & 1 + \sin x & b \\ a & 1 & b + \sin x \end{vmatrix}, x \neq 0, \lim_{x \to 0} f(x) = \lambda + \mu a + \nu b\). Then \((\lambda + \mu + \nu)^2\) is equal to:
(1) 16
(2) 25
(3) 9
(4) 36 | 1 |
3df5204a-ffbf-4d19-980a-8e2fb65f7d87 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 19 | If the equation of the parabola with vertex \(V\left(\frac{3}{2}, 3\right)\) and the directrix \(x + 2y = 0\) is \(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\), then \(\alpha + \beta + \gamma\) is equal to: (1) 7
(2) 9
(3) 8
(4) 6
\(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\) | 2 |
fb786c03-50bb-45a1-8a07-5ba97cb76d37 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 20 | If $\alpha > \beta > \gamma > 0$, then the expression $\cot^{-1} \left\{ \beta + \frac{(1+\beta^2)}{(\alpha-\beta)} \right\} + \cot^{-1} \left\{ \gamma + \frac{(1+\gamma^2)}{(\beta-\gamma)} \right\} + \cot^{-1} \left\{ \alpha + \frac{(1+\alpha^2)}{(\gamma-\alpha)} \right\}$ is equal to:
(1) $\pi$
(2) 0
(3) $\pi - (\alpha + \beta + \gamma)$
(4) $3\pi$ | 1 |
9b4a5341-8d99-48ab-9a32-ddf854a5b148 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 21 | Let $P$ be the image of the point $Q(7, -2, 5)$ in the line $L : \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and $R(5, p, q)$ be a point on $L$. Then the square of the area of $\triangle PQR$ is ________. | 957 |
d4ee3e0f-52dd-4c48-802d-7ebfc9b8846e | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 22 | If $\int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = x\sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e |x + \frac{1}{2} + \sqrt{x^2 + x + 1}| + C$, where $C$ is the constant of integration, then $\alpha + 2\beta$ is equal to ________. | 16 |
46d3b2e8-b00d-42e1-ac03-81431e08fd4a | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 23 | Let $y = y(x)$ be the solution of the differential equation $2 \cos x \frac{dy}{dx} = \sin 2x - 4y \sin x$, $x \in (0, \frac{\pi}{2})$. If $y \left( \frac{\pi}{4} \right) = 0$, then $y' \left( \frac{\pi}{4} \right) + y \left( \frac{\pi}{4} \right)$ is equal to ________. | 1 |
e9b2988b-7ac7-4528-a0e5-fa70df795c0b | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 24 | Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to ________. | 392 |
984ecf93-7530-44c2-b9fc-40badc1010a5 | JEE Main 2025 (24 Jan Shift 2) | Mathematics | 25 | Let $H_1 : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $H_2 : -\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_1 = \sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25e_2^2$ is equal to ________. | 55 |
f0c34e35-de11-40b6-b844-4aceda9c9c64 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 1 | 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
(1) area of triangle ABO is \( \frac{11}{3} \)
(2) ABO is an obtuse angled isosceles triangle
(3) area of triangle ABO is \( \frac{11}{4} \)
(4) ABO is a scalene triangle | 2 |
386008a7-9f2d-417a-bc5e-9da6c7c9c48a | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 2 | Let \( f : \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = (2 + 3a)x^2 + \left( \frac{2a+7}{2} \right)x + b, a \neq 1. \) If \( f(x + y) = f(x) + f(y) + 1 - \frac{1}{2}xy \), then the value of \( 28 \sum_{i=1}^{5} |f(i)| \) is
(1) 545
