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2022 53 questions

Q22022 · 29 Jul 2022 · Shift 1
If $z = 2 + 3 i$ , then $z^{5} + (\overset{z}{ˉ})^{5}$ is equal to :
1. A.244
2. B.224
3. C.245
4. D.265
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Answer: A
Q152022 · 29 Jul 2022 · Shift 2
Let ${a_{n}}_{n = 0}^{\infty}$ be a sequence such that $a_{0} = a_{1} = 0$ and $a_{n + 2} = 3 a_{n + 1} - 2 a_{n} + 1, \forall n \geq 0$ . Then $a_{25} a_{23} - 2 a_{25} a_{22} - 2 a_{23} a_{24} + 4 a_{22} a_{24}$ is equal to
1. A.483
2. B.528
3. C.575
4. D.624
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Answer: B
Q162022 · 29 Jul 2022 · Shift 2
$\sum_{r = 1}^{20} (r^{2} + 1) (r!)$ is equal to
1. A. $22! - 21!$
2. B. $22! - 2 (21!)$
3. C. $21! - 2 (20!)$
4. D. $21! - 20!$
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Answer: B
Q222022 · 29 Jul 2022 · Shift 2· Numerical answer
Let $X = 111$ and $A = - 1 00 210 36 - 1$ . For $k \in N$ , if $X^{'} A^{k} X = 33$ , then $k$ is equal to ______.
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Answer: 10
Q292022 · 29 Jul 2022 · Shift 2· Numerical answer
If $\sum_{k = 1}^{10} k^{2} (^{10} C_{k})^{2} = 22000 L$ , then $L$ is equal to ____.
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Answer: 221
Q112022 · 28 Jul 2022 · Shift 1
The remainder when $7^{2022} + 3^{2022}$ is divided by 5 is :
1. A.0
2. B.2
3. C.3
4. D.4
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Answer: C
Q202022 · 28 Jul 2022 · Shift 1
Consider the sequence $a_{1}, a_{2}, a_{3}, \dots$ such that $a_{1} = 1, a_{2} = 2$ and $a_{n + 2} = \frac{2}{a _{n + 1}} + a_{n}$ for $n = 1, 2, 3, \dots$ . If $(\frac{a _{1} + \frac{1}{a _{2}}}{a _{3}}) \cdot (\frac{a _{2} + \frac{1}{a _{3}}}{a _{4}}) \cdot (\frac{a _{3} + \frac{1}{a _{4}}}{a _{5}}) \dots (\frac{a _{30} + \frac{1}{a _{31}}}{a _{32}}) = 2^{α} (^{61} C_{31})$ , then $α$ is equal to :
1. A. $- 30$
2. B. $- 31$
3. C. $- 60$
4. D. $- 61$
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Answer: C
Q292022 · 28 Jul 2022 · Shift 1· Numerical answer
The sum of all real values of $x$ for which $\frac{3 x ^{2} - 9 x + 17}{x ^{2} + 3 x + 10} = \frac{5 x ^{2} - 7 x + 19}{3 x ^{2} + 5 x + 12}$ is equal to ____.
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Answer: 6
Q192022 · 28 Jul 2022 · Shift 2
A horizontal park is in the shape of a triangle OAB with $AB = 16$ . A vertical lamp post OP is erected at the point O such that $∠ PAO = ∠ PBO = 1 5^{\circ}$ and $∠ PCO = 4 5^{\circ}$ , where C is the midpoint of AB. Then $(OP)^{2}$ is equal to :
1. A. $\frac{32}{3} (3 - 1)$
2. B. $\frac{32}{3} (2 - 3)$
3. C. $\frac{16}{3} (3 - 1)$
4. D. $\frac{16}{3} (2 - 3)$
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Answer: B
Q272022 · 28 Jul 2022 · Shift 2· Numerical answer
If $1 + (2 +^{49} C_{1} +^{49} C_{2} + \dots +^{49} C_{49}) (^{50} C_{2} +^{50} C_{4} + \dots +^{50} C_{50})$ is equal to $2^{n} \cdot m$ , where m is odd, then $n + m$ is equal to _________.
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Answer: 99
Q292022 · 28 Jul 2022 · Shift 2· Numerical answer
If $\frac{6}{3 ^{12}} + \frac{10}{3 ^{11}} + \frac{20}{3 ^{10}} + \frac{40}{3 ^{9}} + \dots + \frac{10240}{3} = 2^{n} \cdot m$ , where m is odd, then $m \cdot n$ is equal to _________.
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Answer: 12
Q42022 · 27 Jul 2022 · Shift 1
Let $A = (1 - 2 2 - 5)$ . Let $α, β \in R$ be such that $α A^{2} + β A = 2 I$ . Then $α + β$ is equal to
1. A. $- 10$
2. B. $- 6$
3. C.6
4. D.10
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Answer: D
Q52022 · 27 Jul 2022 · Shift 1
The remainder when $(2021)^{2022} + (2022)^{2021}$ is divided by 7 is
1. A.0
2. B.1
3. C.2
4. D.6
Show answer
Answer: A
Q282022 · 27 Jul 2022 · Shift 1· Numerical answer
Let $S = {z \in C : z^{2} + \overset{z}{ˉ} = 0}$ . Then $\sum_{z \in S} (Re (z) + Im (z))$ is equal to _________.
