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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Medium

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Showing 1-30 of 294 questions
1 Medium
1 of 294
Understanding +4 | -1
Statement I: If slope of the tangent line is zero, then tangent line is perpendicular to the X\mathrm{X}-axis.

Statement II: If θ→π2\theta \rightarrow \frac{\pi}{2}, then tangent line is parallel to the Y-axis.
View answer and explanation
Correct option: B
Only statement II is true
Option B is correct because statement II is true, while statement I is false; a zero slope means the tangent is parallel to the x-axis.
2 Medium
2 of 294
Application +4 | -1
If sin⁡x+cos⁡x=72\sin x+\cos x=\frac{\sqrt{7}}{2}, where x∈[0,π4]x \in\left[0, \frac{\pi}{4}\right], then tan⁡x2\tan \frac{x}{2} is equal to
View answer and explanation
Correct option: B
7−23\frac{\sqrt{7}-2}{3}
Option B (7−23\frac{{\sqrt{{7}}-2}}{{3}}) is correct because it is the solution to the quadratic equation derived from the given trigonometric equation and the constraint on x.
3 Medium
3 of 294
Analyzing +4 | -1
The value of tan⁡π5+2tan⁡2π5+4cot⁡4π5\tan \frac{\pi}{5}+2 \tan \frac{2 \pi}{5}+4 \cot \frac{4 \pi}{5} is
View answer and explanation
Correct option: A
cot⁡π5\cot \frac{\pi}{5}
Option A (cot⁡π5\cot \frac{\pi}{5}) is correct because simplifying the expression using trigonometric identities leads to cot⁡π5\cot \frac{\pi}{5}.
4 Medium
4 of 294
Application +4 | -1
If 0<x<π0 < x < \pi and cos⁡x+sin⁡x=12\cos x+\sin x=\frac{1}{2}, then tan⁡x\tan x is equal to
View answer and explanation
Correct option: B
−(4+7)3-\frac{(4+\sqrt{7})}{3}
Option B (−(4+7)3-\frac{(4+\sqrt{7})}{3}) is correct because it is the value of tan⁡x\tan x calculated using the given equation and the half-angle formula.
5 Medium
5 of 294
Application +4 | -1
General solution of sin⁡  x+cos⁡  x=mina∈R  {1,   a2−4a+6}\sin \,\,x + \cos \,\,x = \begin{array}{cccccccccccccccccccc} {min}\\ {a \in R} \end{array}\,\,\left\{ {1,\,\;{a^2} - 4a + 6} \right\} is

View answer and explanation
Correct option: D
nπ+(−1)n  π4−π4n\pi + {\left( { - 1} \right)^n}\,\,\frac{\pi }{4} - \frac{\pi }{4}
Option D (nπ+(−1)n  π4−π4n\pi + {(- 1)^n}\,\,\frac{\pi }{4} - \frac{\pi }{4}) is correct because it represents the general solution of the given trigonometric equation.
6 Medium
6 of 294
Application +4 | -1
Two sides of a triangle are given by the roots of the equation x2−23x+2=0x^2-2 \sqrt{3} x+2=0. The angle between the sides is π3\frac{\pi}{3}. The perimeter of the triangle is
View answer and explanation
Correct option: B
23+62 \sqrt{3}+\sqrt{6}
Option B (23+62 \sqrt{3}+\sqrt{6}) is correct. By using the given quadratic equation to find the two sides and the given angle, the third side and hence the perimeter can be calculated.
7 Medium
7 of 294
Analyzing +4 | -1
If the equation cos⁡4θ+sin⁡2θ+λ=0\cos ^4 \theta+\sin ^2 \theta+\lambda=0 has real solution for θ\theta, then λ\lambda lies in the interval
View answer and explanation
Correct option: A
[−1,−34]\left[-1,-\frac{3}{4}\right]
Option B (32\frac{3}{2}) is correct because after calculating the eccentricities and using the given relationship, the length of the latus rectum of the ellipse can be found to be 3/2.
8 Medium
8 of 294
Analyzing +4 | -1
For x∈(0,π)x \in(0, \pi), the equation sin⁡x+2sin⁡2x−sin⁡3x=3\sin x+2 \sin 2 x-\sin 3 x=3 has
View answer and explanation
Correct option: D
No solution
Option D (No solution) is correct because the equation simplifies to a form where sin⁡x\sin x must be 1 and cos⁡x\cos x must be 1/2 simultaneously, which is impossible within the given interval.
