Clear Notion
Login

Practice

Beyond Advanced · Trigonometry

Your Score

0 / 280

Correct

0 / 70

Attempted

0 / 70

1
Real numbers x,yx,y satisfy sin⁡x+sin⁡y=13\sin x+\sin y=\dfrac13 and cos⁡x−cos⁡y=15\cos x-\cos y=\dfrac15. Which of the following statements are correct?
Not attempted
A
sin⁡(x+y)\sin(x+y) is uniquely determined by the given data
B
cos⁡(x+y)=208225\cos(x+y)=\dfrac{208}{225}
C
cos⁡(x−y)=817\cos(x-y)=\dfrac{8}{17}
D
tan⁡x−y2=−35\tan\dfrac{x-y}{2}=-\dfrac35
2
Match each quantity in List-I with its value in List-II. List-IList-IIP) cos⁡2π7+cos⁡22π7+cos⁡23π71) 54Q) cos⁡π7−cos⁡2π7+cos⁡3π72) 12R) tan⁡π7tan⁡2π7tan⁡3π73) 7S) cos⁡2π7cos⁡4π7+cos⁡4π7cos⁡8π7+cos⁡8π7cos⁡2π74) −125) 18\begin{array}{ll}\textbf{List-I} & \textbf{List-II}\\ \text{P) } \cos^2\frac{\pi}{7}+\cos^2\frac{2\pi}{7}+\cos^2\frac{3\pi}{7} & \text{1) } \frac54\\ \text{Q) } \cos\frac{\pi}{7}-\cos\frac{2\pi}{7}+\cos\frac{3\pi}{7} & \text{2) } \frac12\\ \text{R) } \tan\frac{\pi}{7}\tan\frac{2\pi}{7}\tan\frac{3\pi}{7} & \text{3) } \sqrt7\\ \text{S) } \cos\frac{2\pi}{7}\cos\frac{4\pi}{7}+\cos\frac{4\pi}{7}\cos\frac{8\pi}{7}+\cos\frac{8\pi}{7}\cos\frac{2\pi}{7} & \text{4) } -\frac12\\ & \text{5) } \frac18\end{array}
Not attempted
A
P-1, Q-2, R-3, S-4
B
P-5, Q-2, R-3, S-1
C
P-1, Q-4, R-3, S-2
D
P-1, Q-5, R-3, S-2
3
Let cos⁡θ=−725\cos\theta=-\dfrac{7}{25} with 5π<θ<11π25\pi<\theta<\dfrac{11\pi}{2}. Which of the following are correct?
Not attempted
A
sin⁡3θ4=255\sin\dfrac{3\theta}{4}=\dfrac{2}{5\sqrt5}
B
tan⁡θ2=−43\tan\dfrac{\theta}{2}=-\dfrac43
C
tan⁡θ4=2\tan\dfrac{\theta}{4}=2
D
cos⁡θ4=15\cos\dfrac{\theta}{4}=\dfrac{1}{\sqrt5}
4
Real numbers α,β,γ\alpha,\beta,\gamma satisfy cos⁡α+cos⁡β+cos⁡γ=0\cos\alpha+\cos\beta+\cos\gamma=0 and sin⁡α+sin⁡β+sin⁡γ=0\sin\alpha+\sin\beta+\sin\gamma=0. Which of the following is NOT necessarily true?
Not attempted
A
cos⁡3α+cos⁡3β+cos⁡3γ=3cos⁡(α+β+γ)\cos3\alpha+\cos3\beta+\cos3\gamma=3\cos(\alpha+\beta+\gamma)
B
cos⁡6α+cos⁡6β+cos⁡6γ=0\cos6\alpha+\cos6\beta+\cos6\gamma=0
C
cos⁡2α+cos⁡2β+cos⁡2γ=0\cos2\alpha+\cos2\beta+\cos2\gamma=0
D
cos⁡(α+β)+cos⁡(β+γ)+cos⁡(γ+α)=0\cos(\alpha+\beta)+\cos(\beta+\gamma)+\cos(\gamma+\alpha)=0
5
Which of the following is the graph of y=sin⁡2x−sin⁡2 ⁣(x−π3)y=\sin^2x-\sin^2\!\left(x-\dfrac{\pi}{3}\right) on [0,π][0,\pi]? (Dashed lines mark y=32y=\dfrac{\sqrt3}{2} and y=12y=\dfrac12.)
Not attempted
A
Sinusoid of period π\pi, amplitude 12\frac12, zeros at π6\frac{\pi}{6} and 2π3\frac{2\pi}{3}
xyOπ/6π/32π/35π/6√3/21/2
B
Sinusoid of period 2π2\pi, amplitude 32\frac{\sqrt3}{2}, zero at π6\frac{\pi}{6}
xyOπ/6π/32π/35π/6√3/21/2
C
Sinusoid of period π\pi, amplitude 32\frac{\sqrt3}{2}, zeros at π6\frac{\pi}{6} and 2π3\frac{2\pi}{3}, value −34-\frac34 at x=0x=0
xyOπ/6π/32π/35π/6√3/21/2
D
Sinusoid of period π\pi, amplitude 32\frac{\sqrt3}{2}, zeros at π3\frac{\pi}{3} and 5π6\frac{5\pi}{6}, value 34\frac34 at x=0x=0
xyOπ/6π/32π/35π/6√3/21/2
6
Which of the following is the graph of y=sin⁡x+cos⁡x−∣sin⁡x−cos⁡x∣y=\sin x+\cos x-\left|\sin x-\cos x\right| on [0,2π][0,2\pi]? (Dashed lines mark y=2y=\sqrt2, y=−1y=-1 and y=−2y=-2.)
