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e0d7c5a9-1325-4f0a-997d-fe2f4c1315b6 | maths | 3d-geometry | sequences-and-series | Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing terms. If $a_1 a_5 = 28$ and $a_2 + a_4 = 29$, then $a_6$ is equal to: | [{"identifier": "A", "content": "628"}, {"identifier": "B", "content": "812"}, {"identifier": "C", "content": "526"}, {"identifier": "D", "content": "784"}] | ["D"] | null | null |
08d4f2e5-79ce-4622-add6-d93266ac5637 | maths | 3d-geometry | differential-equations | Let $x = x(y)$ be the solution of the differential equation $y^2 \, dx + (x - \frac{1}{y}) \, dy = 0$. If $x | [{"identifier": "A", "content": "= 1$, then $x \\left( \\frac{1}{3} \\right)$ is:"}, {"identifier": "A", "content": "$\\frac{1}{3} + e$"}, {"identifier": "B", "content": "$3 + e$"}, {"identifier": "C", "content": "$3 - e$"}, {"identifier": "D", "content": "$\\frac{3}{2} + e$"}] | ["C"] | null | null |
204278cb-f634-49b7-aab5-39caf9d1b0c0 | maths | 3d-geometry | probability | Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is equal to: | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "14"}, {"identifier": "C", "content": "13"}, {"identifier": "D", "content": "11"}] | ["B"] | null | null |
e988b658-42ea-4ae7-bbdf-be615cbcb52c | maths | 3d-geometry | exponential-and-logarithm | The product of all solutions of the equation $e^{5 \log x^2 + 3} = x^8, x > 0$, is: | [{"identifier": "A", "content": "$e^{8/5}$"}, {"identifier": "B", "content": "$e^{6/5}$"}, {"identifier": "C", "content": "$e^{2}$"}, {"identifier": "D", "content": "$e$"}] | ["A"] | null | null |
f0f1ad4e-2071-4dce-8938-4c1fadada37b | maths | 3d-geometry | coordinate-geometry | Let the triangle PQR be the image of the triangle with vertices $(1, 3), (3, 1)$ and $(2, 4)$ in the line $x + 2y = 2$. If the centroid of $\triangle PQR$ is the point $(\alpha, \beta)$, then $15(\alpha - \beta)$ is equal to: | [{"identifier": "A", "content": "19"}, {"identifier": "B", "content": "24"}, {"identifier": "C", "content": "21"}, {"identifier": "D", "content": "22"}] | ["D"] | null | null |
664b04f0-da33-499a-a662-520d29573c7c | maths | 3d-geometry | calculus-integration | Let for $f(x) = 7 \tan^8 x + 7 \tan^6 x - 3 \tan^4 x - 3 \tan^2 x$, $I_1 = \int_{0}^{\pi/4} f(x) \, dx$ and $I_2 = \int_{0}^{\pi/4} x f(x) \, dx$. Then $7I_1 + 12I_2$ is equal to: | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "$2\\pi$"}, {"identifier": "D", "content": "$\\pi$"}] | ["B"] | null | null |
89bc55ee-62e6-4c46-b1ff-ec9357df79a0 | maths | 3d-geometry | conic-sections | Let the parabola $y = x^2 + px - 3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(\alpha, \beta)$ passes through the points P, Q and R, then the area of $\triangle PQR$ is: | [{"identifier": "A", "content": "7"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "5"}] | ["C"] | null | null |
d747393b-66be-4c6a-a129-fad2f2e01985 | maths | 3d-geometry | jee-mathematics | Let $L_1 : \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $L_2 : \frac{x-3}{2} = \frac{y-4}{3} = \frac{z-5}{4}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$? | [{"identifier": "A", "content": "$\\left( \\frac{14}{5}, -3, \\frac{22}{3} \\right)$"}, {"identifier": "B", "content": "$\\left( -\\frac{5}{3}, -7, 1 \\right)$"}, {"identifier": "C", "content": "$\\left( 2, 3, \\frac{1}{2} \\right)$"}, {"identifier": "D", "content": "$\\left( \\frac{5}{3}, -1, \\frac{1}{2} \\right)$"}] | ["A"] | null | null |
c485593e-f8a4-4b04-9036-b4b0f7085acc | maths | 3d-geometry | jee-mathematics | Let $f(x)$ be a real differentiable function such that $f | [{"identifier": "@", "content": "= 1$ and $f(x + y) = f(x)f(y) + f'(x)f(y)$ for all $x, y \\in \\mathbb{R}$. Then $\\sum_{n=1}^{100} \\log_2 f(n)$ is equal to:"}, {"identifier": "A", "content": "2525"}, {"identifier": "B", "content": "5220"}, {"identifier": "C", "content": "2384"}, {"identifier": "D", "content": "2406"}] | ["A"] | null | null |
1ef41510-bd7f-4907-ad8b-7d4fcc50395d | maths | 3d-geometry | jee-mathematics | From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is: | [] | ["A"] | null | null |
aeb18745-57be-4719-b72d-ec3323bc24ed | maths | 3d-geometry | jee-mathematics | Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of \(16 \left( \sec^{-1} x \right)^2 + \left( \cosec^{-1} x \right)^2 \) is: | [{"identifier": "A", "content": "\\(24\\pi^2\\)"}, {"identifier": "B", "content": "\\(22\\pi^2\\)"}, {"identifier": "C", "content": "\\(31\\pi^2\\)"}, {"identifier": "D", "content": "\\(18\\pi^2\\)"}] | ["B"] | null | null |
d0973693-11dc-4dc4-b641-a23af7fd1fe1 | maths | 3d-geometry | jee-mathematics | Let \(f : \mathbb{R} \rightarrow \mathbb{R}\) be a twice differentiable function such that \(f(x + y) = f(x)f(y)\) for all \(x, y \in \mathbb{R}\). If \(f' | [{"identifier": "@", "content": "= 4a\\) and \\(f\\) satisfies \\(f''(x) - 3af'(x) - f(x) = 0, a > 0\\), then the area of the region \\(R = \\{(x, y) \\mid 0 \\leq y \\leq f(ax), 0 \\leq x \\leq 2\\}\\) is:"}, {"identifier": "A", "content": "\\(e^2 - 1\\)"}, {"identifier": "B", "content": "\\(e^2 + 1\\)"}, {"identifier": "C", "content": "\\(e^4 + 1\\)"}, {"identifier": "D", "content": "\\(e^4 - 1\\)"}] | ["A"] | null | null |
