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Trigonometric Functions and Equations

31 questions · Mathematics · JEE Advanced

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JEE Advanced
Mathematics
Trigonometric Functions and Equations

Trigonometric Functions and Equations

31 questions · Mathematics · JEE Advanced

Let $α = \frac{1}{s i n 6 0 ^{\circ} s i n 6 1 ^{\circ}} + \frac{1}{s i n 6 2 ^{\circ} s i n 6 3 ^{\circ}} + \dots + \frac{1}{s i n 11 8 ^{\circ} s i n 11 9 ^{\circ}}$ Then the value of $(\frac{cosec 1 ^{\circ}}{α})^{2}$ is …2025 · Shift 2 · Q31 · Numerical
Let $\frac{π}{2}$ $(sin \frac{11 x}{2}) (sin 6 x - cos 6 x) + (cos \frac{11 x}{2}) (sin 6 x + cos 6 x)$ $ is equal to :2024 · Shift 1 · Q20 · MCQ
Consider an obtuse angled triangle $A B C$ in which the difference between the largest and the smallest angle is $\frac{π}{2}$ and whose sides are in arithmetic progression. Suppose that the vertices of this triangle lie on a circle of…2023 · Shift 2 · Q32 · Numerical
Consider the following lists : The correct option is: Includes table2022 · Shift 1 · Q33 · MCQ
Let $α$ and $β$ be real numbers such that $- \frac{π}{4} < β < 0 < α < \frac{π}{4}$ . If $sin (α + β) = \frac{1}{3}$ and $cos (α - β) = \frac{2}{3}$ , then the greatest integer less than or equal to $(\frac{s i n α}{c o s β} + \frac{c o s β}{s i n α} + \frac{c o s α}{s i n β} + \frac{s i n β}{c o s α})^{2}$ …2022 · Shift 2 · Q19 · Numerical
Let $P QR S$ be a quadrilateral in a plane, where $QR = 1, ∠ P QR = ∠ QR S = 7 0^{\circ}, ∠ P QS = 1 5^{\circ}$ and $∠ P R S = 4 0^{\circ}$ . If $∠ R P S = θ^{\circ}, P Q = α$ and $P S = β$ , then the interval(s)…2022 · Shift 2 · Q27 · Multiple correct
For non-negative integers n, let $f (n) = \frac{k = 0 \sum n s i n ( \frac{k + 1}{n + 2} π ) s i n ( \frac{k + 2}{n + 2} π )}{k = 0 \sum n s i n ^{2} ( \frac{k + 1}{n + 2} π )}$ …2019 · Shift 2 · Q19 · Multiple correct
Let f(x) = sin( $π$ cos x) and g(x) = cos(2 $π$ sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order: X = {x : f(x) = 0}, Y = {x : f'(x) = 0} Z = {x : g(x) = 0}, W =… Includes diagram2019 · Shift 2 · Q33 · MCQ
Let f(x) = sin( $π$ cos x) and g(x) = cos(2 $π$ sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order : X = {x : f(x) = 0}, Y = {x : f'(x) = 0} Z = {x : g(x) = 0}, W =… Includes diagram2019 · Shift 2 · Q34 · MCQ
In a $Δ$ PQR = 30 $^{\circ}$ and the sides PQ and QR have lengths 10 $3$ and 10, respectively. Then, which of the following statement(s) is(are) TRUE?2018 · Shift 1 · Q20 · Multiple correct
Let f : R $\to$ R be a differentiable function with f(0) = 1 and satisfying the equation f(x + y) = f(x) f'(y) + f'(x) f(y) for all x, y $\in$ R. Then, the value of loge(f(4)) is ...........2018 · Shift 2 · Q29 · Numerical
Let $α$ and $β$ be non zero real numbers such that $2 (cos β - cos α) + cos α cos β = 1$ . Then which of the following is/are true?2017 · Shift 2 · Q29 · Multiple correct
Let O be the origin and $O X$ , $O Y$ , $O Z$ be three unit vectors in the directions of the sides $QR$ , $R P$ , $P Q$ respectively, of a…2017 · Shift 2 · Q33 · MCQ
