LFM Pure

isi-entrance 2026 QB8 Trigonometric Equation Solving via Identities View

Let $\frac { \tan ( \alpha - \beta + \gamma ) } { \tan ( \alpha + \beta - \gamma ) } = \frac { \tan \beta } { \tan \gamma }$. Then
(A) $\sin ( \beta - \gamma ) = \sin ( \alpha - \beta )$.
(B) $\sin ( \alpha - \gamma ) = \sin ( \beta - \gamma )$.
(C) $\sin ( \beta - \gamma ) = 0$.
(D) $\sin 2 \alpha + \sin 2 \beta + \sin 2 \gamma = 0$

isi-entrance 2026 QB10 Qualitative Reasoning about Double Angle Signs or Inequalities View

Let $\frac { \tan 3 \theta } { \tan \theta } = k$. Then
(A) $k \in ( 1 / 3,3 )$
(B) $k \notin ( 1 / 3,3 )$
(C) $\frac { \sin 3 \theta } { \sin \theta } = \frac { 2 k } { k - 1 }$.
(D) $\frac { \sin 3 \theta } { \sin \theta } > \frac { 2 k } { k - 1 }$

isi-entrance 2026 Q16 Ordering or Comparing Trigonometric Expressions View

Suppose $x , y \in ( 0 , \pi / 2 )$ and $x \neq y$. Which of the following statements is true?
(a) $2 \sin ( x + y ) < \sin 2 x + \sin 2 y$ for all $x , y$.
(b) $2 \sin ( x + y ) > \sin 2 x + \sin 2 y$ for all $x , y$.
(c) There exist $x , y$ such that $2 \sin ( x + y ) = \sin 2 x + \sin 2 y$.
(d) None of the above.

jee-advanced 2001 Q32 Horizontal Launch or Dropped Object Problems View

32. A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of 300 . After some time, the angle of depression becomes 600. The distance (in metres) travelled by the car during this time is :
(A) $100 \sqrt { 3 }$
(B) $( 200 \sqrt { } 3 ) / 3$
(C) $( 100 \sqrt { } 3 ) / 3$
(D) $200 \sqrt { } 3$

jee-advanced 2017 Q45 Half-Angle Formula Evaluation View

Let $\alpha$ and $\beta$ be nonzero real numbers such that $2 ( \cos \beta - \cos \alpha ) + \cos \alpha \cos \beta = 1$. Then which of the following is/are true?
[A] $\tan \left( \frac { \alpha } { 2 } \right) + \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[B] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) + \tan \left( \frac { \beta } { 2 } \right) = 0$
[C] $\tan \left( \frac { \alpha } { 2 } \right) - \sqrt { 3 } \tan \left( \frac { \beta } { 2 } \right) = 0$
[D] $\sqrt { 3 } \tan \left( \frac { \alpha } { 2 } \right) - \tan \left( \frac { \beta } { 2 } \right) = 0$

jee-advanced 2020 Q6 Simplification of Trigonometric Expressions with Specific Angles View

The value of the limit $$\lim_{x \rightarrow \frac{\pi}{2}} \frac{4\sqrt{2}(\sin 3x + \sin x)}{\left(2\sin 2x \sin\frac{3x}{2} + \cos\frac{5x}{2}\right) - \left(\sqrt{2} + \sqrt{2}\cos 2x + \cos\frac{3x}{2}\right)}$$ is $\_\_\_\_$

jee-advanced 2022 Q1 3 marks Multi-Step Composite Problem Using Identities View

Let $\alpha$ and $\beta$ be real numbers such that $- \frac { \pi } { 4 } < \beta < 0 < \alpha < \frac { \pi } { 4 }$. If $\sin ( \alpha + \beta ) = \frac { 1 } { 3 }$ and $\cos ( \alpha - \beta ) = \frac { 2 } { 3 }$, then the greatest integer less than or equal to
$$\left( \frac { \sin \alpha } { \cos \beta } + \frac { \cos \beta } { \sin \alpha } + \frac { \cos \alpha } { \sin \beta } + \frac { \sin \beta } { \cos \alpha } \right) ^ { 2 }$$
is $\_\_\_\_$ .

jee-main 2012 Q67 Trigonometric Equation Solving via Identities View

Suppose $\theta$ and $\phi ( \neq 0 )$ are such that $\sec ( \theta + \phi )$, $\sec \theta$ and $\sec ( \theta - \phi )$ are in A.P. If $\cos \theta = k \cos \left( \frac { \phi } { 2 } \right)$ for some $k$, then $k$ is equal to
(1) $\pm \sqrt { 2 }$
(2) $\pm 1$
(3) $\pm \frac { 1 } { \sqrt { 2 } }$
(4) $\pm 2$

jee-main 2012 Q67 Simplification of Trigonometric Expressions with Specific Angles View

