Important question-chapter 3,4

11th Standard

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Maths

Use blue pen Only

Time : 00:50:00 Hrs
Total Marks : 70

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. If cos280+sin280=k3, then cos 170 is equal to

(a)

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

(b)

-$\frac { { k }^{ 3 } }{ \sqrt { 2 } }$

(c)

±$\frac { { k }^{ 3 } }{ \sqrt { 2 } }$

(d)

-$\frac { { k }^{ 3 } }{ \sqrt { 3 } }$

3. $\frac { cos6x+6cos4x+15cos2x+10 }{ cos5x+5cos3x+10cosx }$ equal to

(a)

cos2x

(b)

cosx

(c)

cos3x

(d)

2cosx

4. cos350+cos850+cos1550=

(a)

0

(b)

$\frac { 1 }{ \sqrt { 3 } }$

(c)

$\frac { 1 }{ \sqrt { 2 } }$

(d)

cos 2750

5. 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

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

(a)

2R sinB

(b)

2 sinB

(c)

0

(d)

2a sinB

7. If 2 sinθ+1=0 and $\sqrt{3}$tanθ=1, then the most general value of θ is

(a)

$n\pi\pm\frac{\pi}{6}$

(b)

$n\pi+(-1)^n\frac{7\pi}{6}$

(c)

$2n\pi+\frac{7\pi}{6}$

(d)

$2n\pi+\frac{11\pi}{6}$

8. Area of triangle ABC is

(a)

$\frac{1}{2}$ab cos C

(b)

$\frac{1}{2}$ab sin C

(c)

$\frac{1}{2}$ab cos B

(d)

$\frac{1}{2}$bc sin B

9. The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines.

(a)

6

(b)

9

(c)

12

(d)

18

10. The product of first n odd natural numbers equals

(a)

2nCn$\times$nPn

(b)

${ \left( \frac { 1 }{ 2 } \right) }^{ n }$2nCn$\times$nPn

(c)

${ \left( \frac { 1 }{ 4 } \right) }^{ n }\times$2nCn$\times$2nPn

(d)

nCn $\times$nPn

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. Two vehicles leave the same place P at the same time moving along two different roads. One vehicle moves at an average speed of 60 km/hr and the other vehicle moves at an average speed of 80 km/hr. After half an hour the vehicle reach the destinations A and B. If AB subtends 60 at the initial point P, then find AB.

14. If sin A = $\frac{3}{5}$ and cos B = $\frac{9}{41}$, 0 < A < $\frac{\pi}{2}$, 0 < B < $\frac{\pi}{2}$. Find the value of cos (A - B)

15. if tan2 $\theta$ = 1 - k2 Show that sec $\theta$ + tan3 $\theta$ cosec $\theta$ = (2 -k2)3/2 also find the value of k for which this result holds

16. Prove that cos (A + B) cos C - cos (B + c) cos A = sin B sin (C - A)

17. If tan x = $\frac{n}{n+1}$ and tan y = $\frac{1}{2n+1}$, find tan (x + y).

18. If the slides of $\triangle$ ABC are a = 4, b = 6, C = 8 then show that 4 cos B + 3 cos C = 2

19. Find the values of sin (-45°).

20. Express each of the following product as a sum or difference.cos 110° sin 55°.

21. Part C

5 x 3 = 15
22. Express the following angles in radian measure  = 1350

23. Find the degree measure corresponding to the following radian measure; $\frac { 2\pi }{ 5 }$

24. Find the values of cos 2A, A lies in the first quadrant, when cos A = $\frac{15}{17}$

25. Prove that $\tan { \left( \frac { \pi }{ 4 } +\theta \right) } -\tan { \left( \frac { \pi }{ 4 } -\theta \right) } =2\tan { 2\theta }$

26. Find the value of $sin\left( 22\frac { 1^{ 0 } }{ 2 } \right)$

27. Part D

5 x 5 = 25
28. Prove that$\frac { sin4x+sin2x }{ cos4x+cos2x } =tan3x$

29. Prove that $\frac { sin(4A-2B)+sin(4B-2A) }{ cos(4A-2B)+cos(4B-2A) } =tan(A+B)$

30. In $\triangle$ABC, Prove the following
$\frac { asin(B-C) }{ { b }^{ 2 }-{ c }^{ 2 } } =\frac { bsin(C-A) }{ { c }^{ 2 }-{ a }^{ 2 } } =\frac { csin(A-B) }{ { a }^{ 2 }-{ b }^{ 2 } }$

31. Express each of the following as a product.
cos 65o + cos 15o

32. Suppose two radar stations located 100 km apart, each detect a fighter aircraft between them. The angle of elevation measured by the first station is 30°, whereas the angle of elevation measured by the second station is 45°. Find the altitude of the aircraft at that instant.