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JEE • MATHEMATICS • TRANSFORMATIONS

Transformations practice · Easy

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Showing 61-90 of 112 questions
61 Easy
61 of 112
Application +4 | -1
The period of the function f(θ)=sin⁡θ3+cos⁡θ2f\left( \mathrm{\theta } \right) = \sin \frac{\mathrm{\theta }}{3} + \cos \frac{\mathrm{\theta }}{2} is
View answer and explanation
Correct option: D
12π12\pi
Option D is correct because the period of the sum of two trigonometric functions is the LCM of their individual periods.
62 Easy
62 of 112
Application +4 | -1
The function f(x)=sin⁡πx2+2cos⁡πx3−tan⁡πx4f\left( x \right) = \sin \frac{{\pi x}}{2} + 2\cos \frac{{\pi x}}{3} - \tan \frac{{\pi x}}{4} is periodic with period
View answer and explanation
Correct option: D
12
Option D (12) is correct because the period of the sum of periodic functions is the least common multiple (LCM) of their individual periods, which in this case is LCM(4, 6, 4) = 12.
63 Easy
63 of 112
Application +4 | -1
The period of the function f(x)=sin⁡(sin⁡x5)f\left( x \right) = \sin \left( {\sin \frac{x}{5}} \right) is
View answer and explanation
Correct option: C
10π10\pi
Option C (10π10\pi) is correct because it is the smallest value for which f(x+10π)=f(x)f(x + 10\pi) = f(x).
64 Easy
64 of 112
Application +4 | -1
If sin⁡  (x+3α)=3 sin⁡(α−x)\sin \,\,\left( {x + 3\alpha } \right) = 3\,\sin \left( {\alpha - x} \right), then
View answer and explanation
Correct option: C
tan⁡  x=tan⁡3  α\tan \,\,x = {\tan ^3}\,\,\alpha
Option C (tan⁡ x=tan⁡3 α\tan \,x = {\tan ^3}\,\alpha) is correct because it is derived by applying componendo and dividendo and trigonometric identities to the given equation.
65 Easy
65 of 112
Application +4 | -1
If     xcos⁡  θ=ycos⁡  (θ−2 π3)=zcos(θ+2 π3),   then   x+y+z  \;\;\frac{x}{{\cos \,\,\mathrm{\theta }}} = \frac{y}{{\cos \,\,\left( {\mathrm{\theta } - \frac{{2\,\pi }}{3}} \right)}} = \frac{\mathrm{z}}{{cos\left( {\mathrm{\theta } + \frac{{2\,\pi }}{3}} \right)}},\;\,\mathrm{then}\;\,x + y + \mathrm{z}\; is equal to

View answer and explanation
Correct option: B
0\mathrm{0}
Option B (0\mathrm{0}) is correct because the sum of the cosines in the denominator simplifies to zero, making the entire expression equal to zero.
66 Easy
66 of 112
Application +4 | -1
In Δ ABC\Delta \,ABC, if sin⁡2A+sin⁡2B+sin⁡2C=2{\sin ^2}A + {\sin ^2}B + {\sin ^2}C = 2, then the triangle is
View answer and explanation
Correct option: A
Right angled, but need not be isosceles
Option A (Right angled, but need not be isosceles) is correct because the given condition sin⁡2A+sin⁡2B+sin⁡2C=2{\sin ^2}A + {\sin ^2}B + {\sin ^2}C = 2 implies that one of the angles is 90 degrees, making the triangle right-angled.
67 Easy
67 of 112
Analyzing +4 | -1
If the equation sin⁡4  θ+cos⁡4  θ=a{\sin ^4}\,\,\theta + {\cos ^4}\,\,\theta = a has a real solution then
View answer and explanation
Correct option: C
12≤a≤1\frac{1}{2} \le a \le 1
Option C (1/2 ≤ a ≤ 1) is correct because the given equation has a real solution only when a lies between 1/2 and 1, inclusive.
68 Easy
68 of 112
Application +4 | -1
If α+β−γ=π\alpha + \beta - \gamma = \pi, then sin⁡2  α+sin⁡2  β−sin⁡2  γ {\sin ^2}\,\,\alpha + {\sin ^2}\,\,\beta - {\sin ^2}\,\,\gamma \, is equal to
View answer and explanation
Correct option: A
2  sin⁡  α  sin⁡  β  cos⁡  γ2\,\,\sin \,\,\alpha \,\,\sin \,\,\beta \,\,\cos \,\,\gamma
Option A (2  sin⁡  α  sin⁡  β  cos⁡  γ2\,\,\sin \,\,\alpha \,\,\sin \,\,\beta \,\,\cos \,\,\gamma) is the correct answer because it is derived using trigonometric identities and the given condition α+β−γ=π\alpha + \beta - \gamma = \pi.