f9fe4e71-6076-4ad0-88d5-6faae33fb0fe | maths | 3d-geometry | jee-mathematics | The area of the region, inside the circle \((x - 2\sqrt{3})^2 + y^2 = 12\) and outside the parabola \(y^2 = 2\sqrt{3}x\) is: | [{"identifier": "A", "content": "\\(3\\pi + 8\\)"}, {"identifier": "B", "content": "\\(6\\pi - 16\\)"}, {"identifier": "C", "content": "\\(3\\pi - 8\\)"}, {"identifier": "D", "content": "\\(6\\pi - 8\\)"}] | ["B"] | null | null |
4a6f25fe-1f9c-446b-a0a2-d20848ea8c84 | maths | 3d-geometry | jee-mathematics | Let the foci of a hyperbola be \((1, 14)\) and \((1, -12)\). If it passes through the point \((1, 6)\), then the length of its latus-rectum is: | [{"identifier": "A", "content": "\\(\\frac{24}{5}\\)"}, {"identifier": "B", "content": "\\(\\frac{25}{9}\\)"}, {"identifier": "C", "content": "\\(\\frac{144}{5}\\)"}, {"identifier": "D", "content": "\\(\\frac{288}{5}\\)"}] | ["D"] | null | null |
3ccbdac9-3078-4d90-bf32-dd8d28c405cb | maths | 3d-geometry | jee-mathematics | If \(\sum_{r=1}^{n} T_r = \frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}\), then \(\lim_{n \to \infty} \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)\) is equal to: | [{"identifier": "A", "content": "\\(0\\)"}, {"identifier": "B", "content": "\\(\\frac{4}{3}\\)"}, {"identifier": "C", "content": "\\(1\\)"}, {"identifier": "D", "content": "\\(\\frac{1}{2}\\)"}] | ["B"] | null | null |
0686a602-5a84-4d9b-ae94-09884e6a8e30 | maths | 3d-geometry | jee-mathematics | A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64(\mu + \sigma^2)\) is: | [{"identifier": "A", "content": "\\(51\\)"}, {"identifier": "B", "content": "\\(64\\)"}, {"identifier": "C", "content": "\\(32\\)"}, {"identifier": "D", "content": "\\(48\\)"}] | ["D"] | null | null |
f3ef4b6f-0c2d-40db-9345-812517d65036 | maths | 3d-geometry | jee-mathematics | The number of non-empty equivalence relations on the set \(\{1, 2, 3\}\) is: | [{"identifier": "A", "content": "\\(6\\)"}, {"identifier": "B", "content": "\\(5\\)"}, {"identifier": "C", "content": "\\(7\\)"}, {"identifier": "D", "content": "\\(4\\)"}] | ["B"] | null | null |
888ff468-a611-426c-8353-8c514f90010a | maths | 3d-geometry | jee-mathematics | A circle \(C\) of radius 2 lies in the second quadrant and touches both the coordinate axes. Let \(r\) be the radius of a circle that has centre at the point \((2, 5)\) and intersects the circle \(C\) at exactly two points. If the set of all possible values of \(r\) is the interval \((\alpha, \beta)\), then \(3\beta - 2\alpha\) is equal to: | [{"identifier": "A", "content": "\\(10\\)"}, {"identifier": "B", "content": "\\(15\\)"}, {"identifier": "C", "content": "\\(12\\)"}, {"identifier": "D", "content": "\\(14\\)"}] | ["B"] | null | null |
d3222c23-1187-43fd-8d44-339a5847e490 | maths | 3d-geometry | jee-mathematics | Let \(A = \{1, 2, 3, \ldots, 10\}\) and \(B = \left\{ \frac{m}{n} : m, n \in A, m < n \text{ and } \gcd(m, n) = 1 \right\}\). Then \(n(B)\) is equal to: | [{"identifier": "A", "content": "\\(36\\)"}, {"identifier": "B", "content": "\\(31\\)"}, {"identifier": "C", "content": "\\(37\\)"}, {"identifier": "D", "content": "\\(29\\)"}] | ["B"] | null | null |
0f735c17-069f-47a0-b141-6d20b9dc2aa5 | maths | 3d-geometry | jee-mathematics | Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z| = 1$ with $\arg(z_1) = \frac{\pi}{4}, \arg(z_2) = 0$ and $\arg(z_3) = \frac{\pi}{4}$. If $|z_1 \bar{z}_2 + z_2 \bar{z}_3 + z_3 \bar{z}_1|^2 = \alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2 + \beta^2$ is: | [{"identifier": "A", "content": "24"}, {"identifier": "B", "content": "29"}, {"identifier": "C", "content": "41"}, {"identifier": "D", "content": "31"}] | ["B"] | null | null |
df79631f-6609-43dd-b1ac-013424516e62 | maths | 3d-geometry | jee-mathematics | Let $A$ be a square matrix of order 3 such that $\det(A) = -2$ and $\det(3 \text{adj}(-6 \text{adj}(3A))) = 2^{m+n} \cdot 3^n, m > n$. Then $4m + 2n$ is equal to ________. | [] | ["b"] | null | null |
239963a0-82b3-415f-8c7c-66dea1cd6c9e | maths | 3d-geometry | jee-mathematics | If $\sum_{r=0}^{5} \frac{1}{2r+1} = \frac{m}{n}, \gcd(m, n) = 1$, then $m - n$ is equal to ________. | [] | ["࠳"] | null | null |
719ddc1e-7de1-4df5-81b0-e414c42d9dde | maths | 3d-geometry | jee-mathematics | Let $\vec{c}$ be the projection vector of $\vec{b} = \lambda \hat{i} + 4\hat{k}, \lambda > 0$, on the vector $\vec{a} = 2\hat{i} + 2\hat{j} + 2\hat{k}$. If $|\vec{a} + \vec{c}| = 7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________. | [] | ["P"] | null | null |
702b65df-7f7b-44c5-b0d9-c278927042f5 | maths | 3d-geometry | jee-mathematics | Let the function, $f(x) = \begin{cases} -3ax^2 - 2, & x < 1 \\ ax^2 + bx, & x \geq 1 \end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a > 1, b \in \mathbb{R}$. If the area of the region enclosed by $y = f(x)$ and the line $y = -20$ is $\alpha + \beta \sqrt{3}, \alpha, \beta \in \mathbb{Z}$, then the value of $\alpha + \beta$ is ________. | [] | ["b"] | null | null |
