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4bb09cfb-2ab1-444e-bea8-964f5d829710 | maths | 3d-geometry | sequences-and-series | 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: | [{"identifier": "A", "content": "\\( \\sqrt{17} \\)"}, {"identifier": "B", "content": "\\( \\sqrt{15} \\)"}, {"identifier": "C", "content": "\\( \\sqrt{14} \\)"}, {"identifier": "D", "content": "\\( \\sqrt{13} \\)"}] | ["C"] | null | null |
e7f78510-f04f-4edc-ae36-a36fa1560105 | maths | 3d-geometry | differential-equations | 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: | [{"identifier": "A", "content": "15"}, {"identifier": "B", "content": "24"}, {"identifier": "C", "content": "18"}, {"identifier": "D", "content": "12"}] | ["D"] | null | null |
99dc6eaa-8767-4188-ba6c-259aa4554208 | maths | 3d-geometry | probability | 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: | [{"identifier": "A", "content": "Both Statement I and Statement II are false"}, {"identifier": "B", "content": "Statement I is true but Statement II is false"}, {"identifier": "C", "content": "Both Statement I and Statement II are true"}, {"identifier": "D", "content": "Statement I is false but Statement II is true"}] | ["B"] | null | null |
9d8b65c8-f334-4258-9396-87e8b05c040f | maths | 3d-geometry | exponential-and-logarithm | 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: | [{"identifier": "A", "content": "48"}, {"identifier": "B", "content": "55"}, {"identifier": "C", "content": "62"}, {"identifier": "D", "content": "47"}] | ["B"] | null | null |
fd4b9955-e499-42a7-b307-dc01f26961e7 | maths | 3d-geometry | coordinate-geometry | 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: | [{"identifier": "A", "content": "22"}, {"identifier": "B", "content": "21"}, {"identifier": "C", "content": "23"}, {"identifier": "D", "content": "24"}] | ["C"] | null | null |
c1114f31-44a0-417b-bb0d-4021c582e91c | maths | 3d-geometry | calculus-integration | 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: | [{"identifier": "A", "content": "21"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "14"}, {"identifier": "D", "content": "6"}] | ["B"] | null | null |
baf77833-80a8-4e67-be37-d8b8e6ea1c76 | maths | 3d-geometry | conic-sections | \( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to: | [{"identifier": "A", "content": "\\( \\frac{2}{3} \\sqrt{e} \\)"}, {"identifier": "B", "content": "\\( \\frac{3e}{5} \\)"}, {"identifier": "C", "content": "\\( \\frac{2e}{3} \\)"}, {"identifier": "D", "content": "\\( \\frac{3e}{5} \\)"}] | ["C"] | null | null |
0e467008-3d2e-4985-a888-c4c4c61ef89a | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "\\( \\sqrt{7} \\)"}, {"identifier": "B", "content": "14"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "7"}] | ["D"] | null | null |
1e9fd5df-4e15-472f-8b30-eb508b1cd5d0 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "\\( \\frac{5}{3} \\sqrt{15} \\)"}, {"identifier": "B", "content": "\\( \\frac{1}{3} \\sqrt{15} \\)"}, {"identifier": "C", "content": "\\( \\frac{2}{3} \\sqrt{15} \\)"}, {"identifier": "D", "content": "\\( \\sqrt{15} \\)"}] | ["C"] | null | null |
f72094ad-9378-463a-8afe-101e4c1feb7c | maths | 3d-geometry | jee-mathematics | The system of equations \( x + 2y + 5z = 9 \), has no solution if:
(x) \( x + 5y + \lambda z = \mu \), | [{"identifier": "A", "content": "\\( \\lambda = 15, \\mu \\neq 17 \\)"}, {"identifier": "B", "content": "\\( \\lambda \\neq 17, \\mu = 18 \\)"}, {"identifier": "C", "content": "\\( \\lambda = 17, \\mu \\neq 18 \\)"}, {"identifier": "D", "content": "\\( \\lambda = 17, \\mu = 18 \\)"}] | ["C"] | null | null |
