LFM Pure

gaokao 2024 Q6 4 marks View

Given $f ( x ) = \sin \omega x , f \left( x _ { 1 } \right) = - 1 , f \left( x _ { 2 } \right) = 1 , \left| x _ { 1 } - x _ { 2 } \right| _ { \text {min} } = \frac { \pi } { 2 }$, then $\omega =$ \_\_\_\_

gaokao 2024 Q7 5 marks View

When $x \in [ 0,2 \pi ]$ , the number of intersection points of the curves $y = \sin x$ and $y = 2 \sin \left( 3 x - \frac { \pi } { 6 } \right)$ is
A. $3$
B. $4$
C. $6$
D. $8$

gaokao 2024 Q9 6 marks View

For functions $f ( x ) = \sin 2 x$ and $g ( x ) = \sin \left( 2 x - \frac { \pi } { 4 } \right)$, the correct statements are
A. $f ( x )$ and $g ( x )$ have the same zeros
B. $f ( x )$ and $g ( x )$ have the same maximum value
C. $f ( x )$ and $g ( x )$ have the same minimum positive period
D. The graphs of $f ( x )$ and $g ( x )$ have the same axes of symmetry

gaokao 2024 Q12 5 marks View

Given $\alpha \in \left[ \frac { \pi } { 6 } , \frac { \pi } { 3 } \right]$, and the terminal sides of $\alpha$ and $\beta$ are symmetric about the origin, then the maximum value of $\cos \beta$ is \_\_\_\_.

gaokao 2025 Q4 5 marks View

If the point $(a, 0)$ $(a > 0)$ is a center of symmetry of the graph of the function $y = 2\tan\left(x - \frac{\pi}{3}\right)$, then the minimum value of $a$ is
A. $\frac{\pi}{4}$
B. $\frac{\pi}{2}$
C. $\frac{\pi}{3}$
D. $\frac{4\pi}{3}$

gaokao 2025 Q15 13 marks View

Given the function $f(x) = \cos(2x + \varphi)$ $(0 \leq \varphi < \pi)$, $f(0) = \frac{1}{2}$.
(1) Find $\varphi$;
(2) Let $g(x) = f(x) + f\left(x - \frac{\pi}{6}\right)$. Find the range and monotonic intervals of $g(x)$.

grandes-ecoles 2010 QI.A.1 Trigonometric Identity Proof or Derivation View

For every integer $n \in \mathbb{N}$, we set $F_n(x) = \cos(n \arccos x)$.
a) Show that the functions $F_n$ are defined on the same domain $D$ which should be specified.
b) Calculate $F_1(x), F_2(x)$ and $F_3(x)$ for all $x \in D$.
c) Calculate $F_n(1), F_n(0)$ and $F_n(-1)$ for all $n \in \mathbb{N}$.
d) Specify the parity properties of $F_n$ as a function of $n$.

iran-konkur 2013 Q103 View

103- The figure shows part of the graph of $y = a\sin\pi\!\left(\dfrac{1}{2} + bx\right)$. What is $a \cdot b$?
[Figure: Graph of a sinusoidal function with amplitude $2$, showing values at $x = 3.5$ and minimum $-2$, with dashed line at $y=2$]

[(1)] $2$
[(2)] $2.5$
[(3)] $3$
[(4)] $3.5$

iran-konkur 2013 Q110 View

110- What is the value of $\tan^{-1}\sqrt{x^2 + x} + \sin^{-1}\sqrt{x^2 + x + 1}$?
(1) $\dfrac{\pi}{4}$ (2) $\dfrac{\pi}{2}$ (3) $\dfrac{3\pi}{4}$ (4) $\pi$

iran-konkur 2022 Q113 View

113. The figure below shows a portion of the graph of $f(x) = a\cos(bx + c)$. If $b > 0$, $0 < c < \pi$, and $\dfrac{ac}{b} = 0$, what is the value?
[Figure: Graph of a cosine function with amplitude $\frac{1}{3}$, showing one full cycle. The graph reaches a minimum of $-\frac{1}{3}$ and passes through key points at $x = \frac{3}{4}$ and $x = \frac{5}{4}$]
(1) $\dfrac{1}{16}$
(2) $1$
(3) $\dfrac{1}{4\pi}$
(4) $\pi$
%% Page 5 Mathematics $\leftarrow$

isi-entrance 2012 Q14 View

Which of the following is true about $\tan(\sin x)$?
(A) $\tan(\sin x) = 1$ has solutions
(B) $\tan(\sin x) \geq 1$ for some $x$
(C) $\tan(\sin x) < 1$ for all $x$
(D) $\tan(\sin x)$ never attains the value $1$

isi-entrance 2012 Q25 View

For $\theta \in (0, \pi/2)$, which of the following is true?
(A) $\cos(\sin\theta) < \cos\theta$
(B) $\cos(\sin\theta) < \sin(\cos\theta)$
(C) $\cos(\sin\theta) > \cos\theta$
(D) $\cos(\sin\theta) > \sin(\cos\theta)$

isi-entrance 2013 Q35 4 marks 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 Q35 4 marks 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 2019 Q11 Solve trigonometric equation for solutions in an interval View