a721a0ed-f380-415a-847b-92fb2201fc16 | maths | 3d-geometry | jee-mathematics | Let $L_1 : z = \frac{-1}{8} = \frac{z+1}{0}$ and $L_2 : z = \frac{-2}{3} = \frac{z+4}{1}, \alpha \in \mathbb{R}$, be two lines, which intersect at the point $B$. If $P$ is the foot of perpendicular from the point $A(1, 1, -1)$ on $L_2$, then the value of $26\alpha(\text{PB})^2$ is ________. | [] | ["Ę"] | null | null |
ec9be866-f5d0-4250-8a11-f61a959bc365 | maths | 3d-geometry | sequences-and-series | Let \( f(x) = \int_0^1 (t^2 - 9t + 20)\,dt, \quad 1 \leq x \leq 5. \) If the range of \( f \) is \([\alpha, \beta]\), then \( 4(\alpha + \beta) \) equals: | [{"identifier": "A", "content": "253"}, {"identifier": "B", "content": "154"}, {"identifier": "C", "content": "125"}, {"identifier": "D", "content": "157"}] | ["D"] | null | null |
35e5cab3-72da-45e2-8679-11c34e004f13 | maths | 3d-geometry | differential-equations | Let \( \vec{a} \) be a unit vector perpendicular to the vectors \( \vec{b} = \hat{i} - 2\hat{j} + 3\hat{k} \) and \( \vec{c} = 2\hat{i} + 3\hat{j} - \hat{k} \), and makes an angle of \( \cos^{-1}\left(-\frac{1}{2}\right) \) with the vector \( \hat{i} + \hat{j} + \hat{k} \). If \( \vec{a} \) makes an angle of \( \frac{\pi}{3} \) with the vector \( \hat{i} + \alpha\hat{j} + \hat{k} \), then the value of \( \alpha \) is: | [{"identifier": "A", "content": "\\( \\sqrt{6} \\)"}, {"identifier": "B", "content": "\\( -\\sqrt{6} \\)"}, {"identifier": "C", "content": "\\( -\\sqrt{3} \\)"}, {"identifier": "D", "content": "\\( \\sqrt{3} \\)"}] | ["B"] | null | null |
368b2e82-01b2-485e-9cde-e017dcb43030 | maths | 3d-geometry | probability | If for the solution curve \( y = f(x) \) of the differential equation \( \frac{dy}{dx} + (\tan x)y = \frac{2 + \sec x}{(1 + 2\sec x)^2} \), \( x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), then \( f\left(\frac{\pi}{4}\right) \) is equal to: | [{"identifier": "A", "content": "\\( \\frac{\\sqrt{3} - 1}{10(4+\\sqrt{3})} \\)"}, {"identifier": "B", "content": "\\( \\frac{\\sqrt{3} - 1}{2\\sqrt{3} - 2} \\)"}, {"identifier": "C", "content": "\\( \\frac{\\sqrt{3} - 1}{10(4+\\sqrt{3})} \\)"}, {"identifier": "D", "content": "\\( \\frac{5 - \\sqrt{3}}{14} \\)"}] | ["D"] | null | null |
034ed0dc-97f8-4959-a406-a0402b39512c | maths | 3d-geometry | exponential-and-logarithm | Let \( P \) be the foot of the perpendicular from the point \( (1, 2, 2) \) on the line \( L : \frac{x-1}{1} = \frac{y+1}{2} = \frac{z-2}{2} \). Let the line \( \vec{r} = (-\hat{i} + \hat{j} - 2\hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k}), \lambda \in \mathbb{R} \), intersect the line \( L \) at \( Q \). Then \( 2(PQ)^2 \) is equal to: | [{"identifier": "A", "content": "25"}, {"identifier": "B", "content": "19"}, {"identifier": "C", "content": "29"}, {"identifier": "D", "content": "27"}] | ["D"] | null | null |
c6fa73d4-22f1-4c73-9a85-06f42c3d90b0 | maths | 3d-geometry | coordinate-geometry | Let \( A = [a_{ij}] \) be a matrix of order \( 3 \times 3 \), with \( a_{ij} = (\sqrt{2})^{i+j} \). If the sum of all the elements in the third row of \( A^2 \) is \( \alpha + \beta\sqrt{2} \), \( \alpha, \beta \in \mathbb{Z} \), then \( \alpha + \beta \) is equal to: | [{"identifier": "A", "content": "280"}, {"identifier": "B", "content": "224"}, {"identifier": "C", "content": "210"}, {"identifier": "D", "content": "168"}] | ["B"] | null | null |
906f9a68-1701-442f-b6d5-09686497c68e | maths | 3d-geometry | calculus-integration | Let the line \( x + y = 1 \) meet the axes of \( x \) and \( y \) at \( A \) and \( B \), respectively. A right angled triangle \( AMN \) is inscribed in the triangle \( OAB \), where \( O \) is the origin and the points \( M \) and \( N \) lie on the lines \( OB \) and \( AB \), respectively. If the area of the triangle \( AMN \) is \( \frac{4}{5} \) of the area of the triangle \( OAB \) and \( AN : NB = \lambda : 1 \), then the sum of all possible value(s) of \( \lambda \) is: | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "\\( \\frac{5}{2} \\)"}, {"identifier": "C", "content": "\\( \\frac{1}{2} \\)"}, {"identifier": "D", "content": "\\( \\frac{13}{6} \\)"}] | ["A"] | null | null |
b7a6bf6c-1bc0-4cc2-95f9-a5d50b065ec6 | maths | 3d-geometry | conic-sections | If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged in a dictionary, then the word at 440th position in this arrangement, is: | [{"identifier": "A", "content": "PRNAUK"}, {"identifier": "B", "content": "PRKANU"}, {"identifier": "C", "content": "PRKAUN"}, {"identifier": "D", "content": "PRNAUK"}] | ["C"] | null | null |
ef21afd8-084b-4d2b-bdaa-3802ac96f2b5 | maths | 3d-geometry | jee-mathematics | If the set of all \( a \in \mathbb{R} \), for which the equation \( 2x^2 + (a - 5)x + 15 = 3a \) has no real root, is the interval \((\alpha, \beta)\), and \( X = \{x \in \mathbb{Z} : \alpha < x < \beta\} \), then \( \sum_{x \in X} x^2 \) is equal to: | [{"identifier": "A", "content": "2109"}, {"identifier": "B", "content": "2129"}, {"identifier": "C", "content": "2119"}, {"identifier": "D", "content": "2139"}] | ["D"] | null | null |
