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2023 63 questions

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
Q102023 · 8 Apr 2023 · Shift 2
If $A = [1 λ 510], A^{- 1} = α A + β I$ and $α + β = - 2$ , then $4 α^{2} + β^{2} + λ^{2}$ is equal to :
1. A. $12$
2. B. $10$
3. C. $19$
4. D. $14$
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Answer: D
Q72023 · 6 Apr 2023 · Shift 1
Let $5 f (x) + 4 f (\frac{1}{x}) = \frac{1}{x} + 3, x > 0$ . Then $18 \int_{1}^{2} f (x) d x$ is equal to :
1. A. $10 lo g_{e} 2 + 6$
2. B. $5 lo g_{e} 2 - 3$
3. C. $10 lo g_{e} 2 - 6$
4. D. $5 lo g_{e} 2 + 3$
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Answer: C
Q132023 · 6 Apr 2023 · Shift 1
If $2 x^{y} + 3 y^{x} = 20$ , then $\frac{d y}{d x}$ at $(2, 2)$ is equal to :
1. A. $- (\frac{3 + l o g _{e} 16}{4 + l o g _{e} 8})$
2. B. $- (\frac{2 + l o g _{e} 8}{3 + l o g _{e} 4})$
3. C. $- (\frac{3 + l o g _{e} 8}{2 + l o g _{e} 4})$
4. D. $- (\frac{3 + l o g _{e} 4}{2 + l o g _{e} 8})$
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Answer: B
Q182023 · 6 Apr 2023 · Shift 1
From the top A of a vertical wall AB of height $30 m$ , the angles of depression of the top P and bottom Q of a vertical tower PQ are $1 5^{\circ}$ and $6 0^{\circ}$ respectively, B and Q are on the same horizontal level. If C is a point on AB such that $CB = PQ$ , then the area (in $m^{2}$ ) of the quadrilateral BCPQ is equal to :
1. A. $200 (3 - 3)$
2. B. $300 (3 - 1)$
3. C. $300 (3 + 1)$
4. D. $600 (3 - 1)$
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Answer: D
Q202023 · 6 Apr 2023 · Shift 1
If $^{2 n} C_{3} :^{n} C_{3} = 10 : 1$ , then the ratio $(n^{2} + 3 n) : (n^{2} - 3 n + 4)$ is :
1. A. $27 : 11$
2. B. $2 : 1$
3. C. $35 : 16$
4. D. $65 : 37$
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Answer: B
Q152023 · 6 Apr 2023 · Shift 2
If $g cd (m, n) = 1$ and $1^{2} - 2^{2} + 3^{2} - 4^{2} + \dots + (2021)^{2} - (2022)^{2} + (2023)^{2} = 1012 m^{2} n$ then $m^{2} - n^{2}$ is equal to :
1. A.220
2. B.200
3. C.240
4. D.180
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Answer: C
Q272023 · 6 Apr 2023 · Shift 2· Numerical answer
If $(20)^{19} + 2 (21) (20)^{18} + 3 (21)^{2} (20)^{17} + \dots + 20 (21)^{19} = k (20)^{19},$ then $k$ is equal to ________.
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Answer: 400
Q282023 · 6 Apr 2023 · Shift 2· Numerical answer
The number of points, where the curve $y = x^{5} - 20 x^{3} + 50 x + 2$ crosses the $x$ -axis, is ________.
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Answer: 5
Q12023 · 1 Feb 2023 · Shift 1
The value of $\frac{1}{1 ! 50 !} + \frac{1}{3 ! 48 !} + \frac{1}{5 ! 46 !} + \dots + \frac{1}{49 ! 2 !} + \frac{1}{51 ! 1 !}$ is :
1. A. $\frac{2 ^{51}}{50 !}$
2. B. $\frac{2 ^{51}}{51 !}$
3. C. $\frac{2 ^{50}}{50 !}$
4. D. $\frac{2 ^{50}}{51 !}$
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Answer: D
Q62023 · 1 Feb 2023 · Shift 1
Let $S = {x : x \in R and (3 + 2)^{x^{2} - 4} + (3 - 2)^{x^{2} - 4} = 10}$ . Then $n (S)$ is equal to
1. A. $6$
2. B. $4$
3. C. $0$
4. D. $2$
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Answer: B
Q142023 · 1 Feb 2023 · Shift 1
$n \to \infty lim [\frac{1}{1 + n} + \frac{1}{2 + n} + \frac{1}{3 + n} + \dots + \frac{1}{2 n}]$ is equal to
1. A. $0$
2. B. $lo g_{e} 2$
3. C. $lo g_{e} (\frac{2}{3})$
4. D. $lo g_{e} (\frac{3}{2})$
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Answer: B
Q212023 · 1 Feb 2023 · Shift 1· Numerical answer
If $\int_{0}^{1} (x^{21} + x^{14} + x^{7}) (2 x^{14} + 3 x^{7} + 6)^{1/7} d x = \frac{1}{l} (11)^{m / n}$ where $l, m, n \in N$ , $m$ and $n$ are coprime then $l + m + n$ is equal to ________.
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Answer: 63
Q242023 · 1 Feb 2023 · Shift 1· Numerical answer
The remainder, when $1 9^{200} + 2 3^{200}$ is divided by 49 , is ________.
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Answer: 29
Q282023 · 1 Feb 2023 · Shift 1· Numerical answer
$A (2, 6, 2), B (- 4, 0, λ), C (2, 3, - 1)$ and $D (4, 5, 0), ∣ λ ∣ \leq 5$ are the vertices of a quadrilateral $A B C D$ . If its area is 18 square units, then $5 - 6 λ$ is equal to ________.
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Answer: 11
Q212023 · 31 Jan 2023 · Shift 1· Numerical answer
The remainder on dividing $5^{99}$ by 11 is ___________.
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Answer: 9
Q32023 · 31 Jan 2023 · Shift 2
The equation $e^{4 x} + 8 e^{3 x} + 13 e^{2 x} - 8 e^{x} + 1 = 0, x \in R$ has :
1. A.two solutions and both are negative
2. B.two solutions and only one of them is negative
3. C.four solutions two of which are negative
4. D.no solution
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Answer: A
Q242023 · 31 Jan 2023 · Shift 2· Numerical answer
If $^{2 n + 1} P_{n - 1} :^{2 n - 1} P_{n} = 11 : 21$ , then $n^{2} + n + 15$ is equal to :
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Answer: 45
Q302023 · 31 Jan 2023 · Shift 2· Numerical answer
The sum $1^{2} - 2 \cdot 3^{2} + 3 \cdot 5^{2} - 4 \cdot 7^{2} + 5 \cdot 9^{2} - \dots + 15 \cdot 2 9^{2}$ is ___________.
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Answer: 6952
Q42023 · 30 Jan 2023 · Shift 1
If $[t]$ denotes the greatest integer $\leq t$ , then the value of $\frac{3 ( e - 1 )}{e} \int_{1}^{2} x^{2} e^{[x] + [x^{3}]} d x$ is :
1. A. $e^{8} - e$
2. B. $e^{7} - 1$
3. C. $e^{9} - e$
4. D. $e^{8} - 1$
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Answer: A