Trigonometric Functions Test 31

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Trigonometric Functions Test 31

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Weekly Quiz Competition

Question 1
1 / -0

The number of solution of $$\sec x \cos 5x+1=0$$ in the interval $$[0,2\pi]$$ is

A
$$5$$

B
$$8$$

C
$$10$$

D
$$12$$
Solution
Given- sec(x)cos(5x)+1=0
cos 5x/cos x + 1 = 0
cos 5x + cos x = 0
2(cos 3x )( cos 2x) = 0 (cosA+cosB=2cos((A+B)/2)cos((A-B)/2)
(i) cos 3x = 0 , 3x = (2 n + 1)π/2 , x = (2n + 1 ) π/6
(ii)cos 2x = 0, 2x = (2 n + 1)π/2 , x = (2 n + 1)π/4
Therefore in [0,2π]
x= π/6, 5π/6, 7π/6, 11π/6, π/4, 3π/4, 5π/4, 7π/4
Question 2
1 / -0

If $$2 \tan ^ { 2 } \theta - 5 \sec \theta = 1$$ has exactly $$7$$ solutions in the interval $$\left[ 0 , \dfrac { n \pi } { 2 } \right] , n \in N$$ then the least and greatest values of $$n$$ are

A
$$6 , 8$$

B
$$12 , 14$$

C
$$13 , 15$$

D
$$15 , 17$$

Solution
Question 3
1 / -0

The value of $$\sqrt { 3 } \cot 20 ^ { \circ } - 4 \cos 20 ^ { \circ }$$ = __________________________.

A
$$1$$

B
$$-1$$

C
$$0$$

D
none of these

Solution
Question 4
1 / -0

If in a triangle $$ABC, \dfrac{b+c}{11}=\dfrac{c+a}{12}=\dfrac{a+b}{13}$$, then $$\cos A$$ is equal to:

A
$$\dfrac{19}{35}$$

B
$$\dfrac{5}{7}$$

C
$$\dfrac{1}{5}$$

D
$$\dfrac{35}{19}$$

Solution

$$\dfrac{b+c}{11}=\dfrac{c+a}{12}=\dfrac{a+b}{13}=k$$.
$$b+c=11\ k,\, c+a=12\ k, \,a+b=13\ k$$.
So $$2(a+b+c)=36\ k$$
$$a+b+c=18\ k$$
$$a=7\ k, b=6\ k, c=5\ k$$
$$\cos A=\dfrac{b^2+c^2-a^2}{2bc}$$
$$=\dfrac{(6k)^2+(5k)^2-(7k)^2}{2\times 6k\times 5k}$$
$$=\dfrac{61\ k^2-49\ k^2}{60\ k^2}=\dfrac{12\ k^2}{60\ k^2}$$
$$\cos A=\dfrac{1}{5}$$
Question 5
1 / -0

The number of real solutions of the equation $$\sin \left( e ^ { x } \right) = 5 ^ { x } + 5 ^ { - x }$$ is ____________________.

A
$$0$$

B
$$1$$

C
$$2$$

D
infinite

Solution
Question 6
1 / -0

The value of $$\sin \dfrac {2\pi}{7}+\sin \dfrac {4\pi}{7}+\sin \dfrac {8\pi}{7}$$ is

A
$$1$$

B
$$\dfrac {\sqrt {7}}{2}$$

C
$$\dfrac {3\sqrt {3}}{4}$$

D
$$\dfrac {\sqrt {15}}{4}$$
Question 7
1 / -0

Directions For Questions

Consider the curve $$x=1-3t^{2}, y=t-3r^{3}$$. If a tangent at point $$(1-3t^{2}, t-3t^{3})$$ inclined at an angel $$\theta$$ to the positive
$$x-axis$$ and another tangent at point $$P(2, 2)$$ cuts the curve
again at $$Q$$.

...view full instructions

The value of $$\tan\theta+\sec\theta$$ is equal to

A
$$3t$$

B
$$t$$

C
$$t-t^{2}$$

D
$$t^{2}-2t$$

Solution
Question 8
1 / -0

If $$\tan^{2}\alpha-\tan^{2}\beta-\dfrac{1}{2}\sin(\alpha-\beta)\sec^{2}\alpha\sec^{2}\beta$$ is zero then value of $$\sin(\alpha+\beta)$$ is $$(\alpha\neq\beta)$$

A
$$1$$

B
$$\dfrac{1}{3}$$

C
$$\dfrac{1}{4}$$

D
$$\dfrac{1}{2}$$

Solution
Question 9
1 / -0

If $$cos\alpha+cos\beta+cos\gamma=sin\alpha+sin\beta+sin\gamma$$ then $$cos^2\alpha+cos^2\beta+cos^2\gamma=?$$

A
$$0$$

B
$$1/2$$

C
$$3/2$$

D
$$2$$

Solution
Question 10
1 / -0

$$tan(\cfrac {1} {2} \sin^{-1} \cfrac {4} {5} + \cfrac {1} {2}cos^{-1} \cfrac {15} {17})$$ is equal to

A
$$\cfrac {9} {22}$$

B
$$\cfrac {42} {13}$$

C
$$\cfrac {2} {3}$$

D
$$\cfrac {44} {15}$$

Solution