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2022 53 questions

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
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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
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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
Q12022 · 30 Jun 2022 · Shift 1
Let $S_{1} = {x \in R - {1, 2} : \frac{( x + 2 ) ( x ^{2} + 3 x + 5 )}{- 2 + 3 x - x ^{2}} \geq 0}$ and $S_{2} = {x \in R : 3^{2 x} - 3^{x + 1} - 3^{x + 2} + 27 \leq 0}$ . Then, $S_{1} \cup S_{2}$ is equal to :
1. A. $(- \infty, - 2] \cup (1, 2)$
2. B. $(- \infty, - 2] \cup [1, 2]$
3. C. $(- 2, 1] \cup [2, \infty)$
4. D. $(- \infty, 2]$
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Answer: B
Q72022 · 30 Jun 2022 · Shift 1
Let $f (x) = 4 x^{3} - 11 x^{2} + 8 x - 5$ , $x \in R$ . Then f :
1. A.has a local minima at $x = \frac{1}{2}$
2. B.has a local minima at $x = \frac{3}{4}$
3. C.is increasing in $(\frac{1}{2}, \frac{3}{4})$
4. D.is decreasing in $(\frac{1}{2}, \frac{4}{3})$
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Answer: D
Q132022 · 30 Jun 2022 · Shift 1
The value of $1 + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \dots + \frac{1}{1 + 2 + 3 + \dots + 11}$ is equal to:
1. A. $\frac{20}{11}$
2. B. $\frac{11}{6}$
3. C. $\frac{241}{132}$
4. D. $\frac{21}{11}$
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Answer: B
Q252022 · 30 Jun 2022 · Shift 1· Numerical answer
A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on $4^{th}$ day is 30, then number of infected students on $8^{th}$ day will be ____ .
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Answer: 90
Q92022 · 27 Jun 2022 · Shift 1
If $\frac{d y}{d x} + \frac{2 ^{x - y} ( 2 ^{y} - 1 )}{2 ^{x} - 1} = 0$ , $x, y > 0$ , $y (1) = 1$ , then $y (2)$ is equal to :
1. A. $2 + lo g_{2} 3$
2. B. $2 + lo g_{3} 2$
3. C. $2 - lo g_{3} 2$
4. D. $2 - lo g_{2} 3$
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Answer: D
Q202022 · 27 Jun 2022 · Shift 1
The boolean expression $(\sim (p \land q)) \lor q$ is equivalent to :
1. A. $q \to (p \land q)$
2. B. $p \to q$
3. C. $p \to (p \to q)$
4. D. $p \to (p \lor q)$
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Answer: D