9 Medium
9 of 294
Analyzing +4 | -1
If cos⁡2B=cos⁡(A+C)cos⁡(A−C)\cos 2 B=\frac{\cos (A+C)}{\cos (A-C)}, then
View answer and explanation
Correct option: B
tan⁡A,tan⁡B\tan A, \tan B and tan⁡C\tan C are in GP
Option B (tan⁡A,tan⁡B\tan A, \tan B and tan⁡C\tan C are in GP) is correct because by applying componendo and dividendo to the given equation, it can be shown that tan⁡2B=tan⁡Atan⁡C\tan^2 B = \tan A \tan C, which is the condition for a geometric progression.
10 Medium
10 of 294
Application +4 | -1
If sin⁡2θ−2cos⁡θ+14=0\sin ^2 \theta-2 \cos \theta+\frac{1}{4}=0, then the general value of θ\theta is
View answer and explanation
Correct option: B
2nπ±π32 n \pi \pm \frac{\pi}{3}
Option B (2nπ±π32 n \pi \pm \frac{{\pi}}{{3}}) is correct because it represents the general solution derived from solving the given trigonometric equation.
11 Medium
11 of 294
Application +4 | -1
If 1 lies between the roots of 3x2−3(sin⁡)θx−2cos⁡2θ=03 x^2-3(\sin ) \theta x-2 \cos ^2 \theta=0 then complete set of values of sin⁡θ\sin \theta is
View answer and explanation
Correct option: C
12<sin⁡θ<1\frac{1}{2} < \sin \theta < 1
Option C (12<sin⁡θ<1\frac{{1}}{{2}} < \sin \theta < 1) is correct because when 1 lies between the roots, f(1) < 0, which leads to the given range for sin⁡θ\sin \theta.
12 Medium
12 of 294
Application +4 | -1
2sin⁡2x+sin⁡22x=2,−π<x<π2 \sin ^{2} x+\sin ^{2} 2 x=2,-\pi < x < \pi, then x=x=
View answer and explanation
Correct option: B
±π4\pm \frac{\pi}{4}
Option B (±π4\pm \frac{{\pi}}{{4}}) is correct because it is a solution to the trigonometric equation within the given interval.
13 Medium
13 of 294
Application +4 | -1
The sum of the solutions in (0,2π)(0,2 \pi) of the equation cos⁡xcos⁡(π3−x)cos⁡(π3+x)=14\cos x \cos \left(\frac{\pi}{3}-x\right) \cos \left(\frac{\pi}{3}+x\right)=\frac{1}{4} is
View answer and explanation
Correct option: C
2π2 \pi
Option C (2π2 \pi) is correct because using the trigonometric identity, the sum of solutions in (0, 2π) is 2π2\pi.
14 Medium
14 of 294
Application +4 | -1
The ordered pair (x,y)(x, y) satisfying 1−tan⁡x1+tan⁡x=tan⁡y\frac{1-\tan x}{1+\tan x}=\tan y and x−y=π6x-y=\frac{\pi}{6} is
View answer and explanation
Correct option: A
(nπ2+π24,nπ2+π24),∈Z\left(\frac{n \pi}{2}+\frac{\pi}{24}, \frac{n \pi}{2}+\frac{\pi}{24}\right), \quad \in \mathbb{Z}
Option C ((nπ2\frac{{nπ}}{{2}} + 5π24\frac{{5π}}{{24}}, nπ2\frac{{nπ}}{{2}} - π24\frac{{π}}{{24}}), n ∈ \mathbb{{Z}}) is correct because it represents the general solution for x and y derived from the given equations using trigonometric identities.
15 Medium
15 of 294
Analyzing +4 | -1
If ntan⁡y=tan⁡x,n∈R+n \tan y=\tan x, n \in R^{+}, then the maximum value of sec⁡2(x−y)\sec ^2(x-y) is equal to
View answer and explanation
Correct option: D
(n+1)24n\frac{(n+1)^2}{4 n}
Option A (nπ−π4n \pi-\frac{{\pi}}{{4}}) is correct because it represents the general solution where the real part of the given complex expression is zero, making it purely imaginary.

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs
16 Medium
16 of 294
Application +4 | -1
A solution (x,y)(x, y) of the system of equation x−y=1/3x-y=1 / 3 and cos⁡2(πx)−sin⁡2(πy)=1/2\cos ^2(\pi x)-\sin ^2(\pi y)=1 / 2 is given by
View answer and explanation
Correct option: C
(13/6,11/6)(13 / 6,11 / 6)
Option C ((13/6,11/6)(13 / 6,11 / 6)) is correct because it satisfies both equations of the given system when n=2n=2.