69 Easy
69 of 112
Application +4 | -1
The maximum value of cos⁡2  (π3−x)−cos⁡2  (π3+x)  {\cos ^2}\,\,\left( {\frac{\pi }{3} - x} \right) - {\cos ^2}\,\,\left( {\frac{\pi }{3} + x} \right)\; is
View answer and explanation
Correct option: C
32\frac{{\sqrt 3 }}{2}
Option C (32\frac{{\sqrt 3 }}{2}) is correct because the maximum value of the expression is achieved when sin⁡2x=1\sin 2x = 1, resulting in 32\frac{{\sqrt 3 }}{2}.
70 Easy
70 of 112
Application +4 | -1
The value of   sin⁡  20∘  sin⁡  40∘  sin⁡  60∘  sin⁡  80∘  \mathrm{\;}\sin \,\,20^\circ \,\,\sin \,\,40^\circ \,\,\sin \,\,60^\circ \,\,\sin \,\,80^\circ \mathrm{\;} is equal to
View answer and explanation
Correct option: C
316\frac{3}{{16}}
Option C (316\frac{3}{{16}}) is correct because the given trigonometric expression simplifies to 316\frac{3}{{16}} using trigonometric identities.
71 Easy
71 of 112
Analyzing +4 | -1
The value of cos⁡  π7+cos⁡  2 π7+cos⁡  3 π7+cos⁡  4 π7+cos⁡  5 π7+cos6π7+cos⁡  7π7\cos \,\,\frac{\pi }{7} + \cos \,\,\frac{{2\,\pi }}{7} + \cos \,\,\frac{{3\,\pi }}{7} + \cos \,\,\frac{{4\,\pi }}{7} + \cos \,\,\frac{{5\,\pi }}{7} + \mathrm{cos}\frac{{6\pi }}{7} + \cos \,\,\frac{{7\pi }}{7}, is
View answer and explanation
Correct option: B
−1- 1
Option B (-1) is correct because the sum of the given cosine series simplifies to -1.
72 Easy
72 of 112
Application +4 | -1
The value of sin⁡  50∘−sin⁡70∘+sin⁡10∘\sin \,\,50^\circ - \sin 70^\circ + \sin 10^\circ is
View answer and explanation
Correct option: A
0
Option A (0) is correct because the expression simplifies to 0 using trigonometric identities.
73 Easy
73 of 112
Analyzing +4 | -1
The range of the function,
f(x)=log⁡5(3+cos⁡(3π4+x)+cos⁡(π4+x)+cos⁡(π4−x)−cos⁡(3π4−x))f(x) = {\log _{\sqrt 5 }}\left( {3 + \cos \left( {\frac{{3\pi }}{4} + x} \right) + \cos \left( {\frac{\pi }{4} + x} \right) + \cos \left( {\frac{\pi }{4} - x} \right) - \cos \left( {\frac{{3\pi }}{4} - x} \right)} \right) is
View answer and explanation
Correct option: D
[0,  2][0,\,\,2]
Option D ([0, 2]) is correct because it represents the range of the function after simplifying the trigonometric expression and applying logarithmic properties.
74 Easy
74 of 112
Understanding +4 | -1
The smallest value of 5  cos⁡  θ+125\,\,\cos \,\,\theta + 12 is
View answer and explanation
Correct option: C
7
Option C (7) is correct because the minimum value of cos θ is -1, so the minimum value of the expression is 5*(-1) + 12 = 7.
75 Easy
75 of 112
Analyzing +4 | -1
If sin⁡  α=sin⁡  β\sin \,\,\alpha = \sin \,\,\beta and cos⁡  α=cos⁡  β\cos \,\,\alpha = \cos \,\,\beta, then
View answer and explanation
Correct option: C
sin⁡  α−β2=0\sin \,\,\frac{{\alpha - \beta }}{2} = 0
Option C (sin⁡α−β2=0\sin \frac{{\alpha - \beta }}{2} = 0) is correct because the given conditions sin⁡α=sin⁡β\sin \alpha = \sin \beta and cos⁡α=cos⁡β\cos \alpha = \cos \beta imply that sin⁡α−β2=0\sin \frac{{\alpha - \beta }}{2} = 0.

