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2026 44 questions

Q52026 · 24 Jan 2026 · Shift 1
The number of the real solutions of the equation: $x ∣ x + 3∣ + ∣ x - 1∣ - 2 = 0$ is
1. A.3
2. B.2
3. C.5
4. D.4
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Answer: A
Q132026 · 24 Jan 2026 · Shift 1
Let $S = \frac{1}{25 !} + \frac{1}{3 ! 23 !} + \frac{1}{5 ! 21 !} + \dots$ up to 13 terms. If $13 S = \frac{2 ^{k}}{n !}, k \in N$ , then $n + k$ is equal to
1. A.50
2. B.52
3. C.49
4. D.51
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Answer: C
Q162026 · 24 Jan 2026 · Shift 1
Let $S = {z \in C : \frac{z - 6 i}{z - 2 i} = 1 and \frac{z - 8 + 2 i}{z + 2 i} = \frac{3}{5}}$ . Then $\sum_{z \in S} ∣ z ∣^{2}$ is equal to :
1. A.413
2. B.398
3. C.385
4. D.423
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Answer: C
Q22026 · 23 Jan 2026 · Shift 1
A rectangle is formed by the lines $x = 0, y = 0, x = 3$ and $y = 4$ . Let the line L be perpendicular to $3 x + y + 6 = 0$ and divide the area of the rectangle into two equal parts. Then the distance of the point $(\frac{1}{2}, - 5)$ from the line $L$ is equal to :
1. A. $10$
2. B. $25$
3. C. $210$
4. D. $310$
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Answer: C
Q122026 · 23 Jan 2026 · Shift 1
Let $a = - \hat{i} + \hat{j} + 2 \hat{k}, b = \hat{i} - \hat{j} - 3 \hat{k}, c = a \times b$ and $d = c \times a$ . Then $(a - b) \cdot d$ is equal to :
1. A.2
2. B.-2
3. C.4
4. D.-4
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Answer: B
Q132026 · 23 Jan 2026 · Shift 1
A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :
1. A.36 days
2. B.24 days
3. C.30 days
4. D.42 days
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Answer: A
Q172026 · 23 Jan 2026 · Shift 1
The value of $\frac{^{100} C _{50}}{51} + \frac{^{100} C _{51}}{52} + \dots + \frac{^{100} C _{100}}{101}$ is:
1. A. $\frac{2 ^{101}}{101}$
2. B. $\frac{2 ^{100}}{101}$
3. C. $\frac{2 ^{100}}{100}$
4. D. $\frac{2 ^{101}}{100}$
Show answer
Answer: B
Q22026 · 23 Jan 2026 · Shift 2
Let $\sum_{k = 1}^{n} a_{k} = α n^{2} + β n$ . If $a_{10} = 59$ and $a_{6} = 7 a_{1}$ , then $α + β$ is equal to :
1. A.3
2. B.5
3. C.7
4. D.12
Show answer
Answer: B
Q32026 · 23 Jan 2026 · Shift 2
If $z = \frac{3}{2} + \frac{i}{2}, i = - 1$ , then $(z^{201} - i)^{8}$ is equal to
1. A.1
2. B.0
3. C.-1
4. D.256
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Answer: D
Q42026 · 22 Jan 2026 · Shift 1
The number of distinct real solutions of the equation $x ∣ x + 4∣ + 3∣ x + 2∣ + 10 = 0$ is
1. A.2
2. B.3
3. C.0
4. D.1
Show answer
Answer: D
Q242026 · 22 Jan 2026 · Shift 1· Numerical answer
Let $α = \frac{- 1 + i 3}{2}$ and $β = \frac{- 1 - i 3}{2}, i = - 1$ . If $(7 - 7 α + 9 β)^{20} + (9 + 7 α - 7 β)^{20} + (- 7 + 9 α + 7 β)^{20} + (14 + 7 α + 7 β)^{20} = m^{10}$ , then $m$ is ____
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Answer: 49
Q22026 · 21 Jan 2026 · Shift 1
If $x^{2} + x + 1 = 0$ , then the value of $(x + \frac{1}{x})^{4} + (x^{2} + \frac{1}{x ^{2}})^{4} + (x^{3} + \frac{1}{x ^{3}})^{4} + \dots + (x^{25} + \frac{1}{x ^{25}})^{4}$ is:
1. A.162
2. B.145
3. C.128
4. D.175
Show answer
Answer: B
Q222026 · 21 Jan 2026 · Shift 1· Numerical answer
Let $S = {(m, n) : m, n \in {1, 2, 3, \dots\dots, 50}}$ . If the number of elements $(m, n)$ in $S$ such that $6^{m} + 9^{n}$ is a multiple of 5 is $p$ and the number of elements $(m, n)$ in $S$ such that $m + n$ is a square of a prime number is $q$ , then $p + q$ is equal to ____ .
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Answer: 1333
Q232026 · 21 Jan 2026 · Shift 1· Numerical answer
Let $a_{1} = 1$ and for $n ⩾ 1$ , $a_{n + 1} = \frac{1}{2} a_{n} + \frac{n ^{2} - 2 n - 1}{n ^{2} ( n + 1 ) ^{2}}$ . Then $\sum_{n = 1}^{\infty} (a_{n} - \frac{2}{n ^{2}})$ is equal to ____ .
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Answer: 2
Q72026 · 21 Jan 2026 · Shift 2
The positive integer n, for which the solutions of the equation $x (x + 2) + (x + 2) (x + 4) + \dots + (x + 2 n - 2) (x + 2 n) = \frac{8 n}{3}$ are two consecutive even integers, is :
1. A.3
2. B.6
3. C.9
4. D.12
Show answer
Answer: A
Q102026 · 21 Jan 2026 · Shift 2
Let the line L pass through the point $(- 3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(- 2, r, 1)$ is $\frac{14}{3}$ , then the sum of all possible values of $r$ is :
1. A.16
2. B.12
3. C.6
4. D.10
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Answer: D
Q192026 · 21 Jan 2026 · Shift 2
The largest $n \in N$ , for which $7^{n}$ divides 101!, is :
1. A.18
2. B.15
3. C.19
4. D.16
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Answer: D
Q232026 · 21 Jan 2026 · Shift 2· Numerical answer
If $(\frac{1}{^{15} C _{0}} + \frac{1}{^{15} C _{1}}) (\frac{1}{^{15} C _{1}} + \frac{1}{^{15} C _{2}}) \dots (\frac{1}{^{15} C _{12}} + \frac{1}{^{15} C _{13}}) = \frac{α ^{13}}{^{14} C _{0} ^{14} C _{1} \dots ^{14} C _{12}}$ , then $30 α$ is equal to ____.
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Answer: 32
Q242026 · 21 Jan 2026 · Shift 2· Numerical answer
If $\int_{0}^{1} 4 cot^{- 1} (1 - 2 x + 4 x^{2}) d x = a tan^{- 1} (2) - b lo g_{e} (5)$ , where $a, b \in N$ , then $(2 a + b)$ is equal to ____.
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Answer: 9