Not attempted
A
Graph with maximum 22, minimum −2-\sqrt2 at x=5π4x=\frac{5\pi}{4}
xyOπ/2π3π/22π√2−1−2
B
Graph with maximum 22\frac{\sqrt2}{2} and minimum −1-1
xyOπ/2π3π/22π√2−1−2
C
Graph equal to −2∣sin⁡x−cos⁡x∣-2\left|\sin x-\cos x\right|, touching 00 at π4\frac\pi4 and 5π4\frac{5\pi}4
xyOπ/2π3π/22π√2−1−2
D
Graph with maximum 2\sqrt2 at x=π4x=\frac\pi4 only, minimum −2-2 at x=πx=\pi and x=3π2x=\frac{3\pi}2, corners at π4\frac\pi4 and 5π4\frac{5\pi}4
xyOπ/2π3π/22π√2−1−2
7
Find the maximum value of f(x)=sin⁡x+sin⁡2xf(x)=\sin x+\sin2x over all real xx. (Take 33=5.7446\sqrt{33}=5.7446.) Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.76
8
Which of the following statements are correct?
Not attempted
A
cos⁡(cos⁡x)+cos⁡(sin⁡x)\cos(\cos x)+\cos(\sin x) has fundamental period π2\dfrac{\pi}{2}
B
sin⁡4x+cos⁡4x\sin^4x+\cos^4x has fundamental period π4\dfrac{\pi}{4}
C
∣sin⁡x∣+∣cos⁡x∣|\sin x|+|\cos x| has fundamental period π2\dfrac{\pi}{2}
D
sin⁡x+sin⁡(2 x)\sin x+\sin\left(\sqrt2\,x\right) is not periodic
9
Which of the following is the graph of y=sin⁡3x−sin⁡xcos⁡3x+cos⁡xy=\dfrac{\sin3x-\sin x}{\cos3x+\cos x} on [0,π][0,\pi] (on its natural domain)? Open circles mark points that are NOT on the graph; the dashed vertical line is an asymptote.
Not attempted
A
y=tan⁡xy=\tan x with points missing at x=π4x=\frac\pi4 and x=3π4x=\frac{3\pi}4
xyOπ/4π/23π/41−1
B
y=tan⁡xy=\tan x with no missing points other than x=π2x=\frac\pi2
xyOπ/4π/23π/41−1
C
y=tan⁡xy=\tan x with points missing at x=π6x=\frac\pi6 and x=5π6x=\frac{5\pi}6
xyOπ/4π/23π/41−1
D
y=tan⁡xy=\tan x with a point missing only at x=π4x=\frac\pi4
xyOπ/4π/23π/41−1
10
The number of values of x∈[0,2π]x\in[0,2\pi] satisfying sin⁡x+sin⁡2x+sin⁡3x+sin⁡4x=0\sin x+\sin2x+\sin3x+\sin4x=0 is
Not attempted
A
1010
B
99
C
77
D
88
11
Which of the following sets equal the solution set of sin⁡3x=sin⁡x\sin3x=\sin x (n∈Zn\in\mathbb{Z})?
Not attempted
A
{x:tan⁡2x=0}\{x:\tan2x=0\}
B
{x:3x=nπ+(−1)nx for some n}\{x: 3x=n\pi+(-1)^nx \text{ for some } n\}
C
{nπ}∪{(2n+1)π4}\{n\pi\}\cup\left\{\dfrac{(2n+1)\pi}{4}\right\}
D
{nπ2}∪{(2n+1)π4}\left\{\dfrac{n\pi}{2}\right\}\cup\left\{\dfrac{(2n+1)\pi}{4}\right\}
12
Match each equation in List-I with the number of its solutions in [0,2π][0,2\pi] in List-II. List-IList-IIP) cos⁡x+cos⁡2x+cos⁡3x=01) 4Q) ∣sin⁡x∣+∣cos⁡x∣=12) 5R) sin⁡4x+cos⁡4x=583) 6S) tan⁡x+cot⁡x=44) 85) 9\begin{array}{ll}\textbf{List-I} & \textbf{List-II}\\ \text{P) } \cos x+\cos2x+\cos3x=0 & \text{1) } 4\\ \text{Q) } |\sin x|+|\cos x|=1 & \text{2) } 5\\ \text{R) } \sin^4x+\cos^4x=\frac58 & \text{3) } 6\\ \text{S) } \tan x+\cot x=4 & \text{4) } 8\\ & \text{5) } 9\end{array}
Not attempted
A
P-3, Q-5, R-1, S-1
B
P-3, Q-2, R-5, S-4
C
P-3, Q-2, R-4, S-1
D
P-1, Q-2, R-4, S-3
13
The general solution of log⁡cos⁡x(sin⁡x)+log⁡sin⁡x(cos⁡x)=2\log_{\cos x}(\sin x)+\log_{\sin x}(\cos x)=2 is (n∈Zn\in\mathbb{Z})
Not attempted
A
x=nπ+π4x=n\pi+\dfrac\pi4
B
x=2nπ±π4x=2n\pi\pm\dfrac\pi4
C
x=nπ2+π4x=\dfrac{n\pi}{2}+\dfrac\pi4
D
x=2nπ+π4x=2n\pi+\dfrac\pi4
14
For the equation sin⁡x+cos⁡x=1+sin⁡xcos⁡x\sin x+\cos x=1+\sin x\cos x, which of the following statements are correct (n∈Zn\in\mathbb{Z})?
