MCQ Questions for Class 11 Engineering Maths Trigonometric Functions Quiz 16

s i n α + s i n β = \frac{1}{2}

$sin \alpha + sin \beta = \dfrac{1}{2}$ and

c o s α + c o s β = \frac{\sqrt{3}}{2}

$cos \alpha + cos \beta = \dfrac{\sqrt{3}}{2}$ then

3 β + α =

$3 \beta + \alpha =$

0%
$0^o$
0%
$60^o$
0%
$120^o$
0%
$90^o$

The value of

csc \frac{π}{13} - \sqrt{3} sec \frac{π}{18}

$\csc \dfrac{\pi}{13}-\sqrt{3} \sec \dfrac{\pi}{18}$ is a

0%
Surd
0%
Rational which is not integral
0%
Negative integer
0%
Natural number

The number of solution of the equation

x^{3} + 2 x^{2} + 5 x + 2 cos x = 0 i n [0, 2 π]

${x^3} + 2{x^2} + 5x + 2\cos x = 0\,in\,[0,2\pi ]$ is

0%
$0$
0%
$1$
0%
$2$
0%
$3$

\frac{{tan}^{2} θ}{{tan}^{2} θ - 1} + \frac{{csc}^{2} θ}{{sec}^{2} θ - {csc}^{2} θ} = \frac{1}{{sin}^{2} θ - {cos}^{2} θ}

$\displaystyle \frac { \tan ^{ 2 }{ \theta } }{ \tan ^{ 2 }{ \theta } -1 } +\frac { \csc ^{ 2 }{ \theta } }{ \sec ^{ 2 }{ \theta } -\csc ^{ 2 }{ \theta } } =\frac { 1 }{ \sin ^{ 2 }{ \theta } -\cos ^{ 2 }{ \theta } }$ .

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True
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False

s e c^{2} (T a n^{- 1} 2) + C o s e c^{2} (C o t^{- 1} 3) =

$sec^{2}(Tan^{-1} 2) +Cosec^{2}(Cot^{-1}3)=$

0%
5
0%
10
0%
15
0%
20

\frac{1}{cosec θ + cot θ} - \frac{1}{sin θ} = \frac{1}{sin θ} - \frac{1}{cosec θ - cot θ}

$\dfrac { 1 }{ \text{cosec}\theta +\cot\theta } -\dfrac { 1 }{ \sin\theta } =\dfrac { 1 }{ \sin\theta } -\dfrac { 1 }{ \text{cosec}\theta -\cot\theta }$

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True
0%
False

If y = acosx + bsinx then find the maximum value of y

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$\sqrt{a^2+b^2}$
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${a^2}+{b^2}$
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$\frac{a+b}{2}$
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$\frac{\sqrt{a^2+b^2}}{2}$

The value of

9 \sum r = 0 {sin}^{2} \frac{π r}{18} i s e q u a l t o

$\sum\limits_{r = 0}^9 {{{\sin }^2}\dfrac{{\pi r}}{{18}}} \;is\;equal\;to\;$

0%
$\dfrac{9}{2}$
0%
$\dfrac{7}{2}$
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5
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None of these

csc θ = \frac{x^{2} - y^{2}}{x^{2} + y^{2}}

$\csc\theta=\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}$ where

x

$x$ and

y

$y$ are two unequal non-zero real numbers, then the number of real values of

θ

$\theta$ is

0%
$0$
0%
$1$
0%
$2$
0%
$infinite$

n - 1 \sum r = 1 {cos}^{2} (\frac{r π}{n}) = ?

