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6a784d07-5204-4c1c-878d-1b5058eb9c9c | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 1 | Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) and \( P = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}, \theta \geq 0. \) If \( B = PAP^T, C = P^TB^TP \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is:
(1) 127
(2) 258
(3) 65
(4) 2049 | 3 |
f165565c-7d21-4031-8cec-9f41449f2a39 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 2 | If the components of \( \vec{a} = \alpha \hat{i} + \beta \hat{j} + \gamma \hat{k} \) along and perpendicular to \( \vec{b} = 3\hat{i} + \hat{j} - \hat{k} \) respectively, are \( \frac{16}{10}(3\hat{i} + \hat{j} - \hat{k}) \) and \( \frac{1}{10}(-4\hat{i} - 5\hat{j} - 17\hat{k}) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:
(1) 26
(2) 18
(3) 23
(4) 16 | 1 |
843fe6c2-b6ac-477f-b561-a71dd5df3469 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 3 | Let \( A, B, C \) be three points in \( xy \)-plane, whose position vectors are given by \( \sqrt{3}\hat{i} + \hat{j} + \sqrt{3}\hat{j} \) and \( \hat{i} + (1 - a)\hat{j} \) respectively with respect to the origin \( O \). If the distance of the point \( C \) from the line bisecting the angle between the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) is \( \frac{a}{\sqrt{2}} \), then the sum of all the possible values of \( a \) is:
(1) 2
(2) 9/2
(3) 1
(4) 0 | 3 |
3f46fd12-9c19-49e9-aebd-3fc5ecc67139 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 4 | Let the coefficients of three consecutive terms \( T_r, T_{r+1}, \) and \( T_{r+2} \) in the binomial expansion of \( (a + b)^{\frac{1}{2}} \) be in a G.P. and let \( p \) be the number of all possible values of \( r \). Let \( q \) be the sum of all rational terms in the binomial expansion of \( (\sqrt{3} + \sqrt{4})^{\frac{1}{2}} \). Then \( p + q \) is equal to:
(1) 283
(2) 287
(3) 295
(4) 299 | 1 |
56cce03c-cae5-4480-ae4e-2ca17508e9aa | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 5 | Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is:
(1) \( (-\infty, -1] \cup [0, \infty) \)
(2) \( (-\infty, -1] \cup [1, \infty) \)
(3) \( (-\infty, \infty) \)
(4) \( (-\infty, \infty) \) \( \setminus \{0\} \) | 3 |
2470b736-8780-4e71-8c6a-ca9e0fcd4a1f | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 6 | Let \( S \) be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set \( S \), one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:
(1) \( \frac{1}{2} \)
(2) \( \frac{1}{4} \)
(3) \( \frac{3}{5} \)
(4) \( \frac{1}{5} \) | 1 |
2459077d-12c1-45e7-83ae-c1d6ea3ab06e | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 7 | If \( \sum_{r=1}^{15} \frac{1}{\sin\left(\frac{r}{2} + \frac{1}{2}\right) \sin\left(\frac{r}{2} + \frac{3}{2}\right)} \) = \( a\sqrt{3} + b \), \( a, b \in \mathbb{Z} \), then \( a^2 + b^2 \) is equal to:
(1) 10
(2) 4
(3) 2
(4) 8 | 4 |
948355f4-6765-49f3-86c3-51028dee123e | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 8 | Let \( f \) be a real valued continuous function defined on the positive real axis such that \( g(x) = \int_0^x t f(t) \, dt \). If \( g(x^2) = x^6 + x^7 \), then value of \( \sum_{r=1}^{15} f(x^3) \) is:
(1) 270
(2) 340
(3) 320
(4) 310 | 4 |
98f67cfb-1e05-4e92-bb4c-49fdebf46689 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 9 | Let \( f : [0, 3] \rightarrow A \) be defined by \( f(x) = 2x^3 - 15x^2 + 36x + 7 \) and \( g : [0, \infty) \rightarrow B \) be defined by \( g(x) = \frac{x^{2025}}{x^{2025} + 1} \). If both the functions are onto and \( S = \{ x \in \mathbb{Z} : x \in A \text{ or } x \in B \} \), then \( n(S) \) is equal to:(1) 29
(2) 30
(3) 31
(4) 36 | 2 |