Not attempted
A
x=3π2x=\dfrac{3\pi}{2} is a solution
B
Its solution set is {nπ+π4±π4:n∈Z}\left\{n\pi+\dfrac\pi4\pm\dfrac\pi4: n\in\mathbb{Z}\right\}
C
Its solutions are exactly x=2nπx=2n\pi and x=2nπ+π2x=2n\pi+\dfrac\pi2
D
It has exactly 22 solutions in [0,2π][0,2\pi]
15
Which of the following is the graph of y=sin⁡−1(sin⁡x)+cos⁡−1(cos⁡x)y=\sin^{-1}(\sin x)+\cos^{-1}(\cos x) for 0≤x≤2π0\le x\le2\pi? (Dashed lines mark y=πy=\pi, y=π2y=\dfrac\pi2 and y=−π2y=-\dfrac\pi2.)
Not attempted
A
Rises as 2x2x to π\pi at π2\frac\pi2, stays at π\pi until x=πx=\pi, falls linearly to 00 at 3π2\frac{3\pi}{2}, then stays 00
xyOπ/2π3π/22πππ/2−π/2
B
Rises as 2x2x to 2π2\pi at x=πx=\pi, then falls back to 00 at 2π2\pi
xyOπ/2π3π/22πππ/2−π/2
C
A zigzag between −π-\pi and π\pi with slopes ±2\pm2
xyOπ/2π3π/22πππ/2−π/2
D
The constant π2\frac\pi2
xyOπ/2π3π/22πππ/2−π/2
16
Find sin⁡−1(sin⁡7)+cos⁡−1(cos⁡8)+tan⁡−1(tan⁡9)\sin^{-1}(\sin7)+\cos^{-1}(\cos8)+\tan^{-1}(\tan9), where all angles are in radians. (Take π=3.1416\pi=3.1416.) Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 2.01
17
The set of all real xx satisfying tan⁡−1x+tan⁡−12x+tan⁡−13x=π2\tan^{-1}x+\tan^{-1}2x+\tan^{-1}3x=\dfrac\pi2 is
Not attempted
A
{0,111}\left\{0,\dfrac{1}{\sqrt{11}}\right\}
B
{111}\left\{\dfrac{1}{\sqrt{11}}\right\}
C
{±111}\left\{\pm\dfrac{1}{\sqrt{11}}\right\}
D
{1}\{1\}
18
Match each expression in List-I with its value in List-II (principal values throughout; sec⁡−1x=cos⁡−11x\sec^{-1}x=\cos^{-1}\frac1x, csc⁡−1x=sin⁡−11x\csc^{-1}x=\sin^{-1}\frac1x). List-IList-IIP) cot⁡−1(−12)+cot⁡−1(−13)1) 5π4Q) sin⁡−145+sin⁡−1513+sin⁡−116652) π2R) sec⁡−1(−2)−csc⁡−1(−2)3) 11π12S) 4tan⁡−115−tan⁡−112394) π45) 3π4\begin{array}{ll}\textbf{List-I} & \textbf{List-II}\\ \text{P) } \cot^{-1}\left(-\frac12\right)+\cot^{-1}\left(-\frac13\right) & \text{1) } \frac{5\pi}{4}\\ \text{Q) } \sin^{-1}\frac45+\sin^{-1}\frac5{13}+\sin^{-1}\frac{16}{65} & \text{2) } \frac{\pi}{2}\\ \text{R) } \sec^{-1}\left(-\sqrt2\right)-\csc^{-1}(-2) & \text{3) } \frac{11\pi}{12}\\ \text{S) } 4\tan^{-1}\frac15-\tan^{-1}\frac1{239} & \text{4) } \frac{\pi}{4}\\ & \text{5) } \frac{3\pi}{4}\end{array}
Not attempted
A
P-1, Q-4, R-3, S-2
B
P-5, Q-2, R-1, S-4
C
P-1, Q-2, R-3, S-4
D
P-5, Q-2, R-3, S-1
19
The set of all real xx satisfying 2sin⁡−1x=cos⁡−1(1−2x2)2\sin^{-1}x=\cos^{-1}\left(1-2x^2\right) is
Not attempted
A
{0}\{0\}
B
[−1,1][-1,1]
C
{0,1}\{0,1\}
D
[0,1][0,1]
20
In triangle ABCABC, ∠A=30∘\angle A=30^\circ, a=BC=5a=BC=5 and b=CA=8b=CA=8. The figure shows that two non-congruent triangles fit these data. The absolute difference of their areas is
Not attempted
ACB₁B₂b = 830°a = 5a = 5
A
1212
B
66
C
2424
D
636\sqrt3
21
The sides of a triangle ABCABC satisfy a4+b4+c4=2c2(a2+b2)a^4+b^4+c^4=2c^2\left(a^2+b^2\right). Which of the following are correct?