$\displaystyle\sum_{r=1}^{n-1} \cos^2\left ( \dfrac{r\pi}{n} \right )=?$

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$\dfrac{n}{2}$
0%
$\dfrac{n-2}{2}$
0%
$\dfrac{n-1}{2}$
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$\dfrac{n-2}{4}$

Explanation

As we know that

cos 2 A = 2 {cos}^{2} A - 1

$\cos 2 A=2 \cos ^{2} A-1$

{cos}^{2} A = \frac{1 + cos 2 A}{2}

$\cos ^{2} A=\frac{1+\cos 2 A}{2}$

say ,

A = \frac{π π}{n}

$A=\frac{\pi \pi}{n}$

\Rightarrow n - 1 \sum r = 1 1 + cos 2 (\frac{r π}{n}) = \frac{1}{2} [n - 1 \sum r = 1 cos \frac{2 π π}{n} + n - 1 \sum n = 1 1]

$\Rightarrow \sum_{r=1}^{n-1} 1+\cos 2\left(\frac{r \pi}{n}\right)=\frac{1}{2}\left[\sum_{r=1}^{n-1} \cos \frac{2 \pi \pi}{n}+\sum_{n=1}^{n-1} 1\right]$

\Rightarrow \frac{1}{2} [cos \frac{2 π}{n} + cos \frac{4 π}{n} + \dots + cos \frac{2 (n - 1)}{n} π + (n - 1)]

$\Rightarrow \frac{1}{2}\left[\cos \frac{2 \pi}{n}+\cos \frac{4 \pi}{n}+\ldots+\cos \frac{2(n-1)}{n} \pi+(n-1)\right]$

As we know that

$\cos \alpha+\cos (\alpha+\beta)+\cos (\alpha+2 \beta)+\cdots+\cos (\alpha+(n-1) \beta)=\frac{\sin \left(\frac{n \beta}{2}\right)}{\sin \frac{\beta}{2}} \cdot \cos \left(\frac{2 \alpha+(n-1) \beta}{2}\right) \\$

$= \frac{1}{2}\left[\frac{\sin \left(\frac{(n-1) \pi}{n}\right)}{\sin \left(\frac{\pi}{n}\right)} \cdot \cos \left(\frac{\frac{4\pi}{n}+(n-2) \frac{2 \pi}{n}}{2}\right)+n-1\right]$

$= \frac{1}{2}\left[\frac{-\sin \left(\frac{n-1}{n} \pi\right)}{\sin \frac{\pi}{n}}+n-1\right]$

$\left.= \frac{1}{2}\left[-\left\{\sin \frac{\sin \left(\pi-\frac{(n-1)}{n}\right) \pi}{\sin \frac{\pi}{n}}\right\}\right\}+n-1\right]$

$= \frac{1}{2}[-\frac{sin\frac{\pi}{n}}{sin\frac{\pi}{n}}+n-1]$

$= \frac{1}{2}(-1+n-1)=\frac{n-2}{2}$

Hence,

$(B)$ is the correct option.

$sinx+{ sin }^{ 2 }x=1,$ then value of

${ cos }^{ 2 }x+{ cos }^{ 4 }x$ is

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1
0%
2
0%
1.5
0%
none of these

$\frac { \cos { \theta } }{ p } =\frac { \sin { \theta } }{ q }$ , then

$\frac { p }{ \sec { 2\theta } } +\frac { q }{ cosec2\theta }$ is equal to

0%
p
0%
q
0%
qp
0%
none of these

The value of

$\sum _{ r=1 }^{ 10 }{ \cos ^{ 3 }{ \frac { r\pi }{ 3 } } } =$ ?

0%
$-\frac { 1 }{ 8 }$
0%
$-\frac { 7 }{ 8 }$
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$-\frac { 9 }{ 8 }$
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None of the above

$x+y=\dfrac { 2\pi }{ 3 } and\quad cosx+cosy=\dfrac { \sqrt { 3 } }{ 2 } ,\quad then\quad x,y\quad is\quad equal\quad to$

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$\dfrac { \pi }{ 3 } ,\dfrac { \pi }{ 6 }$
0%
$\dfrac { \pi }{ 2 } ,\dfrac { \pi }{ 6 }$
0%
$\dfrac { \pi }{ 4 } ,\dfrac { \pi }{ 3 }$
0%
no solution

$\sqrt{\dfrac{1+ cosx }{1-cos x}}$

0%
$\dfrac{1+ cosx }{1-cos x}$
0%
$cosec x+ cot x$
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$cosec x- cot x$
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$cosec x$

