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JEE • MATHEMATICS • TRANSFORMATIONS

Transformations practice · Mixed difficulty

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Showing 91-120 of 165 questions
91 Medium
91 of 165
Application +4 | -1
The maximum value of 3 cos⁡ θ+5 sin⁡(θ−π6)3\,\cos \,\theta + 5\,\sin \left( {\theta - \frac{\pi }{6}} \right) for any real value of θ\theta is
View answer and explanation
Correct option: D
19\sqrt {19}
Option C (19\sqrt {19}) is correct. The maximum value is found by converting the expression into the form a*cos(theta) + b*sin(theta) and then using the formula sqrt(a2 + b2).
92 Medium
92 of 165
Application +4 | -1
Minimum value of 13  sin⁡  θ−4  cos⁡  θ+7\frac{1}{{3\,\,\sin \,\,\theta - 4\,\,\cos \,\,\theta + 7}} is
View answer and explanation
Correct option: A
112\frac{1}{{12}}
Option A, 112\frac{1}{{12}}, is correct because it is the minimum value of the given expression, which occurs when the denominator is maximized.
93 Medium
93 of 165
Analyzing +4 | -1
If an+1=12  (1+an){a_{n + 1}} = \sqrt {\frac{1}{2}\,\,\left( {1 + {a_n}} \right)}, then cos⁡  (1−a02a1 a2 a3…to    ∞)=\cos \,\,\left( {\frac{{\sqrt {1 - a_0^2} }}{{{a_1}\,{a_2}\,{a_3} \ldots \mathrm{to\;}\,\,\infty }}} \right) =
View answer and explanation
Correct option: C
a0{a_0}
Option C (a0{a_0}) is correct because the infinite product simplifies to a value that allows the cosine to be evaluated, resulting in a0{a_0}.
94 Easy
94 of 165
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.
95 Easy
95 of 165
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.
96 Easy
96 of 165
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.
97 Easy
97 of 165
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.
98 Easy
98 of 165
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}.
99 Medium
99 of 165
Analyzing +4 | -1
In a triangle ABC,  sin⁡A−cos⁡  B=cos⁡  CABC,\,\,\sin A - \cos \,\,B = \cos \,\,C, then angle BB is
View answer and explanation
Correct option: A
π2\frac{\pi }{2}
Option A (π2\frac{\pi }{2}) is correct because substituting sin⁡A=cos⁡B+cos⁡C\sin A = \cos B + \cos C and using trigonometric identities leads to B=π2B = \frac{\pi}{2}.
100 Medium
100 of 165
Application +4 | -1
If A+B+C=πA+B+C=\pi, then the value of cos⁡2A+cos⁡2B+cos⁡2C\cos ^2 A+\cos ^2 B+\cos ^2 C is
View answer and explanation
Correct option: D
1−2cos⁡Acos⁡Bcos⁡C1-2 \cos A \cos B \cos C
Option D (1−2cos⁡Acos⁡Bcos⁡C1-2 \cos A \cos B \cos C) is correct because by using trigonometric identities and the given condition A+B+C=πA+B+C=\pi, we can simplify the expression.
101 Easy
101 of 165
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.
102 Medium
102 of 165
Application +4 | -1
The value of cos⁡4  (π8)+cos⁡4  (3 π8)+cos⁡4  (5 π8)+cos⁡4  (7π8){\cos ^4}\,\,\left( {\frac{\pi }{8}} \right) + {\cos ^4}\,\,\left( {\frac{{3\,\pi }}{8}} \right) + {\cos ^4}\,\,\left( {\frac{{5\,\pi }}{8}} \right) + {\cos ^4}\,\,\left( {\frac{{7\pi }}{8}} \right) is
View answer and explanation
Correct option: C
32\frac{3}{2}
Option C (32\frac{3}{2}) is correct because it is the result of simplifying the given trigonometric expression using trigonometric identities and angle transformations.
103 Easy
103 of 165
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.
104 Easy
104 of 165
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.
105 Easy
105 of 165
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.

