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6a784d07-5204-4c1c-878d-1b5058eb9c9c | maths | 3d-geometry | sequences-and-series | 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: | [{"identifier": "A", "content": "127"}, {"identifier": "B", "content": "258"}, {"identifier": "C", "content": "65"}, {"identifier": "D", "content": "2049"}] | ["C"] | null | null |
f165565c-7d21-4031-8cec-9f41449f2a39 | maths | 3d-geometry | differential-equations | 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: | [{"identifier": "A", "content": "26"}, {"identifier": "B", "content": "18"}, {"identifier": "C", "content": "23"}, {"identifier": "D", "content": "16"}] | ["A"] | null | null |
843fe6c2-b6ac-477f-b561-a71dd5df3469 | maths | 3d-geometry | probability | 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: | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "9/2"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "0"}] | ["C"] | null | null |
3f46fd12-9c19-49e9-aebd-3fc5ecc67139 | maths | 3d-geometry | exponential-and-logarithm | 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: | [{"identifier": "A", "content": "283"}, {"identifier": "B", "content": "287"}, {"identifier": "C", "content": "295"}, {"identifier": "D", "content": "299"}] | ["A"] | null | null |
56cce03c-cae5-4480-ae4e-2ca17508e9aa | maths | 3d-geometry | coordinate-geometry | Let \( [x] \) denote the greatest integer less than or equal to \( x \). Then the domain of \( f(x) = \sec^{-1}(2[x] + 1) \) is: | [{"identifier": "A", "content": "\\( (-\\infty, -1] \\cup [0, \\infty) \\)"}, {"identifier": "B", "content": "\\( (-\\infty, -1] \\cup [1, \\infty) \\)"}, {"identifier": "C", "content": "\\( (-\\infty, \\infty) \\)"}, {"identifier": "D", "content": "\\( (-\\infty, \\infty) \\) \\( \\setminus \\{0\\} \\)"}] | ["C"] | null | null |
2470b736-8780-4e71-8c6a-ca9e0fcd4a1f | maths | 3d-geometry | calculus-integration | 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: | [{"identifier": "A", "content": "\\( \\frac{1}{2} \\)"}, {"identifier": "B", "content": "\\( \\frac{1}{4} \\)"}, {"identifier": "C", "content": "\\( \\frac{3}{5} \\)"}, {"identifier": "D", "content": "\\( \\frac{1}{5} \\)"}] | ["A"] | null | null |
2459077d-12c1-45e7-83ae-c1d6ea3ab06e | maths | 3d-geometry | conic-sections | 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: | [{"identifier": "A", "content": "10"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "8"}] | ["D"] | null | null |
948355f4-6765-49f3-86c3-51028dee123e | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "270"}, {"identifier": "B", "content": "340"}, {"identifier": "C", "content": "320"}, {"identifier": "D", "content": "310"}] | ["D"] | null | null |
98f67cfb-1e05-4e92-bb4c-49fdebf46689 | maths | 3d-geometry | jee-mathematics | 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: | [] | ["B"] | null | null |
7af307a7-d2ac-4ef8-b48f-a2e6059c95d4 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "$\\frac{4}{15}$"}, {"identifier": "B", "content": "$\\frac{1}{3}$"}, {"identifier": "C", "content": "$\\frac{2}{5}$"}, {"identifier": "D", "content": "$\\frac{4}{5}$"}] | ["A"] | null | null |
8b5b6bb0-7a08-49a4-af8b-3fea283b05bd | maths | 3d-geometry | jee-mathematics | Let $f : \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f | [{"identifier": "B", "content": "= 1$. 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 $F'"}, {"identifier": "B", "content": "+ \\int_{x}^{2} F(x)\\,dx$ is equal to:"}, {"identifier": "A", "content": "11"}, {"identifier": "B", "content": "13"}, {"identifier": "C", "content": "15"}, {"identifier": "D", "content": "9"}] | ["A"] | null | null |
33f5a9c1-f95c-4b38-b86d-9636f14834e2 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "540"}, {"identifier": "B", "content": "675"}, {"identifier": "C", "content": "1350"}, {"identifier": "D", "content": "135"}] | ["B"] | null | null |