The value of $\cos 255 ^ { \circ } + \sin 195 ^ { \circ }$ is
(1) $\frac { \sqrt { 3 } - 1 } { 2 \sqrt { 2 } }$
(2) $\frac { \sqrt { 3 } - 1 } { \sqrt { 2 } }$
(3) $- \frac { \sqrt { 3 } - 1 } { \sqrt { 2 } }$
(4) $\frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }$

jee-main 2015 Q79 Multi-Step Composite Problem Using Identities View

If $\tan\left(\frac{\pi}{4} + \frac{\theta}{2}\right) = \tan^3\left(\frac{\pi}{4} + \frac{\alpha}{2}\right)$, then $\sin\theta = $:
(1) $\frac{\sin\alpha(3 + \sin^2\alpha)}{1 + 3\sin^2\alpha}$
(2) $\frac{\sin\alpha(3 - \sin^2\alpha)}{1 + 3\sin^2\alpha}$
(3) $\frac{\sin\alpha(3 + \cos^2\alpha)}{1 + 3\cos^2\alpha}$
(4) $\frac{\sin\alpha(3 - \cos^2\alpha)}{1 + 3\cos^2\alpha}$

jee-main 2016 Q67 Function Analysis via Identity Transformation View

If $A > 0 , B > 0$ and $A + B = \frac { \pi } { 6 }$, then the minimum positive value of $( \tan A + \tan B )$ is :
(1) $\sqrt { 3 } - \sqrt { 2 }$
(2) $4 - 2 \sqrt { 3 }$
(3) $\frac { 2 } { \sqrt { 3 } }$
(4) $2 - \sqrt { 3 }$

jee-main 2017 Q90 Trigonometric Equation Solving via Identities View

If $5 ( \tan ^ { 2 } x - \cos ^ { 2 } x ) = 2 \cos 2 x + 9$, then the value of $\cos 4 x$ is:
(1) $- \frac { 7 } { 9 }$
(2) $- \frac { 3 } { 5 }$
(3) $\frac { 1 } { 3 }$
(4) $\frac { 2 } { 9 }$

jee-main 2019 Q65 Trigonometric Equation Constraint Deduction View

If $\sin ^ { 4 } \alpha + 4 \cos ^ { 4 } \beta + 2 = 4 \sqrt { 2 } \sin \alpha \cos \beta , \alpha , \beta \in [ 0 , \pi ]$, then $\cos ( \alpha + \beta ) - \cos ( \alpha - \beta )$ is equal to
(1) - 1
(2) $- \sqrt { 2 }$
(3) $\sqrt { 2 }$
(4) 0

jee-main 2019 Q67 Addition/Subtraction Formula Evaluation View

If $\cos(\alpha + \beta) = \frac{3}{5}$, $\sin(\alpha - \beta) = \frac{5}{13}$ and $0 < \alpha, \beta < \frac{\pi}{4}$, then $\tan 2\alpha$ is equal to:
(1) $\frac{21}{16}$
(2) $\frac{63}{52}$
(3) $\frac{33}{52}$
(4) $\frac{63}{16}$

jee-main 2020 Q55 Simplification of Trigonometric Expressions with Specific Angles View

The value of $\cos ^ { 3 } \left( \frac { \pi } { 8 } \right) \cdot \cos \left( \frac { 3 \pi } { 8 } \right) + \sin ^ { 3 } \left( \frac { \pi } { 8 } \right) \cdot \sin \left( \frac { 3 \pi } { 8 } \right)$ is:
(1) $\frac { 1 } { \sqrt { 2 } }$
(2) $\frac { 1 } { 2 \sqrt { 2 } }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 1 } { 4 }$

jee-main 2020 Q56 Half-Angle Formula Evaluation View

If $L = \sin^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$ and $M = \cos^2\left(\frac{\pi}{16}\right) - \sin^2\left(\frac{\pi}{8}\right)$
(1) $L = -\frac{1}{2\sqrt{2}} + \frac{1}{2}\cos\frac{\pi}{8}$
(2) $L = \frac{1}{4\sqrt{2}} - \frac{1}{4}\cos\frac{\pi}{8}$
(3) $M = \frac{1}{4\sqrt{2}} + \frac{1}{4}\cos\frac{\pi}{8}$
(4) $M = \frac{1}{2\sqrt{2}} + \frac{1}{2}\cos\frac{\pi}{8}$

