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2025 40 questions

Q32025 · 8 Apr 2025 · Shift 2
If $\frac{1}{1 ^{4}} + \frac{1}{2 ^{4}} + \frac{1}{3 ^{4}} + \dots \infty = \frac{π ^{4}}{90}$ , $\frac{1}{1 ^{4}} + \frac{1}{3 ^{4}} + \frac{1}{5 ^{4}} + \dots \infty = α$ , $\frac{1}{2 ^{4}} + \frac{1}{4 ^{4}} + \frac{1}{6 ^{4}} + \dots \infty = β$ , then $\frac{α}{β}$ is equal to :
1. A. $23$
2. B. $14$
3. C. $18$
4. D. $15$
Show answer
Answer: D
Q42025 · 8 Apr 2025 · Shift 2
Let $a$ be the length of a side of a square $OABC$ with $O$ being the origin. Its side $OA$ makes an acute angle $α$ with the positive $x$ -axis and the equations of its diagonals are $(3 + 1) x + (3 - 1) y = 0$ and $(3 - 1) x - (3 + 1) y + 83 = 0$ . Then $a^{2}$ is equal to :
1. A. $48$
2. B. $16$
3. C. $24$
4. D. $32$
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Answer: A
Q82025 · 8 Apr 2025 · Shift 2
The sum of the squares of the roots of $∣ x - 2 ∣^{2} + ∣ x - 2∣ - 2 = 0$ and the squares of the roots of $x^{2} - 2∣ x - 3∣ - 5 = 0$ , is
1. A. $24$
2. B. $26$
3. C. $36$
4. D. $30$
Show answer
Answer: C
Q72025 · 4 Apr 2025 · Shift 1
$1 + 3 + 5^{2} + 7 + 9^{2} + \dots$ upto $40$ terms is equal to
1. A. $40870$
2. B. $41880$
3. C. $43890$
4. D. $33980$
Show answer
Answer: B
Q92025 · 4 Apr 2025 · Shift 1
The value of $\int_{- 1}^{1} \frac{( 1 + ∣ x ∣ - x ) e ^{x} + ( ∣ x ∣ - x ) e ^{- x}}{e ^{x} + e ^{- x}} d x$ is equal to
1. A. $1 + \frac{2 2}{3}$
2. B. $1 - \frac{2 2}{3}$
3. C. $2 + \frac{2 2}{3}$
4. D. $3 - \frac{2 2}{3}$
Show answer
Answer: A
Q152025 · 4 Apr 2025 · Shift 1
Let $A$ and $B$ be two distinct points on the line $L : \frac{x - 6}{3} = \frac{y - 7}{2} = \frac{z - 7}{- 2}$ . Both $A$ and $B$ are at a distance $217$ from the foot of perpendicular drawn from the point $(1, 2, 3)$ on the line $L$ . If $O$ is the origin, then $OA \cdot OB$ is equal to
1. A. $49$
2. B. $21$
3. C. $47$
4. D. $62$
Show answer
Answer: C
Q132025 · 4 Apr 2025 · Shift 2
If $1^{2} \cdot (^{15} C_{1}) + 2^{2} \cdot (^{15} C_{2}) + 3^{2} \cdot (^{15} C_{3}) + \dots + 1 5^{2} \cdot (^{15} C_{15}) = 2^{m} \cdot 3^{n} \cdot 5^{k}$ , where $m, n, k \in N$ , then $m + n + k$ is equal to :
1. A.20
2. B.19
3. C.18
4. D.21
Show answer
Answer: B
Q242025 · 4 Apr 2025 · Shift 2· Numerical answer
If $α$ is a root of the equation $x^{2} + x + 1 = 0$ and $\sum_{k = 1}^{n} (α^{k} + \frac{1}{α ^{k}})^{2} = 20$ , then n is equal to ____ .