17 Medium
17 of 294
Application +4 | -1
If x=sin⁡2θcos⁡θx=\sin ^2 \theta \cos \theta and y=cos⁡2θsin⁡θy=\cos ^2 \theta \sin \theta, then which of the following is true ? 0<θ<π2,x>y>00 < \theta < \frac{\pi}{2}, x > y > 0
View answer and explanation
Correct option: B
(x2y)23+(y2x)23=1\left(\frac{x^2}{y}\right)^{\frac{2}{3}}+\left(\frac{y^2}{x}\right)^{\frac{2}{3}}=1
Option B ((x2y)23+(y2x)23=1\left(\frac{{x^2}}{{y}}\right)^{\frac{{2}}{{3}}}+\left(\frac{{y^2}}{{x}}\right)^{\frac{{2}}{{3}}}=1) is correct because it is the result of simplifying the given equations using trigonometric identities.
18 Medium
18 of 294
Application +4 | -1
The value of 3[sin⁡4(3π2−α)+sin⁡4(3π+α)]3\left[\sin ^4\left(\frac{3 \pi}{2}-\alpha\right)+\sin ^4(3 \pi+\alpha)\right] −2[sin⁡6(π2+α)+sin⁡6(5π−α)]-2\left[\sin ^6\left(\frac{\pi}{2}+\alpha\right)+\sin ^6(5 \pi-\alpha)\right] is
View answer and explanation
Correct option: B
1
The correct answer is 1 because the expression simplifies to 1 using trigonometric identities and reduction formulas.
19 Medium
19 of 294
Application +4 | -1
The solutions of the equation 4  cos⁡2  x+6  sin⁡2  x=54\,\,{\cos ^2}\,\,x + 6\,\,{\sin ^2}\,\,x = 5 are
View answer and explanation
Correct option: A
x=nπ  ±  π4x = n\pi \,\, \pm \,\,\frac{\pi }{4}
Option A (x=nπ ± π4x = n\pi \, \pm \, \frac{\pi }{4}) is correct because solving the trigonometric equation 4cos⁡2x+6sin⁡2x=54{\cos ^2}x + 6{\sin ^2}x = 5 leads to sin⁡x=±12\sin x = \pm \frac{1}{\sqrt{2}}, which gives the general solution x=nπ±π4x = n\pi \pm \frac{\pi}{4}.
20 Medium
20 of 294
Application +4 | -1
sin⁡3θ2cos⁡2θ+1=12\frac{\sin 3 \theta}{2 \cos 2 \theta+1}=\frac{1}{2} if -
View answer and explanation
Correct option: C
θ=nπ+(−1)nπ6\theta={n} \pi+(-1)^{{n}} \frac{\pi}{6}
Option C (θ=nπ+(−1)nπ6\theta={n} \pi+(-1)^{{n}} \frac{\pi}{6}) is correct because it represents the general solution of the simplified trigonometric equation sin⁡θ=12\sin \theta = \frac{1}{2}.
21 Medium
21 of 294
Analyzing +4 | -1
If the expression nsin⁡2θ+2ncos⁡(θ+α)sin⁡αsin⁡θn \sin ^2 \theta+2 n \cos (\theta+\alpha) \sin \alpha \sin \theta +cos⁡2(α+θ)+\cos 2(\alpha+\theta) is independent of θ\theta, then the value of nn is
View answer and explanation
Correct option: B
2
Option A (2) is correct because setting n=2 makes the expression independent of theta, as the theta-dependent terms cancel out.
22 Medium
22 of 294
Application +4 | -1
The solution set of (2cos⁡x−1)  (3+2 cos⁡x)=0\left( {2\cos x - 1} \right)\,\,\left( {3 + 2\,\cos x} \right) = 0 in the interval 0≤x≤2  π0 \le x \le 2\;\pi, is
View answer and explanation
Correct option: B
{π3,  5  π3}\left\{ {\frac{\pi }{3},\,\,\frac{{5\;\pi }}{3}} \right\}
Option B ({π3,  5  π3}\left\{ {\frac{\pi }{3},\,\,\frac{{5\;\pi }}{3}} \right\}) is correct because these are the only solutions to the equation within the given interval.
23 Medium
23 of 294
Application +4 | -1
If sin⁡x+3cos⁡x=2\sin x+\sqrt{3} \cos x=\sqrt{2}, then xx is equal to
View answer and explanation
Correct option: B
2nπ+5π12,2nπ−π122 n \pi+\frac{5 \pi}{12}, 2 n \pi-\frac{\pi}{12}
Option B (2nπ+5π12,2nπ−π122 n \pi+\frac{{5 \pi}}{{12}}, 2 n \pi-\frac{{\pi}}{{12}}) is correct because it represents the general solution for x after solving the trigonometric equation.