8767ea4d-8340-48c8-855d-b38999732170 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "11"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "9"}] | ["A"] | null | null |
1c0b8f27-c697-4c95-8608-edc4eac9c44f | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "= \\frac{\\pi}{2} \\]. Then\n\\[ \\cos(x"}, {"identifier": "B", "content": ") \\] is equal to:"}, {"identifier": "A", "content": "\\( 1 - 2(\\log_2 2)^2 \\)"}, {"identifier": "B", "content": "\\( 1 - 2(\\log_2 2) \\)"}, {"identifier": "C", "content": "\\( 2(\\log_2 2)^2 - 1 \\)"}, {"identifier": "D", "content": "\\( 2(\\log_2 2)^2 - 1 \\)"}] | ["D"] | null | null |
b99eecd0-86cd-4151-9d7a-05d422815525 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "196\u03c0"}, {"identifier": "B", "content": "256\u03c0"}, {"identifier": "C", "content": "225\u03c0"}, {"identifier": "D", "content": "128\u03c0"}] | ["B"] | null | null |
fc5d4114-d3a7-440c-ab06-28e802c7e40f | maths | 3d-geometry | jee-mathematics | The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is: | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "6"}] | ["B"] | null | null |
0c203c20-5dc4-4021-aaa0-252c32704f27 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "-1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "0"}] | ["A"] | null | null |
2760158c-5cbb-4b11-b37f-f567861c194e | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "\\( \\frac{x^2}{12} \\)"}, {"identifier": "B", "content": "\\( \\frac{x^2}{4} \\)"}, {"identifier": "C", "content": "\\( \\frac{x^2}{16} \\)"}, {"identifier": "D", "content": "\\( \\frac{x^2}{8} \\)"}] | ["C"] | null | null |
c2ee8f45-0547-4c65-a11b-0105f122e588 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "\\(7/10\\)"}, {"identifier": "B", "content": "\\(4/5\\)"}, {"identifier": "C", "content": "\\(23/30\\)"}, {"identifier": "D", "content": "\\(3/5\\)"}] | ["B"] | null | null |
d128d6b6-a0f1-4ff4-b83f-13c828d9b444 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "\\(x^2 + y^2 - 10x + 9 = 0\\)"}, {"identifier": "B", "content": "\\(x^2 + y^2 - 6x + 5 = 0\\)"}, {"identifier": "C", "content": "\\(x^2 + y^2 - 4x + 3 = 0\\)"}, {"identifier": "D", "content": "\\(x^2 + y^2 - 8x + 7 = 0\\)"}] | ["B"] | null | null |
4a48d53e-0787-43ac-aa0c-e9efba9739d7 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "18"}, {"identifier": "B", "content": "13"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "20"}] | ["B"] | null | null |
31fb90ca-bd84-4ca9-9b71-f32add59351d | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "6"}] | ["C"] | null | null |
f3c8fe19-bd8c-4f37-9e11-2b5930e3c8c6 | maths | 3d-geometry | jee-mathematics | The variance of the numbers 8, 21, 34, 47, \ldots, 320 is | [] | ["C"] | null | null |
0fb44c34-5e78-4474-a9c9-44ee4d1a0246 | maths | 3d-geometry | jee-mathematics | 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 | [] | ["Ț"] | null | null |
c355123b-d38e-40f8-942f-38ad395686a8 | maths | 3d-geometry | jee-mathematics | 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 | [] | ["䏀"] | null | null |
9c014fa2-7f21-45b7-9d92-468f1b26fbdc | maths | 3d-geometry | jee-mathematics | 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 | [] | ["O"] | null | null |
25c19494-d9a2-408c-ade3-38e0a55aa8f2 | maths | 3d-geometry | jee-mathematics | 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 | [] | ["_"] | null | null |
716dfedd-e03b-472b-b096-06aa266e154c | maths | 3d-geometry | sequences-and-series | 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 | [{"identifier": "A", "content": "$3 + \\sqrt{3}$"}, {"identifier": "B", "content": "$4$"}, {"identifier": "C", "content": "$4 - \\sqrt{3}$"}, {"identifier": "D", "content": "$3$"}] | ["B"] | null | null |