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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
76 Easy
76 of 112
Application +4 | -1
If   α+β−γ=π,   then  sin⁡2  α+sin⁡2  β−sin⁡2  γ  \;\alpha + \beta - \gamma = \pi ,\,\;\mathrm{then}\,\,{\sin ^2}\,\,\alpha + {\sin ^2}\,\,\beta - {\sin ^2}\,\,\gamma \; is equal to

View answer and explanation
Correct option: A
2  sin⁡  α  sin⁡β  cos⁡  γ2\,\,\sin \,\,\alpha \,\,\sin \beta \,\,\cos \,\,\gamma
Option A, 2 sin⁡ α sin⁡β cos⁡ γ2\,\sin \,\alpha \,\sin \beta \,\cos \,\gamma, is correct because it is the simplified form of the given expression using trigonometric identities and the given condition   α+β−γ=π\;\alpha + \beta - \gamma = \pi.
77 Easy
77 of 112
Application +4 | -1
If cos⁡  x=3  cos⁡  y\cos \,\,x = 3\,\,\cos \,\,y, then 2  tan⁡  y−x22\,\,\tan \,\,\frac{{y - x}}{2} is equal to
View answer and explanation
Correct option: D
cot⁡  (x+y2)\cot \,\,\left( {\frac{{x + y}}{2}} \right)
Option D (cot⁡  (x+y2)\cot \,\,\left( {\frac{{x + y}}{2}} \right)) is correct because it is derived by applying componendo and dividendo and using trigonometric identities.
78 Easy
78 of 112
Analyzing +4 | -1
If in a Δ ABC,  cos⁡  A+2  cos⁡  B+cos⁡  C=2\Delta \,ABC,\,\,\cos \,\,A + 2\,\,\cos \,\,B + \cos \,\,C = 2, then a,  b,  ca,\,\,b,\,\,c are in
View answer and explanation
Correct option: A
A.P.
Option A (A.P.) is correct because the given condition leads to the relationship a + c = 2b, which defines an arithmetic progression.
79 Easy
79 of 112
Application +4 | -1
If A+B+C+D=πA+B+C+D=\pi, then the value of Σcos⁡Acos⁡C−Σsin⁡Asin⁡C=\Sigma \cos A \cos C-\Sigma \sin A \sin C=
View answer and explanation
Correct option: D
0
Option D (0) is correct because the given expression simplifies to cos⁡(A+C)+cos⁡(A+D)+cos⁡(B+C)+cos⁡(B+D)+cos⁡(A+B)+cos⁡(C+D)=0\cos(A+C) + \cos(A+D) + \cos(B+C) + \cos(B+D) + \cos(A+B) + \cos(C+D) = 0 given A+B+C+D=πA+B+C+D = \pi.
80 Easy
80 of 112
Application +4 | -1
The value of sin⁡  π14  sin⁡  3 π14  sin⁡  5 π14  sin⁡  7π14\sin \,\,\frac{\pi }{{14}}\,\,\sin \,\,\frac{{3\,\pi }}{{14}}\,\,\sin \,\,\frac{{5\,\pi }}{{14}}\,\,\sin \,\,\frac{{7\pi }}{{14}}, is
View answer and explanation
Correct option: C
1 / 8\mathrm{1}\,\mathrm{/}\,\mathrm{8}
Option C (1/8) is correct because the product of sines can be simplified using trigonometric identities to obtain the value 1/8.
81 Easy
81 of 112
Application +4 | -1
The minimum value of cos⁡  2θ+cos⁡  θ\cos \,\,2\theta + \cos \,\,\theta for real values of θ\theta, is
View answer and explanation
Correct option: A
− 9/8- \,9/8
Option A (− 9/8- \,9/8) is correct because the minimum value of cos⁡ 2θ+cos⁡θ\cos \,2\theta + \cos \theta is − 9/8- \,9/8 when θ\theta is a real number.
82 Easy
82 of 112
Application +4 | -1
sin⁡  12∘  sin⁡  48∘  sin⁡  54∘\sin \,\,12^\circ \,\,\sin \,\,48^\circ \,\,\sin \,\,54^\circ is equal to
View answer and explanation
Correct option: C
1 / 8\mathrm{1}\,\mathrm{/}\,\mathrm{8}
Option C (1 / 8\mathrm{1}\,\mathrm{/}\,\mathrm{8}) is correct because by using product-to-sum identities, the expression simplifies to 18\frac{1}{8}.
