Trigonometry · JEE Main previous year questions

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2019 20 questions

Q22019 · 12 Apr 2019 · Shift 1
For $x \in (0, 3/2)$ , let $f (x) = x$ , $g (x) = tan x$ and $h (x) = \frac{1 - x ^{2}}{1 + x ^{2}}$ . If $ϕ (x) = ((hof) og) (x)$ , then $ϕ (\frac{π}{3})$ is equal to :
1. A. $tan \frac{7 π}{12}$
2. B. $tan \frac{11 π}{12}$
3. C. $tan \frac{π}{12}$
4. D. $tan \frac{5 π}{12}$
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Answer: B
Q172019 · 12 Apr 2019 · Shift 1
A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is :
1. A. $\frac{25}{3}$
2. B.25
3. C. $253$
4. D. $\frac{25}{3}$
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Answer: D
Q212019 · 12 Apr 2019 · Shift 1
The value of $sin^{- 1} (\frac{12}{13}) - sin^{- 1} (\frac{3}{5})$ is equal to :
1. A. $π - sin^{- 1} (\frac{63}{65})$
2. B. $\frac{π}{2} - sin^{- 1} (\frac{56}{65})$
3. C. $\frac{π}{2} - cos^{- 1} (\frac{9}{65})$
4. D. $π - cos^{- 1} (\frac{33}{65})$
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Answer: B
Q242019 · 12 Apr 2019 · Shift 1
The number of solutions of the equation $1 + sin^{4} x = cos^{2} 3 x$ , $x \in [- \frac{5 π}{2}, \frac{5 π}{2}]$ is :
1. A.5
2. B.3
3. C.7
4. D.4
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Answer: A
Q132019 · 12 Apr 2019 · Shift 2
The derivative of $tan^{- 1} (\frac{s i n x - c o s x}{s i n x + c o s x})$ , with respect to $\frac{x}{2}$ , where $(x \in (0, \frac{π}{2}))$ is :
1. A.1
2. B.2
3. C. $\frac{2}{3}$
4. D. $\frac{1}{2}$
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Answer: B
Q162019 · 12 Apr 2019 · Shift 2
A value of $θ \in (0, \frac{π}{3})$ , for which $1 + cos^{2} θ cos^{2} θ cos^{2} θ sin^{2} θ 1 + sin^{2} θ sin^{2} θ 4 cos 6 θ 4 cos 6 θ 1 + 4 cos 6 θ = 0$ , is :
1. A. $\frac{π}{18}$
2. B. $\frac{π}{9}$
3. C. $\frac{7 π}{24}$
4. D. $\frac{7 π}{36}$
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Answer: B
Q272019 · 12 Apr 2019 · Shift 2
If [x] denotes the greatest integer $\leq x$ , then the system of linear equations $[sin θ] x + [- cos θ] y = 0$ , $[cot θ] x + y = 0$
1. A.has a unique solution if $θ \in (\frac{π}{2}, \frac{2 π}{3})$ and have infinitely many solutions if $θ \in (π, \frac{7 π}{6})$
2. B.have infinitely many solutions if $θ \in (\frac{π}{2}, \frac{2 π}{3})$ and has a unique solution if $θ \in (π, \frac{7 π}{6})$
3. C.have infinitely many solutions if $θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$
4. D.has a unique solution if $θ \in (\frac{π}{2}, \frac{2 π}{3}) \cup (π, \frac{7 π}{6})$
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Answer: B
Q72019 · 10 Apr 2019 · Shift 1
If $Δ_{1} = x - sin θ cos θ sin θ - x 1 cos θ 1 x$ and $Δ_{2} = x - sin 2 θ cos 2 θ sin 2 θ - x 1 cos 2 θ 1 x, x \neq = 0$ ; then for all $θ \in (0, \frac{π}{2})$ :
1. A. $Δ_{1} - Δ_{2} = x (cos 2 θ - cos 4 θ)$
2. B. $Δ_{1} + Δ_{2} = - 2 x^{3}$
3. C. $Δ_{1} + Δ_{2} = - 2 (x^{3} + x - 1)$
4. D. $Δ_{1} - Δ_{2} = - 2 x^{3}$
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Answer: B
Q162019 · 10 Apr 2019 · Shift 1
Let A (3, 0, -1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G divides BM in the ratio, 2 : 1, then $cos (∠ GOA)$ (O being the origin) is equal to :
1. A. $\frac{1}{15}$
2. B. $\frac{1}{6 10}$
3. C. $\frac{1}{30}$
4. D. $\frac{1}{2 15}$
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Answer: A
Q282019 · 10 Apr 2019 · Shift 1
