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JEE • MATHEMATICS • TRIGONOMETRIC EQUATIONS

Trigonometric equations practice · Mixed difficulty

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331 Hard
331 of 581
Analyzing +4 | -1
The number of solutions of the equation x3+x2+4x+2  sin⁡  x=0  in   0≤  x  ≤  2π{x^3} + {x^2} + 4x + 2\,\,\sin \,\,x = 0\,\,\mathrm{in}\,\;0 \le \,\,x\,\, \le \,\,2\pi is
View answer and explanation
Correct option: B
One
Option B (One) is correct because x=0 is the only solution that satisfies the given equation in the specified interval.
332 Easy
332 of 581
Analyzing +4 | -1
The maximum value of sin⁡(cos⁡x)\sin (\cos x) is
View answer and explanation
Correct option: A
sin⁡1\sin 1
Option A (sin⁡1\sin 1) is correct because the maximum value of cos⁡x\cos x is 1, and sin⁡x\sin x is increasing on [−1,1][-1, 1].
333 Easy
333 of 581
Application +4 | -1
If 3sin⁡θ+4cos⁡θ=53 \sin \theta+4 \cos \theta=5, then the value of 3cos⁡θ−4sin⁡θ3 \cos \theta-4 \sin \theta is equal to
View answer and explanation
Correct option: A
00
Option A (0) is correct because by squaring both sides of the given equation and using trigonometric identities, we find that (3cos⁡θ−4sin⁡θ)2=0(3 \cos \theta - 4 \sin \theta)^2 = 0.
334 Medium
334 of 581
Analyzing +4 | -1
If sin⁡α=15\sin \alpha=\frac{1}{\sqrt{5}} and sin⁡β=35\sin \beta=\frac{3}{5}, then β−α\beta-\alpha lies in the interval
View answer and explanation
Correct option: A
(0,π4)∪(3π4,π)\left(0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \pi\right)
Option A ((0,π4)∪(3π4,π)\left(0, \frac{{\pi}}{{4}}\right) \cup\left(\frac{{3 \pi}}{{4}}, \pi\right)) is correct because the value of sin⁡(β−α)\sin(\beta - \alpha) is positive and less than 12\frac{{1}}{{\sqrt{{2}}}}, which restricts the interval for β−α\beta - \alpha.
335 Medium
335 of 581
Analyzing +4 | -1
The equation 3sin⁡x+cos⁡x=4\sqrt{3} \sin x+\cos x=4 has
View answer and explanation
Correct option: D
No solution
Option D (No solution) is correct because the maximum value of 3sin⁡x+cos⁡x\sqrt{3} \sin x + \cos x is 2, which is less than 4.
336 Medium
336 of 581
Analyzing +4 | -1
If −π2≤x≤π2- \frac{\pi }{2} \le x \le \frac{\pi }{2}, then the two curves y=cos⁡xy = \cos x and y=sin⁡3xy = \sin 3x intersect at
View answer and explanation
Correct option: A
(π4, 12)   and   (π8,  cos⁡  π8)\left( {\frac{\pi }{4},\,\frac{1}{{\sqrt 2 }}} \right)\;\,\mathrm{and\;}\,\left( {\frac{\pi }{8},\,\,\cos \,\,\frac{\pi }{8}} \right)
Option A is correct because the intersection points are found by equating the two functions and solving for x, then finding the corresponding y values.
337 Hard
337 of 581
Application +4 | -1
Solution for xx and yy when x2+2xsin⁡xy+1=0x^2+2 x \sin x y+1=0 is
View answer and explanation
Correct option: A
x=±1,y=2nπ−π/2;n∈Zx= \pm 1, y=2 n \pi-\pi / 2 ; \mathrm{n} \in \mathbb{Z}
Option A (x=±1,y=2nπ−π/2;n∈Zx= \pm 1, y=2 n \pi-\pi / 2 ; \mathrm{{n}} \in \mathbb{{Z}}) is correct. The equation is satisfied only when x=±1x = \pm 1 and sin⁡(xy)=−1\sin(xy) = -1, leading to the given solution.