8ea3fe25-f9be-4a1f-bda3-27f56bb14b73 | maths | 3d-geometry | differential-equations | 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: | [{"identifier": "A", "content": "$17$"}, {"identifier": "B", "content": "$21$"}, {"identifier": "C", "content": "$56$"}, {"identifier": "D", "content": "$42$"}] | ["B"] | null | null |
2a299a1a-0c06-4a60-ac2f-6bdaa40b0fc8 | maths | 3d-geometry | probability | 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 | [{"identifier": "A", "content": "$\\frac{1 - \\sqrt{3}}{\\sqrt{2}}$"}, {"identifier": "B", "content": "$\\frac{3 - 2\\sqrt{2}}{2\\sqrt{3}}$"}, {"identifier": "C", "content": "$\\frac{3 - 2\\sqrt{2}}{3\\sqrt{2}}$"}, {"identifier": "D", "content": "$\\frac{1 - 2\\sqrt{2}}{\\sqrt{3}}$"}] | ["B"] | null | null |
ba619dfa-a9be-4aef-8a40-71019abd7612 | maths | 3d-geometry | exponential-and-logarithm | 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 | [{"identifier": "A", "content": "$57$"}, {"identifier": "B", "content": "$59$"}, {"identifier": "C", "content": "$55$"}, {"identifier": "D", "content": "$56$"}] | ["A"] | null | null |
d46a273c-4f2e-406e-8349-7ebd51134569 | maths | 3d-geometry | coordinate-geometry | 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: | [{"identifier": "A", "content": "$20$"}, {"identifier": "B", "content": "$10$"}, {"identifier": "C", "content": "$35$"}, {"identifier": "D", "content": "$70$"}] | ["C"] | null | null |
1df5bc52-fb7d-4f2b-85ce-28e0f668fd1e | maths | 3d-geometry | calculus-integration | 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 | [{"identifier": "A", "content": "$14$"}, {"identifier": "B", "content": "$21$"}, {"identifier": "C", "content": "$28$"}, {"identifier": "D", "content": "$7$"}] | ["A"] | null | null |
36122fa4-0d5f-4c76-bd83-d0725f7934a9 | maths | 3d-geometry | conic-sections | 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 | [{"identifier": "A", "content": "$5$"}, {"identifier": "B", "content": "$5\\sqrt{5}$"}, {"identifier": "C", "content": "$5\\sqrt{6}$"}, {"identifier": "D", "content": "$10$"}] | ["B"] | null | null |
9fb4b7b3-0406-4aa9-b10b-3911e5b9f686 | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "$84$"}, {"identifier": "B", "content": "$113$"}, {"identifier": "C", "content": "$91$"}, {"identifier": "D", "content": "$101$"}] | ["C"] | null | null |
fe6d35b5-ea23-494d-b95f-7df08f41b8a3 | maths | 3d-geometry | jee-mathematics | 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 | [] | ["A"] | null | null |
70403eb0-a5a1-4b96-aa65-89a15344632f | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "7"}, {"identifier": "D", "content": "4"}] | ["C"] | null | null |
c826c0a1-6d4c-48c3-8879-78bd54e24c1d | maths | 3d-geometry | jee-mathematics | The area of the region \(\{(x, y) : x^2 + 4x + 2 \leq y \leq |x + 2|\}\) is equal to | [{"identifier": "A", "content": "7"}, {"identifier": "B", "content": "5"}, {"identifier": "C", "content": "24/5"}, {"identifier": "D", "content": "20/3"}] | ["D"] | null | null |
0b48edf8-5c2a-437c-9ccd-a97230a90e3d | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "20"}, {"identifier": "B", "content": "90"}, {"identifier": "C", "content": "45"}, {"identifier": "D", "content": "25"}] | ["D"] | null | null |
0eb94d01-6c3e-47e4-84b4-ea1767364fb9 | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "6"}] | ["C"] | null | null |