(2) 715
(3) 735
(4) 675 | 4 |
3abb0ea3-69b6-4fe5-bc92-c24c1341f8de | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 3 | 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
(1) \( \frac{73}{8} \)
(2) \( \frac{25}{9} \)
(3) \( \frac{16}{8} \)
(4) \( \frac{75}{8} \) | 1 |
ce69f97f-67ad-4564-96b5-d19b006f6e1b | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 4 | The sum of all local minimum values of the function
\[
f(x) = \begin{cases}
1 - 2x, & x < -1 \\
\frac{1}{3}(7 + 2|x|), & -1 \leq x \leq 2 \\
\frac{1}{12}(x - 4)(x - 5), & x > 2
\end{cases}
\]
is
(1) \( \frac{137}{72} \)
(2) \( \frac{131}{72} \)
(3) \( \frac{137}{72} \)
(4) \( \frac{167}{72} \) | 1 |
7a54308e-bc39-4255-b8dc-4b50afb12022 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 5 | 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 \( t \) is a parameter. If \( (3x - 1)^2 + (3y)^2 = \alpha \), is the locus of the centroid of triangle \( ABC \), then \( \alpha \) equals
(1) 6
(2) 18
(3) 8
(4) 20 | 4 |
bf5f71d9-d0bb-4ce3-b1a3-bf2218c08673 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 6 | 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, then the ordered pair \((2a, b^2)\) is equal to
(1) \( (\gamma, \beta^2 - 4\alpha) \)
(2) \( (\alpha, \beta^2 + 4\gamma) \)
(3) \( (\gamma, \beta^2 + 4\alpha) \)
(4) \( (\alpha, \beta^2 - 4\gamma) \) | 4 |
fcb2d563-8ac4-452f-be89-ce259c8146c1 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 7 | If \( f(x) = \frac{x^2}{2x^2 + \sqrt{2}}, x \in \mathbb{R}, \) then \( \sum_{k=1}^{81} f \left( \frac{k}{82} \right) \) is equal to
(1) \( 1.81\sqrt{2} \)
(2) \( 41 \)
(3) \( 82 \)
(4) \( \frac{81}{2} \) | 4 |
d323f007-a281-4f15-8c52-e3b78f1b9fb8 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 8 | Two number \( k_1 \) and \( k_2 \) are randomly chosen from the set of natural numbers. Then, the probability that the value of \( i^{k_1} + j^{k_2}, (i = \sqrt{-1}) \) is non-zero, equals:(1) \( \frac{1}{2} \)
(2) \( \frac{3}{4} \)
(3) \( \frac{1}{7} \)
(4) \( \frac{2}{3} \) | 2 |
2400477c-a743-4733-95d2-a4f76884f8f5 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 9 | If the image of the point \((4, 4, 3)\) in the line \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{3}\) is \((\alpha, \beta, \gamma)\), then \(\alpha + \beta + \gamma\) is equal to
\[(1)\ 9 \quad (2)\ 12 \quad (3)\ 7 \quad (4)\ 8.
\] | 1 |
13d54bd0-0372-4112-a983-0f35933d161b | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 10 | \(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to:
\[(1)\ 1 \quad (2)\ 0 \quad (3)\ \frac{32}{65} \quad (4)\ \frac{33}{65}.
\] | 2 |
669b1282-46df-4444-b097-abc0e94d1c6f | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 11 | 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 \(\frac{9\sqrt{2}}{2}\),
\[(1)\ \text{both are true} \quad (2)\ \text{only (S2) is true} \quad (3)\ \text{only (S1) is true} \quad (4)\ \text{both are false}.
\] | 3 |
e8e525f0-719e-4737-b196-ec4c9442a18e | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 12 | The area (in sq. units) of the region \(\{ (x, y) : 0 \leq y \leq 2|x| + 1, 0 \leq y \leq x^2 + 1, |x| \leq 3 \}\) is
\[(1)\ \frac{80}{3} \quad (2)\ \frac{44}{3} \quad (3)\ \frac{32}{3} \quad (4)\ \frac{17}{3}.