338 Easy
338 of 581
Application +4 | -1
The root of the equation 1−cos⁡  θ=sin⁡  θ .  sin⁡  θ21 - \cos \,\,\theta = \sin \,\,\theta \,.\,\,\sin \,\,\frac{\theta }{2} is
View answer and explanation
Correct option: B
2k  π,   k∈I2k\;\pi ,\;\,k \in I
Option B (2k  π,   k∈I2k\;\pi ,\;\,k \in I) is correct because it represents the general solution of the given trigonometric equation.
339 Medium
339 of 581
Application +4 | -1
If α\alpha is a root of 25cos⁡2θ+5cos⁡θ−12=025 \cos ^2 \theta+5 \cos \theta-12=0, π2<α<π\frac{\pi}{2} < \alpha < \pi, then sin⁡2α\sin 2 \alpha is equal to
View answer and explanation
Correct option: B
−2425-\frac{24}{25}
Option B (−2425-\frac{24}{25}) is correct because solving the quadratic equation for cos⁡α\cos \alpha and using the given range for α\alpha allows us to find sin⁡α\sin \alpha and then sin⁡2α\sin 2\alpha.
340 Medium
340 of 581
Analyzing +4 | -1
If 0≤x≤3π,0≤y≤3π0 \leq x \leq 3 \pi, 0 \leq y \leq 3 \pi and cos⁡xsin⁡y=1\cos x \sin y=1, then the possible number of values of the ordered pair (x,y)(x, y) is
View answer and explanation
Correct option: A
6
Option B (3) is correct because there are three ordered pairs (x, y) that satisfy the given equation within the specified intervals.
341 Medium
341 of 581
Analyzing +4 | -1
If α,β,γ\alpha, \beta, \gamma and δ\delta are the solutions of the equation tan⁡(θ+π4)=3tan⁡3θ\tan \left(\theta+\frac{\pi}{4}\right)=3 \tan 3 \theta, no two of which have equal tangents, then the value of tan⁡α+tan⁡β+tan⁡γ+tan⁡δ\tan \alpha+\tan \beta+\tan \gamma+\tan \delta is
View answer and explanation
Correct option: D
0
Option D (0) is correct because the sum of the roots of the quartic equation in tan⁡θ\tan \theta is 0.
342 Easy
342 of 581
Understanding +4 | -1
In △ABC\triangle A B C, the value of the expression cosec⁡A(sin⁡Bcos⁡C+cos⁡Bsin⁡C)\operatorname{cosec} A(\sin B \cos C+\cos B \sin C) is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the expression simplifies to cosec⁡Asin⁡(B+C)=cosec⁡Asin⁡(π−A)=cosec⁡Asin⁡A=1\operatorname{cosec} A \sin(B+C) = \operatorname{cosec} A \sin(\pi - A) = \operatorname{cosec} A \sin A = 1.
343 Easy
343 of 581
Application +4 | -1
The number of values of xx in [0,   2π]\left[ {0,\;\,2\pi } \right] satisfying the equation 3  cos⁡  2x−10  cos⁡  x+7=03\,\,\cos \,\,2x - 10\,\,\cos \,\,x + 7 = 0 is

View answer and explanation
Correct option: D
4\mathrm{4}
Option D (4) is correct because solving the trigonometric equation yields four solutions for x in the given interval [0, 2π].
344 Easy
344 of 581
Analyzing +4 | -1
If 2cos⁡2x2sin⁡2x=x2+x−22 \cos ^2 \frac{x}{2} \sin ^2 x=x^2+x^{-2},0<x≤π20 < x \leq \frac{\pi}{2}, then xx is equal to
View answer and explanation
Correct option: A
ϕ\phi
Option A (ϕ\phi) is correct because the LHS is always less than 2, while the RHS is always greater than or equal to 2, so there is no solution.
345 Medium
345 of 581
Application +4 | -1
If A+B=π4A+B=\frac{\pi}{4}, where A,B∈R+A, B \in R^{+}, then the minimum value of (1+tan⁡A)(1+tan⁡B)(1+\tan A)(1+\tan B) is
View answer and explanation
Correct option: A
2
Option A (2) is correct because the minimum value of the expression is found by using the given condition and trigonometric identities to simplify it to a constant value of 2.

