#### Important 5 mark questions paper

11th Standard

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Maths

Use blue pen Only

Time : 00:50:00 Hrs
Total Marks : 75

Part A

15 x 5 = 75
1. Find all the angle between 0o and 360o which satisfy the equation $\sin ^{ 2 }{ \theta } =\frac { 3 }{ 4 }$

2. Show that $\sin ^{ 2 }{ \frac { \pi }{ 18 } } +\sin ^{ 2 }{ \frac { \pi }{ 9 } } +\sin ^{ 2 }{ \frac { 7\pi }{ 18 } } +\sin ^{ 2 }{ \frac { 4\pi }{ 9 } } =2$

3. Show that$\frac { sin8x\quad cosx-sin6x\quad cos3x }{ cos2x\quad cosx-sin3x\quad sin4x } =tan2x$

4. Show that $\frac { (cos\theta -cos3\theta )(sin8\theta +sin2\theta ) }{ (sin5\theta -sin\theta )(cos4\theta -cos6\theta ) } =1$

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

6. Prove that $\frac { cot(180+\theta )sin(90-\theta )cos(-\theta ) }{ sin(270+\theta )tan(-\theta )cosec(360+\theta ) } ={ cos }^{ 2 }\theta cot\theta$

7. Find the values of other five trigonometric functions for the following
Cos $\theta$ = -$\frac { 1 }{ 2 }$ $\theta$ lies in the III quadrant

8. In $\triangle$ABC, 60° prove that b + c = 2a cos $\left( \frac { B-C }{ 2 } \right)$

9. In $\triangle$ABC, Prove the following a sin $\left( \frac { A }{ 2 } +B \right)$=(b+c) sin $\frac { A }{ 2 }$

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

11. In a $\triangle$ABC, if $\frac { sin\quad A }{ sin\quad C } =\frac { sin(A-B) }{ sin(B-c) }$,prove that a2,b2,c2are in arithmetic progression

12. If A + B + C = $\pi$, prove the following
i. cos A + cos B + cos C = 1 + 4 sin $({A\over 2})$ sin $({B\over 2})$ sin $({C\over 2})$
ii. sin $({A\over 2})sin({B\over2})sin({C\over 2})\le{1\over 8}$
iii. 1 < cos A + cos B + cos C $\le\frac{3}{2}$

13. Solve$\sqrt{3}$  sin $\theta$ - cos $\theta$ =$\sqrt{2}$

14. In a triangle  ABC, prove that ${a^2+b^2\over a^2+c^2}={1+cos(A-B)cos C\over 1+cos (A-C)cos B}$

15. 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.