7af307a7-d2ac-4ef8-b48f-a2e6059c95d4 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 10 | Bag $B_1$ contains 6 white and 4 blue balls, Bag $B_2$ contains 4 white and 6 blue balls, and Bag $B_3$ contains 5 white and 5 blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability, that the ball is drawn from Bag $B_2$, is:
(1) $\frac{4}{15}$
(2) $\frac{1}{3}$
(3) $\frac{2}{5}$
(4) $\frac{4}{5}$ | 1 |
8b5b6bb0-7a08-49a4-af8b-3fea283b05bd | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 11 | Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(2) = 1$. If $F(x) = xf(x)$ for all $x \in \mathbb{R}$, $\int_{x}^{2} x F'(x)\,dx = 6$ and $\int_{x}^{2} x^2 F''(x)\,dx = 40$, then $F'(2) + \int_{x}^{2} F(x)\,dx$ is equal to:
(1) 11
(2) 13
(3) 15
(4) 9 | 1 |
33f5a9c1-f95c-4b38-b86d-9636f14834e2 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 12 | For positive integers $n$, if $4a_n = (n^2 + 5n + 6)$ and $S_n = \sum_{k=1}^{n} \left( \frac{1}{ak} \right)$, then the value of $507S_{2025}$ is:
(1) 540
(2) 675
(3) 1350
(4) 135 | 2 |
6a8fac3c-20ca-4da7-be4b-00fdf3d812a1 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 13 | Let $f : \mathbb{R} \setminus \{0\} \to (-\infty, 1)$ be a polynomial of degree 2, satisfying $f(x) f\left( \frac{1}{x} \right) = f(x) + f\left( \frac{1}{x} \right)$. If $f(K) = -2K$, then the sum of squares of all possible values of $K$ is:
(1) 7
(2) 6
(3) 1
(4) 9 | 2 |
a1f3abd3-97f9-4a96-8965-6bddf9ac8b33 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 14 | If $A$ and $B$ are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{y^2} - \frac{y^2}{x^2} = 1$ and a point $P$ moves on the line $2x - 3y + 4 = 0$, then the centroid of $\triangle PAB$ lies on the line:
(1) $x + 9y = 36$
(2) $4x - 9y = 12$
(3) $6x - 9y = 20$
(4) $9x - 9y = 32$ | 3 |
b9a37e33-f873-4d50-b919-9864d6fcfec7 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 15 | If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f(0) = -6$, then $f(1)$ is equal to:
(1) $4 \log_e 2 - 2$
(2) $2 - \log_e x$
(3) $\log_e 2 + 2$
(4) $4 \log_e 2 + 2$ | 1 |
3b235e61-18ba-47aa-a627-bf5a37bdb464 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 16 | The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is:
(1) $2 \left( \frac{\pi}{2} - \frac{1}{3} \right)$
(2) $\frac{\pi}{2} - \frac{1}{3}$
(3) $\frac{\pi}{2} - \frac{1}{3}$
(4) $\frac{1}{3} \left( \frac{\pi}{2} - \frac{1}{3} \right)$ | 2 |
a94bc062-64c8-4846-a7f8-6ceb620ae4ee | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 17 | The square of the distance of the point $\left( \frac{15}{7}, \frac{22}{7}, 7 \right)$ from the line $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ in the direction of the vector $\hat{i} + 4\hat{j} + 7\hat{k}$ is:
(1) 54
(2) 44
(3) 41
(4) 66 | 4 |
47c8b9a2-16cc-4302-be4c-ae7c30ea8d6c | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 18 | If the midpoint of a chord of the ellipse $\frac{x^2}{9} + \frac{y^2}{4} = 1$ is $(\sqrt{2}, 4/3)$, and the length of the chord is $\frac{2\sqrt{5}}{3}$, then $\alpha$ is:
(1) 20
(2) 22
(3) 18
(4) 26 | 2 |
911e6993-1f84-423b-9702-25e971cbe392 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 19 | If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\alpha\gamma + \beta\delta$ is equal to:
(1) $-2$
(2) $6$
(3) $-6$
(4) $2$ | 4 |
1f39939e-71c6-4a74-8bea-f6c2d235d37d | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 20 | Two equal sides of an isosceles triangle are along $-x + 2y = 4$ and $x + y = 4$. If $m$ is the slope of its third side, then the sum, of all possible distinct values of $m$, is:
(1) $-2\sqrt{10}$
(2) $12$
(3) $6$
(4) $-6$ | 3 |
9e5b117e-fb2e-45ad-8e64-5c166daba7ad | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 21 | Let $A$ and $B$ be the two points of intersection of the line $y = 5 = 0$ and the mirror image of the parabola $y^2 = 4x$ with respect to the line $x + y + 4 = 0$. If $d$ denotes the distance between $A$ and $B$, and $a$ denotes the area of $\triangle SAB$, where $S$ is the focus of the parabola $y^2 = 4x$, then the value of $(a + d)$ is | 14 |