In the range $0 \leq x \leq 2 \pi$, the equation $\cos ( \sin ( x ) ) = \frac { 1 } { 2 }$ has
(A) 0 solutions.
(B) 2 solutions.
(C) 4 solutions.
(D) infinitely many solutions.

isi-entrance 2021 Q27 Extremal Value of Trigonometric Expression View

If the maximum and minimum values of $\sin ^ { 6 } x + \cos ^ { 6 } x$, as $x$ takes all real values, are $a$ and $b$, respectively, then $a - b$ equals
(A) $\frac { 1 } { 2 }$.
(B) $\frac { 2 } { 3 }$.
(C) $\frac { 3 } { 4 }$.
(D) 1 .

isi-entrance 2026 Q4 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) $\pi / 2$.

jee-advanced 2000 Q21 Inscribed/Circumscribed Circle Computations View

21. The incentre of the triangle with vertices $( 1 , \sqrt { 3 } ) , ( 0,0 )$ and $( 2,0 )$ is :
(A) $( 1 , \sqrt { } 3 / 2 )$
(B) $( 2 / 3,1 / \sqrt { } 3 )$
(C) $( 2 / 3 , \sqrt { } 3 / 2 )$
(D) $( 1,1 / \sqrt { } 3 )$

jee-advanced 2015 Q50 View

If $\alpha = 3 \sin ^ { - 1 } \left( \frac { 6 } { 11 } \right)$ and $\beta = 3 \cos ^ { - 1 } \left( \frac { 4 } { 9 } \right)$, where the inverse trigonometric functions take only the principal values, then the correct option(s) is(are)
(A) $\quad \cos \beta > 0$
(B) $\quad \sin \beta < 0$
(C) $\quad \cos ( \alpha + \beta ) > 0$
(D) $\quad \cos \alpha < 0$

jee-main 2007 Q102 Inverse trigonometric equation View

If $\sin ^ { - 1 } \left( \frac { x } { 5 } \right) + \operatorname { cosec } ^ { - 1 } \left( \frac { 5 } { 4 } \right) = \frac { \pi } { 2 }$ then a value of $x$ is
(1) 1
(2) 3
(3) 4
(4) 5

jee-main 2012 Q71 Chord Properties and Midpoint Problems View

The chord $PQ$ of the parabola $y ^ { 2 } = x$, where one end $P$ of the chord is at point $( 4 , - 2 )$, is perpendicular to the axis of the parabola. Then the slope of the normal at $Q$ is
(1) $-4$
(2) $- \frac { 1 } { 4 }$
(3) $4$
(4) $\frac { 1 } { 4 }$

jee-main 2013 Q69 View

Let $x \in ( 0,1 )$. The set of all $x$ such that $\sin ^ { -1 } x > \cos ^ { -1 } x$, is the interval:
(1) $\left( \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$
(2) $\left( \frac { 1 } { \sqrt { 2 } } , 1 \right)$
(3) $( 0,1 )$
(4) $\left( 0 , \frac { \sqrt { 3 } } { 2 } \right)$

jee-main 2014 Q76 Inverse trigonometric equation View

The principal value of $\tan ^ { - 1 } \left( \cot \frac { 43 \pi } { 4 } \right)$ is
(1) $\frac { \pi } { 4 }$
(2) $- \frac { \pi } { 4 }$
(3) $\frac { 3 \pi } { 4 }$
(4) $- \frac { 3 \pi } { 4 }$

jee-main 2014 Q77 Function Properties from Symmetry or Parity View

The function $f ( x ) = | \sin 4 x | + | \cos 2 x |$, is a periodic function with a fundamental period
(1) $\pi$
(2) $2 \pi$
(3) $\frac { \pi } { 4 }$
(4) $\frac { \pi } { 2 }$

jee-main 2022 Q63 Solve trigonometric equation for solutions in an interval View

Let $S = \left\{ \theta \in [ 0,2 \pi ] : 8 ^ { 2 \sin ^ { 2 } \theta } + 8 ^ { 2 \cos ^ { 2 } \theta } = 16 \right\}$. Then $n ( S ) + \sum _ { \theta \in \mathrm { S } } \left( \sec \left( \frac { \pi } { 4 } + 2 \theta \right) \operatorname { cosec } \left( \frac { \pi } { 4 } + 2 \theta \right) \right)$ is equal to:
(1) 0
(2) $- 2$
(3) $- 4$
(4) 12

Load questions...

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