83 Easy
83 of 112
Application +4 | -1
The period of the function f(x)=cosec23x+cot⁡4xf\left( x \right) = \mathrm{cose}{\mathrm{c}^2}3x + \cot 4x is
View answer and explanation
Correct option: D
π\pi
Option D (π\pi) is correct because it is the least common multiple of the periods of the individual functions, which are π3\frac{\pi}{3} and π4\frac{\pi}{4}.
84 Easy
84 of 112
Application +4 | -1
Maximum value of sin⁡θ+cos⁡θ\sin \theta + \cos \theta in [0,   π2]\left[ {0,\;\,\frac{\pi }{2}} \right] is

View answer and explanation
Correct option: A
2\sqrt 2
Option A (2\sqrt 2) is correct because the maximum value of sin⁡θ+cos⁡θ\sin \theta + \cos \theta is 2\sqrt 2 and occurs at θ=π4\theta = \frac{\pi}{4} within the given interval.
85 Easy
85 of 112
Application +4 | -1
The value of sin⁡  55∘−cos⁡  55∘sin⁡  10∘\frac{{\sin \,\,55^\circ - \cos \,\,55^\circ }}{{\sin \,\,10^\circ }} is
View answer and explanation
Correct option: D
2\sqrt 2
Option D (2\sqrt{2}) is correct because the expression simplifies to 2\sqrt{2} after applying the sine subtraction formula and using the value of cos⁡45∘\cos 45^\circ.
86 Easy
86 of 112
Analyzing +4 | -1
The value of sin⁡  12∘  sin⁡  24∘  sin⁡  48∘  sin⁡  84∘\sin \,\,12^\circ \,\,\sin \,\,24^\circ \,\,\sin \,\,48^\circ \,\,\sin \,\,84^\circ, is
View answer and explanation
Correct option: A
cos⁡  20∘  cos⁡  40∘  cos⁡  60∘  cos⁡   80∘\cos \,\,20^\circ \,\,\cos \,\,40^\circ \,\,\cos \,\,60^\circ \,\,\cos \,\,\,80^\circ
Option A (cos⁡ 20∘ cos⁡ 40∘ cos⁡ 60∘ cos⁡ 80∘\cos \,20^\circ \,\cos \,40^\circ \,\cos \,60^\circ \,\cos \,80^\circ) is correct because it is equal to the given expression after applying trigonometric identities.
87 Easy
87 of 112
Application +4 | -1
If sin⁡  (x+y)sin⁡  (x−y)=a+ba−b\frac{{\sin \,\,\left( {x + y} \right)}}{{\sin \,\,\left( {x - y} \right)}} = \frac{{a + b}}{{a - b}}, then tan⁡  xtan⁡  y\frac{{\tan \,\,x}}{{\tan \,\,y}} is equal to
View answer and explanation
Correct option: B
ab\frac{a}{b}
Option B (ab\frac{a}{b}) is correct because it is derived by applying the componendo and dividendo rule and simplifying the trigonometric ratios.
88 Easy
88 of 112
Application +4 | -1
If sin⁡A=nsin  B\sin A = nsin\;B, then n−1n+1  tan⁡  A+B2\frac{{n - 1}}{{n + 1}}\,\,\tan \,\,\frac{{A + B}}{2} is equal to
View answer and explanation
Correct option: B
tan⁡  A−B2\tan \,\,\frac{{A - B}}{2}
Option B tan⁡  A−B2\tan \,\,\frac{{A - B}}{2} is correct because applying componendo and dividendo to n1=sin⁡Asin⁡B\frac{n}{1} = \frac{{\sin A}}{{\sin B}} and simplifying leads to the given expression being equal to tan⁡  A−B2\tan \,\,\frac{{A - B}}{2}.
89 Easy
89 of 112
Analyzing +4 | -1
If A+B+C=180∘A + B + C = 180^\circ, then ∑  tan⁡  A2  tan⁡  B2\sum \,\,\tan \,\,\frac{A}{2}\,\,\tan \,\,\frac{B}{2} is
View answer and explanation
Correct option: B
1
Option B (1) is correct because the given condition A+B+C=180∘A + B + C = 180^\circ leads to the trigonometric identity ∑tan⁡A2tan⁡B2=1\sum \tan \frac{A}{2} \tan \frac{B}{2} = 1.
90 Easy
90 of 112
Application +4 | -1
If 5  cos⁡x+12  cos⁡y=135\,\,\cos x + 12\,\,\cos y = 13, then the maximum value of 5  sin⁡x+12  sin⁡y5\,\,\sin x + 12\,\,\sin y is
View answer and explanation
Correct option: B
120\sqrt {120}
Option B is correct because it represents the maximum value obtained by using the given equation and trigonometric identities.
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