1957dd69-b21f-4609-ac4b-d610195a1745 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 22 | The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is | 64 |
7e9ca87b-0353-425d-b767-83cf4cff50f1 | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 23 | If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y(2) = \frac{x^2 - 8}{4}$, then $y(0)$ is equal to | 4 |
e8d84166-bcf8-4796-855c-8b642125ce9b | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 24 | The interior angles of a polygon with $n$ sides, are in an A.P. with common difference $6^\circ$. If the largest interior angle of the polygon is $219^\circ$, then $n$ is equal to | 20 |
45a8920f-53e0-4308-ba55-52e6797819cf | JEE Main 2025 (28 Jan Shift 2) | Mathematics | 25 | Let $f(x) = \lim_{x \to \infty} \sum_{r=0}^{n} \left(\frac{\tan(x/2^{r+1}) + \tan^2(x/2^{r+1})}{1 - \tan^2(x/2^{r+1})}\right)$. Then $\lim_{x \to 0} \frac{x - e^{-f(x)}}{x - f(x)}$ is equal to | 1 |
87886fd7-10e5-4ed6-acb0-e2f288266f84 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 1 | Let \( x_1, x_2, \ldots, x_{10} \) be ten observations such that \( \sum_{i=1}^{10} (x_i - 2) = 30, \) \( \sum_{i=1}^{10} (x_i - \beta)^2 = 98, \beta > 2, \) and their variance is \( \frac{4}{5}. \) If \( \mu \) and \( \sigma^2 \) are respectively the mean and the variance of \( 2(x_1 - 1) + 4\beta, \) \( 2(x_2 - 1) + 4\beta, \ldots, 2(x_{10} - 1) + 4\beta, \) then \( \frac{\partial \mu}{\partial \beta} \) is equal to:
(1) 100 (2) 120 (3) 110 (4) 90 | 1 |
4a3af956-2936-4814-a7c5-ad78d38791b4 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 2 | Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its 11th term is:
(1) 90 (2) 84 (3) 122 (4) 108 | 1 |
842577b9-21fa-44e9-8071-82c1a6bb6254 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 3 | The number of solutions of the equation \( \left( \frac{9}{\sqrt{x}} - \frac{9}{\sqrt{x}} + 2 \right) \left( \frac{2}{\sqrt{x}} - \frac{7}{\sqrt{x}} + 3 \right) = 0 \) is:
(1) 2 (2) 3 (3) 1 (4) 4 | 4 |
22a47813-1150-4bfb-8c1e-7779eb2311f2 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 4 | Define a relation \( R \) on the interval \( [0, \frac{\pi}{4}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1. \) Then \( R \) is:
(1) both reflexive and transitive but not symmetric (2) an equivalence relation
(3) reflexive but neither symmetric nor transitive (4) both reflexive and symmetric but not transitive | 2 |
9abb4939-e62b-4c95-b6ae-baa8b0f237b8 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 5 | Two parabolas have the same focus \( (4, 3) \) and their directrices are the \( x \)-axis and the \( y \)-axis, respectively. If these parabolas intersect at the points \( A \) and \( B, \) then \( (AB)^2 \) is equal to:
(1) 392 (2) 384 (3) 192 (4) 96 | 3 |
d70df5b1-fbd0-4589-b004-9693965e1ff7 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 6 | Let \( P \) be the set of seven digit numbers with sum of their digits equal to 11. If the numbers in \( P \) are formed by using the digits 1, 2 and 3 only, then the number of elements in the set \( P \) is:
(1) 173 (2) 164 (3) 158 (4) 161 | 4 |
421fc75d-2e8a-492d-804b-2cb033503d1a | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 7 | Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + 7\hat{j} + 3\hat{k}. \) Let \( L_1 : \vec{r} = (-\hat{i} + 2\hat{j} + \hat{k}) + \lambda \vec{a}, \lambda \in \mathbb{R} \) and \( L_2 : \vec{r} = (\hat{i} + \hat{k}) + \mu \vec{b}, \mu \in \mathbb{R} \) be two lines. If the line \( L_3 \) passes through the point of intersection of \( L_1 \) and \( L_2, \) and is parallel to \( \vec{a} + \vec{b}, \) then \( L_3 \) passes through the point:
(1) (5, 17, 4) (2) (2, 8, 5)
(3) (8, 26, 12) (4) (-1, -1, 1) | 3 |
b858d9a3-549d-424b-a0f5-3f33c1789778 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 8 | Let \( \vec{a} = 2\hat{i} - \hat{j} + 3\hat{k}, \vec{b} = 3\hat{i} - 5\hat{j} + \hat{k} \) and \( \vec{c} \) be a vector such that \( \vec{a} \times \vec{c} = \vec{c} \times \vec{b} \) and \( (\vec{a} + \vec{c}) \cdot (\vec{b} + \vec{c}) = 168. \) Then the maximum value of \( |\vec{c}|^2 \) is:
(1) 462 (2) 77 (3) 154 (4) 308 | 4 |
a7ad8cd4-2aac-4155-9246-182dfb07b87d | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 9 | The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to:
(1) 3 \log_e 4 (2) 4 \log_e 3
(3) 6 \log_e 4 (4) 2 \log_e 3 | 2 |
9f3f478f-3e56-42e5-9b3e-6ad35534d3e1 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 10 | Let the ellipse \( E_1 : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( E_2 : \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1, A < B \) have same eccentricity \( \frac{1}{\sqrt{3}} \). Let the product of their lengths of latus rectums be \( \frac{32}{\sqrt{3}} \), and the distance between the foci of \( E_1 \) be 4. If \( E_1 \) and \( E_2 \) meet at \( A, B, C \) and \( D \), then the area of the quadrilateral \( ABCD \) equals:
\[
\begin{align*}
(1) & \quad 4\sqrt{6} \\
(2) & \quad 6\sqrt{6} \\
(3) & \quad 18\sqrt{6}/5 \\
(4) & \quad 24\sqrt{6}/5 \\
\end{align*}
\] | 4 |
b1f11888-ceb0-442f-810a-0a51a834e287 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 11 | Let \( A = [a_{ij}] = \begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix} \). If \( A_{ij} \) is the cofactor of \( a_{ij} \), \( C_{ij} = \sum_{k=1}^{2} a_{ik}A_{jk}, 1 \leq i, j \leq 2 \), and \( C = [C_{ij}] \), then \( 8|C| \) is equal to:
\[
\begin{align*}
(1) & \quad 288 \\
(2) & \quad 222 \\
(3) & \quad 242 \\
(4) & \quad 262 \\
\end{align*}
\] | 3 |
3dd9de4c-0aa8-46af-be7b-f79cd0ee13e2 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 12 | Let \( |z_1 - 8 - 2i| \leq 1 \) and \( |z_2 - 2 + 6i| \leq 2, z_1, z_2 \in \mathbb{C} \). Then the minimum value of \( |z_1 - z_2| \) is:
\[
\begin{align*}
(1) & \quad 13 \\
(2) & \quad 10 \\
(3) & \quad 3 \\
(4) & \quad 7 \\
\end{align*}
\] | 4 |
87d55516-470a-4c57-9713-3f320ed53247 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 13 | Let \( L_1 : \frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \) and \( L_2 : \frac{x+1}{2} = \frac{y-2}{3} = \frac{z}{1} \) 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 \( L_1 \), then \( |5\alpha - 11\beta - 8\gamma| \) equals:
\[
\begin{align*}
(1) & \quad 20 \\
(2) & \quad 18 \\
(3) & \quad 25 \\
(4) & \quad 16 \\
\end{align*}
\] | 3 |
f131eebf-78ee-4273-b473-b776fb3280b4 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 14 | Let \( M \) and \( m \) respectively be the maximum and the minimum values of
\[
f(x) = \begin{bmatrix} 1 + \sin^2 x & \cos^2 x & 4\sin 4x \\ \sin^2 x & 1 + \cos^2 x & 4\sin 4x \\ \sin^2 x & \cos^2 x & 1 + 4\sin 4x \end{bmatrix}, x \in \mathbb{R}
\]
Then \( M^4 - m^4 \) is equal to:
\[
\begin{align*}
(1) & \quad 1280 \\
(2) & \quad 1295 \\
(3) & \quad 1215 \\
(4) & \quad 1040 \\
\end{align*}
\] | 1 |
cbc72177-b566-4494-87df-ab52acb84bfc | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 15 | Let \( ABC \) be a triangle formed by the lines \( 7x - 6y + 3 = 0, x + 2y - 31 = 0 \) and \( 9x - 2y - 19 = 0 \). Let the point \((h, k)\) be the image of the centroid of \( \Delta ABC \) in the line \( 3x + 6y - 53 = 0 \). Then \( h^2 + k^2 + hk \) is equal to:
\[
\begin{align*}
(1) & \quad 47 \\
(2) & \quad 36 \\
(3) & \quad 47 \\
(4) & \quad 40 \\
\end{align*}
\] | 2 |