1eced83e-3ea5-474f-9a8a-033ba116a923 | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "\\(I(19, 27)\\)"}, {"identifier": "B", "content": "\\(I(9, 1)\\)"}, {"identifier": "C", "content": "\\(I(1, 13)\\)"}, {"identifier": "D", "content": "\\(I(9, 13)\\)"}] | ["D"] | null | null |
4a14b483-58b6-4bca-9bf7-426433b14c59 | maths | 3d-geometry | jee-mathematics | \(A\) 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 | [{"identifier": "A", "content": "\\(\\frac{9}{17}\\)"}, {"identifier": "B", "content": "\\(\\frac{9}{17}\\)"}, {"identifier": "C", "content": "\\(\\frac{9}{17}\\)"}, {"identifier": "D", "content": "\\(\\frac{8}{17}\\)"}] | ["B"] | null | null |
0c6f607d-54d7-466d-be1c-db64fc917a2e | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "92"}, {"identifier": "B", "content": "118"}, {"identifier": "C", "content": "102"}, {"identifier": "D", "content": "108"}] | ["B"] | null | null |
a857fdf6-b070-4f2d-a94c-7f592eeb3779 | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "@", "content": "= 0\\). Then \\(y(\\sqrt{3})\\) is equal to"}, {"identifier": "A", "content": "\\(\\sqrt{15} \\div 2\\)"}, {"identifier": "B", "content": "\\(\\frac{1}{2} \\sqrt{\\frac{3}{2}}\\)"}, {"identifier": "C", "content": "\\(2\\sqrt{2}\\)"}, {"identifier": "D", "content": "\\(\\sqrt{\\frac{14}{3}}\\)"}] | ["B"] | null | null |
74a5b313-5de1-4b42-afac-e4b11c1d75e0 | maths | 3d-geometry | jee-mathematics | \(\lim_{x \to 0} \csc x \left(\sqrt{2 \cos^2 x + 3 \cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)\) is: | [] | ["D"] | null | null |
4eabea93-8e5f-433c-914f-3757e0201d82 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "\\( \\frac{90}{11} \\)"}, {"identifier": "B", "content": "\\( \\frac{85}{11} \\)"}, {"identifier": "C", "content": "\\( \\frac{61}{12} \\)"}, {"identifier": "D", "content": "\\( \\frac{567}{121} \\)"}] | ["D"] | null | null |
a093afec-18f3-4da4-ae18-f21d8f60edb8 | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "\\( \\sqrt{\\frac{11}{6}} \\)"}, {"identifier": "B", "content": "\\( \\frac{1}{3\\sqrt{2}} \\)"}, {"identifier": "C", "content": "\\( 16 \\)"}, {"identifier": "D", "content": "\\( 18 \\)"}] | ["A"] | null | null |
aa607395-f954-4395-99bd-bf683bb6e0f6 | maths | 3d-geometry | jee-mathematics | 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 ______. | [] | ["ᑀ"] | null | null |
f5866438-d51e-4568-989c-3d999deac59e | maths | 3d-geometry | jee-mathematics | 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 ______. | [] | ["N"] | null | null |
d29e5e0c-48f5-4303-b50d-3a8ff8eb8710 | maths | 3d-geometry | jee-mathematics | 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______. | [] | ["l"] | null | null |
7fca30a7-2f14-4248-8bf0-1b687fba8e51 | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "B", "content": "\\) is equal to ______."}] | ["S"] | null | null |
599bb4f3-4909-4595-9cd4-3822f2ceae1c | maths | 3d-geometry | jee-mathematics | The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is______. | [] | ["½"] | null | null |
ef0e84d8-2585-4467-b206-30704260a218 | maths | 3d-geometry | sequences-and-series | 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: | [{"identifier": "A", "content": "8750"}, {"identifier": "B", "content": "9100"}, {"identifier": "C", "content": "8925"}, {"identifier": "D", "content": "8575"}] | ["C"] | null | null |
42221b8f-55a0-4bd7-a037-6e5173757fe9 | maths | 3d-geometry | differential-equations | If the system of equations $2x + \lambda y + 5z = 5$ has infinitely many solutions, then $\lambda + \mu$ is equal to:
$14x + 3y + \mu z = 33$ | [{"identifier": "A", "content": "13"}, {"identifier": "B", "content": "10"}, {"identifier": "C", "content": "12"}, {"identifier": "D", "content": "11"}] | ["C"] | null | null |