24 Medium
24 of 294
Application +4 | -1
If sin⁡xsin⁡y=12,cos⁡xcos⁡y=32\frac{\sin x}{\sin y}=\frac{1}{2}, \frac{\cos x}{\cos y}=\frac{3}{2}, where x,y∈(0,π2)x, y \in\left(0, \frac{\pi}{2}\right), then the value of tan⁡(x+y)\tan (x+y) is
View answer and explanation
Correct option: D
15\sqrt{15}
Option D (15\sqrt{15}) is correct because by using the given ratios and trigonometric identities, we can find the value of tan⁡x\tan x and then calculate tan⁡(x+y)\tan(x+y).
25 Medium
25 of 294
Understanding +4 | -1
The equation asin⁡x+bcos⁡x=ca \sin x+b \cos x=c, where ∣c∣>a2+b2|c|>\sqrt{a^2+b^2} has
View answer and explanation
Correct option: C
no solution
Option B (0) is correct because lim⁡x→0xsin⁡1x=0\lim_{x \rightarrow 0} x \sin \frac{1}{x} = 0 due to the Squeeze Theorem.
26 Medium
26 of 294
Analyzing +4 | -1
The most general solution of

21+ ∣ cos⁡x ∣+ cos⁡2x+ ∣ cos⁡3x ∣+…∞=4{2^{1 + \,\left| {\,\cos x\,} \right| + \,{{\cos }^2}x + \,\left| {\,{{\cos }^3}x\,} \right| + \ldots \infty }} = 4 is given by
View answer and explanation
Correct option: A
x=nπ  ±  π3,  n∈Zx = n\pi \,\, \pm \,\,\frac{\pi }{3},\,\,n \in Z
Option A (x = nπ ± π/3, n ∈ Z) is correct because solving the equation leads to cos x = ±1/2, which gives the general solution x = nπ ± π/3.
27 Medium
27 of 294
Analyzing +4 | -1
Let f(θ)=3(sin⁡4(3π2−θ)+sin⁡4(3π+θ))−2(1−sin⁡22θ)f(\theta)=3\left(\sin ^{4}\left(\frac{3 \pi}{2}-\theta\right)+\sin ^{4}(3 \pi+\theta)\right)-2\left(1-\sin ^{2} 2 \theta\right) and S={θ∈[0,π]:f′(θ)=−32}S=\left\{\theta \in[0, \pi]: f^{\prime}(\theta)=-\frac{\sqrt{3}}{2}\right\}. If 4β=∑θ∈Sθ4 \beta=\sum_{\theta \in S} \theta, then f(β)f(\beta) is equal to
View answer and explanation
Correct option: C
54\frac{5}{4}
The correct answer is C (54\frac{5}{4}). This involves simplifying the trigonometric function, finding its derivative, solving the equation f′(θ)=−32f'(\theta) = -\frac{\sqrt{3}}{2}, and then evaluating f(β)f(\beta).
28 Medium
28 of 294
Analyzing +4 | -1
If r>0,−π≤θ≤πr > 0,-\pi \leq \theta \leq \pi and r,θr, \theta satisfy rsin⁡θ=3r \sin \theta=3 and r=4(1+sin⁡θ)r=4(1+\sin \theta), then the number of possible solutions of the pair (r,θ)(r, \theta) is
View answer and explanation
Correct option: A
2
Option A (2) is correct because there are two possible solutions for the pair (r,θ)(r, \theta) after solving the system of equations.
29 Medium
29 of 294
Analyzing +4 | -1
Statement-1: The number of common solutions of the trigonometric equations 2sin⁡2θ−cos⁡2θ=02 \sin ^{2} \theta-\cos 2 \theta=0 and 2cos⁡2θ−3sin⁡θ=02 \cos ^{2} \theta-3 \sin \theta=0 in the interval [0,2π][0,2 \pi] is two.

Statement-2: The number of solutions of the equation, 2 cos⁡2θ−3sin⁡θ=0\cos ^{2} \theta-3 \sin \theta=0 in the interval [0,π][0, \pi] is two.
View answer and explanation
Correct option: B
Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for statement-1.
Statement-1 is true because the equations have two common solutions, and Statement-2 is true because the equation has two solutions in the interval, but Statement-2 doesn't explain Statement-1.
30 Medium
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Application +4 | -1
If 0<β<α<π4,cos⁡(α+β)=350 < \beta < \alpha < \frac{\pi}{4}, \cos (\alpha+\beta)=\frac{3}{5} and cos⁡(α−β)=45\cos (\alpha-\beta)=\frac{4}{5}, then sin⁡2α\sin 2 \alpha is equal to
View answer and explanation
Correct option: B
1
Option B (1) is correct because using the formula sin⁡2α=sin⁡(((α+β)+(α−β)))\sin 2\alpha = \sin(((\alpha + \beta) + (\alpha - \beta))) and the given values, we get sin⁡2α=1\sin 2\alpha = 1.
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