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Answer: 0
Q52022 · 27 Jul 2022 · Shift 2
Let $f (x) = 2 + ∣ x ∣ - ∣ x - 1∣ + ∣ x + 1∣, x \in R$ . Consider (S1) : $f^{'} (- \frac{3}{2}) + f^{'} (- \frac{1}{2}) + f^{'} (\frac{1}{2}) + f^{'} (\frac{3}{2}) = 2$ (S2) : $\int_{- 2}^{2} f (x) d x = 12$ Then,
1. A.both (S1) and (S2) are correct
2. B.both (S1) and (S2) are wrong
3. C.only (S1) is correct
4. D.only (S2) is correct
Show answer
Answer: D
Q242022 · 27 Jul 2022 · Shift 2· Numerical answer
For the curve $C : (x^{2} + y^{2} - 3) + (x^{2} - y^{2} - 1)^{5} = 0$ , the value of $3 y^{'} - y^{3} y^{''}$ , at the point $(α, α)$ , $α > 0$ , on $C$ , is equal to ____.
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Answer: 16
Q252022 · 27 Jul 2022 · Shift 2· Numerical answer
Let $f (x) = min {[x - 1], [x - 2], \dots, [x - 10]}$ where $[t]$ denotes the greatest integer $\leq t$ . Then $\int_{0}^{10} f (x) d x + \int_{0}^{10} (f (x))^{2} d x + \int_{0}^{10} ∣ f (x) ∣ d x$ is equal to ____.
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Answer: 385
Q292022 · 27 Jul 2022 · Shift 2· Numerical answer
$\frac{2 ^{3} - 1 ^{3}}{1 \times 7} + \frac{4 ^{3} - 3 ^{3} + 2 ^{3} - 1 ^{3}}{2 \times 11} + \frac{6 ^{3} - 5 ^{3} + 4 ^{3} - 3 ^{3} + 2 ^{3} - 1 ^{3}}{3 \times 15} + \dots + \frac{3 0 ^{3} - 2 9 ^{3} + 2 8 ^{3} - 2 7 ^{3} + \dots + 2 ^{3} - 1 ^{3}}{15 \times 63}$ is equal to ____.
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Answer: 120
Q22022 · 26 Jul 2022 · Shift 1
Let O be the origin and A be the point $z_{1} = 1 + 2 i$ . If B is the point $z_{2}$ , $Re (z_{2}) < 0$ , such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?
1. A. $ar g z_{2} = π - tan^{- 1} 3$
2. B. $ar g (z_{1} - 2 z_{2}) = - tan^{- 1} \frac{4}{3}$
3. C. $∣ z_{2} ∣ = 10$
4. D. $∣2 z_{1} - z_{2} ∣ = 5$
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Answer: D
Q272022 · 26 Jul 2022 · Shift 1· Numerical answer
Let Q and R be two points on the line $\frac{x + 1}{2} = \frac{y + 2}{3} = \frac{z - 1}{2}$ at a distance $26$ from the point $P (4, 2, 7)$ . Then the square of the area of the triangle $P QR$ is ____ .
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Answer: 153
Q42022 · 25 Jul 2022 · Shift 2
The remainder when $(11)^{1011} + (1011)^{11}$ is divided by 9 is
1. A.1
2. B.4
3. C.6
4. D.8
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Answer: D
Q82022 · 25 Jul 2022 · Shift 2
Let the point $P (α, β)$ be at a unit distance from each of the two lines $L_{1} : 3 x - 4 y + 12 = 0$ , and $L_{2} : 8 x + 6 y + 11 = 0$ . If $P$ lies below $L_{1}$ and above $L_{2}$ , then $100 (α + β)$ is equal to :
1. A. $- 14$
2. B. $42$
3. C. $- 22$
4. D. $14$
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Answer: D
Q152022 · 25 Jul 2022 · Shift 2
The sum $\sum_{n = 1}^{21} \frac{3}{( 4 n - 1 ) ( 4 n + 3 )}$ is equal to
1. A. $\frac{7}{87}$
2. B. $\frac{7}{29}$
3. C. $\frac{14}{87}$
4. D. $\frac{21}{29}$
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Answer: B
Q162022 · 25 Jul 2022 · Shift 2
$lim_{n \to \infty} \frac{1}{2 ^{n}} (\frac{1}{1 - \frac{1}{2 ^{n}}} + \frac{1}{1 - \frac{2}{2 ^{n}}} + \frac{1}{1 - \frac{3}{2 ^{n}}} + \dots + \frac{1}{1 - \frac{2 ^{n} - 1}{2 ^{n}}})$ is equal to
1. A. $\frac{1}{2}$
2. B.1
3. C.2
4. D. $- 2$
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Answer: C
Q232022 · 25 Jul 2022 · Shift 2· Numerical answer
Let $A = 100 a 10 a b 1, a, b \in R$ . If for some $n \in N$ , $A^{n} = 10048102160961$ then $n + a + b$ is equal to ___________.
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Answer: 24