6a8fac3c-20ca-4da7-be4b-00fdf3d812a1 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "7"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "9"}] | ["B"] | null | null |
a1f3abd3-97f9-4a96-8965-6bddf9ac8b33 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "$x + 9y = 36$"}, {"identifier": "B", "content": "$4x - 9y = 12$"}, {"identifier": "C", "content": "$6x - 9y = 20$"}, {"identifier": "D", "content": "$9x - 9y = 32$"}] | ["C"] | null | null |
b9a37e33-f873-4d50-b919-9864d6fcfec7 | maths | 3d-geometry | jee-mathematics | If $f(x) = \int_{\frac{1}{x}}^{x^{1/4}(1+x^{1/4})} \frac{1}{dx}$, $f | [{"identifier": "@", "content": "= -6$, then $f"}, {"identifier": "A", "content": "$ is equal to:"}, {"identifier": "A", "content": "$4 \\log_e 2 - 2$"}, {"identifier": "B", "content": "$2 - \\log_e x$"}, {"identifier": "C", "content": "$\\log_e 2 + 2$"}, {"identifier": "D", "content": "$4 \\log_e 2 + 2$"}] | ["A"] | null | null |
3b235e61-18ba-47aa-a627-bf5a37bdb464 | maths | 3d-geometry | jee-mathematics | The area of the region bounded by the curves $x \left( 1 + y^2 \right) = 1$ and $y^2 = 2x$ is: | [{"identifier": "A", "content": "$2 \\left( \\frac{\\pi}{2} - \\frac{1}{3} \\right)$"}, {"identifier": "B", "content": "$\\frac{\\pi}{2} - \\frac{1}{3}$"}, {"identifier": "C", "content": "$\\frac{\\pi}{2} - \\frac{1}{3}$"}, {"identifier": "D", "content": "$\\frac{1}{3} \\left( \\frac{\\pi}{2} - \\frac{1}{3} \\right)$"}] | ["B"] | null | null |
a94bc062-64c8-4846-a7f8-6ceb620ae4ee | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "54"}, {"identifier": "B", "content": "44"}, {"identifier": "C", "content": "41"}, {"identifier": "D", "content": "66"}] | ["D"] | null | null |
47c8b9a2-16cc-4302-be4c-ae7c30ea8d6c | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "20"}, {"identifier": "B", "content": "22"}, {"identifier": "C", "content": "18"}, {"identifier": "D", "content": "26"}] | ["B"] | null | null |
911e6993-1f84-423b-9702-25e971cbe392 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "$-2$"}, {"identifier": "B", "content": "$6$"}, {"identifier": "C", "content": "$-6$"}, {"identifier": "D", "content": "$2$"}] | ["D"] | null | null |
1f39939e-71c6-4a74-8bea-f6c2d235d37d | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "$-2\\sqrt{10}$"}, {"identifier": "B", "content": "$12$"}, {"identifier": "C", "content": "$6$"}, {"identifier": "D", "content": "$-6$"}] | ["C"] | null | null |
9e5b117e-fb2e-45ad-8e64-5c166daba7ad | maths | 3d-geometry | jee-mathematics | 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 | [] | ["N"] | null | null |
1957dd69-b21f-4609-ac4b-d610195a1745 | maths | 3d-geometry | jee-mathematics | The number of natural numbers, between 212 and 999, such that the sum of their digits is 15, is | [] | [""] | null | null |
7e9ca87b-0353-425d-b767-83cf4cff50f1 | maths | 3d-geometry | jee-mathematics | If $y = y(x)$ is the solution of the differential equation, $\sqrt{4 - x^2} \frac{dy}{dx} = \left(\sin^{-1}\left(\frac{x}{2}\right)\right)^2 - y \sin^{-1}\left(\frac{x}{2}\right)$, $-2 \leq x \leq 2$, $y | [{"identifier": "B", "content": "= \\frac{x^2 - 8}{4}$, then $y"}, {"identifier": "@", "content": "$ is equal to"}] | ["D"] | null | null |
e8d84166-bcf8-4796-855c-8b642125ce9b | maths | 3d-geometry | jee-mathematics | 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 | [] | ["T"] | null | null |
45a8920f-53e0-4308-ba55-52e6797819cf | maths | 3d-geometry | jee-mathematics | 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 | [] | ["A"] | null | null |