68edd102-6049-4277-a9d4-e831373e20f9 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 16 | The value of \( \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^4 + 4k^2 + 11k + 5}{(k+3)!} \right) \) is:
\[
\begin{align*}
(1) & \quad 4/3 \\
(2) & \quad 2 \\
(3) & \quad 7/3 \\
(4) & \quad 5/3 \\
\end{align*}
\] | 4 |
284e0696-65cb-4558-a145-131420581840 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 17 | The least value of \( n \) for which the number of integral terms in the Binomial expansion of \( (\sqrt{7} + \sqrt{11})^n \) is 183, is:
\[
\begin{align*}
(1) & \quad 2184 \\
(2) & \quad 2196 \\
(3) & \quad 2148 \\
(4) & \quad 2172 \\
\end{align*}
\] | 1 |
7547e409-5d97-4179-9041-5f85e5a66c5e | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 18 | Let \( y = y(x) \) be the solution of the differential equation
\[
\cos x \left( \log_e (\cos x) \right)^2 dy + (\sin x - 3y \sin x \log_e (\cos x)) dx = 0, \quad x \in \left( 0, \frac{\pi}{2} \right).\]
If \( y \left( \frac{\pi}{6} \right) = \frac{-1}{\log_2 2} \), then \( y \left( \frac{\pi}{8} \right) \) is equal to:
\[
\begin{align*}
(1) & \frac{1}{\log_2 (5) - \log_2 (4)} \\
(2) & \frac{2}{\log_2 (3) - \log_2 (4)} \\
(3) & \frac{1}{\log_2 (4) - \log_2 (3)} \\
(4) & \frac{1}{\log_2 (4)}
\end{align*}
\] | 1 |
d87ae9e7-3636-4854-a874-65795666e575 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 19 | Let the line \( x + y = 1 \) meet the circle \( x^2 + y^2 = 4 \) at the points A and B. If the line perpendicular to AB and passing through the mid point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ADBC is equal to:
\[
\begin{align*}
(1) & \sqrt{14} \\
(2) & 3\sqrt{7} \\
(3) & 2\sqrt{14} \\
(4) & 5\sqrt{7}
\end{align*}
\] | 3 |
dd7bf750-08ca-4eee-8b1b-3aa7880e708f | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 20 | Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \quad y \geq |x|, \quad |y| \leq |x - 1|\} \) be A. Then \( 6A \) is equal to:
\[
\begin{align*}
(1) & 16 \\
(2) & 12 \\
(3) & 14 \\
(4) & 18
\end{align*}
\] | 3 |
1c2e865d-ac5b-42e3-bca5-9a6f1c08d311 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 21 | Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1}(2x + 1) \} \). Then \( \sum_{x \in S} (2x - 1)^2 \) is equal to ______. | 5 |
162f99d2-c3c1-40b7-9898-27bf925f81b1 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 22 | Let \( f : (0, \infty) \rightarrow \mathbb{R} \) be a twice differentiable function. If for some \( a \neq 0, \int_0^1 f(\lambda x) d\lambda = a f(x) \), \( f(1) = 1 \) and \( f(16) = \frac{1}{8} \), then \( 16 - f' \left( \frac{1}{16} \right) \) is equal to ______. | 112 |
0b67c9d8-fb39-44a3-b259-978ebd5b5085 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 23 | The number of 6-letter words, with or without meaning, that can be formed using the letters of the word MATHS such that any letter that appears in the word must appear at least twice, is ______. | 1,405 |
92a4b8da-f6bb-4de1-b0cb-e42c14b3dcd6 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 24 | Let \( S = \{ m \in \mathbb{Z} : A^m + A^m = 3I - A^{-6} \} \), where \( A = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix} \). Then \( n(S) \) is equal to ______. | 2 |
a9460d26-8af6-4c4c-9ce8-487a3c0312f2 | JEE Main 2025 (29 Jan Shift 1) | Mathematics | 25 | Let \([t]\) be the greatest integer less than or equal to \( t \). Then the least value of \( p \in \mathbb{N} \) for which
\[
\lim_{x \to \infty} \left( x \left( \left\lfloor \frac{1}{x} \right\rfloor + \left\lfloor \frac{2}{x} \right\rfloor + \cdots + \left\lfloor \frac{p}{x} \right\rfloor \right) - x^2 \left( \left\lfloor \frac{1}{x^2} \right\rfloor + \left\lfloor \frac{2}{x^2} \right\rfloor + \cdots + \left\lfloor \frac{p}{x^2} \right\rfloor \right) \right) \geq 1
\]
is equal to ______. | 24 |
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