Not attempted
A
If a=ba=b, then ∠C\angle C must be 45∘45^\circ
B
sin⁡C=12\sin C=\dfrac{1}{\sqrt2}
C
∠C\angle C can be 135∘135^\circ
D
∠C\angle C can be 45∘45^\circ
22
In a triangle ABCABC, acos⁡A+bcos⁡B=ccos⁡Ca\cos A+b\cos B=c\cos C. Then the triangle is necessarily
Not attempted
A
equilateral
B
right-angled at AA or at BB
C
right-angled at CC
D
isosceles
23
Match each condition on a triangle ABCABC in List-I with what it implies about the triangle in List-II. List-IList-IIP) acos⁡A=bcos⁡B1) isosceles or right-angledQ) sin⁡2A+sin⁡2B+sin⁡2C=22) right-angledR) cos⁡A+cos⁡B+cos⁡C=323) equilateralS) tan⁡A+tan⁡B+tan⁡C>04) acute-angled5) obtuse-angled\begin{array}{ll}\textbf{List-I} & \textbf{List-II}\\ \text{P) } a\cos A=b\cos B & \text{1) isosceles or right-angled}\\ \text{Q) } \sin^2A+\sin^2B+\sin^2C=2 & \text{2) right-angled}\\ \text{R) } \cos A+\cos B+\cos C=\frac32 & \text{3) equilateral}\\ \text{S) } \tan A+\tan B+\tan C>0 & \text{4) acute-angled}\\ & \text{5) obtuse-angled}\end{array}
Not attempted
A
P-1, Q-4, R-3, S-2
B
P-3, Q-2, R-1, S-4
C
P-1, Q-2, R-3, S-4
D
P-2, Q-2, R-3, S-5
24
For the triangle with sides a=7a=7, b=8b=8, c=9c=9, match each quantity in List-I with its value in List-II (OO is the circumcentre, II the incentre, r1r_1 the exradius opposite the side aa). List-IList-IIP) circumradius R1) 21510Q) inradius r2) 5R) r13) 1255S) OI24) 21205) 45\begin{array}{ll}\textbf{List-I} & \textbf{List-II}\\ \text{P) circumradius } R & \text{1) } \frac{21\sqrt5}{10}\\ \text{Q) inradius } r & \text{2) } \sqrt5\\ \text{R) } r_1 & \text{3) } \frac{12\sqrt5}{5}\\ \text{S) } OI^2 & \text{4) } \frac{21}{20}\\ & \text{5) } 4\sqrt5\end{array}
Not attempted
A
P-1, Q-2, R-5, S-3
B
P-3, Q-2, R-1, S-4
C
P-1, Q-4, R-3, S-2
D
P-1, Q-2, R-3, S-4
25
An observer OO is 2020 m above the surface of a still lake. The angle of elevation of a cloud from OO is 30∘30^\circ and the angle of depression of its reflection in the lake is 60∘60^\circ (figure not to scale). Which of the following are correct?
Not attempted
lakeOcloudimage20 m30°60°
A
The distance from OO to the cloud is 4040 m
B
The cloud is 4040 m above the lake
C
The distance from OO to the reflection is 40340\sqrt3 m
D
The horizontal distance from OO to the cloud is 203\dfrac{20}{\sqrt3} m
26
Let P=∏k=03(1+2cos⁡2⋅3kπ30)=(1+2cos⁡12∘)(1+2cos⁡36∘)(1+2cos⁡108∘)(1+2cos⁡324∘)P=\prod_{k=0}^{3}\left(1+2\cos\dfrac{2\cdot 3^{k}\pi}{30}\right)=\left(1+2\cos 12^\circ\right)\left(1+2\cos 36^\circ\right)\left(1+2\cos 108^\circ\right)\left(1+2\cos 324^\circ\right). Then PP equals
Not attempted
A
1+54sin⁡6∘\dfrac{1+\sqrt5}{4\sin 6^\circ}
B
5−14sin⁡6∘\dfrac{\sqrt5-1}{4\sin 6^\circ}
C
−1+54sin⁡6∘-\dfrac{1+\sqrt5}{4\sin 6^\circ}
D
1+5324sin⁡6∘\dfrac{1+\sqrt5}{324\sin 6^\circ}
27
Find the value of S=∑k=1∞3ksin⁡3 ⁣(π2⋅3k)\displaystyle S=\sum_{k=1}^{\infty} 3^{k}\sin^{3}\!\left(\frac{\pi}{2\cdot 3^{k}}\right). (Take π=3.1416\pi=3.1416.) Give SS correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 0.43
28
Real numbers α,β,γ\alpha,\beta,\gamma are such that tan⁡α,tan⁡β,tan⁡γ\tan\alpha,\tan\beta,\tan\gamma are all defined and non-zero, and tan⁡α+tan⁡β+tan⁡γ=tan⁡αtan⁡βtan⁡γ\tan\alpha+\tan\beta+\tan\gamma=\tan\alpha\tan\beta\tan\gamma. Which of the following MUST be true?
Not attempted
A
tan⁡2α+tan⁡2β+tan⁡2γ=tan⁡2αtan⁡2βtan⁡2γ\tan2\alpha+\tan2\beta+\tan2\gamma=\tan2\alpha\tan2\beta\tan2\gamma, whenever tan⁡2α,tan⁡2β,tan⁡2γ\tan2\alpha,\tan2\beta,\tan2\gamma are all defined
B
cot⁡αcot⁡β+cot⁡βcot⁡γ+cot⁡γcot⁡α=1\cot\alpha\cot\beta+\cot\beta\cot\gamma+\cot\gamma\cot\alpha=1
C
sin⁡2α+sin⁡2β+sin⁡2γ=4sin⁡αsin⁡βsin⁡γ\sin2\alpha+\sin2\beta+\sin2\gamma=4\sin\alpha\sin\beta\sin\gamma
D
α+β+γ\alpha+\beta+\gamma is an integer multiple of π\pi
29
Find S=∑k=18tan⁡kπ24 tan⁡(k+1)π24\displaystyle S=\sum_{k=1}^{8}\tan\frac{k\pi}{24}\,\tan\frac{(k+1)\pi}{24}. (Use 2=1.4142\sqrt2=1.4142, 3=1.7321\sqrt3=1.7321, 6=2.4495\sqrt6=2.4495.) Give SS correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 9.34
30
Let x=2cos⁡2π9x=2\cos\dfrac{2\pi}{9}, y=2cos⁡4π9y=2\cos\dfrac{4\pi}{9}, z=2cos⁡8π9z=2\cos\dfrac{8\pi}{9}. Then x2y+y2z+z2xx^2y+y^2z+z^2x equals
Not attempted
A
99
B
66
C
−3-3
D
33
31
For real θ\theta with all terms defined, let f(θ)=tan⁡θ+tan⁡(θ+π3)+tan⁡(θ+2π3)f(\theta)=\tan\theta+\tan\left(\theta+\dfrac{\pi}{3}\right)+\tan\left(\theta+\dfrac{2\pi}{3}\right). Which of the following are correct?
