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2023 63 questions

Q82023 · 13 Apr 2023 · Shift 1
$0 \int \infty \frac{6}{e ^{3 x} + 6 e ^{2 x} + 11 e ^{x} + 6} d x =$
1. A. $lo g_{e} (\frac{256}{81})$
2. B. $lo g_{e} (\frac{64}{27})$
3. C. $lo g_{e} (\frac{32}{27})$
4. D. $lo g_{e} (\frac{512}{81})$
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Answer: C
Q112023 · 13 Apr 2023 · Shift 1
Fractional part of the number $\frac{4 ^{2022}}{15}$ is equal to
1. A. $\frac{8}{15}$
2. B. $\frac{4}{15}$
3. C. $\frac{1}{15}$
4. D. $\frac{14}{15}$
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Answer: C
Q42023 · 13 Apr 2023 · Shift 2
Let $S = {z \in C : \overset{z}{ˉ} = i (z^{2} + Re (\overset{z}{ˉ}))}$ . Then $z \in S \sum ∣ z ∣^{2}$ is equal to :
1. A. $\frac{7}{2}$
2. B. $4$
3. C. $3$
4. D. $\frac{5}{2}$
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Answer: B
Q242023 · 13 Apr 2023 · Shift 2· Numerical answer
The remainder, when $7^{103}$ is divided by 17, is _________
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Answer: 12
Q282023 · 13 Apr 2023 · Shift 2· Numerical answer
Let $[α]$ denote the greatest integer $\leq α$ . Then $[1] + [2] + [3] + \dots + [120]$ is equal to __________
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Answer: 825
Q132023 · 12 Apr 2023 · Shift 1
If $\frac{1}{n + 1}^{n} C_{n} + \frac{1}{n}^{n} C_{n - 1} + \dots + \frac{1}{2}^{n} C_{1} +^{n} C_{0} = \frac{1023}{10}$ then $n$ is equal to :
1. A. $9$
2. B. $6$
3. C. $7$
4. D. $8$
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Answer: A
Q142023 · 12 Apr 2023 · Shift 1
Let $< a_{n} >$ be a sequence such that $a_{1} + a_{2} + \dots + a_{n} = \frac{n ^{2} + 3 n}{( n + 1 ) ( n + 2 )}$ . If $28 \sum_{k = 1}^{10} \frac{1}{a _{k}} = p_{1} p_{2} p_{3} \dots p_{m}$ , where $p_{1}, p_{2}, \dots, p_{m}$ are the first $m$ prime numbers, then $m$ is equal to
1. A. $5$
2. B. $7$
3. C. $6$
4. D. $8$
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Answer: C
Q12023 · 11 Apr 2023 · Shift 1
Let R be a rectangle given by the lines $x = 0, x = 2, y = 0$ and $y = 5$ . Let $A (α, 0)$ and $B (0, β), α \in [0, 2]$ and $β \in [0, 5]$ , be such that the line segment AB divides the area of the rectangle R in the ratio $4 : 1$ . Then, the mid-point of AB lies on a :
1. A.hyperbola
2. B.straight line
3. C.parabola
4. D.circle
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Answer: A
Q102023 · 11 Apr 2023 · Shift 1
Let $f (x) = [x^{2} - x] + ∣ - x + [x] ∣$ , where $x \in R$ and $[t]$ denotes the greatest integer less than or equal to $t$ . Then, $f$ is :
1. A.continuous at $x = 0$ , but not continuous at $x = 1$
2. B.continuous at $x = 0$ and $x = 1$
3. C.continuous at $x = 1$ , but not continuous at $x = 0$
4. D.not continuous at $x = 0$ and $x = 1$
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Answer: C
Q212023 · 11 Apr 2023 · Shift 1· Numerical answer
Let $A = 0 a 1 10 c 230$ , where $a, c \in R$ . If $A^{3} = A$ and the positive value of $a$ belongs to the interval $(n - 1, n]$ , where $n \in N$ , then $n$ is equal to _________.
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Answer: 2
Q232023 · 11 Apr 2023 · Shift 1· Numerical answer
For $m, n > 0$ , let $α (m, n) = \int_{0}^{2} t^{m} (1 + 3 t)^{n} d t$ . If $11 α (10, 6) + 18 α (11, 5) = p (14)^{6}$ , then $p$ is equal to _________.
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Answer: 32
Q302023 · 11 Apr 2023 · Shift 1· Numerical answer
Let $S = 109 + \frac{108}{5} + \frac{107}{5 ^{2}} + \dots + \frac{2}{5 ^{107}} + \frac{1}{5 ^{108}}$ . Then the value of $(16 S - (25)^{- 54})$ is equal to _________.
