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2024 47 questions

Q252024 · 1 Feb 2024 · Shift 2· Numerical answer
Let $f : (0, \infty) \to R$ and $F (x) = \int_{0}^{x} t f (t) d t$ . If $F (x^{2}) = x^{4} + x^{5}$ , then $\sum_{r = 1}^{12} f (r^{2})$ is equal to ________ .
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Answer: 219
Q262024 · 1 Feb 2024 · Shift 2· Numerical answer
Let $A B C$ be an isosceles triangle in which $A$ is at $(- 1, 0)$ , $∠ A = \frac{2 π}{3}$ , $A B = A C$ and $B$ is on the positve $x$ -axis. If $BC = 43$ and the line $BC$ intersects the line $y = x + 3$ at $(α, β)$ , then $\frac{β ^{4}}{α ^{2}}$ is ________ .
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Answer: 36
Q42024 · 31 Jan 2024 · Shift 1
Let $α, β, γ, δ \in Z$ and let $A (α, β), B (1, 0), C (γ, δ)$ and $D (1, 2)$ be the vertices of a parallelogram $A B C D$ . If $A B = 10$ and the points A and C lie on the line $3 y = 2 x + 1$ , then $2 (α + β + γ + δ)$ is equal to
1. A.8
2. B.5
3. C.12
4. D.10
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Answer: A
Q162024 · 31 Jan 2024 · Shift 2
If for some $m, n;^{6} C_{m} + 2 (^{6} C_{m + 1}) +^{6} C_{m + 2} >^{8} C_{3}$ and $^{n - 1} P_{3} :^{n} P_{4} = 1 : 8$ , then $^{n} P_{m + 1} +^{n + 1} C_{m}$ is equal to
1. A.380
2. B.376
3. C.372
4. D.384
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Answer: C
Q72024 · 30 Jan 2024 · Shift 1
Let $(α, β, γ)$ be the foot of perpendicular from the point $(1, 2, 3)$ on the line $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}$ . Then $19 (α + β + γ)$ is equal to :
1. A.99
2. B.102
3. C.101
4. D.100
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Answer: C
Q82024 · 30 Jan 2024 · Shift 1
A line passing through the point $A (9, 0)$ makes an angle of $3 0^{\circ}$ with the positive direction of $x$ -axis. If this line is rotated about A through an angle of $1 5^{\circ}$ in the clockwise direction, then its equation in the new position is :
1. A. $\frac{y}{3 + 2} + x = 9$
2. B. $\frac{x}{3 + 2} + y = 9$
3. C. $\frac{x}{3 - 2} + y = 9$
4. D. $\frac{y}{3 - 2} + x = 9$
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Answer: D
Q142024 · 30 Jan 2024 · Shift 1
Let $A (2, 3, 5)$ and $C (- 3, 4, - 2)$ be opposite vertices of a parallelogram $A B C D$ . If the diagonal $B D = \hat{i} + 2 \hat{j} + 3 \hat{k}$ , then the area of the parallelogram is equal to :
1. A. $\frac{1}{2} 410$
2. B. $\frac{1}{2} 306$
3. C. $\frac{1}{2} 586$
4. D. $\frac{1}{2} 474$
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Answer: D
Q282024 · 30 Jan 2024 · Shift 1· Numerical answer
Let $α = 1^{2} + 4^{2} + 8^{2} + 1 3^{2} + 1 9^{2} + 2 6^{2} + \dots$ upto 10 terms and $β = \sum_{n = 1}^{10} n^{4}$ . If $4 α - β = 55 k + 40$ , then k is equal to _________.
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Answer: 353
Q82024 · 29 Jan 2024 · Shift 1
Let $A = 100 0 α β 0 β α$ and $∣2 A ∣^{3} = 2^{21}$ where $α, β \in Z$ . Then a value of $α$ is
1. A. $9$
2. B. $17$
3. C. $3$
4. D. $5$
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Answer: D
Q152024 · 29 Jan 2024 · Shift 1
If $z = \frac{1}{2} - 2 i$ is such that $∣ z + 1∣ = α z + β (1 + i)$ , $i = - 1$ and $α, β \in R$ , then $α + β$ is equal to
1. A. $2$
2. B. $- 4$
3. C. $3$
4. D. $- 1$
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Answer: C
Q252024 · 29 Jan 2024 · Shift 1· Numerical answer
If $\frac{^{11} C _{1}}{2} + \frac{^{11} C _{2}}{3} + \dots + \frac{^{11} C _{9}}{10} = \frac{n}{m}$ with $g cd (n, m) = 1$ , then $n + m$ is equal to _______.