Not attempted
A
tan⁡θ tan⁡(θ+π3)tan⁡(θ+2π3)=−tan⁡3θ\tan\theta\,\tan\left(\theta+\dfrac{\pi}{3}\right)\tan\left(\theta+\dfrac{2\pi}{3}\right)=-\tan3\theta
B
f(θ)=3tan⁡3θf(\theta)=3\tan3\theta
C
tan⁡2θ+tan⁡2(θ+π3)+tan⁡2(θ+2π3)=9tan⁡23θ+3\tan^2\theta+\tan^2\left(\theta+\dfrac{\pi}{3}\right)+\tan^2\left(\theta+\dfrac{2\pi}{3}\right)=9\tan^23\theta+3
D
ff is periodic with fundamental period π3\dfrac{\pi}{3}
32
The value of ∑k=042ktan⁡2kπ96\displaystyle\sum_{k=0}^{4}2^{k}\tan\frac{2^{k}\pi}{96} is
Not attempted
A
cot⁡π96−163\cot\dfrac{\pi}{96}-\dfrac{16}{\sqrt3}
B
323−cot⁡π9632\sqrt3-\cot\dfrac{\pi}{96}
C
cot⁡π96−323\cot\dfrac{\pi}{96}-\dfrac{32}{\sqrt3}
D
cot⁡π96−323\cot\dfrac{\pi}{96}-32\sqrt3
33
Find the number of real θ∈(0,π)\theta\in(0,\pi) satisfying cos⁡θ cos⁡2θ cos⁡4θ cos⁡8θ=116\cos\theta\,\cos2\theta\,\cos4\theta\,\cos8\theta=\dfrac{1}{16}.
Not attempted
Your answer: —
Correct answer: 15
34
Find the minimum value of g(x)=∣sin⁡x∣+∣sin⁡(x+1)∣+∣sin⁡(x+2)∣g(x)=|\sin x|+|\sin(x+1)|+|\sin(x+2)| over all real xx (angles in radians). (Take sin⁡1=0.8415\sin1=0.8415, sin⁡2=0.9093\sin2=0.9093.) Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.68
35
Find the maximum value of 2sin⁡xcos⁡y+3sin⁡y+6cos⁡xcos⁡y2\sin x\cos y+3\sin y+6\cos x\cos y over all real x,yx,y.
Not attempted
Your answer: —
Correct answer: 7
36
As xx and yy range over all real numbers, the set of values of cos⁡xcos⁡ycos⁡(x+y)\cos x\cos y\cos(x+y) is
Not attempted
A
[−1, 1]\left[-1,\,1\right]
B
[−18, 18]\left[-\dfrac18,\,\dfrac18\right]
C
[−12, 1]\left[-\dfrac12,\,1\right]
D
[−18, 1]\left[-\dfrac18,\,1\right]
37
Let f(x)=sin⁡(cos⁡x)f(x)=\sin(\cos x) and g(x)=cos⁡(sin⁡x)g(x)=\cos(\sin x) for real xx (all angles in radians). Which of the following are correct?
Not attempted
A
The minimum value of gg is cos⁡1\cos1
B
The fundamental period of ff is 2π2\pi and that of gg is π\pi
C
f(x)<g(x)f(x)<g(x) for every real xx
D
The maximum value of ff is sin⁡1\sin1
38
For which of the following pairs (α,β)(\alpha,\beta) is the function h(x)=sin⁡2x+sin⁡2(x+α)+sin⁡2(x+β)h(x)=\sin^2x+\sin^2(x+\alpha)+\sin^2(x+\beta) constant on R\mathbb{R}?
Not attempted
A
(α,β)=(π6,5π6)(\alpha,\beta)=\left(\dfrac\pi6,\dfrac{5\pi}6\right)
B
(α,β)=(2π3,4π3)(\alpha,\beta)=\left(\dfrac{2\pi}3,\dfrac{4\pi}3\right)
C
(α,β)=(π3,π3)(\alpha,\beta)=\left(\dfrac\pi3,\dfrac{\pi}3\right)
D
(α,β)=(π3,2π3)(\alpha,\beta)=\left(\dfrac\pi3,\dfrac{2\pi}3\right)
39
Match each function in List-I with the description (fundamental period; range) in List-II. List-IList-IIP) sin⁡4x+cos⁡4x1) period π2; range [12,1]Q) sin⁡x sin⁡(π3−x)sin⁡(π3+x)2) period 2π3; range [−14,14]R) tan⁡x tan⁡(π3−x)tan⁡(π3+x)3) period π2; range [1,2]S) ∣sin⁡x∣+∣cos⁡x∣4) period π3; range R5) period π; range [12,1]\begin{array}{ll}\textbf{List-I} & \textbf{List-II}\\ \text{P) } \sin^4x+\cos^4x & \text{1) period } \frac{\pi}{2}\text{; range } \left[\frac12,1\right]\\ \text{Q) } \sin x\,\sin\left(\frac\pi3-x\right)\sin\left(\frac\pi3+x\right) & \text{2) period } \frac{2\pi}{3}\text{; range } \left[-\frac14,\frac14\right]\\ \text{R) } \tan x\,\tan\left(\frac\pi3-x\right)\tan\left(\frac\pi3+x\right) & \text{3) period } \frac{\pi}{2}\text{; range } \left[1,\sqrt2\right]\\ \text{S) } |\sin x|+|\cos x| & \text{4) period } \frac{\pi}{3}\text{; range } \mathbb{R}\\ & \text{5) period } \pi\text{; range } \left[\frac12,1\right]\end{array}
Not attempted
A
P-1, Q-2, R-4, S-3
B
P-5, Q-2, R-4, S-1
C
P-1, Q-4, R-2, S-3
D
P-3, Q-2, R-4, S-1
40
Find the minimum value of 27sin⁡x+8cos⁡x\dfrac{27}{\sin x}+\dfrac{8}{\cos x} for 0<x<π20<x<\dfrac{\pi}{2}. (Take 13=3.6056\sqrt{13}=3.6056.) Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 46.87
41
The figure shows the curve y=sin⁡x+∣cos⁡x∣y=\sin x+|\cos x| and the line y=1y=1 for 0≤x≤π0\le x\le\pi. Find the total area of the region(s) enclosed between them on [0,π][0,\pi] (shaded). (Take π=3.1416\pi=3.1416.) Give the answer correct to two decimal places.