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346 Easy
346 of 581
Analyzing +4 | -1
If θ∈  [0,  5π]\theta \in \,\,\left[ {0,\,\,5\pi } \right] and r∈Rr \in R such that 2  sin⁡  θ=r4−2r2+32\,\,\sin \,\,\theta = {r^4} - 2{r^2} + 3 then the maximum number of values of the pair (r,  θ)\left( {r,\,\,\theta } \right) is
View answer and explanation
Correct option: A
6\mathrm{6}
Option A (6) is correct because the equation 2 sin θ = r4 - 2r2 + 3 has solutions only when both sides equal 2, leading to 6 possible pairs of (r, θ).
347 Easy
347 of 581
Analyzing +4 | -1
If sin⁡2θ−2sin⁡θ−1=0\sin ^2 \theta-2 \sin \theta-1=0 is to be satisfied for exactly 4 distinct values of θ∈[0,nπ],n∈N\theta \in[0, n \pi], n \in N, then the least value of nn is
View answer and explanation
Correct option: C
4
Option C (4) is correct because the equation sin⁡2θ−2sin⁡θ−1=0\sin^2 \theta - 2 \sin \theta - 1 = 0 has exactly 4 distinct values of θ\theta in the interval [0,nπ][0, n\pi] when n = 4.
348 Medium
348 of 581
Application +4 | -1
If 3  cos⁡ x≠2  sin⁡x3\,\,\cos \,x \ne 2\,\,\sin x, then the general solution of sin⁡2x−cos⁡  2x=2−sin⁡  2x{\sin ^2}x - \cos \,\,2x = 2 - \sin \,\,2x is
View answer and explanation
Correct option: C
(4n±1)π2,   n∈Z\left( {4n \pm 1} \right)\frac{\pi }{2},\,\;n \in Z
Option C ((4n±1)π2,; n∈Z\left( {4n \pm 1} \right)\frac{\pi }{{2}},\\;\,n \in Z) is correct because solving the trigonometric equation leads to this general solution.
349 Easy
349 of 581
Application +4 | -1
The values of θ\theta satisfying sin⁡  7  θ=sin⁡  4  θ−sin⁡  θ\sin \,\,7\;\theta = \sin \,\,4\;\theta - \sin \,\,\theta and 0<θ<π20 < \theta < \frac{\pi }{2} are
View answer and explanation
Correct option: A
π9,  π4\frac{\pi }{9},\,\,\frac{\pi }{4}
Option A (π9,  π4\frac{\pi }{9},\,\,\frac{\pi }{4}) is correct because these are the solutions to the trigonometric equation within the given range, found by using trigonometric identities and solving for θ\theta.
350 Medium
350 of 581
Application +4 | -1
If cos⁡θ=12(x+1x)\cos \theta=\frac{1}{2}\left(x+\frac{1}{x}\right), then 12(x2+1x2)\frac{1}{2}\left(x^2+\frac{1}{x^2}\right) is equal to
View answer and explanation
Correct option: B
cos⁡2θ\cos 2 \theta
Option B (cos⁡2θ\cos 2 \theta) is correct because it is derived from the given equation by squaring and using the double angle formula for cosine.
351 Easy
351 of 581
Application +4 | -1
The values of θ\theta in the interval (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) satisfying the equation (3)sec⁡2θ=tan⁡4θ+2tan⁡2θ(\sqrt{3})^{\sec ^2 \theta}=\tan ^4 \theta+2 \tan ^2 \theta is
View answer and explanation
Correct option: A
π4\frac{\pi}{4}
Option A (π4\frac{\pi}{4}) is correct because it is the only value in the given interval that satisfies the provided trigonometric equation.
352 Easy
352 of 581
Analyzing +4 | -1
Total number of solutions of cos⁡  x=1−sin⁡2x\cos \,\,x = \sqrt {1 - \sin 2x} in [0,   2π]\left[ {0,\;\,2\mathrm{\pi }} \right] is equal to
View answer and explanation
Correct option: A
2
Option A (2) is correct because there are two solutions to the given equation within the specified interval.
353 Medium
353 of 581
Analyzing +4 | -1
Let α,  β\mathrm{\alpha },\,\,\mathrm{\beta } be any two positive values of xx for which 2cos⁡  x,  ∣cos⁡  x∣     and   1−3  cos⁡2x2\cos \,\,x,\;|\cos \,\,x|\;\;\,\mathrm{and}\;\,1 - 3\,\,{\cos ^2}x are in GP. The minimum value of ∣α−β∣\left| {\mathrm{\alpha } - \mathrm{\beta }} \right| is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the minimum value of |α−β\alpha - \beta| is π6\frac{\pi}{6}, which is not among the given options.
