#### Important questions -Trigonometry

11th Standard

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Maths

Use blue pen Only

Time : 01:00:00 Hrs
Total Marks : 45

Part A

10 x 1 = 10
1. $\frac { 1 }{ cos{ 80 }^{ 0 } } -\frac { \sqrt { 3 } }{ sin{ 80 }^{ 0 } }$=

(a)

$\sqrt{2}$

(b)

$\sqrt{3}$

(c)

2

(d)

4

2. Let fk(x)=$\frac { 1 }{ k }$[sinkx+coskx] where x$\in$R and k≥1. Then f4(x)-f6(x)=

(a)

$\frac { 1 }{ 4 }$

(b)

$\frac { 1 }{ 12 }$

(c)

$\frac { 1 }{ 6 }$

(d)

$\frac { 1 }{ 3 }$

3. If sinα+cosα=b, then sin2α is equal to

(a)

b2-1, if b≤$\sqrt { 2 }$

(b)

b2-1, if b>$\sqrt { 2 }$

(c)

b2-1, if b≥$\sqrt { 2 }$

(d)

b2-1, if b≥$\sqrt { 2 }$

4. If cosec x+cotx=$\frac { 11 }{ 2 }$ then tanx=

(a)

$\frac { 21 }{ 22 }$

(b)

$\frac { 15 }{ 16 }$

(c)

$\frac { 44 }{ 117 }$

(d)

$\frac { 117 }{ 44 }$

5. The value of sin2$\frac { 5\pi }{ 12 } -sin^{ 2 }\frac { \pi }{ 12 }$ is

(a)

$\frac { 1 }{ 2 }$

(b)

$\frac { \sqrt { 3 } }{ 2 }$

(c)

1

(d)

0

6. sin$(22{1\over 2}^o)$is

(a)

${\sqrt{2-\sqrt{2}}}\over2$

(b)

${2\sqrt{2}-1\over 4\sqrt{2}}$

(c)

${\sqrt{2-\sqrt{2}\over 2}}$

(d)

none of these

7. In $\triangle$ABC, $\hat{C}$ = 90° then a cosA + b cosB is:

(a)

2R sinB

(b)

2 sinB

(c)

0

(d)

2a sinB

8. The value of tan 1° tan 2° tan 3°...tan 89° is

(a)

$\infty$

(b)

0

(c)

1

(d)

$\sqrt{3}$

9. If cosθ+$\sqrt{3}$sinθ=2 and θ∈[0,2π] then θ is

(a)

$\frac{\pi}{3}$

(b)

$\frac{5\pi}{3}$

(c)

$\frac{2\pi}{3}$

(d)

$\frac{4\pi}{3}$

10. The numerical value of tan-11+tan-12+tan-13=

(a)

$\pi$

(b)

$\frac{\pi}{2}$

(c)

0

(d)

$\frac{\pi}{4}$

11. Part B

10 x 2 = 20
12. A fighter jet has to hit a small target by flying a horizontal distance. When the target is sighted, the pilot measures the angle of depression to be 300. If after 100km, the target has an angle of depression of 450, how far is the target from the fighter jet at that instant?

13. Suppose that a satellite in space, an earth station and the centre of earth all in the same plane. Let r be the radius of earth and R be the distance from the centre of earth to the satellite. Let d be the distance from the earth station to the satellite. Let 30 be the angle of elevation from the earth station to the satellite. If the line segment connecting earth station and satellite substends angle at the centre of earth, then prove that d=$\sqrt { 1+\left( \frac { r }{ R } \right) ^{ 2 }-2\frac { r }{ R } cos\alpha }$ .

14. If $\frac { cos^{ 4 }\alpha }{ { cos }^{ 2 }\beta } +\frac { { sin }^{ 4 }\alpha }{ { sin }^{ 2 }\beta } =1$ prove that $\frac { { cos }^{ 4 }\beta }{ { cos }^{ 2 }\alpha } +\frac { { sin }^{ 4 }\beta }{ { sin }^{ 2 }\alpha } =1$

15. Prove that sin (A + B) sin (A - B) = sin2 A - sin2 B.

16. If A+B+C=$\frac { \pi }{ 2 }$ ,prove the following cos2A+cos2B+cos2C=1+4 sinA sinB sinC

17. Prove that $\frac { cos(2\pi +x)cosec(2\pi +x)tan\left( \frac { \pi }{ 2 } +x \right) }{ sec\left( \frac { \pi }{ 2 } +x \right) cos.cot(\pi +x) }$=1

18. Find the general solution of $\sqrt { 3 }$ sec2x=2

19. Find the value of sin 34° + cos 64° - cos 4°.

20. Find the principal value of sin-1$({\sqrt{3}\over2})$

21. Find the values of cot(660°).

22. Part C

5 x 3 = 15
23. Express the following angles in radian measure
300

24. Find the degree measure corresponding to the following radian measure; $\frac { \pi }{ 3 }$

25. A circular metallic plate of radius 8 cm and thickness 6 mm is melted and molded into a pie ( s sector of the circle with thickness) of radius 16 cm and thickness 4 mm. find the angle of the sector

26. Show that $\cot { \left( 7\frac { 1° }{ 2 } \right) } =\sqrt { 2 } +\sqrt { 3 } +\sqrt { 4 } +\sqrt { 6 }$

27. In a $\triangle$ABC, prove that (b + c) cos A +(c + a) cos B + (a + b) cos C = a + b + c