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Answer: 2175
Q252023 · 11 Apr 2023 · Shift 2· Numerical answer
Let $S = {z \in C - {i, 2 i} : \frac{z ^{2} + 8 i z - 15}{z ^{2} - 3 i z - 2} \in R}$ . If $α - \frac{13}{11} i \in S$ , $α \in R - {0}$ , then $242 α^{2}$ is equal to _________.
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Answer: 1680
Q262023 · 11 Apr 2023 · Shift 2· Numerical answer
The number of points, where the curve $f (x) = e^{8 x} - e^{6 x} - 3 e^{4 x} - e^{2 x} + 1$ , $x \in R$ cuts $x$ -axis, is equal to _________.
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Answer: 2
Q282023 · 10 Apr 2023 · Shift 1· Numerical answer
Let a, b, c be three distinct positive real numbers such that $(2 a)^{l o g_{e} a} = (b c)^{l o g_{e} b}$ and $b^{l o g_{e} 2} = a^{l o g_{e} c}$ . Then, $6 a + 5 b c$ is equal to _________.
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Answer: 8
Q12023 · 10 Apr 2023 · Shift 2
Let $S = {z = x + i y : \frac{2 z - 3 i}{4 z + 2 i} is a real number}$ . Then which of the following is NOT correct?
1. A. $y + x^{2} + y^{2} \neq = - \frac{1}{4}$
2. B. $(x, y) = (0, - \frac{1}{2})$
3. C. $x = 0$
4. D. $y \in (- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$
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Answer: B
Q32023 · 10 Apr 2023 · Shift 2
Let the number $(22)^{2022} + (2022)^{22}$ leave the remainder $α$ when divided by 3 and $β$ when divided by 7. Then $(α^{2} + β^{2})$ is equal to :
1. A.13
2. B.10
3. C.20
4. D.5
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Answer: D
Q142023 · 10 Apr 2023 · Shift 2
If $S_{n} = 4 + 11 + 21 + 34 + 50 + \dots$ to $n$ terms, then $\frac{1}{60} (S_{29} - S_{9})$ is equal to :
1. A.227
2. B.226
3. C.220
4. D.223
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Answer: D
Q302023 · 10 Apr 2023 · Shift 2· Numerical answer
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Answer: 6
Q142023 · 8 Apr 2023 · Shift 1
Let $P = [\frac{3}{2} - \frac{1}{2} \frac{1}{2} \frac{3}{2}], A = [1011]$ and $Q = P A P^{T}$ . If $P^{T} Q^{2007} P = [a c b d]$ , then $2 a + b - 3 c - 4 d$ equal to :
1. A. $2004$
2. B. $2006$
3. C. $2007$
4. D. $2005$
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Answer: D
Q182023 · 8 Apr 2023 · Shift 1
Let $S_{K} = \frac{1 + 2 + \dots + K}{K}$ and $\sum_{j = 1}^{n} S_{j}^{2} = \frac{n}{A} (B n^{2} + C n + D)$ , where $A, B, C, D \in N$ and $A$ has least value. Then
1. A. $A + B + C + D$ is divisible by $5$
2. B. $A + C + D$ is not divisible by $B$
3. C. $A + B = 5 (D - C)$
4. D. $A + B$ is divisible by D
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Answer: D
Q262023 · 8 Apr 2023 · Shift 1· Numerical answer
If $a_{α}$ is the greatest term in the sequence $a_{n} = \frac{n ^{3}}{n ^{4} + 147}, n = 1, 2, 3, \dots$ , then $α$ is equal to ___________.
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Answer: 5
Q292023 · 8 Apr 2023 · Shift 1· Numerical answer
The largest natural number $n$ such that $3^{n}$ divides $66!$ is ___________.
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Answer: 31
Q22023 · 8 Apr 2023 · Shift 2
$2 5^{190} - 1 9^{190} - 8^{190} + 2^{190}$ is divisible by :
1. A. $14$ but not by $34$
2. B.neither $14$ nor $34$
3. C.both $14$ and $34$
4. D. $34$ but not by $14$
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Answer: D
Q82023 · 8 Apr 2023 · Shift 2
The area of the quadrilateral $ABCD$ with vertices $A (2, 1, 1), B (1, 2, 5), C (- 2, - 3, 5)$ and $D (1, - 6, - 7)$ is equal to :
1. A. $48$
2. B. $838$
3. C. $54$
4. D. $938$
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Answer: B