ABC is a triangular park with AB = AC = 100 metres. A vertical tower is situated at the mid-point of BC. If the angles of elevation of the top of the tower at A and B are $cot^{- 1} (32)$ and $cosec^{- 1} (22)$ respectively, then the height of the tower (in metres) is :
1. A. $\frac{100}{3 3}$
2. B.25
3. C.20
4. D. $105$
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Answer: C
Q182019 · 9 Apr 2019 · Shift 1
The value of $cos^{2} 1 0^{\circ} - cos 1 0^{\circ} cos 5 0^{\circ} + cos^{2} 5 0^{\circ}$ is
1. A. $\frac{3}{2} + cos 2 0^{\circ}$
2. B. $\frac{3}{4}$
3. C. $\frac{3}{2} (1 + cos 2 0^{\circ})$
4. D. $\frac{3}{2}$
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Answer: B
Q52019 · 9 Apr 2019 · Shift 2
The value of $sin 1 0^{\circ} sin 3 0^{\circ} sin 5 0^{\circ} sin 7 0^{\circ}$ is :-
1. A. $\frac{1}{36}$
2. B. $\frac{1}{16}$
3. C. $\frac{1}{32}$
4. D. $\frac{1}{18}$
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Answer: B
Q92019 · 9 Apr 2019 · Shift 2
A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $tan^{- 1} (\frac{1}{2})$ . Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10m; is :-
1. A. $\frac{1}{15 π}$
2. B. $\frac{1}{5 π}$
3. C. $\frac{1}{10 π}$
4. D. $\frac{2}{π}$
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Answer: B
Q62019 · 12 Jan 2019 · Shift 2
If $sin^{4} α + 4 cos^{4} β + 2 = 42 sin α cos β$ ; $α, β \in [0, π]$ , then $cos (α + β) - cos (α - β)$ is equal to :
1. A. $- 2$
2. B.0
3. C. $- 1$
4. D. $2$
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Answer: A
Q272019 · 12 Jan 2019 · Shift 2
If the angle of elevation of a cloud from a point P which is 25 m above a lake be $3 0^{\circ}$ and the angle of depression of reflection of the cloud in the lake from P be $6 0^{\circ}$ , then the height of the cloud (in meters) from the surface of the lake is :
1. A.45
2. B.42
3. C.50
4. D.60
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Answer: C
Q192019 · 11 Jan 2019 · Shift 2
All x satisfying the inequality $(cot^{- 1} x)^{2} - 7 (cot^{- 1} x) + 10 > 0$ , lie in the interval :
1. A. $(cot 2, \infty)$
2. B. $(- \infty, cot 5) \cup (cot 2, \infty)$
3. C. $(cot 5, cot 4)$
4. D. $(- \infty, cot 5) \cup (cot 4, cot 2)$
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Answer: A
Q302019 · 11 Jan 2019 · Shift 2
Given $\frac{b + c}{11} = \frac{c + a}{12} = \frac{a + b}{13}$ for a $Δ ABC$ with usual notation. If $\frac{c o s A}{α} = \frac{c o s B}{β} = \frac{c o s C}{γ}$ , then the ordered triad $(α, β, γ)$ has a value :
1. A. $(19, 7, 25)$
2. B. $(7, 19, 25)$
3. C. $(5, 12, 13)$
4. D. $(3, 4, 5)$
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Answer: B
Q162019 · 9 Jan 2019 · Shift 1
If $cos^{- 1} (\frac{2}{3 x}) + cos^{- 1} (\frac{3}{4 x}) = \frac{π}{2}$ $(x > \frac{3}{4})$ , then x is equal to :
1. A. $\frac{145}{10}$
2. B. $\frac{145}{11}$
3. C. $\frac{145}{12}$
4. D. $\frac{146}{12}$
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Answer: C
Q182019 · 9 Jan 2019 · Shift 1
For any $θ \in (\frac{π}{4}, \frac{π}{2})$ , the expression $3 (cos θ - sin θ)^{4} + 6 (sin θ + cos θ)^{2} + 4 sin^{6} θ$ equals :
1. A. $13 - 4 cos^{2} θ + 6 sin^{2} θ cos^{2} θ$
2. B. $13 - 4 cos^{6} θ$
3. C. $13 - 4 cos^{2} θ + 6 cos^{4} θ$
4. D. $13 - 4 cos^{4} θ + 2 sin^{2} θ cos^{2} θ$
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Answer: B
Q202019 · 9 Jan 2019 · Shift 1
If $θ$ denotes the acute angle between the curves, $y = 10 - x^{2}$ and $y = 2 + x^{2}$ at a point of their intersection, the $∣ tan θ ∣$ is equal to :
1. A. $\frac{8}{15}$
2. B. $\frac{4}{9}$
3. C. $\frac{7}{17}$
4. D. $\frac{8}{17}$
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Answer: A