Not attempted
xyOπ/2π1y = sin x + |cos x|y = 1
Your answer: —
Correct answer: 0.86
42
The maximum value of sin⁡xcos⁡y+sin⁡ycos⁡z+sin⁡zcos⁡x\sin x\cos y+\sin y\cos z+\sin z\cos x, over all real x,y,zx,y,z, is
Not attempted
A
332\dfrac{3\sqrt3}{2}
B
32\dfrac32
C
334\dfrac{3\sqrt3}{4}
D
33
43
Let SS be the sum of all solutions of sin⁡5xcos⁡3x=sin⁡9xcos⁡7x\sin5x\cos3x=\sin9x\cos7x in the interval [0,π2]\left[0,\dfrac{\pi}{2}\right]. Find Sπ\dfrac{S}{\pi}.
Not attempted
Your answer: —
Correct answer: 2.25
44
The complete solution set of tan⁡x+tan⁡2x+tan⁡3x=tan⁡xtan⁡2xtan⁡3x\tan x+\tan2x+\tan3x=\tan x\tan2x\tan3x is (m,n∈Zm,n\in\mathbb{Z})
Not attempted
A
x=nπ6, n≢3(mod6)x=\dfrac{n\pi}{6},\ n\not\equiv3\pmod 6
B
x=mπx=m\pi
C
x=mπ3x=\dfrac{m\pi}{3}
D
x=nπ6x=\dfrac{n\pi}{6}
45
Consider the system x+y=2π3x+y=\dfrac{2\pi}{3}, sin⁡x+sin⁡y=32\sin x+\sin y=\dfrac32 in real numbers x,yx,y. Which of the following are correct (k∈Zk\in\mathbb{Z})?
Not attempted
A
tan⁡xtan⁡y\tan x\tan y is defined for every solution
B
For some solution, x−y=11π3x-y=\dfrac{11\pi}{3}
C
For some solution, x−y=5π3x-y=\dfrac{5\pi}{3}
D
All solutions are (π2+2kπ, π6−2kπ)\left(\dfrac\pi2+2k\pi,\ \dfrac\pi6-2k\pi\right) and (π6+2kπ, π2−2kπ)\left(\dfrac\pi6+2k\pi,\ \dfrac\pi2-2k\pi\right)
46
The figure shows y=∣cos⁡x∣y=|\cos x| (solid) and y=2−∣x∣y=2^{-|x|} (dashed) for −2π≤x≤2π-2\pi\le x\le2\pi. Find the number of real solutions of ∣cos⁡x∣=2−∣x∣|\cos x|=2^{-|x|} in [−2π,2π][-2\pi,2\pi].
Not attempted
xyO−2π−ππ2π1solid: y = |cos x|dashed: y = 2^(−|x|)
Your answer: —
Correct answer: 9
47
Find the number of real x∈[0,2π]x\in[0,2\pi] satisfying sin⁡2027x+cos⁡2028x=1\sin^{2027}x+\cos^{2028}x=1.
Not attempted
Your answer: —
Correct answer: 4
48
The figure shows y=sin⁡x+3cos⁡xy=\sin x+\sqrt3\cos x (solid) and y=2sin⁡2xy=2\sin2x (dashed) on [0,2π][0,2\pi]. The set of x∈(0,2π)x\in(0,2\pi) with sin⁡x+3cos⁡x≥2sin⁡2x\sin x+\sqrt3\cos x\ge2\sin2x is a union of intervals of total length kπk\pi. Find kk. Give kk correct to two decimal places.
Not attempted
xyOπ/2π3π/22π2−2
Your answer: —
Correct answer: 1.22
49
The number of x∈[0,2π]x\in[0,2\pi] satisfying 2sin⁡x+2cos⁡x=2 1−122^{\sin x}+2^{\cos x}=2^{\,1-\frac{1}{\sqrt2}} is
Not attempted
A
00
B
22
C
44
D
11
50
Find ∑n=1∞tan⁡−14nn4+5\displaystyle\sum_{n=1}^{\infty}\tan^{-1}\frac{4n}{n^4+5}. (Take π=3.1416\pi=3.1416 and tan⁡−12=1.1071\tan^{-1}2=1.1071.) Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 1.25
51
Let g(x)=cos⁡−11−x21+x2+sin⁡−12x1+x2g(x)=\cos^{-1}\dfrac{1-x^2}{1+x^2}+\sin^{-1}\dfrac{2x}{1+x^2} for real xx. Which of the following are correct?
