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2021 44 questions

Q202021 · 18 Mar 2021 · Shift 1
$\frac{1}{3 ^{2} - 1} + \frac{1}{5 ^{2} - 1} + \frac{1}{7 ^{2} - 1} + \dots + \frac{1}{( 201 ) ^{2} - 1}$ is equal to
1. A. $\frac{101}{404}$
2. B. $\frac{25}{101}$
3. C. $\frac{101}{408}$
4. D. $\frac{99}{400}$
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Answer: B
Q232021 · 18 Mar 2021 · Shift 1· Numerical answer
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Answer: 4
Q262021 · 18 Mar 2021 · Shift 1· Numerical answer
If $f (x) = \int \frac{5 x ^{8} + 7 x ^{6}}{( x ^{2} + 1 + 2 x ^{7} ) ^{2}} d x$ , $(x \geq 0)$ , $f (0) = 0$ and $f (1) = \frac{1}{K}$ , then the value of K is
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Answer: 4
Q272021 · 18 Mar 2021 · Shift 1· Numerical answer
A square ABCD has all its vertices on the curve $x^{2} y^{2} = 1$ . The midpoints of its sides also lie on the same curve. Then, the square of area of ABCD is _______.
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Answer: 80
Q132021 · 18 Mar 2021 · Shift 2
Let $f : R - {3} \to R - {1}$ be defined by $f (x) = \frac{x - 2}{x - 3}$ . Let $g : R \to R$ be given as $g (x) = 2 x - 3$ . Then, the sum of all the values of x for which $f^{- 1} (x) + g^{- 1} (x) = \frac{13}{2}$ is equal to :
1. A.3
2. B.5
3. C.2
4. D.7
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Answer: B
Q212021 · 18 Mar 2021 · Shift 2· Numerical answer
If $\sum_{r = 1}^{10} r! (r^{3} + 6 r^{2} + 2 r + 5) = α (11!)$ , then the value of $α$ is equal to ____.
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Answer: 160
Q202021 · 16 Mar 2021 · Shift 1
Let $[x]$ denote greatest integer less than or equal to x. If for $n \in N$ , $(1 - x + x^{3})^{n} = \sum_{j = 0}^{3 n} a_{j} x^{j}$ , then $\sum_{j = 0}^{[\frac{3 n}{2}]} a_{2 j} + 4 \sum_{j = 0}^{[\frac{3 n - 1}{2}]} a_{2 j + 1}$ is equal to :
1. A. $2^{n} - 1$
2. B.n
3. C.2
4. D.1
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Answer: D
Q42021 · 26 Feb 2021 · Shift 1
The value of $\int_{- π /2}^{π /2} \frac{c o s ^{2} x}{1 + 3 ^{x}} d x$ is :
1. A. $2 π$
2. B. $\frac{π}{2}$
3. C. $4 π$
4. D. $\frac{π}{4}$
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Answer: D
Q72021 · 26 Feb 2021 · Shift 1
The value of $\sum_{n = 1}^{100} \int_{n - 1}^{n} e^{x - [x]} d x$ , where $[x]$ is the greatest integer $\leq x$ , is :
1. A. $100 e$
2. B. $100 (e - 1)$
3. C. $100 (1 + e)$
4. D. $100 (1 - e)$
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Answer: B
Q102021 · 26 Feb 2021 · Shift 1
The value of $(a + 1) (a + 2) (a + 2) (a + 3) (a + 3) (a + 4) a + 2 a + 3 a + 4 111$ is :
1. A. $- 2$
2. B. $0$
3. C. $(a + 2) (a + 3) (a + 4)$
4. D. $(a + 1) (a + 2) (a + 3)$
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Answer: A
Q222021 · 26 Feb 2021 · Shift 1· Numerical answer
The number of solutions of the equation $lo g_{4} (x - 1) = lo g_{2} (x - 3)$ is _________.
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Answer: 1
Q302021 · 26 Feb 2021 · Shift 1· Numerical answer
Let $m, n \in N$ and $g cd (2, n) = 1$ . If $30 (300) + 29 (301) + \dots\dots + 2 (3028) + 1 (3029) = n \cdot 2^{m}$ , then $n + m$ is equal to _________. (Here $(n k) =^{n} C_{k}$ )
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Answer: 45
Q152021 · 26 Feb 2021 · Shift 2
Let $F_{1} (A, B, C) = (A \land \sim B) \lor [\sim C \land (A \lor B)] \lor \sim A$ and $F_{2} (A, B) = (A \lor B) \lor (B \to \sim A)$ be two logical expressions. Then :
1. A.Both $F_{1}$ and $F_{2}$ are not tautologies
2. B. $F_{1}$ and $F_{2}$ both are tautologies
3. C. $F_{1}$ is not a tautology but $F_{2}$ is a tautology
4. D. $F_{1}$ is a tautology but $F_{2}$ is not a tautology
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Answer: C
Q232021 · 26 Feb 2021 · Shift 2· Numerical answer
Let $α$ and $β$ be two real numbers such that $α + β = 1$ and $α β = - 1$ . Let $p_{n} = (α)^{n} + (β)^{n}$ , $p_{n - 1} = 11$ and $p_{n + 1} = 29$ for some integer $n \geq 1$ . Then, the value of $p_{n}^{2}$ is _________.
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Answer: 324
Q292021 · 26 Feb 2021 · Shift 2· Numerical answer
Let the normals at all the points on a given curve pass through a fixed point $(a, b)$ . If the curve passes through $(3, - 3)$ and $(4, - 22)$ , and given that $a - 22 b = 3$ , then $(a^{2} + b^{2} + ab)$ is equal to _________.
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Answer: 9
Q62021 · 25 Feb 2021 · Shift 2
Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ for $n \geq 1$ , then the value of $\frac{a _{10} - 2 a _{8}}{3 a _{9}}$ is :
1. A.3
2. B.2
3. C.4
4. D.1
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Answer: B
Q92021 · 25 Feb 2021 · Shift 2
If $α, β \in R$ are such that $1 - 2 i$ (here $i^{2} = - 1$ ) is a root of $z^{2} + α z + β = 0$ , then $(α - β)$ is equal to :
1. A. $- 7$
2. B.7
3. C.3
4. D. $- 3$
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Answer: A
Q212021 · 25 Feb 2021 · Shift 2· Numerical answer
If the remainder when x is divided by 4 is 3, then the remainder when $(2020 + x)^{2022}$ is divided by 8 is ______ .
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Answer: 1
Q302021 · 25 Feb 2021 · Shift 2· Numerical answer
If the curves $x = y^{4}$ and $x y = k$ cut at right angles, then $(4 k)^{6}$ is equal to ______ .
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Answer: 4