6d3d3618-c7da-4b3a-8383-66b5b182ab6b | maths | 3d-geometry | probability | 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: | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "2"}] | ["B"] | null | null |
c8897713-8999-444b-84d6-3a54ba0b823d | maths | 3d-geometry | exponential-and-logarithm | The area of the region enclosed by the curves $y = e^x$, $y = |e^x - 1|$ and y-axis is: | [{"identifier": "A", "content": "$1 - \\log_e 2$"}, {"identifier": "B", "content": "$\\log_e 2$"}, {"identifier": "C", "content": "$1 + \\log_e 2$"}, {"identifier": "D", "content": "$2 \\log_e 2 - 1$"}] | ["A"] | null | null |
35b0bdd6-4e7e-4ed5-86e3-484752574845 | maths | 3d-geometry | coordinate-geometry | The equation of the chord, of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$, whose mid-point is $(3, 1)$, is:
$25x + 101y = 176$ | [{"identifier": "A", "content": "$48x + 25y = 169$"}, {"identifier": "B", "content": "$5x + 16y = 31$"}, {"identifier": "C", "content": "$4x + 122y = 134$"}, {"identifier": "D", "content": "$4x + 122y = 134$"}] | ["A"] | null | null |
f602ed0f-1b06-458b-8ca3-2bf6c12b4f42 | maths | 3d-geometry | calculus-integration | 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: | [{"identifier": "A", "content": "44"}, {"identifier": "B", "content": "22"}, {"identifier": "C", "content": "33"}, {"identifier": "D", "content": "55"}] | ["C"] | null | null |
9e392e8e-9769-4730-8c3c-be055a34abcb | maths | 3d-geometry | conic-sections | 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 $f | [{"identifier": "A", "content": "= 4$. Then $2f"}, {"identifier": "B", "content": "$ is equal to:"}, {"identifier": "A", "content": "39"}, {"identifier": "B", "content": "19"}, {"identifier": "C", "content": "29"}, {"identifier": "D", "content": "23"}] | ["A"] | null | null |
9ea00ba8-51c4-4d92-b9fa-4f615e9937b2 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "$\\frac{6}{7}$"}, {"identifier": "B", "content": "6"}, {"identifier": "C", "content": "$\\frac{1}{7}$"}, {"identifier": "D", "content": "1"}] | ["B"] | null | null |
9f7ee42f-a422-42c0-a0cd-ff4f14da835a | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "7"}] | ["B"] | null | null |
ff241288-2eaf-4e9a-8c75-5e9c5bbf48ec | maths | 3d-geometry | jee-mathematics | 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: | [] | ["C"] | null | null |
01a34c0e-d55e-41c9-881a-c5007a131d39 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "6"}] | ["A"] | null | null |
b8677182-5ae1-4aed-ba9a-f5b13c00f01a | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "\\(2\\sqrt{14}\\)"}, {"identifier": "B", "content": "\\(\\mathbf{v}\\)"}, {"identifier": "C", "content": "\\(\\sqrt{7}\\)"}, {"identifier": "D", "content": "\\(2\\sqrt{7}\\)"}] | ["A"] | null | null |
92212052-fd2a-4fdb-9567-d15d7e04b3e2 | maths | 3d-geometry | jee-mathematics | The number of real solution(s) of the equation \(x^2 + 3x + 2 = \min\{\vert x - 3\vert, \vert x + 2\vert\}\) is: | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["C"] | null | null |
66422792-577f-48e3-8577-6ce01f4feeb0 | maths | 3d-geometry | jee-mathematics | The function \(f : (-\infty, \infty) \rightarrow (-\infty, 1), \text{ defined by } f(x) = \frac{x^2 - 2x}{x^2 + 2}\) is: | [{"identifier": "A", "content": "Neither one-one nor onto"}, {"identifier": "B", "content": "Onto but not one-one"}, {"identifier": "C", "content": "Both one-one and onto"}, {"identifier": "D", "content": "One-one but not onto"}] | ["D"] | null | null |
