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2025 40 questions

Q72025 · 28 Jan 2025 · Shift 1
If $\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{96 x ^{2} c o s ^{2} x}{( 1 + e ^{x} )} d x = π (α π^{2} + β), α, β \in Z$ , then $(α + β)^{2}$ equals
1. A.196
2. B.100
3. C.64
4. D.144
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Answer: B
Q232025 · 28 Jan 2025 · Shift 1· Numerical answer
If $α = 1 + \sum_{r = 1}^{6} (- 3)^{r - 1}^{12} C_{2 r - 1}$ , then the distance of the point $(12, 3)$ from the line $α x - 3 y + 1 = 0$ is ____.
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Answer: 5
Q52025 · 24 Jan 2025 · Shift 1
Let $f (x) = \frac{2 ^{x + 2} + 16}{2 ^{2 x + 1} + 2 ^{x + 4} + 32}$ . Then the value of $8 (f (\frac{1}{15}) + f (\frac{2}{15}) + \dots + f (\frac{59}{15}))$ is equal to
1. A. $108$
2. B. $92$
3. C. $118$
4. D. $102$
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Answer: C
Q102025 · 24 Jan 2025 · Shift 1
Let in a $△ A B C$ , the length of the side $A C$ be $6$ , the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ . Then the area (in sq. units) of $△ A B C$ is:
1. A. $42$
2. B. $17$
3. C. $56$
4. D. $21$
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Answer: D
Q112025 · 24 Jan 2025 · Shift 1
Let the line passing through the point $(- 1, 2, 1)$ and parallel to the line $\frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4}$ 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
1. A. $56$
2. B. $5$
3. C. $55$
4. D. $10$
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Answer: C
Q112025 · 24 Jan 2025 · Shift 2
If $7 = 5 + \frac{1}{7} (5 + α) + \frac{1}{7 ^{2}} (5 + 2 α) + \frac{1}{7 ^{3}} (5 + 3 α) + \dots\dots\dots\dots \infty$ , then the value of $α$ is :
1. A. $\frac{1}{7}$
2. B.1
3. C. $\frac{6}{7}$
4. D.6
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Answer: D
Q182025 · 24 Jan 2025 · Shift 2
The number of real solution(s) of the equation $x^{2} + 3 x + 2 = min {∣ x - 3∣, ∣ x + 2∣}$ is :
1. A.2
2. B.3
3. C.1
4. D.0
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Answer: A
Q62025 · 23 Jan 2025 · Shift 2
If the square of the shortest distance between the lines $\frac{x - 2}{1} = \frac{y - 1}{2} = \frac{z + 3}{- 3}$ and $\frac{x + 1}{2} = \frac{y + 3}{4} = \frac{z + 5}{- 5}$ is $\frac{m}{n}$ , where $m, n$ are coprime numbers, then $m + n$ is equal to :
1. A.14
2. B.6
3. C.21
4. D.9
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Answer: D
Q112025 · 23 Jan 2025 · Shift 2
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 $\frac{x}{1} = \frac{y - 2}{2} = \frac{z + 3}{3}$ is :
1. A. $17$
2. B. $13$
3. C. $15$
4. D. $14$
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Answer: D
Q182025 · 23 Jan 2025 · Shift 2
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 $81 cm^{3} / min$ and the thickness of the ice-cream layer decreases at the rate of $\frac{1}{4 π} cm/min$ . The surface area (in $cm^{2}$ ) of the chocolate ball (without the ice-cream layer) is :
1. A. $128 π$
2. B. $196 π$
3. C. $225 π$
4. D. $256 π$
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Answer: D
Q52025 · 22 Jan 2025 · Shift 1
Let $A = {1, 2, 3, \dots, 10}$ and $B = {\frac{m}{n} : m, n \in A, m < n and g cd (m, n) = 1}$ . Then $n (B)$ is equal to :
1. A.29
2. B.31
3. C.37
4. D.36
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Answer: B
Q72025 · 22 Jan 2025 · Shift 1
The product of all solutions of the equation $e^{5 (l o g_{e} x)^{2} + 3} = x^{8}, x > 0$ , is :
1. A. $e^{2}$
2. B. $e$
3. C. $e^{6/5}$
4. D. $e^{8/5}$
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Answer: D
Q202025 · 22 Jan 2025 · Shift 1
If $\sum_{r = 1}^{n} T_{r} = \frac{( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ( 2 n + 5 )}{64}$ , then $lim_{n \to \infty} \sum_{r = 1}^{n} (\frac{1}{T _{r}})$ is equal to :
1. A. $\frac{2}{3}$
2. B. $\frac{1}{3}$
3. C.1
4. D.0
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Answer: A
Q232025 · 22 Jan 2025 · Shift 1· Numerical answer
If $\sum_{r = 0}^{5} \frac{^{11} C _{2 r + 1}}{2 r + 2} = \frac{m}{n}$ , $g cd (m, n) = 1$ , then $m - n$ is equal to _________.
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Answer: 2035
Q242025 · 22 Jan 2025 · Shift 1· Numerical answer
Let $L_{1} : \frac{x - 1}{3} = \frac{y - 1}{- 1} = \frac{z + 1}{0}$ and $L_{2} : \frac{x - 2}{2} = \frac{y}{0} = \frac{z + 4}{α}, α \in R$ , 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 $26 α (PB)^{2}$ is _________.
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Answer: 216