Trigonometry · JEE Main previous year questions

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2024 31 questions

Q82024 · 9 Apr 2024 · Shift 2
If $lo g_{e} y = 3 sin^{- 1} x$ , then $(1 - x^{2}) y^{''} - x y^{'}$ at $x = \frac{1}{2}$ is equal to
1. A. $9 e^{π /2}$
2. B. $9 e^{π /6}$
3. C. $3 e^{π /2}$
4. D. $3 e^{π /6}$
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Answer: A
Q272024 · 9 Apr 2024 · Shift 2· Numerical answer
Let the set of all values of $p$ , for which $f (x) = (p^{2} - 6 p + 8) (sin^{2} 2 x - cos^{2} 2 x) + 2 (2 - p) x + 7$ does not have any critical point, be the interval $(a, b)$ . Then $16 ab$ is equal to ____.
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Answer: 252
Q292024 · 9 Apr 2024 · Shift 2· Numerical answer
Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 sin^{- 1} x + 3 cos^{- 1} x = \frac{2 π}{5}$ , is ____.
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Answer: 0
Q162024 · 8 Apr 2024 · Shift 1
If $sin x = - \frac{3}{5}$ , where $π < x < \frac{3 π}{2}$ , then $80 (tan^{2} x - cos x)$ is equal to
1. A. $109$
2. B. $108$
3. C. $19$
4. D. $18$
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Answer: A
Q222024 · 8 Apr 2024 · Shift 1· Numerical answer
If the range of $f (θ) = \frac{s i n ^{4} θ + 3 c o s ^{2} θ}{s i n ^{4} θ + c o s ^{2} θ}, θ \in R$ is $[α, β]$ , then the sum of the infinite G.P., whose first term is $64$ and the common ratio is $\frac{α}{β}$ , is equal to ________.
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Answer: 96
Q292024 · 8 Apr 2024 · Shift 1· Numerical answer
Let the area of the region enclosed by the curve $y = min {sin x, cos x}$ and the $x$ axis between $x = - π$ to $x = π$ be $A$ . Then $A^{2}$ is equal to ________.
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Answer: 16
Q152024 · 8 Apr 2024 · Shift 2
The sum of all possible values of $θ \in [- π, 2 π]$ , for which $\frac{1 + i c o s θ}{1 - 2 i c o s θ}$ is purely imaginary, is equal to :
1. A. $4 π$
2. B. $3 π$
3. C. $2 π$
4. D. $5 π$
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Answer: B
Q182024 · 8 Apr 2024 · Shift 2
If the value of $\frac{3 c o s 3 6 ^{\circ} + 5 s i n 1 8 ^{\circ}}{5 c o s 3 6 ^{\circ} - 3 s i n 1 8 ^{\circ}}$ is $\frac{a 5 - b}{c}$ , where $a, b, c$ are natural numbers and $g cd (a, c) = 1$ , then $a + b + c$ is equal to :
1. A.54
2. B.52
3. C.50
4. D.40
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Answer: B
Q122024 · 6 Apr 2024 · Shift 2
Let $A B C$ be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle $A B C$ and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process, then :
1. A. $P^{2} = 723 Q$
2. B. $P^{2} = 363 Q$
3. C. $P = 363 Q^{2}$
4. D. $P^{2} = 63 Q$
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Answer: B
Q222024 · 6 Apr 2024 · Shift 2· Numerical answer
In a triangle $ABC, BC = 7, AC = 8, AB = α \in N$ and $cos A = \frac{2}{3}$ . If $49 cos (3 C) + 42 = \frac{m}{n}$ , where $g cd (m, n) = 1$ , then $m + n$ is equal to ____.
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Answer: 39
Q292024 · 5 Apr 2024 · Shift 2· Numerical answer
The number of solutions of $sin^{2} x + (2 + 2 x - x^{2}) sin x - 3 (x - 1)^{2} = 0$ , where $- π \leq x \leq π$ , is _________.
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Answer: 2
Q182024 · 4 Apr 2024 · Shift 1
If the system of equations $x + (2 sin α) y + (2 cos α) z = 0$ $x + (cos α) y + (sin α) z = 0$ $x + (sin α) y - (cos α) z = 0$ has a non-trivial solution, then $α \in (0, \frac{π}{2})$ is equal to :
1. A. $\frac{5 π}{24}$
2. B. $\frac{11 π}{24}$
3. C. $\frac{7 π}{24}$
4. D. $\frac{3 π}{4}$
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Answer: A
Q272024 · 4 Apr 2024 · Shift 1· Numerical answer
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $m$ and $n$ respectively be the least and the most number of students who studied all the three subjects. Then $m + n$ is equal to ____.