jee-main 2021 Q64 Trigonometric Equation Solving via Identities View

If $0 < x , y < \pi$ and $\cos x + \cos y - \cos ( x + y ) = \frac { 3 } { 2 }$, then $\sin x + \cos y$ is equal to:
(1) $\frac { 1 } { 2 }$
(2) $\frac { \sqrt { 3 } } { 2 }$
(3) $\frac { 1 - \sqrt { 3 } } { 2 }$
(4) $\frac { 1 + \sqrt { 3 } } { 2 }$

jee-main 2022 Q62 Simplification of Trigonometric Expressions with Specific Angles View

$16 \sin \left( 20 ^ { \circ } \right) \sin \left( 40 ^ { \circ } \right) \sin \left( 80 ^ { \circ } \right)$ is equal to
(1) $\sqrt { 3 }$
(2) $2 \sqrt { 3 }$
(3) 3
(4) $4 \sqrt { 3 }$

jee-main 2022 Q63 Roots of Unity and Cyclotomic Properties View

The value of $\cos \left( \frac { 2 \pi } { 7 } \right) + \cos \left( \frac { 4 \pi } { 7 } \right) + \cos \left( \frac { 6 \pi } { 7 } \right)$ is equal to
(1) $- 1$
(2) $- \frac { 1 } { 2 }$
(3) $- \frac { 1 } { 3 }$
(4) $- \frac { 1 } { 4 }$

jee-main 2022 Q64 Simplification of Trigonometric Expressions with Specific Angles View

The value of $2\sin\frac{\pi}{22}\sin\frac{3\pi}{22}\sin\frac{5\pi}{22}\sin\frac{7\pi}{22}\sin\frac{9\pi}{22}$ is equal to:
(1) $\frac{1}{16}$
(2) $\frac{5}{16}$
(3) $\frac{7}{16}$
(4) $\frac{3}{16}$

jee-main 2022 Q66 Simplification of Trigonometric Expressions with Specific Angles View

The value of $2\sin 12^\circ - \sin 72^\circ$ is
(1) $\frac{\sqrt{5}(1 - \sqrt{3})}{4}$
(2) $\frac{1 - \sqrt{5}}{8}$
(3) $\frac{\sqrt{3}(1 - \sqrt{5})}{2}$
(4) $\frac{\sqrt{3}(1 - \sqrt{5})}{4}$

jee-main 2023 Q64 Trigonometric Identity Simplification View

$96 \cos\frac{\pi}{33} \cos\frac{2\pi}{33} \cos\frac{4\pi}{33} \cos\frac{8\pi}{33} \cos\frac{16\pi}{33}$ is equal to
(1) 3
(2) 1
(3) 4
(4) 2

jee-main 2023 Q73 Convexity and inflection point analysis View

Let $g(x) = f(x) + f(1 - x)$ and $f ^ { \prime \prime } (x) > 0 , x \in (0,1)$. If $g$ is decreasing in the interval $(0 , \alpha)$ and increasing in the interval $(\alpha , 1)$, then $\tan ^ { - 1 } (2\alpha) + \tan ^ { - 1 } \left( \frac { 1 } { \alpha } \right) + \tan ^ { - 1 } \left( \frac { \alpha + 1 } { \alpha } \right)$ is equal to
(1) $\pi$
(2) $\frac { 5\pi } { 4 }$
(3) $\frac { 3\pi } { 4 }$
(4) $\frac { 3\pi } { 2 }$

jee-main 2024 Q63 Determine Parameters from Conditions on Coefficients or Terms View

Suppose $28 - p,\ p,\ 70 - \alpha,\ \alpha$ are the coefficient of four consecutive terms in the expansion of $(1 + x)^n$. Then the value of $2\alpha - 3p$ equals
(1) 7
(2) 10
(3) 4
(4) 6

jee-main 2025 Q7 Telescoping Sum of Trigonometric Terms View

If $\sum _ { r = 1 } ^ { 13 } \left\{ \frac { 1 } { \sin \left( \frac { \pi } { 4 } + ( r - 1 ) \frac { \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { r \pi } { 6 } \right) } \right\} = a \sqrt { 3 } + b , a , b \in \mathbf { Z }$, then $a ^ { 2 } + b ^ { 2 }$ is equal to :
(1) 10
(2) 4
(3) 2
(4) 8

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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