Not attempted
A
gg is differentiable at x=0x=0
B
g(x)=πg(x)=\pi for every x≥1x\ge1
C
gg is continuous on R\mathbb{R}
D
g(x)=0g(x)=0 for every x∈[−1,0]x\in[-1,0]
52
The figure shows y=sin⁡−1(sin⁡x)y=\sin^{-1}(\sin x) (solid) and y=x5y=\dfrac{x}{5} (dashed). Find the number of real solutions of sin⁡−1(sin⁡x)=x5\sin^{-1}(\sin x)=\dfrac x5.
Not attempted
xyO−5π/2−ππ5π/2π/2−π/2
Your answer: —
Correct answer: 5
53
All angles are in radians. Which of the following statements are correct?
Not attempted
A
cos⁡−1(sin⁡5)=5−3π2\cos^{-1}(\sin5)=5-\dfrac{3\pi}{2}
B
sin⁡−1(sin⁡5)+cos⁡−1(cos⁡5)=0\sin^{-1}(\sin5)+\cos^{-1}(\cos5)=0
C
tan⁡−1(tan⁡5)+cot⁡−1(cot⁡5)=π2\tan^{-1}(\tan5)+\cot^{-1}(\cot5)=\dfrac\pi2
D
sin⁡−1(cos⁡5)=3π2−5\sin^{-1}(\cos5)=\dfrac{3\pi}{2}-5
54
Find the sum of the squares of all real solutions of cos⁡−1x−sin⁡−1x=cos⁡−1(3 x)\cos^{-1}x-\sin^{-1}x=\cos^{-1}\left(\sqrt3\,x\right).
Not attempted
Your answer: —
Correct answer: 0.5
55
Which of the following is the graph of y=tan⁡−11+x1−xy=\tan^{-1}\dfrac{1+x}{1-x}, x≠1x\ne1? (Dashed lines mark y=±π2y=\pm\dfrac\pi2 and y=±π4y=\pm\dfrac\pi4.)
Not attempted
A
Increasing on (−∞,1)(-\infty,1) from −π4-\frac\pi4 towards π2\frac\pi2; jumps down at x=1x=1 and increases on (1,∞)(1,\infty) from −π2-\frac\pi2 towards −π4-\frac\pi4
xy−11π/2π/4−π/4−π/2
B
The continuous increasing curve y=π4+tan⁡−1xy=\frac\pi4+\tan^{-1}x from −π4-\frac\pi4 to 3π4\frac{3\pi}{4}
xy−11π/2π/4−π/4−π/2
C
Increasing curve with a jump at x=−1x=-1, tending to π4\frac\pi4 at both ends
xy−11π/2π/4−π/4−π/2
D
Decreasing curve with a jump at x=−1x=-1, tending to −π4-\frac\pi4 at both ends
xy−11π/2π/4−π/4−π/2
56
Let h(x)=sin⁡−1(3x−4x3)h(x)=\sin^{-1}\left(3x-4x^3\right) for −1≤x≤1-1\le x\le1. The figure shows y=3x−4x3y=3x-4x^3 on [−1,1][-1,1]. Which of the following are correct?
Not attempted
xyO−1−1/21/211−1y = 3x − 4x³
A
h(x)=3sin⁡−1xh(x)=3\sin^{-1}x for ∣x∣≤12|x|\le\dfrac12
B
h(x)=π−3sin⁡−1xh(x)=\pi-3\sin^{-1}x for 12<x≤1\dfrac12<x\le1
C
hh attains the value π2\dfrac\pi2 at exactly two points of [−1,1][-1,1]
D
h(x)=3sin⁡−1x+πh(x)=3\sin^{-1}x+\pi for −1≤x<−12-1\le x<-\dfrac12
57
The number of ordered pairs (x,y)(x,y) of non-zero integers satisfying tan⁡−11x+tan⁡−11y=tan⁡−113\tan^{-1}\dfrac1x+\tan^{-1}\dfrac1y=\tan^{-1}\dfrac13 is
Not attempted
A
1616
B
88
C
44
D
66
58
Which of the following are equal to 2tan⁡−1(−2)2\tan^{-1}(-2)?
Not attempted
A
−cos⁡−1(−35)-\cos^{-1}\left(-\dfrac35\right)
B
tan⁡−143\tan^{-1}\dfrac43
C
−π−sin⁡−1(−45)-\pi-\sin^{-1}\left(-\dfrac45\right)
D
π+tan⁡−143\pi+\tan^{-1}\dfrac43
59
In every triangle ABCABC (circumradius RR), which of the following hold?
Not attempted
A
If acos⁡A=bcos⁡Ba\cos A=b\cos B, then a=ba=b
B
acos⁡A+bcos⁡B+ccos⁡C=4Rsin⁡Asin⁡Bsin⁡Ca\cos A+b\cos B+c\cos C=4R\sin A\sin B\sin C
C
a(cos⁡B+cos⁡C)=2(b+c)sin⁡2A2a(\cos B+\cos C)=2(b+c)\sin^2\dfrac A2
D
b2−c2a2=sin⁡(B−C)sin⁡(B+C)\dfrac{b^2-c^2}{a^2}=\dfrac{\sin(B-C)}{\sin(B+C)}
60
The exradii of a triangle are r1=8r_1=8, r2=12r_2=12, r3=24r_3=24. Which of the following are correct?
Not attempted
A
The circumradius is 1010
B
The triangle is right-angled
C
The inradius is 44
D
The area is 9696
61
In triangle ABCABC, BC=8BC=8 and ∠BAC=60∘\angle BAC=60^\circ. Its incentre is II, and the line AIAI meets the circumcircle again at DD (see figure). Find DIDI. (Take 3=1.7321\sqrt3=1.7321.) Give the answer correct to two decimal places.
