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2022 16 questions

Q222022 · 29 Jul 2022 · Shift 1· Numerical answer
Let $a_{1}, a_{2}, a_{3}, \dots$ be an A.P. If $\sum_{r = 1}^{\infty} \frac{a _{r}}{2 ^{r}} = 4$ , then $4 a_{2}$ is equal to ____.
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Answer: 16
Q142022 · 28 Jul 2022 · Shift 1
Let $α, β$ and $γ$ be three positive real numbers. Let $f (x) = α x^{5} + β x^{3} + γ x, x \in R$ and $g : R \to R$ be such that $g (f (x)) = x$ for all $x \in R$ . If $a_{1}, a_{2}, a_{3}, \dots, a_{n}$ be in arithmetic progression with mean zero, then the value of $f (g (\frac{1}{n} i = 1 \sum n f (a_{i})))$ is equal to :
1. A.0
2. B.3
3. C.9
4. D.27
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Answer: A
Q262022 · 28 Jul 2022 · Shift 1· Numerical answer
For $p, q \in R$ , consider the real valued function $f (x) = (x - p)^{2} - q, x \in R$ and $q > 0$ . Let $a_{1}, a_{2}, a_{3}$ and $a_{4}$ be in an arithmetic progression with mean $p$ and positive common difference. If $∣ f (a_{i}) ∣ = 500$ for all $i = 1, 2, 3, 4$ , then the absolute difference between the roots of $f (x) = 0$ is ____.
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Answer: 50
Q272022 · 28 Jul 2022 · Shift 1· Numerical answer
Let $x_{1}, x_{2}, x_{3}, \dots, x_{20}$ be in geometric progression with $x_{1} = 3$ and the common ratio $\frac{1}{2}$ . A new data is constructed replacing each $x_{i}$ by $(x_{i} - i)^{2}$ . If $\overset{x}{ˉ}$ is the mean of new data, then the greatest integer less than or equal to $\overset{x}{ˉ}$ is ____.
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Answer: 142
Q62022 · 27 Jul 2022 · Shift 1
Suppose $a_{1}, a_{2}, \dots, a_{n}, \dots$ be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is $5 : 17$ and $110 < a_{15} < 120$ , then the sum of the first ten terms of the progression is equal to
1. A.290
2. B.380
3. C.460
4. D.510
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Answer: B
Q132022 · 27 Jul 2022 · Shift 1
Let $A (1, 1), B (- 4, 3), C (- 2, - 5)$ be vertices of a triangle $A B C$ , $P$ be a point on side $B C$ , and $Δ_{1}$ and $Δ_{2}$ be the areas of triangles $A P B$ and $A B C$ , respectively. If $Δ_{1} : Δ_{2} = 4 : 7$ , then the area enclosed by the lines $A P, A C$ and the $x$ -axis is :
1. A. $\frac{1}{4}$
2. B. $\frac{3}{4}$
3. C. $\frac{1}{2}$
4. D.1
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Answer: C
Q62022 · 27 Jul 2022 · Shift 2
Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be $5$ . Let the sum of its first five terms be $\frac{98}{25}$ . Then the sum of the first $21$ terms of an AP, whose first term is $10 ar$ , $n^{th}$ term is $a_{n}$ and the common difference is $10 ar^{2}$ , is equal to :
1. A. $21 a_{11}$
2. B. $22 a_{11}$
3. C. $15 a_{16}$
4. D. $14 a_{16}$
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Answer: A
Q52022 · 26 Jul 2022 · Shift 1
Consider two G.Ps. $2, 2^{2}, 2^{3}, \dots$ and $4, 4^{2}, 4^{3}, \dots$ of 60 and n terms respectively. If the geometric mean of all the $60 + n$ terms is $(2)^{\frac{225}{8}}$ , then $\sum_{k = 1}^{n} k (n - k)$ is equal to :
1. A.560
2. B.1540
3. C.1330
4. D.2600
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Answer: C
Q282022 · 26 Jul 2022 · Shift 1· Numerical answer
The series of positive multiples of 3 is divided into sets : ${3}$ , ${6, 9, 12}$ , ${15, 18, 21, 24, 27}$ , \dots . Then the sum of the elements in the $1 1^{th}$ set is equal to ____ .
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Answer: 6993
Q292022 · 30 Jun 2022 · Shift 1· Numerical answer
Let for $f (x) = a_{0} x^{2} + a_{1} x + a_{2}$ , $f^{'} (0) = 1$ and $f^{'} (1) = 0$ . If $a_{0}$ , $a_{1}$ , $a_{2}$ are in an arithmetico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then $f (4)$ is equal to ____ .
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Answer: 2
Q52022 · 28 Jun 2022 · Shift 2
If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :
1. A.21
2. B.22
3. C.23
4. D.24
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Answer: C
Q252022 · 28 Jun 2022 · Shift 2· Numerical answer
Let for $n = 1, 2, \dots, 50$ , $S_{n}$ be the sum of the infinite geometric progression whose first term is $n^{2}$ and whose common ratio is $\frac{1}{( n + 1 ) ^{2}}$ . Then the value of $\frac{1}{26} + \sum_{n = 1}^{50} (S_{n} + \frac{2}{n + 1} - n - 1)$ is equal to ________.
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Answer: 41651
Q282022 · 27 Jun 2022 · Shift 1· Numerical answer
If the sum of the first ten terms of the series $\frac{1}{5} + \frac{2}{65} + \frac{3}{325} + \frac{4}{1025} + \frac{5}{2501} + \dots$ is $\frac{m}{n}$ , where $m$ and $n$ are co-prime numbers, then $m + n$ is equal to ____.
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Answer: 276
Q22022 · 27 Jun 2022 · Shift 2
If $a_{1}, a_{2}, a_{3} \dots\dots$ and $b_{1}, b_{2}, b_{3} \dots\dots$ are A.P., and $a_{1} = 2, a_{10} = 3, a_{1} b_{1} = 1 = a_{10} b_{10}$ , then $a_{4} b_{4}$ is equal to -
1. A. $\frac{35}{27}$
2. B.1
3. C. $\frac{27}{28}$
4. D. $\frac{28}{27}$
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Answer: D
Q262022 · 26 Jun 2022 · Shift 2· Numerical answer
If $a_{1} (> 0), a_{2}, a_{3}, a_{4}, a_{5}$ are in a G.P., $a_{2} + a_{4} = 2 a_{3} + 1$ and $3 a_{2} + a_{3} = 2 a_{4}$ , then $a_{2} + a_{4} + 2 a_{5}$ is equal to ________.
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Answer: 40
Q302022 · 25 Jun 2022 · Shift 1· Numerical answer
The greatest integer less than or equal to the sum of first 100 terms of the sequence $\frac{1}{3}$ , $\frac{5}{9}$ , $\frac{19}{27}$ , $\frac{65}{81}$ , $\dots$ is equal to .
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Answer: 98