LFM Pure

isi-entrance 2014 Q3 Solve trigonometric equation for solutions in an interval View

Find the number of solutions of $\sec x + \tan x = 2\cos x$ in $[0, 2\pi]$.
(A) 0 (B) 1 (C) 2 (D) 3

isi-entrance 2016 Q20 4 marks Solve trigonometric equation for solutions in an interval View

In the triangle $A B C$, the angle $\angle B A C$ is a root of the equation $$\sqrt { 3 } \cos x + \sin x = 1 / 2$$ Then the triangle $A B C$ is
(A) obtuse angled
(B) right angled
(C) acute angled but not equilateral
(D) equilateral

isi-entrance 2016 Q33 4 marks Solve trigonometric equation for solutions in an interval View

The set of all solutions of the equation $\cos 2\theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n\pi + \frac{\pi}{2}$, where $n$ is any integer
(C) $\theta = 2n\pi$ or $\theta = 2n\pi - \frac{\pi}{2}$ or $\theta = n\pi - \frac{\pi}{4}$, where $n$ is any integer
(D) $\theta = 2n\pi$ or $\theta = n\pi + \frac{\pi}{4}$, where $n$ is any integer

isi-entrance 2016 Q33 4 marks Solve trigonometric equation for solutions in an interval View

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(A) $\theta = 0$
(B) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer
(C) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer
(D) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer

isi-entrance 2016 Q35 4 marks Inverse trigonometric equation View

The value of $$\sin ^ { - 1 } \cot \left[ \sin ^ { - 1 } \left\{ \frac { 1 } { 2 } \left( 1 - \sqrt { \frac { 5 } { 6 } } \right) \right\} + \cos ^ { - 1 } \sqrt { \frac { 2 } { 3 } } + \sec ^ { - 1 } \sqrt { \frac { 8 } { 3 } } \right]$$ is
(A) 0
(B) $\pi / 6$
(C) $\pi / 4$
(D) $\pi / 2$

isi-entrance 2016 Q45 4 marks Inverse trigonometric equation View

The number of solutions of the equation $\sin^{-1} x = 2 \tan^{-1} x$ is
(A) 1
(B) 2
(C) 3
(D) 5

isi-entrance 2016 Q45 4 marks Inverse trigonometric equation View

The number of solutions of the equation $\sin ^ { - 1 } x = 2 \tan ^ { - 1 } x$ is
(A) 1
(B) 2
(C) 3
(D) 5

isi-entrance 2018 Q1 Locus or solution set characterization of a trigonometric relation View

Find all pairs $( x , y )$ with $x , y$ real, satisfying the equations: $$\sin \left( \frac { x + y } { 2 } \right) = 0 , \quad | x | + | y | = 1$$

isi-entrance 2018 Q19 Locus or solution set characterization of a trigonometric relation View

For a real number $\alpha$, let $S _ { \alpha }$ denote the set of those real numbers $\beta$ that satisfy $\alpha \sin ( \beta ) = \beta \sin ( \alpha )$. Then which of the following statements is true?
(A) For any $\alpha , S _ { \alpha }$ is an infinite set.
(B) $S _ { \alpha }$ is a finite set if and only if $\alpha$ is not an integer multiple of $\pi$.
(C) There are infinitely many numbers $\alpha$ for which $S _ { \alpha }$ is the set of all real numbers.
(D) $S _ { \alpha }$ is always finite.

isi-entrance 2018 Q22 Solve trigonometric equation for solutions in an interval View

The number of solutions of the equation $\sin ( 7 x ) + \sin ( 3 x ) = 0$ with $0 \leq x \leq 2 \pi$ is
(A) 9
(B) 12
(C) 15
(D) 18.

isi-entrance 2026 Q2 Solve trigonometric equation for solutions in an interval View

The set of all solutions of the equation $\cos 2 \theta = \sin \theta + \cos \theta$ is given by
(a) $\theta = 0$.
(b) $\theta = n \pi + \frac { \pi } { 2 }$, where $n$ is any integer.
(c) $\theta = 2 n \pi$ or $\theta = 2 n \pi - \frac { \pi } { 2 }$ or $\theta = n \pi - \frac { \pi } { 4 }$, where $n$ is any integer.
(d) $\theta = 2 n \pi$ or $\theta = n \pi + \frac { \pi } { 4 }$, where $n$ is any integer.

isi-entrance 2026 Q13 Inverse trigonometric equation View

The number of solutions of the equation $\sin ^ { - 1 } x = 2 \tan ^ { - 1 } x$ is
(a) 1 .
(B) 2 .
(C) 3 .
(D) 5 .