\] | 2 |
3c8d4be8-c367-4856-bdc1-bfe37e60d677 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 13 | The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is
\[(1)\ 3(3 - \sqrt{2}) \quad (2)\ 6(3 - \sqrt{2}) \quad (3)\ 6(2 - \sqrt{2}) \quad (4)\ 3(2 - \sqrt{2})
\] | 3 |
577660f5-d79b-4d00-9af8-d50b7849743f | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 14 | Let \(T_r\) be the \(r^{th}\) term of an A.P. If for some \(m, T_m = \frac{1}{25}, T_{25} = \frac{1}{25}\), and \(20 \sum_{r=1}^{25} T_r = 13\), then
\(5m \sum_{r=m}^{m+2} T_r\) is equal to
\[(1)\ 98 \quad (2)\ 126 \quad (3)\ 142 \quad (4)\ 112.
\] | 2 |
d688540d-d072-44a1-b481-a459af38593f | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 15 | 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
\[(1)\ \frac{28}{75} \quad (2)\ \frac{18}{25} \quad (3)\ \frac{26}{75} \quad (4)\ \frac{14}{25}.
\] | 1 |
403a27b0-6d1e-49c4-924d-8722f6a2915f | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 16 | Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f(2) = 3\). Then \(f(6)\) is equal to
\[(1)\ 1 \quad (2)\ 3 \quad (3)\ 6 \quad (4)\ 2.
\] | 1 |
f15c71bb-d3e0-42f2-b7da-a93f795013ef | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 17 | If \(\int \frac{9x^2 \cos \pi x}{(1 + x^2)^2} dx = \pi (\alpha x^2 + \beta), \alpha, \beta \in \mathbb{Z}\), then \((\alpha + \beta)^2\) equals
\[(1)\ 64 \quad (2)\ 196 \quad (3)\ 144 \quad (4)\ 100.
\] | 4 |
bec8e309-4e7f-4767-9f29-a2c36dac2786 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 18 | 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 \(\sum_{k=1}^{100} a_k\) is equal to : (1) \( 3a_{99} - 100 \)
(2) \( 3a_{100} + 100 \)
(3) \( 3a_{99} + 100 \)
(4) \( 3a_{100} - 100 \)
| 2 |
4dfa6e24-a0d7-4203-b672-58d09c63870b | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 19 | 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 digits should not be more than 8, is
(1) 4608
(2) 5720
(3) 5719
(4) 4607 | 4 |
01291000-d3a5-41f5-aac9-1a90237dadf5 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 20 | The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is:
(1) reflexive and symmetric but not transitive
(2) an equivalence relation
(3) symmetric and transitive but not reflexive
(4) reflexive and transitive but not symmetric | 2 |
5df8a224-1e66-4ab3-a733-446cb3d5df54 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 21 | Let \( f(x) = \begin{cases} 3x, & x < 0 \\ \min\{1 + x + |x|, x + 2|x|\}, & 0 \leq x \leq 2 \text{ where } [.] \text{ denotes greatest integer function} \end{cases} \). If \( \alpha \) and \( \beta \) are the number of points, where \( f \) is not continuous and is not differentiable, respectively, then \( \alpha + \beta \) equals
\( \frac{5}{2} \) | 5 |
70401e79-9f72-4a97-9c2d-6ff0463d1a59 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 22 | 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
\( \frac{5}{2} \) | 1,613 |
4141f89c-4f78-485e-a9bc-6f3ab89dc31c | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 23 | If \( \alpha = 1 + \sum_{r=1}^{6} (-3)^{r-1} 12C_{2r-1} \), then the distance of the point \( (12, \sqrt{3}) \) from the line \( \alpha x - \sqrt{3}y + 1 = 0 \) is
\( \frac{5}{2} \) | 5 |
70575064-0b51-4839-8155-1945b70779e0 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 24 | Let \( E_1 : \frac{x^2}{3} + \frac{y^2}{4} = 1 \) 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 \( \frac{5}{\pi} \left( \sum_{i=1}^{\infty} A_i \right) \) is equal to
\( \frac{5}{\pi} \) | 54 |
25f7b0a3-85b7-4842-bc13-f36f8a07d9a2 | JEE Main 2025 (28 Jan Shift 1) | Mathematics | 25 | 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
\( \frac{5}{2} \) | 6 |
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