871bdfa7-de89-4ac4-a10f-ae86f6d7ca3e | maths | 3d-geometry | jee-mathematics | Let \( A = [a_{ij}] \) be a \( 2 \times 2 \) matrix such that \( a_{ij} \in \{0, 1\} \) for all \( i \) and \( j \). Let the random variable \( X \) denote the possible values of the determinant of the matrix \( A \). Then, the variance of \( X \) is: | [] | ["C"] | null | null |
dd382693-5904-47f1-adae-2d25135a2a52 | maths | 3d-geometry | jee-mathematics | Let the function \( f(x) = (x^2 + 1) \left|x^2 - ax + 2 \right| + \cos |x| \) be not differentiable at the two points \( x = \alpha = 2 \) and \( x = \beta \). Then the distance of the point \((\alpha, \beta)\) from the line \(12x + 5y + 10 = 0\) is equal to: | [{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "2"}] | ["C"] | null | null |
5eb60c4b-a5ad-4f28-9aa2-d26655ee2b96 | maths | 3d-geometry | jee-mathematics | Let the area enclosed between the curves \( |y| = 1 - x^2 \) and \( x^2 + y^2 = 1 \) be \( \alpha \). If \( 9\alpha = \beta \pi + \gamma \), \( \beta, \gamma \) are integers, then the value of \(|\beta - \gamma|\) equals. | [{"identifier": "A", "content": "27"}, {"identifier": "B", "content": "33"}, {"identifier": "C", "content": "15"}, {"identifier": "D", "content": "18"}] | ["B"] | null | null |
be10c500-adb2-4a82-b9a7-c768ae0f1c50 | maths | 3d-geometry | jee-mathematics | The remainder, when \( 7^{10^3} \) is divided by 23, is equal to: | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "17"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "14"}] | ["D"] | null | null |
b45bc7e2-fdf6-47e8-b2b9-962e11c3f8f4 | maths | 3d-geometry | jee-mathematics | If \( \alpha x + \beta y = 109 \) is the equation of the chord of the ellipse \( \frac{x^2}{\alpha} + \frac{y^2}{\beta} = 1 \), whose mid point is \( \left( \frac{1}{2}, \frac{1}{4} \right) \), then \( \alpha + \beta \) is equal to: | [{"identifier": "A", "content": "58"}, {"identifier": "B", "content": "46"}, {"identifier": "C", "content": "37"}, {"identifier": "D", "content": "72"}] | ["A"] | null | null |
dcddf043-98de-4a82-8127-0f309a5b5588 | maths | 3d-geometry | jee-mathematics | If the domain of the function \( \log_5 (18x - x^2 - 77) \) is \( (\alpha, \beta) \) and the domain of the function \( \log(x-1) \left( \frac{2x^2 + 3x - 2}{x^2 - 3x - 4} \right) \) is \( (\gamma, \delta) \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to: | [{"identifier": "A", "content": "195"}, {"identifier": "B", "content": "179"}, {"identifier": "C", "content": "186"}, {"identifier": "D", "content": "174"}] | ["C"] | null | null |
67f0e924-1075-4881-be6e-0567d5b1cb9c | maths | 3d-geometry | jee-mathematics | Let a circle \( C \) pass through the points \( (4, 2) \) and \( (0, 2) \), and its centre lie on \( 3x + 2y + 2 = 0 \). Then the length of the chord, of the circle \( C \), whose mid-point is \( (1, 2) \), is: | [{"identifier": "A", "content": "\\( \\sqrt{3} \\)"}, {"identifier": "B", "content": "\\( 2\\sqrt{2} \\)"}, {"identifier": "C", "content": "\\( 2\\sqrt{3} \\)"}, {"identifier": "D", "content": "\\( 4\\sqrt{2} \\)"}] | ["C"] | null | null |
a622af54-b6f8-49d7-a20d-cb4055c187a8 | maths | 3d-geometry | jee-mathematics | Let a straight line \( L \) pass through the point \( P(2, -1, 3) \) and be perpendicular to the lines \( \frac{x-1}{2} = \frac{y+1}{1} = \frac{z-3}{-2} \) and \( \frac{x-3}{1} = \frac{y-2}{-1} = \frac{z+2}{4} \). If the line \( L \) intersects the \( yz \)-plane at the point \( Q \), then the distance between the points \( P \) and \( Q \) is: | [{"identifier": "A", "content": "\\( \\sqrt{10} \\)"}, {"identifier": "B", "content": "\\( 2\\sqrt{3} \\)"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["D"] | null | null |
bf95aebc-42f7-48ac-8f31-59073d69fc4e | maths | 3d-geometry | jee-mathematics | Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains \( n \) white balls and 3 black balls. One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2. If the probability, that the ball drawn is white, is \( \frac{29}{45} \), then \( n \) is equal to: | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "4"}] | ["A"] | null | null |
f2175a59-df34-4c91-9c7d-d9880d79d3a5 | maths | 3d-geometry | jee-mathematics | Let $\alpha, \beta (\alpha \neq \beta)$ be the values of $m$, for which the equations $x + y + z = 1, x + 2y + 4z = m$ and $x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of $\sum_{n=1}^{10} (n^\alpha + n^\beta)$ is equal to: | [{"identifier": "A", "content": "3080"}, {"identifier": "B", "content": "560"}, {"identifier": "C", "content": "3410"}, {"identifier": "D", "content": "440"}] | ["D"] | null | null |
c74e8625-9c18-429e-9232-354a6a6c0d5e | maths | 3d-geometry | jee-mathematics | Let $S = N \cup \{0\}$. Define a relation $R$ from $S$ to $R$ by $R = \{ (x, y) : \log_e y = x \log_e \left( \frac{2}{3} \right), x \in S, y \in R \}$
Then, the sum of all the elements in the range of $R$ is equal to: | [{"identifier": "A", "content": "$\\frac{10}{9}$"}, {"identifier": "B", "content": "$\\frac{5}{2}$"}, {"identifier": "C", "content": "$\\frac{\\sqrt{3}}{2}$"}, {"identifier": "D", "content": "$\\frac{1}{3}$"}] | ["D"] | null | null |