Show answer
Answer: 11
Q32025 · 3 Apr 2025 · Shift 1
The sum $1 + 3 + 11 + 25 + 45 + 71 + \dots$ upto 20 terms, is equal to
1. A.7240
2. B.8124
3. C.7130
4. D.6982
Show answer
Answer: A
Q102025 · 3 Apr 2025 · Shift 1
Let a line passing through the point $(4, 1, 0)$ intersect the line $L_{1} : \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4}$ at the point $A (α, β, γ)$ and the line $L_{2} : x - 6 = y = - z + 4$ at the point $B (a, b, c)$ . Then $1 α a 0 β b 1 γ c$ is equal to
1. A.16
2. B.6
3. C.8
4. D.12
Show answer
Answer: C
Q162025 · 3 Apr 2025 · Shift 1
If $\sum_{r = 1}^{9} (\frac{r + 3}{2 ^{r}}) \cdot^{9} C_{r} = α (\frac{3}{2})^{9} - β$ , $α, β \in N$ , then $(α + β)^{2}$ is equal to
1. A.27
2. B.81
3. C.18
4. D.9
Show answer
Answer: B
Q172025 · 3 Apr 2025 · Shift 1
Let $z \in C$ be such that $\frac{z ^{2} + 3 i}{z - 2 + i} = 2 + 3 i$ . Then the sum of all possible values of $z^{2}$ is :
1. A. $- 19 + 2 i$
2. B. $- 19 - 2 i$
3. C. $19 - 2 i$
4. D. $19 + 2 i$
Show answer
Answer: B
Q32025 · 3 Apr 2025 · Shift 2
Each of the angles $β$ and $γ$ that a given line makes with the positive $y$ - and $z$ -axes, respectively, is half of the angle that this line makes with the positive $x$ -axes. Then the sum of all possible values of the angle $β$ is
1. A. $\frac{π}{2}$
2. B. $π$
3. C. $\frac{3 π}{4}$
4. D. $\frac{3 π}{2}$
Show answer
Answer: C
Q72025 · 3 Apr 2025 · Shift 2
Consider the lines $x (3 λ + 1) + y (7 λ + 2) = 17 λ + 5$ , $λ$ being a parameter, all passing through a point $P$ . One of these lines (say $L$ ) is farthest from the origin. If the distance of $L$ from the point $(3, 6)$ is $d$ , then the value of $d^{2}$ is
1. A. $10$
2. B. $20$
3. C. $15$
4. D. $30$
Show answer
Answer: B
Q142025 · 3 Apr 2025 · Shift 2
The distance of the point $(7, 10, 11)$ from the line $\frac{x - 4}{1} = \frac{y - 4}{0} = \frac{z - 2}{3}$ along the line $\frac{x - 9}{2} = \frac{y - 13}{3} = \frac{z - 17}{6}$ is
1. A. $16$
2. B. $12$
3. C. $18$
4. D. $14$
Show answer
Answer: D
Q192025 · 3 Apr 2025 · Shift 2
The sum $1 + \frac{1 + 3}{2 !} + \frac{1 + 3 + 5}{3 !} + \frac{1 + 3 + 5 + 7}{4 !} + \dots$ upto $\infty$ terms, is equal to
1. A. $3 e$
2. B. $2 e$
3. C. $4 e$
4. D. $6 e$
Show answer
Answer: B
Q232025 · 3 Apr 2025 · Shift 2· Numerical answer
Let $(1 + x + x^{2})^{10} = a_{0} + a_{1} x + a_{2} x^{2} + \dots + a_{20} x^{20}$ . If $(a_{1} + a_{3} + a_{5} + \dots + a_{19}) - 11 a_{2} = 121 k$ , then $k$ is equal to_________.
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Answer: 239
Q22025 · 2 Apr 2025 · Shift 1
Let the vertices $Q$ and $R$ of the triangle $PQR$ lie on the line $\frac{x + 3}{5} = \frac{y - 1}{2} = \frac{z + 4}{3}$ , $QR = 5$ and the coordinates of the point $P$ be $(0, 2, 3)$ . If the area of the triangle $P QR$ is $\frac{m}{n}$ then :
1. A. $2 m - 521 n = 0$
2. B. $m - 521 n = 0$
3. C. $5 m - 212 n = 0$
4. D. $5 m - 221 n = 0$
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Answer: A
Q112025 · 2 Apr 2025 · Shift 1
The largest $n \in N$ such that $3^{n}$ divides $50!$ is :
1. A.22
2. B.20
3. C.21
4. D.23
Show answer
Answer: A
Q172025 · 2 Apr 2025 · Shift 1
The number of sequences of ten terms, whose terms are either 0 or 1 or 2 , that contain exactly five 1 s and exactly three 2 s , is equal to :
1. A.360
2. B.2520
3. C.1820
4. D.45
Show answer
Answer: B
Q172025 · 2 Apr 2025 · Shift 2
If $\sum_{r = 0}^{10} (\frac{1 0 ^{r + 1} - 1}{1 0 ^{r}}) \cdot^{11} C_{r + 1} = \frac{α ^{11} - 1 1 ^{11}}{1 0 ^{10}}$ , then $α$ is equal to :
1. A.11
2. B.20
3. C.24
4. D.15
Show answer
Answer: B
Q102025 · 29 Jan 2025 · Shift 2
The remainder, when $7^{103}$ is divided by 23, is equal to:
1. A. $9$
2. B. $6$
3. C. $14$
4. D. $17$
Show answer
Answer: C
Q232025 · 29 Jan 2025 · Shift 2· Numerical answer
Let integers $a, b \in [- 3, 3]$ be such that $a + b \neq = 0$ . Then the number of all possible ordered pairs (a, b), for which $\frac{z - a}{z + b} = 1$ and $z + 1 ω ω^{2} ω z + ω^{2} 1 ω^{2} 1 z + ω = 1$ , $z \in C$ , where $ω$ and $ω^{2}$ are the roots of $x^{2} + x + 1 = 0$ , is equal to ____.
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Answer: 10
Q22025 · 28 Jan 2025 · Shift 1
If $f (x) = \frac{2 ^{x}}{2 ^{x} + 2}$ , $x \in R$ , then $\sum_{k = 1}^{81} f (\frac{k}{82})$ is equal to
1. A.82
2. B. $812$
3. C.41
4. D. $\frac{81}{2}$
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Answer: D
Q52025 · 28 Jan 2025 · Shift 1
Let $⟨ a_{n} ⟩$ be a sequence such that $a_{0} = 0, a_{1} = \frac{1}{2}$ and $2 a_{n + 2} = 5 a_{n + 1} - 3 a_{n}, n = 0, 1, 2, 3, \dots$ . Then $\sum_{k = 1}^{100} a_{k}$ is equal to
1. A. $3 a_{100} + 100$
2. B. $3 a_{100} - 100$
3. C. $3 a_{99} - 100$
4. D. $3 a_{99} + 100$
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Answer: B