87886fd7-10e5-4ed6-acb0-e2f288266f84 | maths | 3d-geometry | sequences-and-series | 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: | [{"identifier": "A", "content": "100"}, {"identifier": "B", "content": "120"}, {"identifier": "C", "content": "110"}, {"identifier": "D", "content": "90"}] | ["A"] | null | null |
4a3af956-2936-4814-a7c5-ad78d38791b4 | maths | 3d-geometry | differential-equations | 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: | [{"identifier": "A", "content": "90"}, {"identifier": "B", "content": "84"}, {"identifier": "C", "content": "122"}, {"identifier": "D", "content": "108"}] | ["A"] | null | null |
842577b9-21fa-44e9-8071-82c1a6bb6254 | maths | 3d-geometry | probability | 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: | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "4"}] | ["D"] | null | null |
22a47813-1150-4bfb-8c1e-7779eb2311f2 | maths | 3d-geometry | exponential-and-logarithm | 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: | [{"identifier": "A", "content": "both reflexive and transitive but not symmetric"}, {"identifier": "B", "content": "an equivalence relation"}, {"identifier": "C", "content": "reflexive but neither symmetric nor transitive"}, {"identifier": "D", "content": "both reflexive and symmetric but not transitive"}] | ["B"] | null | null |
9abb4939-e62b-4c95-b6ae-baa8b0f237b8 | maths | 3d-geometry | coordinate-geometry | 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: | [{"identifier": "A", "content": "392"}, {"identifier": "B", "content": "384"}, {"identifier": "C", "content": "192"}, {"identifier": "D", "content": "96"}] | ["C"] | null | null |
d70df5b1-fbd0-4589-b004-9693965e1ff7 | maths | 3d-geometry | calculus-integration | 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: | [{"identifier": "A", "content": "173"}, {"identifier": "B", "content": "164"}, {"identifier": "C", "content": "158"}, {"identifier": "D", "content": "161"}] | ["D"] | null | null |
421fc75d-2e8a-492d-804b-2cb033503d1a | maths | 3d-geometry | conic-sections | 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: | [{"identifier": "A", "content": "(5, 17, 4)"}, {"identifier": "B", "content": "(2, 8, 5)"}, {"identifier": "C", "content": "(8, 26, 12)"}, {"identifier": "D", "content": "(-1, -1, 1)"}] | ["C"] | null | null |
b858d9a3-549d-424b-a0f5-3f33c1789778 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "462"}, {"identifier": "B", "content": "77"}, {"identifier": "C", "content": "154"}, {"identifier": "D", "content": "308"}] | ["D"] | null | null |
a7ad8cd4-2aac-4155-9246-182dfb07b87d | maths | 3d-geometry | jee-mathematics | The integral \( 80 \int_0^\pi \left( \frac{\sin \theta + \cos \theta}{9 + 16 \sin 2\theta} \right) d\theta \) is equal to: | [{"identifier": "A", "content": "3 \\log_e 4"}, {"identifier": "B", "content": "4 \\log_e 3"}, {"identifier": "C", "content": "6 \\log_e 4"}, {"identifier": "D", "content": "2 \\log_e 3"}] | ["B"] | null | null |
9f3f478f-3e56-42e5-9b3e-6ad35534d3e1 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 4\\sqrt{6} \\\\"}, {"identifier": "B", "content": "& \\quad 6\\sqrt{6} \\\\"}, {"identifier": "C", "content": "& \\quad 18\\sqrt{6}/5 \\\\"}, {"identifier": "D", "content": "& \\quad 24\\sqrt{6}/5 \\\\\n\\end{align*}\n\\]"}] | ["D"] | null | null |
b1f11888-ceb0-442f-810a-0a51a834e287 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 288 \\\\"}, {"identifier": "B", "content": "& \\quad 222 \\\\"}, {"identifier": "C", "content": "& \\quad 242 \\\\"}, {"identifier": "D", "content": "& \\quad 262 \\\\\n\\end{align*}\n\\]"}] | ["C"] | null | null |
3dd9de4c-0aa8-46af-be7b-f79cd0ee13e2 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 13 \\\\"}, {"identifier": "B", "content": "& \\quad 10 \\\\"}, {"identifier": "C", "content": "& \\quad 3 \\\\"}, {"identifier": "D", "content": "& \\quad 7 \\\\\n\\end{align*}\n\\]"}] | ["D"] | null | null |