83fc3287-ea27-433e-9432-41a5ccc19e93 | maths | 3d-geometry | jee-mathematics | If $\sin x + \sin^2 x = 1, x \in \left(0, \frac{\pi}{2}\right)$, then
$(\cos^{12} x + x \tan^{12} x) + 3 (\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x) + (\cos^6 x + \tan^6 x)$ is equal to: | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "$\\frac{4}{3}$"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "2"}] | ["D"] | null | null |
0b94f03a-6e37-44d4-a32e-639768bcf41a | maths | 3d-geometry | jee-mathematics | If $24 \int_0^\frac{\pi}{3} (\sin 4x - \frac{1}{12}) + (2 \sin x) \ dx = 2\pi + \alpha$, where $[\cdot]$ denotes the greatest integer function, then $\alpha$ is equal to ________. | [] | ["L"] | null | null |
9d8c9921-7ad2-44e5-9db4-f0e957421b1c | maths | 3d-geometry | jee-mathematics | Let $a_1, a_2, \ldots, a_{2024}$ be an Arithmetic Progression such that $a_1 + (a_5 + a_{10} + a_{15} + \ldots + a_{2020}) + a_{2024} = 2233$. Then $a_1 + a_2 + a_3 + \ldots + a_{2024}$ is equal to ________. | [] | ["⮼"] | null | null |
96f0e2bd-4dc5-475b-96c4-95e9dd872243 | maths | 3d-geometry | jee-mathematics | If $\lim_{x \to 0} \left( \int_0^1 (3x + 5)^4 \ dx \right)^{\frac{1}{5}} = \frac{a}{5^\alpha} \left( \frac{6}{5} \right)^\frac{\beta}{5}$, then $\alpha$ is equal to ________. | [] | [""] | null | null |
d1f5c97d-292a-41fb-bcdc-9b2129e4281c | maths | 3d-geometry | jee-mathematics | Let $y^2 = 12x$ be the parabola and $S$ be its focus. Let $PQ$ be a focal chord of the parabola such that $(\text{SP})(\text{SQ}) = 144$. Let $C$ be the circle described taking $PQ$ as a diameter. If the equation of a circle $C$ is $64x^2 + 64y^2 - 16x - 64\sqrt{3}y = \beta$, then $\beta - \alpha$ is equal to ________. | [] | ["հ"] | null | null |
608cb694-9676-4e36-80d5-45ee0c8e75c6 | maths | 3d-geometry | jee-mathematics | Let integers $a, b \in [-3, 3]$ be such that $a + b \neq 0$. Then the number of all possible ordered pairs $(a, b)$, for which $|\frac{x - a}{x + b}| = 1$ and $\begin{vmatrix} z + 1 & \omega & \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix} = 1, z \in \mathbb{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to ________. | [] | ["J"] | null | null |
d0af9cd1-adf9-44e6-b6d2-687330d7feaf | maths | 3d-geometry | sequences-and-series | For a $3 \times 3$ matrix $M$, let trace ($M$) denote the sum of all the diagonal elements of $M$. Let $A$ be a $3 \times 3$ matrix such that $|A| = \frac{1}{2}$ and trace ($A$) = 3. If $B = \text{adj(adj}(2A))$, then the value of $|B| + \text{trace (B)}$ equals:
1. 56
2. 132
3. 174
4. 280 | [] | ["D"] | null | null |
543e2004-2c62-40f4-be96-639a9ead7604 | maths | 3d-geometry | differential-equations | In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways, in which these girls and boys can stand in a queue such that all the girls stand together, all the boys stand together, but $B_1$ and $B_2$ are not adjacent to each other, is:
1. 96
2. 144
3. 120
4. 72 | [] | ["B"] | null | null |
f2b30fa8-86b8-4027-91a1-dc1625cab997 | maths | 3d-geometry | probability | Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x + \sqrt{x^3 - 1})^5 + (x - \sqrt{x^3 - 1})^5, x > 1$. If $u$ and $v$ satisfy the equations $\alpha u + \beta v = 18$ and $\gamma u + \delta v = 20$, then $u + v$ equals:
1. 5
2. 4
3. 3
4. 8 | [] | ["A"] | null | null |
64133d32-f5f1-47e1-97c0-7f990cae9160 | maths | 3d-geometry | exponential-and-logarithm | Let a line pass through two distinct points $P(-2, -1, 3)$ and $Q$, and be parallel to the vector $3\hat{i} + 2\hat{j} + 2\hat{k}$. If the distance of the point $Q$ from the point $R(1, 3, 3)$ is 5, then the area of the triangle $\Delta PQR$ is equal to:
1. 148
2. 136
3. 144
4. 140 | [] | ["B"] | null | null |
1213b036-8cd7-477e-9318-558044637164 | maths | 3d-geometry | coordinate-geometry | If $A$ and $B$ are two events such that $P(A \cap B) = 0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equation $12x^2 - 7x + 1 = 0$, then the value of $\frac{P(A \cup B)}{P(A \cap B)}$ is:
1. $\frac{4}{3}
2. \frac{7}{4}
3. \frac{5}{3}
4. \frac{3}{4}$ | [] | ["D"] | null | null |
b346e320-6ede-4419-8068-12e151527048 | maths | 3d-geometry | calculus-integration | If $\int e^x \left( \frac{x^2 - 1}{\sqrt{1-x^2}} + \frac{x^2 - 1}{\sqrt{1-x^2}} \right) dx = g(x) + C$, where $C$ is the constant of integration, then $g \left( \frac{1}{2} \right)$ equals:
1. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$
2. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$
3. $\frac{\pi}{4} \sqrt{\frac{e}{3}}$
4. $\frac{\pi}{6} \sqrt{\frac{e}{3}}$ | [] | ["B"] | null | null |
8b63e5f8-b075-4142-a8de-2419cd1f1a99 | maths | 3d-geometry | conic-sections | The area of the region enclosed by the curves $y = x^2 - 4x + 4$ and $y^2 = 16 - 8x$ is:
1. $\frac{8}{3}$
2. $\frac{4}{3}$
3. 8
4. $\frac{3}{2}$ | [] | ["A"] | null | null |
353189b4-26ef-42bc-8232-2c2905826d29 | maths | 3d-geometry | jee-mathematics | Let $f(x) = \int_0^x t^2 \frac{t^2 - 8 + 16}{t^2} dt, x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are:
1. 2 and 3
2. 2 and 1
3. 3 and 2
4. 1 and 3 | [] | ["A"] | null | null |