8b3060d6-97c3-427e-802a-831bef7af864 | maths | 3d-geometry | jee-mathematics | In an arithmetic progression, if \(S_{10} = 1030\) and \(S_{12} = 57\), then \(S_{30} - S_{10}\) is equal to: | [{"identifier": "A", "content": "525"}, {"identifier": "B", "content": "510"}, {"identifier": "C", "content": "515"}, {"identifier": "D", "content": "505"}] | ["C"] | null | null |
37be8789-ed18-4591-910c-57c0774c29c8 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "22"}, {"identifier": "B", "content": "20"}, {"identifier": "C", "content": "21"}, {"identifier": "D", "content": "19"}] | ["C"] | null | null |
8999ad58-5192-4896-bbd5-131e2f36c4a7 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "420"}, {"identifier": "B", "content": "360"}, {"identifier": "C", "content": "160"}, {"identifier": "D", "content": "280"}] | ["B"] | null | null |
5dc239df-f2e9-4f64-956d-7cf33138be50 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "16"}, {"identifier": "B", "content": "25"}, {"identifier": "C", "content": "9"}, {"identifier": "D", "content": "36"}] | ["A"] | null | null |
3df5204a-ffbf-4d19-980a-8e2fb65f7d87 | maths | 3d-geometry | jee-mathematics | 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:
\(\alpha x^2 + \beta y^2 - \gamma xy - 30x - 60y + 225 = 0\) | [] | ["B"] | null | null |
fb786c03-50bb-45a1-8a07-5ba97cb76d37 | maths | 3d-geometry | jee-mathematics | 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: | [{"identifier": "A", "content": "$\\pi$"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "$\\pi - (\\alpha + \\beta + \\gamma)$"}, {"identifier": "D", "content": "$3\\pi$"}] | ["A"] | null | null |
9b4a5341-8d99-48ab-9a32-ddf854a5b148 | maths | 3d-geometry | jee-mathematics | 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 ________. | [] | ["Ͻ"] | null | null |
d4ee3e0f-52dd-4c48-802d-7ebfc9b8846e | maths | 3d-geometry | jee-mathematics | 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 ________. | [] | ["P"] | null | null |
46d3b2e8-b00d-42e1-ac03-81431e08fd4a | maths | 3d-geometry | jee-mathematics | 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 ________. | [] | ["A"] | null | null |
e9b2988b-7ac7-4528-a0e5-fa70df795c0b | maths | 3d-geometry | jee-mathematics | 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 ________. | [] | ["Lj"] | null | null |
984ecf93-7530-44c2-b9fc-40badc1010a5 | maths | 3d-geometry | jee-mathematics | 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 ________. | [] | ["w"] | null | null |
f0c34e35-de11-40b6-b844-4aceda9c9c64 | maths | 3d-geometry | sequences-and-series | 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 | [{"identifier": "A", "content": "area of triangle ABO is \\( \\frac{11}{3} \\)"}, {"identifier": "B", "content": "ABO is an obtuse angled isosceles triangle"}, {"identifier": "C", "content": "area of triangle ABO is \\( \\frac{11}{4} \\)"}, {"identifier": "D", "content": "ABO is a scalene triangle"}] | ["B"] | null | null |
386008a7-9f2d-417a-bc5e-9da6c7c9c48a | maths | 3d-geometry | differential-equations | 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 | [{"identifier": "A", "content": "545"}, {"identifier": "B", "content": "715"}, {"identifier": "C", "content": "735"}, {"identifier": "D", "content": "675"}] | ["D"] | null | null |
3abb0ea3-69b6-4fe5-bc92-c24c1341f8de | maths | 3d-geometry | probability | 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 | [{"identifier": "A", "content": "\\( \\frac{73}{8} \\)"}, {"identifier": "B", "content": "\\( \\frac{25}{9} \\)"}, {"identifier": "C", "content": "\\( \\frac{16}{8} \\)"}, {"identifier": "D", "content": "\\( \\frac{75}{8} \\)"}] | ["A"] | null | null |
