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2026 15 questions

Q62026 · 6 Apr 2026 · Shift 1
The sum of the first ten terms of an A.P. is $160$ and the sum of the first two terms of a G.P. is $8$ . If the first term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:
1. A. $\frac{34}{9}$
2. B. $\frac{34}{13}$
3. C. $\frac{32}{9}$
4. D. $\frac{32}{13}$
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Answer: A
Q222026 · 6 Apr 2026 · Shift 2· Numerical answer
If $(1 - x^{3})^{10} = \sum_{r = 0}^{10} a_{r} x^{r} (1 - x)^{30 - 2 r}$ , then $\frac{9 a _{9}}{a _{10}}$ is equal to ____ .
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Answer: 30
Q22026 · 5 Apr 2026 · Shift 1
Let the sum of the first $n$ terms of an A.P. be $3 n^{2} + 5 n$ . Then the sum of squares of the first $10$ terms of the A.P. is:
1. A.10220
2. B.12860
3. C.15220
4. D.19780
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Answer: C
Q52026 · 5 Apr 2026 · Shift 2
If the sum of the first 10 terms of the series $\frac{1}{1 + 1 ^{4} \times 4} + \frac{2}{1 + 2 ^{4} \times 4} + \frac{3}{1 + 3 ^{4} \times 4} + \frac{4}{1 + 4 ^{4} \times 4} + \dots$ is $\frac{m}{n}$ , $g cd (m, n) = 1$ , then $m + n$ is equal to :
1. A. $256$
2. B. $264$
3. C. $276$
4. D. $284$
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Answer: C
Q62026 · 5 Apr 2026 · Shift 2
Let $A_{1}, A_{2}, A_{3}, \dots, A_{39}$ be $39$ arithmetic means between the numbers $59$ and $159$ . Then the mean of $A_{25}, A_{28}, A_{31}$ and $A_{36}$ is equal to :
1. A. $129$
2. B. $136$
3. C. $131.50$
4. D. $134$
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Answer: D
Q72026 · 4 Apr 2026 · Shift 1
The first term of an A.P. of 30 non-negative terms is $\frac{10}{3}$ . If the sum of this A.P. is the cube of its last term, then its common difference is:
1. A. $\frac{5}{87}$
2. B. $\frac{25}{83}$
3. C. $\frac{15}{29}$
4. D. $\frac{5}{29}$
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Answer: A
Q52026 · 2 Apr 2026 · Shift 1
Let $A$ be the set of first $101$ terms of an A.P., whose first term is $1$ and the common difference is $5$ and let $B$ be the set of first $71$ terms of an A.P., whose first term is $9$ and the common difference is $7$ . Then the number of elements in $A \cap B$ , which are divisible by $3$ , is :
1. A.4
2. B.5
3. C.6
4. D.7
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Answer: B
Q42026 · 2 Apr 2026 · Shift 2
Gas 'A' undergoes change from state 'X' to state 'Y'. In this process, the heat absorbed and work done by the gas is 10 J and 18 J respectively. Now gas is brought back to state 'X' by another process during which 6 J of heat is evolved. In the reverse process of 'Y' to 'X',
1. A.18 J of the work is done by the gas 'A'.
2. B.2 J of the work is done by the gas 'A'.
3. C.12 J of the work is done on the gas 'A' by the surrounding.
4. D.14 J of the work is done on the gas 'A' by the surrounding.
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Answer: D
Q142026 · 28 Jan 2026 · Shift 1
The common difference of the A.P.: $a_{1}, a_{2}, \dots, a_{m}$ is $13$ more than the common difference of the A.P.: $b_{1}, b_{2}, \dots, b_{n}$ . If $b_{31} = - 277, b_{43} = - 385$ and $a_{78} = 327$ , then $a_{1}$ is equal to
1. A.21
2. B.19
3. C.24
4. D.16
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Answer: B
Q52026 · 28 Jan 2026 · Shift 2
Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}, a > 2$ . If $α$ is such that $a, 4, α, b$ are in A.P., then the equation $α x^{2} - a x + 2 (α - 2 b) = 0$ has :
1. A.one root in $(1, 4)$ and another in $(- 2, 0)$
2. B.one root in $(0, 2)$ and another in $(- 4, - 2)$
3. C.both roots in the interval $(- 2, 0)$
4. D.complex roots of magnitude less than 2
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Answer: A
Q142026 · 24 Jan 2026 · Shift 1
Let $729, 81, 9, 1, \dots$ be a sequence and $P_{n}$ denote the product of the first $n$ terms of this sequence. If $2 \sum_{n = 1}^{40} (P_{n})^{\frac{1}{n}} = \frac{3 ^{α} - 1}{3 ^{β}}$ and $g cd (α, β) = 1$ , then $α + β$ is equal to
1. A.73
2. B.74
3. C.75
4. D.76
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Answer: A
Q172026 · 24 Jan 2026 · Shift 1
Consider an A.P.: $a_{1}, a_{2}, \dots, a_{n}; a_{1} > 0$ . If $a_{2} - a_{1} = \frac{- 3}{4}$ , $a_{n} = \frac{1}{4} a_{1}$ , and $\sum_{i = 1}^{n} a_{i} = \frac{525}{2}$ , then $\sum_{i = 1}^{17} a_{i}$ is equal to
1. A.238
2. B.136
3. C.476
4. D.952
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Answer: A
Q162026 · 23 Jan 2026 · Shift 1
The sum of all possible values of $n \in N$ , so that the coefficients of $x$ , $x^{2}$ and $x^{3}$ in the expansion of $(1 + x^{2})^{2} (1 + x)^{n}$ , are in arithmetic progression is :
1. A.12
2. B.9
3. C.3
4. D.7
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Answer: B
Q142026 · 22 Jan 2026 · Shift 1
If the sum of the first four terms of an A.P. is 6 and the sum of its first six terms is 4 , then the sum of its first twelve terms is
1. A.-26
2. B.-20
3. C.-24
4. D.-22
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Answer: D
Q122026 · 21 Jan 2026 · Shift 2
Let $a_{1}, \frac{a _{2}}{2}, \frac{a _{3}}{2 ^{2}}, \dots, \frac{a _{10}}{2 ^{9}}$ be a G.P. of common ratio $\frac{1}{2}$ . If $a_{1} + a_{2} + \dots + a_{10} = 62$ , then $a_{1}$ is equal to:
1. A. $2 - 1$
2. B. $2 (2 - 1)$
3. C. $2 - 2$
4. D. $2 (2 - 2)$
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Answer: B