6050b2c9-2f29-428b-b1a2-645018469c78 | maths | 3d-geometry | jee-mathematics | Let $P(4, 4\sqrt{3})$ be a point on the parabola $y^2 = 4ax$ and $PQ$ be a focal chord of the parabola. If $M$ and $N$ are the foot of perpendiculars drawn from $P$ and $Q$ respectively on the directrix of the parabola, then the area of the quadrilateral PQMN is equal to: | [] | ["D"] | null | null |
69f6b4c7-326e-4b13-8b97-e761e52bfd89 | maths | 3d-geometry | jee-mathematics | Let \( \mathbf{a} \) and \( \mathbf{b} \) be two unit vectors such that the angle between them is \( \frac{\pi}{3} \). If \( \lambda \mathbf{a} + 2\mathbf{b} \) and \( 3\mathbf{a} - \lambda \mathbf{b} \) are perpendicular to each other, then the number of values of \( \lambda \) in \([-1, 3]\) is:
1. 2
2. 1
3. 0
4. 3 | [] | ["C"] | null | null |
f663e627-ddd3-4c39-b7b9-111eee8e8269 | maths | 3d-geometry | jee-mathematics | If \( \lim_{x \to \infty} \left( \left( \frac{x}{1-x} \right) \left( \frac{1-x}{x+2} \right) \right)^x = \alpha \), then the value of \( \log_x \alpha \) equals:
1. \( e^{-1} \)
2. \( e^2 \)
3. \( e^4 \)
4. \( e^6 \) | [] | ["D"] | null | null |
437d1d0a-9050-4f22-aae3-8dc8b2421add | maths | 3d-geometry | jee-mathematics | Let \( A = \{1, 2, 3, 4\} \) and \( B = \{1, 4, 9, 16\} \). Then the number of many-one functions \( f : A \to B \) such that \( 1 \in f(A) \) is equal to:
1. 151
2. 139
3. 163
4. 127 | [] | ["A"] | null | null |
f4001f0f-9413-4ab1-a8f0-30257539b847 | maths | 3d-geometry | jee-mathematics | Suppose that the number of terms in an A.P. is \( 2k, k \in N \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55 and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
1. 6
2. 5
3. 8
4. 4 | [] | ["B"] | null | null |
271e1d93-bac3-4328-89f9-785cd030d496 | maths | 3d-geometry | jee-mathematics | The perpendicular distance, of the line \( \frac{x-1}{2} = \frac{y+2}{-1} = \frac{z+3}{2} \) from the point \( P(2, -10, 1) \), is:
1. \( 4\sqrt{3} \)
2. \( 5\sqrt{2} \)
3. \( 4\sqrt{3} \)
4. \( 3\sqrt{5} \) | [] | ["D"] | null | null |
feab3b23-dde2-40ee-b7d3-26b1a5a545a3 | maths | 3d-geometry | jee-mathematics | The system of linear equations:
\[
\begin{align*}
x + y + 2z &= 6 \\
-2x + 3y + az &= a + 1 \\
7a + 3b &= 0
\end{align*}
\]
If the system of linear equations: \( 2x + 3y + az = a + 1 \) where \( a, b \in \mathbb{R} \), has infinitely many solutions, then \( 7a + 3b \) is equal to:
1. 16
2. 12
3. 22
4. 9 | [] | ["A"] | null | null |
08e5060e-dc0a-454f-b789-9af233997119 | maths | 3d-geometry | jee-mathematics | If \( x = f(y) \) is the solution of the differential equation \( (1 + y^2) + \left( x - 2e^{\tan^{-1} y} \right) \frac{dy}{dx} = 0 \), \( y \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) with \( f | [{"identifier": "@", "content": "= 1 \\), then \\( f \\left( \\frac{1}{\\sqrt{3}} \\right) \\) is equal to:\n\n1. \\( e^{\\pi/12} \\)\n2. \\( e^{\\pi/4} \\)\n3. \\( e^{\\pi/3} \\)\n4. \\( e^{\\pi/6} \\)"}] | ["D"] | null | null |
0a58d51f-1bc6-488d-8536-9c62d69bbd89 | maths | 3d-geometry | jee-mathematics | Let \( \alpha_\theta \) and \( \beta_\theta \) be the distinct roots of \( 2x^2 + (\cos \theta)x - 1 = 0, \theta \in (0, 2\pi) \). If \( m \) and \( M \) are the minimum and the maximum values of \( \alpha^4_\theta + \beta^4_\theta \), then \( 16(M + m) \) equals:
1. 24
2. 25
3. 17
4. 27 | [] | ["B"] | null | null |
ba446548-3c5e-434a-a4f9-ff93210e0167 | maths | 3d-geometry | jee-mathematics | The sum of all values of \( \theta \in [0, 2\pi] \) satisfying \( 2\sin^2 \theta = \cos 2\theta \) and \( 2\cos^2 \theta = 3\sin \theta \) is | [] | ["C"] | null | null |
7061bd67-c3b7-4aca-9919-70c9af344083 | maths | 3d-geometry | jee-mathematics | Let the curve \( z(1 + i) + \bar{z}(1 - i) = 4 \), \( z \in \mathbb{C} \), divide the region \( |z - 3| \leq 1 \) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals: | [{"identifier": "A", "content": "\\( 1 + \\frac{\\pi}{2} \\)"}, {"identifier": "B", "content": "\\( 1 + \\frac{\\pi}{3} \\)"}, {"identifier": "C", "content": "\\( 1 + \\frac{\\pi}{6} \\)"}, {"identifier": "D", "content": "\\( 1 + \\frac{\\pi}{4} \\)"}] | ["A"] | null | null |
4108dd5f-fa57-4844-b135-30ab0af558d1 | maths | 3d-geometry | jee-mathematics | Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to: | [{"identifier": "A", "content": "10"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "7"}] | ["C"] | null | null |
632756dd-ef72-4259-b346-9f43bb1fa419 | maths | 3d-geometry | jee-mathematics | If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______. | [] | ["ȑ"] | null | null |
ccc20cc0-9a53-4c5b-8326-1b1a7627d330 | maths | 3d-geometry | jee-mathematics | Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______. | [] | ["C"] | null | null |
488b8dc9-101e-496c-8d75-9e350b0b7330 | maths | 3d-geometry | jee-mathematics | Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______. | [] | ["Ñ"] | null | null |
436d0a30-99f4-4a87-a9a3-3acecad4694f | maths | 3d-geometry | jee-mathematics | Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f | [{"identifier": "@", "content": "= 0 \\). If \\( \\int_{-1/2}^{1/2} f(x)dx = 2\\pi - \\alpha \\) then \\( \\alpha^2 \\) is equal to _______."}] | ["["] | null | null |