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Answer: 2041
Q262024 · 29 Jan 2024 · Shift 1· Numerical answer
If the points of intersection of two distinct conics $x^{2} + y^{2} = 4 b$ and $\frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1$ lie on the curve $y^{2} = 3 x^{2}$ , then $33$ times the area of the rectangle formed by the intersection points is _______.
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Answer: 432
Q92024 · 29 Jan 2024 · Shift 2
Let $P (3, 2, 3), Q (4, 6, 2)$ and $R (7, 3, 2)$ be the vertices of $△ PQR$ . Then, the angle $∠ QPR$ is
1. A. $cos^{- 1} (\frac{7}{18})$
2. B. $\frac{π}{6}$
3. C. $cos^{- 1} (\frac{1}{18})$
4. D. $\frac{π}{3}$
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Answer: D
Q102024 · 29 Jan 2024 · Shift 2
Let $y = lo g_{e} (\frac{1 - x ^{2}}{1 + x ^{2}}), - 1 < x < 1$ . Then at $x = \frac{1}{2}$ , the value of $225 (y^{'} - y^{''})$ is equal to
1. A.732
2. B.736
3. C.742
4. D.746
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Answer: B
Q282024 · 29 Jan 2024 · Shift 2· Numerical answer
Remainder when $6 4^{3 2^{32}}$ is divided by 9 is equal to _________.
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Answer: 1
Q72024 · 27 Jan 2024 · Shift 1
The distance of the point $(7, - 2, 11)$ from the line $\frac{x - 6}{1} = \frac{y - 4}{0} = \frac{z - 8}{3}$ along the line $\frac{x - 5}{2} = \frac{y - 1}{- 3} = \frac{z - 5}{6}$ , is :
1. A.12
2. B.18
3. C.21
4. D.14
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Answer: D
Q152024 · 27 Jan 2024 · Shift 1
$^{n - 1} C_{r} = (k^{2} - 8)^{n} C_{r + 1}$ if and only if :
1. A. $22 < k < 23$
2. B. $22 < k \leq 3$
3. C. $23 < k < 33$
4. D. $23 < k \leq 32$
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Answer: B
Q212024 · 27 Jan 2024 · Shift 1· Numerical answer
If $8 = 3 + \frac{1}{4} (3 + p) + \frac{1}{4 ^{2}} (3 + 2 p) + \frac{1}{4 ^{3}} (3 + 3 p) + \dots\dots \infty$ , then the value of $p$ is ____.
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Answer: 9
Q162024 · 27 Jan 2024 · Shift 2
Let $α = \frac{( 4 !)!}{( 4 ! ) ^{3!}}$ and $β = \frac{( 5 !)!}{( 5 ! ) ^{4!}}$ . Then :
1. A. $α \in N$ and $β \in N$
2. B. $α \in N$ and $β \in / N$
3. C. $α \in / N$ and $β \in N$
4. D. $α \in / N$ and $β \in / N$
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Answer: A
Q182024 · 27 Jan 2024 · Shift 2
The values of $α$ , for which $11 2 α + 3 \frac{3}{2} \frac{1}{3} 3 α + 1 α + \frac{3}{2} α + \frac{1}{3} 0 = 0$ , lie in the interval
1. A. $(- 2, 1)$
2. B. $(- \frac{3}{2}, \frac{3}{2})$
3. C. $(- 3, 0)$
4. D. $(0, 3)$
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Answer: C
Q232024 · 27 Jan 2024 · Shift 2· Numerical answer
The lines $\frac{x - 2}{2} = \frac{y}{- 2} = \frac{z - 7}{16}$ and $\frac{x + 3}{4} = \frac{y + 2}{3} = \frac{z + 2}{1}$ intersect at the point $P$ . If the distance of P from the line $\frac{x + 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{1}$ is $l$ , then $14 l^{2}$ is equal to _______.
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Answer: 108
Q262024 · 27 Jan 2024 · Shift 2· Numerical answer
If the sum of squares of all real values of $α$ , for which the lines $2 x - y + 3 = 0, 6 x + 3 y + 1 = 0$ and $α x + 2 y - 2 = 0$ do not form a triangle is $p$ , then the greatest integer less than or equal to $p$ is _______.
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Answer: 32