jee-advanced 1999 Q11 Inverse trigonometric equation View

11. The number of real solutions of $\tan - 1 \sqrt { } ( x ( x + 1 ) ) + \sin - 1 \sqrt { } ( x 2 + x + 1 ) = \pi / 2$ is:
(A) zero
(B) one
(C) two
(D) infinite

jee-advanced 2001 Q31 Tangent Lines and Tangent Lengths View

31. Let $P Q$ and $R S$ be tangents at the extremities of the diameter $P R$ of a circle of radius $r$. If PS and RQ intersect at a point X on the circumference of the circle, then 2 r equals :
(A) $\sqrt { } ( \mathrm { PQ } . \mathrm { RS } )$
(B) $( P Q + R S ) / 2$
(C) $( 2 \mathrm { PQ } \cdot \mathrm { RS } ) / ( \mathrm { PQ } + \mathrm { RS } )$
(D) $\sqrt { } ( ( P Q 2 + R S 2 ) / 2 )$

jee-advanced 2002 Q11 Solve trigonometric equation for solutions in an interval View

11. The number of integral values of $k$ for which the equation $7 \cos x + 5 \sin x = 2 k +$ 1 has a solution is
(A) 4
(B) 8
(C) 10
(D) 12

jee-advanced 2007 Q51 Solve trigonometric equation for solutions in an interval View

The number of solutions of the pair of equations $$2\sin^2\theta - \cos 2\theta = 0$$ $$2\cos^2\theta - 3\sin\theta = 0$$ in the interval $[0, 2\pi]$ is
(A) 0
(B) 1
(C) 2
(D) 4

jee-advanced 2007 Q63 Find or Apply an Inverse Function Formula View

The number of solutions of the equation $\sin^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right) = \sin^{-1}\left(\frac{x-1}{\sqrt{1+(x-1)^2}}\right) - \sin^{-1}\left(\frac{1}{\sqrt{1+(x-1)^2}}\right)$ is
(A) 0
(B) 1
(C) 2
(D) infinite

jee-advanced 2009 Q28 Solve trigonometric equation for solutions in an interval View

For $0<\theta<\frac{\pi}{2}$, the solution(s) of $$\sum_{m=1}^{6}\operatorname{cosec}\left(\theta+\frac{(m-1)\pi}{4}\right)\operatorname{cosec}\left(\theta+\frac{m\pi}{4}\right)=4\sqrt{2}$$ is(are)
(A) $\frac{\pi}{4}$
(B) $\frac{\pi}{6}$
(C) $\frac{\pi}{12}$
(D) $\frac{5\pi}{12}$

jee-advanced 2010 Q47 Solve trigonometric equation for solutions in an interval View

The number of values of $\theta$ in the interval $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ such that $\theta \neq \frac { \mathrm { n } \pi } { 5 }$ for $\mathrm { n } = 0 , \pm 1 , \pm 2$ and $\tan \theta = \cot 5 \theta$ as well as $\sin 2 \theta = \cos 4 \theta$ is

jee-advanced 2010 Q55 Solve trigonometric equation for solutions in an interval View

The number of all possible values of $\theta$, where $0 < \theta < \pi$, for which the system of equations $$\begin{gathered} ( y + z ) \cos 3 \theta = ( x y z ) \sin 3 \theta \\ x \sin 3 \theta = \frac { 2 \cos 3 \theta } { y } + \frac { 2 \sin 3 \theta } { z } \\ ( x y z ) \sin 3 \theta = ( y + 2 z ) \cos 3 \theta + y \sin 3 \theta \end{gathered}$$ have a solution $\left( x _ { 0 } , y _ { 0 } , z _ { 0 } \right)$ with $y _ { 0 } z _ { 0 } \neq 0$, is

jee-advanced 2014 Q46 Solve trigonometric equation for solutions in an interval View

For $x \in (0, \pi)$, the equation $\sin x + 2\sin 2x - \sin 3x = 3$ has
(A) infinitely many solutions
(B) three solutions
(C) one solution
(D) no solution

jee-advanced 2015 Q41 Solve trigonometric equation for solutions in an interval View

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 $[ 0,2 \pi ]$ is

jee-advanced 2016 Q37 Evaluate trigonometric expression given a constraint View

Let $-\frac{\pi}{6} < \theta < -\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2 - 2x\sec\theta + 1 = 0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2 + 2x\tan\theta - 1 = 0$. If $\alpha_1 > \beta_1$ and $\alpha_2 > \beta_2$, then $\alpha_1 + \beta_2$ equals
(A) $2(\sec\theta - \tan\theta)$
(B) $2\sec\theta$
(C) $-2\tan\theta$
(D) $0$

jee-advanced 2016 Q39 Finite Equally-Likely Probability Computation View

Let $S = \left\{x \in (-\pi, \pi) : x \neq 0, \pm\frac{\pi}{2}\right\}$. The sum of all distinct solutions of the equation $\sqrt{3}\sec x + \operatorname{cosec} x + 2(\tan x - \cot x) = 0$ in the set $S$ is equal to
(A) $-\frac{7\pi}{9}$
(B) $-\frac{2\pi}{9}$
(C) $0$
(D) $\frac{5\pi}{9}$

jee-advanced 2018 Q13 Trigonometric Equation Solving via Identities View

Let $a , b , c$ be three non-zero real numbers such that the equation $$\sqrt { 3 } a \cos x + 2 b \sin x = c , x \in \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac { \pi } { 3 }$. Then, the value of $\frac { b } { a }$ is $\_\_\_\_$.

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164