Not attempted
ABCID
Your answer: —
Correct answer: 4.62
62
Over all triangles, find the least possible value of r1+r2+r3r\dfrac{r_1+r_2+r_3}{r}, where rr is the inradius and r1,r2,r3r_1,r_2,r_3 are the exradii.
Not attempted
Your answer: —
Correct answer: 9
63
In a triangle ABCABC with inradius rr, circumradius RR and exradius r1r_1 (opposite AA), r:R:r1=2:5:12r:R:r_1=2:5:12. Then the triangle is
Not attempted
A
equilateral
B
obtuse-angled at AA
C
right-angled at AA
D
right-angled at BB or CC
64
A vertical pole 2020 m tall stands on top of a vertical tower 1818 m tall on level ground. An observer whose eyes are 22 m above the ground stands at horizontal distance xx from the tower (see figure). Find the value of xx (in m) for which the pole subtends the greatest angle at the observer's eye.
Not attempted
tower 18 mpole 20 mobserver (eye 2 m)x
Your answer: —
Correct answer: 24
65
FF is the foot and TT the top of a vertical tower. Points AA and BB on level ground are 100100 m apart and ∠AFB=120∘\angle AFB=120^\circ (see figure). The angles of elevation of TT from AA and BB are 45∘45^\circ and 30∘30^\circ. The height of the tower (in m) is
Not attempted
ABFTAB = 100 m120°45°30°
A
1004−3\dfrac{100}{\sqrt{4-\sqrt3}}
B
1002+3\dfrac{100}{2+\sqrt3}
C
5050
D
1004+3\dfrac{100}{\sqrt{4+\sqrt3}}
66
Two vertical poles of heights 66 m and 1010 m stand at points AA and BB of level ground, 1212 m apart. The set of points PP on the ground from which the tops of the two poles have equal angles of elevation is a circle (see figure). Find its radius in metres. Give the answer correct to two decimal places.
Not attempted
6 m10 m12 mPAB
Your answer: —
Correct answer: 11.25
67
A man walks on level ground straight towards the foot FF of a vertical tower FTFT. At AA the angle of elevation of TT is θ\theta; after walking 3030 m to BB it is 2θ2\theta; after a further 1212 m, at CC, it is 3θ3\theta (see figure; CC is still short of FF). The height of the tower (in m) is
Not attempted
ABCTF30 m12 mθ2θ3θ
A
1572\dfrac{15\sqrt7}{2}
B
1574\dfrac{15\sqrt7}{4}
C
15315\sqrt3
D
15152\dfrac{15\sqrt{15}}{2}
68
Positive real numbers x,y,zx,y,z satisfy xy+yz+zx=1xy+yz+zx=1. Find the maximum value of 1−x21+x2+1−y21+y2+2⋅1−z21+z2\dfrac{1-x^2}{1+x^2}+\dfrac{1-y^2}{1+y^2}+2\cdot\dfrac{1-z^2}{1+z^2}. Give the answer correct to two decimal places.
Not attempted
Your answer: —
Correct answer: 2.25
69
Find the exact value of sec⁡2π7+sec⁡22π7+sec⁡23π7\sec^2\dfrac{\pi}{7}+\sec^2\dfrac{2\pi}{7}+\sec^2\dfrac{3\pi}{7}.
Not attempted
Your answer: —
Correct answer: 24
70
A point PP inside an equilateral triangle ABCABC satisfies PA=5PA=5, PB=7PB=7, PC=8PC=8 (see figure). Find AB2AB^2.
Not attempted
ABCP578
Your answer: —
Correct answer: 129
Rotate the triangle APBAPB about AA through 60∘60^\circ so that BB goes to CC; the three given lengths then become the sides of a triangle, and an equilateral triangle appears next to it.
Rotate about AA by 60∘60^\circ taking BB to CC; let PP go to P′P'. Then AP′=AP=5AP'=AP=5, ∠PAP′=60∘\angle PAP'=60^\circ, so triangle APP′APP' is equilateral: PP′=5PP'=5. Also CP′=BP=7CP'=BP=7. So triangle PP′CPP'C has sides 5,7,85,7,8, with area 10⋅5⋅3⋅2=103\sqrt{10\cdot5\cdot3\cdot2}=10\sqrt3. The rotation maps triangle APBAPB onto AP′CAP'C, so [APB]+[APC]=[AP′C]+[APC]=[APP′]+[PP′C]=34⋅25+103[APB]+[APC]=[AP'C]+[APC]=[APP']+[PP'C]=\dfrac{\sqrt3}{4}\cdot25+10\sqrt3. Rotating about BB and about CC in the same way gives [BPC]+[BPA]=34⋅49+103[BPC]+[BPA]=\dfrac{\sqrt3}{4}\cdot49+10\sqrt3 and [CPA]+[CPB]=34⋅64+103[CPA]+[CPB]=\dfrac{\sqrt3}{4}\cdot64+10\sqrt3 (each time the small triangle again has sides 5,7,85,7,8). Adding: 2[ABC]=34(25+49+64)+3032[ABC]=\dfrac{\sqrt3}{4}(25+49+64)+30\sqrt3, so [ABC]=6934+153=12934[ABC]=\dfrac{69\sqrt3}{4}+15\sqrt3=\dfrac{129\sqrt3}{4}. Since [ABC]=34AB2[ABC]=\dfrac{\sqrt3}{4}AB^2, AB2=129AB^2=129. (Check with coordinates: A=(0,0)A=(0,0), B=(129,0)B=(\sqrt{129},0) gives P≈(4.62,1.91)P\approx(4.62,1.91) and PC=8PC=8 exactly.) Directly applying the cosine rule in the three small triangles leads to three equations in three unknown angles that are hard to close without the rotation idea.