ce69f97f-67ad-4564-96b5-d19b006f6e1b | maths | 3d-geometry | exponential-and-logarithm | 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 | [{"identifier": "A", "content": "\\( \\frac{137}{72} \\)"}, {"identifier": "B", "content": "\\( \\frac{131}{72} \\)"}, {"identifier": "C", "content": "\\( \\frac{137}{72} \\)"}, {"identifier": "D", "content": "\\( \\frac{167}{72} \\)"}] | ["A"] | null | null |
7a54308e-bc39-4255-b8dc-4b50afb12022 | maths | 3d-geometry | coordinate-geometry | 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 | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "18"}, {"identifier": "C", "content": "8"}, {"identifier": "D", "content": "20"}] | ["D"] | null | null |
bf5f71d9-d0bb-4ce3-b1a3-bf2218c08673 | maths | 3d-geometry | calculus-integration | 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 | [{"identifier": "A", "content": "\\( (\\gamma, \\beta^2 - 4\\alpha) \\)"}, {"identifier": "B", "content": "\\( (\\alpha, \\beta^2 + 4\\gamma) \\)"}, {"identifier": "C", "content": "\\( (\\gamma, \\beta^2 + 4\\alpha) \\)"}, {"identifier": "D", "content": "\\( (\\alpha, \\beta^2 - 4\\gamma) \\)"}] | ["D"] | null | null |
fcb2d563-8ac4-452f-be89-ce259c8146c1 | maths | 3d-geometry | conic-sections | 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 | [{"identifier": "A", "content": "\\( 1.81\\sqrt{2} \\)"}, {"identifier": "B", "content": "\\( 41 \\)"}, {"identifier": "C", "content": "\\( 82 \\)"}, {"identifier": "D", "content": "\\( \\frac{81}{2} \\)"}] | ["D"] | null | null |
d323f007-a281-4f15-8c52-e3b78f1b9fb8 | maths | 3d-geometry | jee-mathematics | 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 | [] | ["B"] | null | null |
2400477c-a743-4733-95d2-a4f76884f8f5 | maths | 3d-geometry | jee-mathematics | 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
\[ | [{"identifier": "A", "content": "\\ 9 \\quad"}, {"identifier": "B", "content": "\\ 12 \\quad"}, {"identifier": "C", "content": "\\ 7 \\quad"}, {"identifier": "D", "content": "\\ 8.\n\\]"}] | ["A"] | null | null |
13d54bd0-0372-4112-a983-0f35933d161b | maths | 3d-geometry | jee-mathematics | \(\cos \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)\) is equal to:
\[ | [{"identifier": "A", "content": "\\ 1 \\quad"}, {"identifier": "B", "content": "\\ 0 \\quad"}, {"identifier": "C", "content": "\\ \\frac{32}{65} \\quad"}, {"identifier": "D", "content": "\\ \\frac{33}{65}.\n\\]"}] | ["B"] | null | null |
669b1282-46df-4444-b097-abc0e94d1c6f | maths | 3d-geometry | jee-mathematics | 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 \(\frac{9\sqrt{2}}{2}\),
\[ | [{"identifier": "A", "content": "\\ \\text{both are true} \\quad"}, {"identifier": "B", "content": "\\ \\text{only (S2) is true} \\quad"}, {"identifier": "C", "content": "\\ \\text{only (S1) is true} \\quad"}, {"identifier": "D", "content": "\\ \\text{both are false}.\n\\]"}] | ["C"] | null | null |
e8e525f0-719e-4737-b196-ec4c9442a18e | maths | 3d-geometry | jee-mathematics | 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
\[ | [{"identifier": "A", "content": "\\ \\frac{80}{3} \\quad"}, {"identifier": "B", "content": "\\ \\frac{44}{3} \\quad"}, {"identifier": "C", "content": "\\ \\frac{32}{3} \\quad"}, {"identifier": "D", "content": "\\ \\frac{17}{3}.\n\\]"}] | ["B"] | null | null |
3c8d4be8-c367-4856-bdc1-bfe37e60d677 | maths | 3d-geometry | jee-mathematics | The sum of the squares of all the roots of the equation \(x^2 + |2x - 3| - 4 = 0\), is
\[ | [{"identifier": "A", "content": "\\ 3(3 - \\sqrt{2}) \\quad"}, {"identifier": "B", "content": "\\ 6(3 - \\sqrt{2}) \\quad"}, {"identifier": "C", "content": "\\ 6(2 - \\sqrt{2}) \\quad"}, {"identifier": "D", "content": "\\ 3(2 - \\sqrt{2})\n\\]"}] | ["C"] | null | null |