354 Easy
354 of 581
Analyzing +4 | -1
The arithmetic mean of the roots of the equation 4cos⁡3x−4cos⁡2x−cos⁡(315 π+x)=14{\cos ^3}x - 4{\cos ^2}x - \cos \left( {315\,\pi + x} \right) = 1 in the interval (0,315)\left( {\mathrm{0, 315}} \right) is equal to
View answer and explanation
Correct option: B
51 π\mathrm{51}\,\mathrm{\pi }
Option B (51 π\mathrm{51}\,\mathrm{\pi }) is correct because it is the arithmetic mean of the roots of the given equation within the specified interval.
355 Medium
355 of 581
Application +4 | -1
If sin⁡x+sin⁡2x=1\sin x+\sin ^2 x=1, then the value of cos⁡12x+3cos⁡10x+3cos⁡8x+cos⁡6x\cos ^{12} x+3 \cos ^{10} x+3 \cos ^8 x+\cos ^6 x is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the given expression simplifies to 1 using the trigonometric identity sin⁡x=cos⁡2x\sin x = \cos^2 x.
356 Medium
356 of 581
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
The correct answer is No solution because the given condition ∣c∣>a2+b2|c| > \sqrt{{a^2+b^2}} implies that the value of c is outside the range of asin⁡x+bcos⁡xa \sin x + b \cos x.
357 Medium
357 of 581
Application +4 | -1
If tan⁡ (πcos⁡  θ)=cot⁡ (π  sin⁡  θ)\tan \,(\mathrm{\pi }\cos \,\,\mathrm{\theta }) = \cot \,(\mathrm{\pi }\,\,\sin \,\,\mathrm{\theta }), then the value of cos⁡  (θ−π4)\cos \,\,\left( {\mathrm{\theta } - \frac{\mathrm{\pi }}{4}} \right) is equal to
View answer and explanation
Correct option: A
12 2\frac{1}{{2\,\sqrt 2 }}
The correct answer is A (122\frac{1}{{2\sqrt 2 }}) because solving the given equation leads to sin⁡θ+cos⁡θ=12\sin \theta + \cos \theta = \frac{1}{2}, which simplifies to the desired expression.
358 Medium
358 of 581
Analyzing +4 | -1
The number of points of intersection of the curves y=cos⁡x,y=sin⁡3xy=\cos x, y=\sin 3 x, if −π2≤x≤π2-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}, is
View answer and explanation
Correct option: A
3
Option A, 3, is correct because there are three points of intersection between the curves y=cos⁡xy=\cos x and y=sin⁡3xy=\sin 3x within the given interval.
359 Easy
359 of 581
Application +4 | -1
The exhaustive range of values of θ\theta in the interval (0,π)(0, \pi) such that the points (3,5)(3,5) and (3sin⁡θ,cos⁡θ)(\sqrt{3} \sin \theta, \cos \theta) lies on the same side of line x+y−1=0x+y-1=0 is
View answer and explanation
Correct option: A
3
The correct answer is that the range of θ is (0, 2π/3). This is found by ensuring that both points lie on the same side of the line x+y-1=0, leading to the inequality sin⁡(θ+π/6)>1/2\sin(\theta + \pi/6) > 1/2.
360 Medium
360 of 581
Analyzing +4 | -1
For 0<ϕ<π20 < \phi < \frac{\pi}{2}, if x=∑n=0∞cos⁡2nϕ,y=∑n=0∞sin⁡2nϕx=\sum\limits_{{n=0}}^{\infty} \cos ^{2 n} \phi, y=\sum\limits_{{n=0}}^{\infty} \sin ^{2 n} \phi and z=∑n=0∞cos⁡2nϕsin⁡2nϕz=\sum\limits_{{n=0}}^{\infty} \cos ^{2 n} \phi \sin ^{2 n} \phi, then
View answer and explanation
Correct option: C
xyz=x+y+zx y z=x+y+z
Option D (xyz=yz+xx y z=y z+x) is incorrect. The correct option is C (xyz=x+y+zx y z=x+y+z) because of the relationship between the sums and the trigonometric functions.
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