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2025 12 questions

Q32025 · 4 Apr 2025 · Shift 1
Let $A = {1, 6, 11, 16, \dots}$ and $B = {9, 16, 23, 30, \dots}$ be the sets consisting of the first $2025$ terms of two arithmetic progressions. Then $n (A \cup B)$ is
1. A. $3814$
2. B. $4003$
3. C. $4027$
4. D. $3761$
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Answer: D
Q82025 · 4 Apr 2025 · Shift 2
If the sum of the first 20 terms of the series $\frac{4 \cdot 1}{4 + 3 \cdot 1 ^{2} + 1 ^{4}} + \frac{4 \cdot 2}{4 + 3 \cdot 2 ^{2} + 2 ^{4}} + \frac{4 \cdot 3}{4 + 3 \cdot 3 ^{2} + 3 ^{4}} + \frac{4 \cdot 4}{4 + 3 \cdot 4 ^{2} + 4 ^{4}} + \dots$ is $\frac{m}{n}$ , where $m$ and $n$ are coprime, then $m + n$ is equal to :
1. A.423
2. B.421
3. C.422
4. D.420
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Answer: B
Q152025 · 4 Apr 2025 · Shift 2
The sum of the infinite series $cot^{- 1} (\frac{7}{4}) + cot^{- 1} (\frac{19}{4}) + cot^{- 1} (\frac{39}{4}) + cot^{- 1} (\frac{67}{4}) + \dots$ is :
1. A. $\frac{π}{2} + cot^{- 1} (\frac{1}{2})$
2. B. $\frac{π}{2} - cot^{- 1} (\frac{1}{2})$
3. C. $\frac{π}{2} - tan^{- 1} (\frac{1}{2})$
4. D. $\frac{π}{2} + tan^{- 1} (\frac{1}{2})$
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Answer: C
Q182025 · 4 Apr 2025 · Shift 2
Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and $q$ respectively. Let d and D be the common differences of $A P^{'} s$ in $A$ and $B$ respectively such that $D = d + 3, d > 0$ . If $\frac{p + q}{p - q} = \frac{19}{5}$ , then $p - q$ is equal to
1. A.540
2. B.450
3. C.600
4. D.630
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Answer: A
Q202025 · 2 Apr 2025 · Shift 1
Let $a_{1}, a_{2}, a_{3}, \dots$ be in an A.P. such that $\sum_{k = 1}^{12} a_{2 k - 1} = - \frac{72}{5} a_{1}, a_{1} \neq = 0$ . If $\sum_{k = 1}^{n} a_{k} = 0$ , then $n$ is :
1. A.18
2. B.17
3. C.11
4. D.10
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Answer: C
Q142025 · 2 Apr 2025 · Shift 2
The number of terms of an A.P. is even; the sum of all the odd terms is 24 , the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$ . Then the number of terms which are integers in the A.P. is :
1. A.6
2. B.4
3. C.8
4. D.10
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Answer: B
Q232025 · 2 Apr 2025 · Shift 2· Numerical answer
If the sum of the first 10 terms of the series $\frac{4 \cdot 1}{1 + 4 \cdot 1 ^{4}} + \frac{4 \cdot 2}{1 + 4 \cdot 2 ^{4}} + \frac{4 \cdot 3}{1 + 4 \cdot 3 ^{4}} + \dots$ is $\frac{m}{n}$ , where $g cd (m, n) = 1$ , then $m + n$ is equal to ____.
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Answer: 441
Q252025 · 29 Jan 2025 · Shift 2· Numerical answer
Let $a_{1}, a_{2}, \dots, a_{2024}$ be an Arithmetic Progression such that $a_{1} + (a_{5} + a_{10} + a_{15} + \dots + a_{2020}) + a_{2024} = 2233$ . Then $a_{1} + a_{2} + a_{3} + \dots + a_{2024}$ is equal to ____.
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Answer: 11132
Q82025 · 28 Jan 2025 · Shift 1
Let $T_{r}$ be the $r^{th}$ term of an A.P. If for some $m, T_{m} = \frac{1}{25}$ , $T_{25} = \frac{1}{20}$ , and $20 \sum_{r = 1}^{25} T_{r} = 13$ , then $5 m \sum_{r = m}^{2 m} T_{r}$ is equal to
1. A.98
2. B.126
3. C.112
4. D.142
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Answer: B
Q162025 · 24 Jan 2025 · Shift 1
Let $S_{n} = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots$ upto $n$ terms. If the sum of the first six terms of an A.P. with first term -p and common difference p is $2026 S_{2025}$ , then the absolute difference between $2 0^{th}$ and $1 5^{th}$ terms of the A.P. is
1. A. $20$
2. B. $45$
3. C. $90$
4. D. $25$
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Answer: D
Q12025 · 24 Jan 2025 · Shift 2
In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$ , then $S_{30} - S_{10}$ is equal to :
1. A.525
2. B.505
3. C.510
4. D.515
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Answer: D
Q242025 · 23 Jan 2025 · Shift 2· Numerical answer
The roots of the quadratic equation $3 x^{2} - p x + q = 0$ are $1 0^{th}$ and $1 1^{th}$ terms of an arithmetic progression with common difference $\frac{3}{2}$ . If the sum of the first 11 terms of this arithmetic progression is 88 , then $q - 2 p$ is equal to _________.
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Answer: 474