Let $S = {x \in (- π, π) : x e 0, \pm \frac{π}{2}} .$ The sum of all distinct solutions of the equation $3 sec x + csc x + 2 (tan x - cot x) = 0$ in the set S is equal to2016 · Shift 1 · Q19 · MCQ
The value of $k = 1 \sum 13 \frac{1}{s i n ( \frac{π}{4} + \frac{( k - 1 ) π}{6} ) s i n ( \frac{π}{4} + \frac{k π}{6} )}$ is equal to2016 · Shift 2 · Q20 · MCQ
The number of distinct solutions of the equation $\frac{5}{4} cos^{2} 2 x + cos^{4} x + sin^{4} x + cos^{6} x + sin^{6} x = 2$ in the interval $[0, 2 π]$ is2015 · Shift 1 · Q22 · Numerical
For $x \in (0, π),$ the equation $sin x + 2 sin 2 x - sin 3 x = 3$ has2014 · Shift 2 · Q22 · MCQ
Let $f (x) = x sin π x, x > 0.$ Then for all natural numbers $n, f^{'} (x)$ vanishes at2013 · Shift 1 · Q21 · Multiple correct
The number of points in $(- \infty \infty),$ for which $x^{2} - x sin x - cos x = 0,$ is2013 · Shift 1 · Q24 · MCQ
Let $θ, φ \in [0, 2 π]$ be such that $2 cos θ (1 - sin φ) = sin^{2} θ (tan \frac{θ}{2} + cot \frac{θ}{2}) cos φ - 1, tan (2 π - θ) > 0$ …2012 · Shift 1 · Q21 · Multiple correct
The positive integer value of $n > 3$ satisfying the equation $\frac{1}{s i n ( \frac{π}{n} )} = \frac{1}{s i n ( \frac{2 π}{n} )} + \frac{1}{s i n ( \frac{3 π}{n} )}$ is2011 · Shift 1 · Q25 · Numerical
Let $P = {θ : sin θ - cos θ = 2 cos θ}$ and $Q = {θ : sin θ + cos θ = 2 sin θ}$ be two sets. Then2011 · Shift 1 · Q40 · MCQ
The number of values of $θ$ in the interval, $(- \frac{π}{2}, \frac{π}{2})$ such that $θ e \frac{nπ}{5}$ for $n = 0, \pm 1, \pm 2$ and $tan θ = cot 5 θ$ as well as $sin 2 θ = cos 4 θ$ …2010 · Shift 1 · Q29 · Numerical
The number of all possible values of $θ$ where $0 < θ < π,$ for which the system of equations $(y + z) cos 3 θ = (x y z) sin 3 θ x sin 3 θ = \frac{2 c o s 3 θ}{y} + \frac{2 s i n 3 θ}{z} (x y z) sin 3 θ = (y + 2 z) cos 3 θ + y s in 3 θ$ …2010 · Shift 1 · Q30 · Numerical
The maximum value of the expression $\frac{1}{s i n ^{2} θ + 3 s i n θ c o s θ + 5 c o s ^{2} θ}$ is2010 · Shift 1 · Q31 · Numerical
Two parallel chords of a circle of radius 2 are at a distance $3 + 1$ apart. If the chords subtend at the center , angles of $\frac{π}{k}$ and $\frac{2 π}{k},$ where $k > 0,$ then the value of $[k]$ is [Note…2010 · Shift 2 · Q24 · Numerical
If $\frac{s i n ^{4} x}{2} + \frac{c o s ^{4} x}{3} = \frac{1}{5},$ then2009 · Shift 1 · Q35 · Multiple correct
For $0 < θ < \frac{π}{2},$ the solution (s) of $m = 1 \sum 6 csc (θ + \frac{( m - 1 ) π}{4}) csc (θ + \frac{mπ}{4}) = 42$ is (are)2009 · Shift 2 · Q21 · Multiple correct
Match the statements/expressions in Column I with the values given in Column II: Includes table2009 · Shift 2 · Q23 · MCQ
Match the Statements/Expressions in Column I with the Statements/Expressions in Column II. Includes table2008 · Shift 2 · Q44 · MCQ
The number of solutions of the pair of equations $2 sin^{2} θ - cos 2 θ = 02 cos^{2} θ - 3 sin θ = 0$ in the interval $[0, 2 π]$ is2007 · Shift 1 · Q31 · MCQ