bc79b31f-ddd7-4251-b859-b65f546db58f | maths | 3d-geometry | jee-mathematics | Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______. | [] | ["\"] | null | null |
33da9282-3bcc-4fd5-a591-c018b712af39 | maths | 3d-geometry | sequences-and-series | If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to
- | [{"identifier": "A", "content": "$-1080$\n-"}, {"identifier": "B", "content": "$-1020$\n-"}, {"identifier": "C", "content": "$-1200$\n-"}, {"identifier": "D", "content": "$-120$"}] | ["A"] | null | null |
4b9c830a-1db7-457c-851b-edd3a124ed9b | maths | 3d-geometry | differential-equations | One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is
- | [{"identifier": "A", "content": "$\\frac{3}{16}$\n-"}, {"identifier": "B", "content": "$\\frac{1}{4}$\n-"}, {"identifier": "C", "content": "$\\frac{3}{8}$\n-"}, {"identifier": "D", "content": "$\\frac{5}{8}$"}] | ["B"] | null | null |
d005ece7-0f50-42ba-ba3d-103b927db487 | maths | 3d-geometry | probability | Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is
- | [{"identifier": "A", "content": "$\\frac{1}{3}(7\\mathbf{i} + 4\\mathbf{j} + 3\\mathbf{k})$\n-"}, {"identifier": "B", "content": "$\\frac{1}{3}(i + 4\\mathbf{j} + 7\\mathbf{k})$\n-"}, {"identifier": "C", "content": "$\\frac{1}{3}(12\\mathbf{i} + 12\\mathbf{j} + \\mathbf{k})$\n-"}, {"identifier": "D", "content": "$\\frac{1}{3}(7\\mathbf{i} + 12\\mathbf{j} + \\mathbf{k})$"}] | ["D"] | null | null |
a67dad96-e96b-4ce7-85be-94ffe8b32a82 | maths | 3d-geometry | exponential-and-logarithm | If $A, B,$ and $(\text{adj} (A^{-1}) + \text{adj} (B^{-1}))$ are non-singular matrices of same order, then the inverse of $A (\text{adj} (A^{-1}) + \text{adj} (B^{-1}))^{-1} B$, is equal to
- | [{"identifier": "A", "content": "$AB^{-1} + A^{-1}B$\n-"}, {"identifier": "B", "content": "$\\text{adj} (B^{-1}) + \\text{adj} (A^{-1})$\n-"}, {"identifier": "C", "content": "$\\frac{AB^{-1}}{|A|} + \\frac{BA^{-1}}{|B|}$\n-"}, {"identifier": "D", "content": "$\\frac{1}{|A|}(\\text{adj}(B) + \\text{adj}(A))$"}] | ["D"] | null | null |
dfdcc2d4-1d85-4c3d-8742-880b5d5a0361 | maths | 3d-geometry | coordinate-geometry | Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12. If the number of students whose marks are less than 12 is 18, then the total number of students is
- | [{"identifier": "A", "content": "52\n-"}, {"identifier": "B", "content": "48\n-"}, {"identifier": "C", "content": "44\n-"}, {"identifier": "D", "content": "40"}] | ["C"] | null | null |
b4c9d3a5-15e8-4d30-acb3-da063db14b28 | maths | 3d-geometry | calculus-integration | Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to
- | [{"identifier": "A", "content": "4\n-"}, {"identifier": "B", "content": "32\n-"}, {"identifier": "C", "content": "8\n-"}, {"identifier": "D", "content": "16"}] | ["C"] | null | null |
625f1ae5-596c-4639-b134-7a15eaf01b76 | maths | 3d-geometry | conic-sections | If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to
- | [{"identifier": "A", "content": "20\n-"}, {"identifier": "B", "content": "5\n-"}, {"identifier": "C", "content": "8\n-"}, {"identifier": "D", "content": "10"}] | ["D"] | null | null |
163a7d1a-cecf-4f07-b8b1-44147f0dbf64 | maths | 3d-geometry | jee-mathematics | If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to | [] | ["B"] | null | null |
c4a0c746-1002-404f-acaf-9bae1604d83c | maths | 3d-geometry | jee-mathematics | Let \( P \) be the foot of the perpendicular from the point \( Q(10, -3, -1) \) on the line \( \frac{x-3}{7} = \frac{y-2}{1} = \frac{z+1}{2} \). Then the area of the right angled triangle \( PQR \), where \( R \) is the point \((3, -2, 1)\), is
\begin{align*} | [{"identifier": "A", "content": "\\ 9\\sqrt{15} & & \\quad"}, {"identifier": "B", "content": "\\ \\sqrt{30} \\\\"}, {"identifier": "C", "content": "\\ 8\\sqrt{15} & & \\quad"}, {"identifier": "D", "content": "\\ 3\\sqrt{30}\n\\end{align*}"}] | ["D"] | null | null |
496fa60a-2674-400d-86ac-c5d00bc38dfb | maths | 3d-geometry | jee-mathematics | Let the arc \( AC \) of a circle subtend a right angle at the centre \( O \). If the point \( B \) on the arc \( AC \), divides the arc \( AC \) such that \( \frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5} \), and \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to
\begin{align*} | [{"identifier": "A", "content": "\\ 2\\sqrt{3} & & \\quad"}, {"identifier": "B", "content": "\\ 2 - \\sqrt{3} \\\\"}, {"identifier": "C", "content": "\\ 5\\sqrt{3} & & \\quad"}, {"identifier": "D", "content": "\\ 2 + \\sqrt{3}\n\\end{align*}"}] | ["B"] | null | null |
9ff44630-babd-4c87-9b8b-2e76dfc56d37 | maths | 3d-geometry | jee-mathematics | Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is
\begin{align*} | [{"identifier": "A", "content": "\\ [0, \\infty) & & \\quad"}, {"identifier": "B", "content": "\\ [1, \\infty) \\\\"}, {"identifier": "C", "content": "\\ (0, \\infty) & & \\quad"}, {"identifier": "D", "content": "\\ \\mathbb{R}\n\\end{align*}"}] | ["D"] | null | null |
6520cadb-f7b1-49f3-870d-ece5223a1e99 | maths | 3d-geometry | jee-mathematics | \((\lambda - 1)x + (\lambda - 4)y + \lambda z = 5 \)