$f(x) = \frac{{{{\cos }^2}x + {{\sin }^4}x}}{{{{\sin }^2}x + {{\cos }^4}x}}$ for

$x \in R$ then

$f(2002)=$

0%
1
0%
2
0%
3
0%
4

${T_n} = \left( {{{\sin }^n}\theta + {{\cos }^n}\theta } \right),\;then\;for\;permissible\;values\;of\;\theta$ ,

$\dfrac{{{T_5} - {T_3}}}{{{T_7} - {T_5}}}$ is always equal to

0%
$\dfrac{{{T_1}}}{{{T_3}}}$
0%
$\dfrac{{{T_2}}}{{{T_4}}}$
0%
$\dfrac{{{T_5}}}{{{T_7}}}$
0%
$\dfrac{{{T_3}}}{{{T_7}}}$

The smallest value of 0 satisfying the equation

$\sqrt { 3(cot0+\quad tan0)\quad =\quad 4 }$ is

0%
$2x/3$
0%
$\pi /3$
0%
$\pi /6$
0%
$\pi /12$

The number of real values of x such that

$\left( {{2^{ - x}} + {2^x} - 2\cos x} \right)\;\left( {{3^{x + \pi }} + {3^{ - x - \pi }} + 2\cos x} \right)\left( {{5^{\pi - x}} - 2\cos x + {5^{x - \pi }}} \right) = 0,is$

0%
1
0%
2
0%
3
0%
infinite

The number of solutions of the equation sin

$\left( \dfrac { \pi x }{ 2\sqrt { 3 } } \right) ={ x }^{ 2 }-2\sqrt { 3 } x+4$ is

0%
0
0%
2
0%
1
0%
none of these

Find the fix point through which the line

$(2cos\Theta +3sin\Theta )x+(3 cos\Theta-5sin\Theta )y-(5cos\Theta -2sin\Theta )=0$ passes for all values of

$\Theta$ -

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(0,0)
0%
(1,1)
0%
(2,1)
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None of these

The principal solutions of

$secx= \frac{2}{\sqrt{3}}$ are _________

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$\frac{\pi }{3},\frac{||\pi }{6}$
0%
$\frac{\pi }{6},\frac{11\pi }{6}$
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$\frac{\pi }{4},\frac{||\pi }{4}$
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$\frac{\pi }{6},\frac{||\pi }{4}$

$\dfrac { \tan\theta }{ 1-\cot\theta } +\dfrac { \cot\theta }{ 1-\tan\theta }$ is equal to

0%
$1+\tan\theta +\cot\theta$
0%
$1-\tan\theta -\cot\theta$
0%
$1+\tan\theta -\cot\theta$
0%
none of these

$cos^{4}\Theta - sin^{4}\Theta$ is equal

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$\frac{1-tan^{2}\Theta }{1+ tan^{2}\Theta }$
0%
$2 cos^{2}\Theta -1$
0%
$1-2 sin^{2}\Theta$
0%
$sin^{2}\Theta - cos^{2}\Theta$

$sin^{6}15^{0}-48sin^{4}15^{0}+18sin^{2}15^{0}=$

0%
1
0%
2
0%
3
0%
-1

Choose the correct answer.

$(1+{ \tan }^{ 2 }\theta ){ \sin }^{ 2 }\theta =$

0%
${ \tan }^{ 2 }\theta$
0%
${ \cot }^{ 2 }\theta$
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${ \sin }^{ 2 }\theta$
0%
${ \cos }^{ 2 }\theta$