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Answer: 45
Q92024 · 4 Apr 2024 · Shift 2
Given that the inverse trigonometric function assumes principal values only. Let $x, y$ be any two real numbers in $[- 1, 1]$ such that $cos^{- 1} x - sin^{- 1} y = α, \frac{- π}{2} \leq α \leq π$ . Then, the minimum value of $x^{2} + y^{2} + 2 x y sin α$ is
1. A.0
2. B. $- 1$
3. C. $\frac{1}{2}$
4. D. $\frac{- 1}{2}$
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Answer: A
Q242024 · 4 Apr 2024 · Shift 2· Numerical answer
Consider a triangle $A B C$ having the vertices $A (1, 2)$ , $B (α, β)$ and $C (γ, δ)$ and angles $∠ A B C = \frac{π}{6}$ and $∠ B A C = \frac{2 π}{3}$ . If the points $B$ and $C$ lie on the line $y = x + 4$ , then $α^{2} + γ^{2}$ is equal to _________.
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Answer: 14
Q42024 · 1 Feb 2024 · Shift 1
If $tan A = \frac{1}{x ( x ^{2} + x + 1 )}, tan B = \frac{x}{x ^{2} + x + 1}$ and $tan C = (x^{- 3} + x^{- 2} + x^{- 1})^{1/2}, 0 < A, B, C < \frac{π}{2}$ , then $A + B$ is equal to :
1. A. $C$
2. B. $π - C$
3. C. $2 π - C$
4. D. $\frac{π}{2} - C$
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Answer: A
Q282024 · 1 Feb 2024 · Shift 1· Numerical answer
If $\int_{- π /2}^{π /2} \frac{8 2 c o s x d x}{( 1 + e ^{s i n x} ) ( 1 + s i n ^{4} x )} = α π + β lo g_{e} (3 + 22)$ , where $α, β$ are integers, then $α^{2} + β^{2}$ equals :
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Answer: 8
Q62024 · 1 Feb 2024 · Shift 2
If $\int_{0}^{\frac{π}{3}} cos^{4} x d x = aπ + b 3$ , where $a$ and $b$ are rational numbers, then $9 a + 8 b$ is equal to :
1. A.2
2. B.1
3. C.3
4. D. $\frac{3}{2}$
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Answer: A
Q122024 · 1 Feb 2024 · Shift 2
The number of solutions of the equation $4 sin^{2} x - 4 cos^{3} x + 9 - 4 cos x = 0; x \in [- 2 π, 2 π]$ is :
1. A.0
2. B.3
3. C.1
4. D.2
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Answer: A
Q232024 · 1 Feb 2024 · Shift 2· Numerical answer
If $y = \frac{( x + 1 ) ( x ^{2} - x )}{x x + x + x} + \frac{1}{15} (3 cos^{2} x - 5) cos^{3} x$ , then $96 y^{'} (\frac{π}{6})$ is equal to :
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Answer: 105
Q22024 · 31 Jan 2024 · Shift 1
For $α, β, γ \neq = 0$ , if $sin^{- 1} α + sin^{- 1} β + sin^{- 1} γ = π$ and $(α + β + γ) (α - γ + β) = 3 α β$ , then $γ$ equals
1. A. $3$
2. B. $\frac{3}{2}$
3. C. $\frac{1}{2}$
4. D. $\frac{3 - 1}{2 2}$
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Answer: B
Q52024 · 31 Jan 2024 · Shift 2
If $a = sin^{- 1} (sin (5))$ and $b = cos^{- 1} (cos (5))$ , then $a^{2} + b^{2}$ is equal to
1. A.25
2. B. $4 π^{2} + 25$
3. C. $8 π^{2} - 40 π + 50$
4. D. $4 π^{2} - 20 π + 50$
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Answer: C
Q72024 · 31 Jan 2024 · Shift 2
The number of solutions, of the equation $e^{s i n x} - 2 e^{- s i n x} = 2$ , is :
1. A.0
2. B.1
3. C.2
4. D.more than 2
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Answer: A
Q62024 · 30 Jan 2024 · Shift 1
If $f (x) = 2 cos^{4} x 3 + 2 cos^{4} x 2 cos^{4} x 2 sin^{4} x 2 sin^{4} x 3 + 2 sin^{4} x 3 + sin^{2} 2 x sin^{2} 2 x sin^{2} 2 x$ , then $\frac{1}{5} f^{'} (0)$ is equal to :
1. A.2
2. B.1
3. C.0
4. D.6
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Answer: C
Q42024 · 29 Jan 2024 · Shift 1
If $α, - \frac{π}{2} < α < \frac{π}{2}$ is the solution of $4 cos θ + 5 sin θ = 1$ , then the value of $tan α$ is
1. A. $\frac{10 - 10}{12}$
2. B. $\frac{10 - 10}{6}$
3. C. $\frac{10 - 10}{12}$
4. D. $\frac{10 - 10}{6}$
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Answer: C