577660f5-d79b-4d00-9af8-d50b7849743f | maths | 3d-geometry | jee-mathematics | 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
\[ | [{"identifier": "A", "content": "\\ 98 \\quad"}, {"identifier": "B", "content": "\\ 126 \\quad"}, {"identifier": "C", "content": "\\ 142 \\quad"}, {"identifier": "D", "content": "\\ 112.\n\\]"}] | ["B"] | null | null |
d688540d-d072-44a1-b481-a459af38593f | maths | 3d-geometry | jee-mathematics | 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
\[ | [{"identifier": "A", "content": "\\ \\frac{28}{75} \\quad"}, {"identifier": "B", "content": "\\ \\frac{18}{25} \\quad"}, {"identifier": "C", "content": "\\ \\frac{26}{75} \\quad"}, {"identifier": "D", "content": "\\ \\frac{14}{25}.\n\\]"}] | ["A"] | null | null |
403a27b0-6d1e-49c4-924d-8722f6a2915f | maths | 3d-geometry | jee-mathematics | Let for some function \(y = f(x), \int_0^x tf(t) dt = x^2 f(x), x > 0\) and \(f | [{"identifier": "B", "content": "= 3\\). Then \\(f"}, {"identifier": "F", "content": "\\) is equal to\n\\["}, {"identifier": "A", "content": "\\ 1 \\quad"}, {"identifier": "B", "content": "\\ 3 \\quad"}, {"identifier": "C", "content": "\\ 6 \\quad"}, {"identifier": "D", "content": "\\ 2.\n\\]"}] | ["A"] | null | null |
f15c71bb-d3e0-42f2-b7da-a93f795013ef | maths | 3d-geometry | jee-mathematics | 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
\[ | [{"identifier": "A", "content": "\\ 64 \\quad"}, {"identifier": "B", "content": "\\ 196 \\quad"}, {"identifier": "C", "content": "\\ 144 \\quad"}, {"identifier": "D", "content": "\\ 100.\n\\]"}] | ["D"] | null | null |
bec8e309-4e7f-4767-9f29-a2c36dac2786 | maths | 3d-geometry | jee-mathematics | 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 | [] | ["B"] | null | null |
4dfa6e24-a0d7-4203-b672-58d09c63870b | maths | 3d-geometry | jee-mathematics | 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 | [{"identifier": "A", "content": "4608"}, {"identifier": "B", "content": "5720"}, {"identifier": "C", "content": "5719"}, {"identifier": "D", "content": "4607"}] | ["D"] | null | null |
01291000-d3a5-41f5-aac9-1a90237dadf5 | maths | 3d-geometry | jee-mathematics | The relation \( R = \{ (x, y) : x, y \in \mathbb{Z} \text{ and } x + y \text{ is even} \} \) is: | [{"identifier": "A", "content": "reflexive and symmetric but not transitive"}, {"identifier": "B", "content": "an equivalence relation"}, {"identifier": "C", "content": "symmetric and transitive but not reflexive"}, {"identifier": "D", "content": "reflexive and transitive but not symmetric"}] | ["B"] | null | null |
5df8a224-1e66-4ab3-a733-446cb3d5df54 | maths | 3d-geometry | jee-mathematics | 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
\( \frac{5}{2} \) | [] | ["E"] | null | null |
70401e79-9f72-4a97-9c2d-6ff0463d1a59 | maths | 3d-geometry | jee-mathematics | 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
\( \frac{5}{2} \) | [] | ["ڍ"] | null | null |
4141f89c-4f78-485e-a9bc-6f3ab89dc31c | maths | 3d-geometry | jee-mathematics | 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
\( \frac{5}{2} \) | [] | ["E"] | null | null |
70575064-0b51-4839-8155-1945b70779e0 | maths | 3d-geometry | jee-mathematics | 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
\( \frac{5}{\pi} \) | [] | ["v"] | null | null |
25f7b0a3-85b7-4842-bc13-f36f8a07d9a2 | maths | 3d-geometry | jee-mathematics | 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
\( \frac{5}{2} \) | [] | ["F"] | null | null |
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