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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
106 Easy
106 of 165
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.
107 Easy
107 of 165
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.
108 Easy
108 of 165
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.
109 Medium
109 of 165
Analyzing +4 | -1
If the equation cos⁡  3x+cos⁡  2x=sin⁡  3x2+sin⁡  x2,   0≤x≤2π\cos \,\,3x + \cos \,\,2x = \sin \,\,\frac{{3x}}{2} + \sin \,\,\frac{x}{2},\,\;0 \le x \le 2\pi, then the number of values of xx is
View answer and explanation
Correct option: D
5
Option D is correct because solving the trigonometric equation yields 5 distinct solutions within the given interval 0 ≤ x ≤ 2π.
110 Medium
110 of 165
Application +4 | -1
The minimum value of f(x)=sin⁡4  x+cos⁡4  x,  0≤  x≤π2f\left( x \right) = {\sin ^4}\,\,x + {\cos ^4}\,\,x,\;0 \le \,\,x \le \frac{\pi }{2} is

View answer and explanation
Correct option: D
12\frac{1}{2}
Option D (12\frac{1}{2}) is correct because the minimum value of the function sin⁡4 x+cos⁡4 x{\sin ^4}\,x + {\cos ^4}\,x in the given range is 12\frac{1}{2}.
111 Medium
111 of 165
Analyzing +4 | -1
The expression 2cos⁡  π13cos⁡  9 π13+cos⁡  3 π13+cos⁡  5 π132\cos \,\,\frac{\pi }{{13}}\cos \,\,\frac{{9\,\pi }}{{13}} + \cos \,\,\frac{{3\,\pi }}{{13}} + \cos \,\,\frac{{5\,\pi }}{{13}} is equal to
View answer and explanation
Correct option: B
0
Option B (0) is correct because by applying trigonometric identities and simplification, the expression reduces to 0.
112 Medium
112 of 165
Analyzing +4 | -1
If A=cos⁡2θ+sin⁡4θA = {\cos ^2}\theta + {\sin ^4}\theta, then for all values of θ,
View answer and explanation
Correct option: D
34≤A≤1\frac{3}{4} \le A \le 1
Option D (34≤A≤1\frac{3}{4} \le A \le 1) is correct because the given expression can be bounded between these values by using trigonometric identities and properties.
113 Medium
113 of 165
Application +4 | -1
Given π2<α<π\frac{\pi}{2} < \alpha < \pi, then the expression 1−sin⁡α1+sin⁡α+1+sin⁡α1−sin⁡α\sqrt{\frac{1-\sin \alpha}{1+\sin \alpha}}+\sqrt{\frac{1+\sin \alpha}{1-\sin \alpha}} is equal to
View answer and explanation
Correct option: B
−2cos⁡α-\frac{2}{\cos \alpha}
Option B (−2cos⁡α-\frac{2}{\cos \alpha}) is correct because, given the range of α\alpha, the expression simplifies to this value using trigonometric identities.
114 Easy
114 of 165
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.
115 Easy
115 of 165
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.
116 Easy
116 of 165
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.
117 Easy
117 of 165
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.
118 Medium
118 of 165
Application +4 | -1
If A+B+C=270∘A + B + C = 270^\circ, then

cos⁡  2A+cos⁡  2B+cos⁡  2C\cos \,\,2A + \cos \,\,2B + \cos \,\,2C is equal to
View answer and explanation
Correct option: C
1−4  sin⁡A  sin⁡B  sin⁡C  1 - 4\,\,\sin A\,\,\sin B\,\,\sin C\;
Option D (1−4 cos⁡A cos⁡B cos⁡C1 - 4\,\cos A\,\cos B\,\cos C) is correct because it is the result of applying trigonometric identities to simplify the given expression with the condition A+B+C=270∘A + B + C = 270^\circ.
119 Easy
119 of 165
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.
120 Easy
120 of 165
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}.
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