87d55516-470a-4c57-9713-3f320ed53247 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 20 \\\\"}, {"identifier": "B", "content": "& \\quad 18 \\\\"}, {"identifier": "C", "content": "& \\quad 25 \\\\"}, {"identifier": "D", "content": "& \\quad 16 \\\\\n\\end{align*}\n\\]"}] | ["C"] | null | null |
f131eebf-78ee-4273-b473-b776fb3280b4 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 1280 \\\\"}, {"identifier": "B", "content": "& \\quad 1295 \\\\"}, {"identifier": "C", "content": "& \\quad 1215 \\\\"}, {"identifier": "D", "content": "& \\quad 1040 \\\\\n\\end{align*}\n\\]"}] | ["A"] | null | null |
cbc72177-b566-4494-87df-ab52acb84bfc | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 47 \\\\"}, {"identifier": "B", "content": "& \\quad 36 \\\\"}, {"identifier": "C", "content": "& \\quad 47 \\\\"}, {"identifier": "D", "content": "& \\quad 40 \\\\\n\\end{align*}\n\\]"}] | ["B"] | null | null |
68edd102-6049-4277-a9d4-e831373e20f9 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 4/3 \\\\"}, {"identifier": "B", "content": "& \\quad 2 \\\\"}, {"identifier": "C", "content": "& \\quad 7/3 \\\\"}, {"identifier": "D", "content": "& \\quad 5/3 \\\\\n\\end{align*}\n\\]"}] | ["D"] | null | null |
284e0696-65cb-4558-a145-131420581840 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\quad 2184 \\\\"}, {"identifier": "B", "content": "& \\quad 2196 \\\\"}, {"identifier": "C", "content": "& \\quad 2148 \\\\"}, {"identifier": "D", "content": "& \\quad 2172 \\\\\n\\end{align*}\n\\]"}] | ["A"] | null | null |
7547e409-5d97-4179-9041-5f85e5a66c5e | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\frac{1}{\\log_2"}, {"identifier": "E", "content": "- \\log_2"}, {"identifier": "D", "content": "} \\\\"}, {"identifier": "B", "content": "& \\frac{2}{\\log_2"}, {"identifier": "C", "content": "- \\log_2"}, {"identifier": "D", "content": "} \\\\"}, {"identifier": "C", "content": "& \\frac{1}{\\log_2"}, {"identifier": "D", "content": "- \\log_2"}, {"identifier": "C", "content": "} \\\\"}, {"identifier": "D", "content": "& \\frac{1}{\\log_2"}, {"identifier": "D", "content": "}\n\\end{align*}\n\\]"}] | ["A"] | null | null |
d87ae9e7-3636-4854-a874-65795666e575 | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& \\sqrt{14} \\\\"}, {"identifier": "B", "content": "& 3\\sqrt{7} \\\\"}, {"identifier": "C", "content": "& 2\\sqrt{14} \\\\"}, {"identifier": "D", "content": "& 5\\sqrt{7}\n\\end{align*}\n\\]"}] | ["C"] | null | null |
dd7bf750-08ca-4eee-8b1b-3aa7880e708f | maths | 3d-geometry | jee-mathematics | 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*} | [{"identifier": "A", "content": "& 16 \\\\"}, {"identifier": "B", "content": "& 12 \\\\"}, {"identifier": "C", "content": "& 14 \\\\"}, {"identifier": "D", "content": "& 18\n\\end{align*}\n\\]"}] | ["C"] | null | null |
1c2e865d-ac5b-42e3-bca5-9a6f1c08d311 | maths | 3d-geometry | jee-mathematics | 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 ______. | [] | ["E"] | null | null |
162f99d2-c3c1-40b7-9898-27bf925f81b1 | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "= 1 \\) and \\( f"}, {"identifier": "P", "content": "= \\frac{1}{8} \\), then \\( 16 - f' \\left( \\frac{1}{16} \\right) \\) is equal to ______."}] | ["°"] | null | null |
0b67c9d8-fb39-44a3-b259-978ebd5b5085 | maths | 3d-geometry | jee-mathematics | 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 ______. | [] | ["ֽ"] | null | null |
92a4b8da-f6bb-4de1-b0cb-e42c14b3dcd6 | maths | 3d-geometry | jee-mathematics | 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 ______. | [] | ["B"] | null | null |
a9460d26-8af6-4c4c-9ce8-487a3c0312f2 | maths | 3d-geometry | jee-mathematics | 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 ______. | [] | ["X"] | null | null |
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