The number of solutions of the equation

$\sin (9x) + \sin (3x) = 0$ in the closed interval

$[0, 2\pi]$ is

0%
$7$
0%
$13$
0%
$19$
0%
$25$

Explanation

$\sin{9x}+\sin{3x}=0,\,\,x\in\left[0,2\pi\right]$

$\Rightarrow\,3\sin{3x}-4{\sin}^{3}{3x}+\sin{3x}=0$

$\Rightarrow\,4\sin{3x}-4{\sin}^{3}{3x}=0$

$\Rightarrow\,4\sin{3x}\left(1-{\sin}^{2}{3x}\right)=0$

$\Rightarrow\,\sin{3x}=0,\,\,\left(1-{\sin}^{2}{3x}\right)=0$

$\Rightarrow\,\sin{3x}=0,\,\,{\sin}^{2}{3x}=1$

$\Rightarrow\,3x=n\pi,\,n\in I$ or

$3x=k\pi+\dfrac{\pi}{2},\,k\in I$

$\Rightarrow\,x=\dfrac{n\pi}{3},\,x=\dfrac{k\pi}{3}+\dfrac{\pi}{6},\,k\in I$

For

$n=0,1,2,3,4,5,6$ we have

$x=0,\dfrac{\pi}{3},\dfrac{2\pi}{3},\pi,\dfrac{4\pi}{3},\dfrac{5\pi}{3},2\pi$

For

$k=0,1,2,3,4,5$ we have

$x=\dfrac{\pi}{6},\,\dfrac{\pi}{3}+\dfrac{\pi}{6},\,\dfrac{2\pi}{3}+\dfrac{\pi}{6},\,\pi+\dfrac{\pi}{6},\,\dfrac{4\pi}{3}+\dfrac{\pi}{6},\,\dfrac{5\pi}{3}+\dfrac{\pi}{6}$

$x=\dfrac{\pi}{6},\,\dfrac{2\pi}{6}+\dfrac{\pi}{6},\,\dfrac{4\pi}{6}+\dfrac{\pi}{6},\,\dfrac{6\pi+\pi}{6},\,\dfrac{8\pi}{6}+\dfrac{\pi}{6},\,\dfrac{10\pi}{6}+\dfrac{\pi}{6}$

$x=\dfrac{\pi}{6},\,\dfrac{3\pi}{6},\,\dfrac{5\pi}{6},\,\dfrac{7\pi}{6},\,\dfrac{9\pi}{6},\,\dfrac{11\pi}{6}$

$x=\dfrac{\pi}{6},\,\dfrac{\pi}{2},\,\dfrac{5\pi}{6},\,\dfrac{7\pi}{6},\,\dfrac{3\pi}{2},\,\dfrac{11\pi}{6}$

Thus, the solutions are

$\left\{0,\dfrac{\pi}{3},\dfrac{2\pi}{3},\pi,\dfrac{4\pi}{3},\dfrac{5\pi}{3},2\pi,\dfrac{\pi}{6},\,\dfrac{\pi}{2},\,\dfrac{5\pi}{6},\,\dfrac{7\pi}{6},\,\dfrac{3\pi}{2},\,\dfrac{11\pi}{6}\right\}$

$\left\{0,\,\dfrac{\pi}{2},\dfrac{\pi}{6},\dfrac{\pi}{3},\dfrac{2\pi}{3},\dfrac{3\pi}{2},\pi,\dfrac{4\pi}{3},\dfrac{5\pi}{3},\dfrac{5\pi}{6},\,\dfrac{7\pi}{6},\dfrac{11\pi}{6},2\pi\right\}$

$=13$ solutions

$sec A=a+(\frac{1}{4a})$ , then sec A + tan A is equal to

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$2a \;or\; \frac{1}{2a}$
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$a or \frac{1}{a}$
0%
$2a or\frac{1}{a}$
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$a or \frac{1}{2a}$

$\displaystyle sin^{-1} x + sin ^{-1} y + sin^{-1} z = 3\pi / 2$ then the value of

$x^{100} + y ^{100} + z^{100} - \dfrac{3}{x^{101} + y^{101} + z^{101}}$ is

0%
$0$
0%
$1$
0%
$2$
0%
$3$

Indicate the relation which is true

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$tan \left | tan^{-1} x \right | = \left | x \right |$
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$cot \left | cot^{-1}x \right | = x$
0%
$tan^{-1} \left | tan\,x \right | = \left | x \right |$
0%
$sin \left | sin^{-1} \right | = \left | x \right |$

CBSE Questions for Class 11 Engineering Maths Trigonometric Functions Quiz 16 - MCQExams.com

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Practice Class 11 Engineering Maths Quiz Questions and Answers

CBSE Questions for Class 11 Engineering Maths Trigonometric Functions Quiz 16 - MCQExams.com

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Practice Class 11 Engineering Maths Quiz Questions and Answers

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