If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to
\begin{align*} | [{"identifier": "A", "content": "\\ 6 & & \\quad"}, {"identifier": "B", "content": "\\ 10 \\\\"}, {"identifier": "C", "content": "\\ 20 & & \\quad"}, {"identifier": "D", "content": "\\ 12\n\\end{align*}"}] | ["D"] | null | null |
57278625-22b7-460b-9c6d-85e0cb9de522 | maths | 3d-geometry | jee-mathematics | The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is
\begin{align*} | [{"identifier": "A", "content": "\\ 36000 & & \\quad"}, {"identifier": "B", "content": "\\ 37000 \\\\"}, {"identifier": "C", "content": "\\ 34000 & & \\quad"}, {"identifier": "D", "content": "\\ 35000\n\\end{align*}"}] | ["A"] | null | null |
e0176c27-17ad-44e9-8200-ea8dc44a25ef | maths | 3d-geometry | jee-mathematics | Let \( R = \{(1, 2), (2, 3), (3, 3)\} \) be a relation defined on the set \( \{1, 2, 3, 4\} \). Then the minimum number of elements, needed to be added in \( R \) so that \( R \) becomes an equivalence relation, is
\begin{align*} | [{"identifier": "A", "content": "\\ 10 & & \\quad"}, {"identifier": "B", "content": "\\ 7 \\\\"}, {"identifier": "C", "content": "\\ 8 & & \\quad"}, {"identifier": "D", "content": "\\ 9\n\\end{align*}"}] | ["B"] | null | null |
c6c9ea0d-2b4a-4d75-9b51-288c394b52ae | maths | 3d-geometry | jee-mathematics | Let the area of a \( \triangle PQR \) with vertices \( P(5, 4), Q(-2, 4) \) and \( R(a, b) \) be 35 square units. If its orthocenter and centroid are \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to
\begin{align*} | [{"identifier": "A", "content": "\\ \\frac{8}{3} & & \\quad"}, {"identifier": "B", "content": "\\ \\frac{7}{3} \\\\"}, {"identifier": "C", "content": "\\ 2 & & \\quad"}, {"identifier": "D", "content": "\\ 3\n\\end{align*}"}] | ["D"] | null | null |
7dbc72dd-8dd6-4ee9-a67b-e3fef13d7cd7 | maths | 3d-geometry | jee-mathematics | The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is
\begin{align*} | [{"identifier": "A", "content": "\\ 2 & & \\quad"}, {"identifier": "B", "content": "\\ \\log_2 2 \\\\"}, {"identifier": "C", "content": "\\ 1 & & \\quad"}, {"identifier": "D", "content": "\\ e^2\n\\end{align*}"}] | ["C"] | null | null |
9e7da6b5-4669-4472-8aaa-cd93a27cf3c1 | maths | 3d-geometry | jee-mathematics | Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), be the equation of a circle with center at \( C \). If the area of the triangle, whose vertices are at the points \( (0, 0) \), \( C \) and \( (\alpha, 0) \) is 11 square units, then \( \alpha^2 \) equals:
- | [{"identifier": "A", "content": "50\n-"}, {"identifier": "B", "content": "100\n-"}, {"identifier": "C", "content": "\\( \\frac{81}{25} \\)\n-"}, {"identifier": "D", "content": "\\( \\frac{121}{25} \\)"}] | ["B"] | null | null |
4a20f27f-5446-4fec-9652-f29e5d09d339 | maths | 3d-geometry | jee-mathematics | The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is
- | [{"identifier": "A", "content": "\\( 2/3 \\)\n-"}, {"identifier": "B", "content": "0\n-"}, {"identifier": "C", "content": "\\( 3/2 \\)\n-"}, {"identifier": "D", "content": "1"}] | ["B"] | null | null |
77f489df-a2bc-4579-95b9-5d61d9d617a7 | maths | 3d-geometry | jee-mathematics | Let \( I(x) = \int \frac{dx}{(x-11)(x+15)} \). If \( I | [{"identifier": "e", "content": "- I"}, {"identifier": "X", "content": "= \\frac{1}{4} \\left( \\frac{1}{\\beta x} - \\frac{1}{c x} \\right) \\), \\( b, c \\in \\mathbb{N} \\), then \\( 3(b + c) \\) is equal to\n\n-"}, {"identifier": "A", "content": "22\n-"}, {"identifier": "B", "content": "39\n-"}, {"identifier": "C", "content": "40\n-"}, {"identifier": "D", "content": "26"}] | ["B"] | null | null |
68c466e8-b8e2-42b5-a836-3bc4dd2e49b5 | maths | 3d-geometry | jee-mathematics | If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to
- | [{"identifier": "A", "content": "\\( x - \\tan^{-1}\\frac{4}{3} \\)\n-"}, {"identifier": "B", "content": "\\( x + \\tan^{-1}\\frac{4}{5} \\)\n-"}, {"identifier": "C", "content": "\\( x - \\tan^{-1}\\frac{5}{12} \\)\n-"}, {"identifier": "D", "content": "\\( x + \\tan^{-1}\\frac{5}{12} \\)"}] | ["C"] | null | null |
bfd1c5c4-ea06-4d15-8e85-2d54fa41bafb | maths | 3d-geometry | jee-mathematics | Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to | [] | ["S"] | null | null |
0c471963-04dd-4a21-987b-535f3ec17da9 | maths | 3d-geometry | jee-mathematics | If the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) has equal roots, where \( a + c = 15 \) and \( b = \frac{36}{5} \), then \( a^2 + c^2 \) is equal to | [] | ["µ"] | null | null |
a5f4fa94-d8e4-4c6e-84f1-c1cf21d2cabf | maths | 3d-geometry | jee-mathematics | If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to | [] | ["^"] | null | null |
75387286-d2e3-4701-a450-6603de1e3ed9 | maths | 3d-geometry | jee-mathematics | The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to | [] | ["ʤ"] | null | null |
4f2b69e7-6590-4ebb-8eea-c25d743cdae4 | maths | 3d-geometry | jee-mathematics | If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to | [] | [""] | null | null |
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