#### Two mark important questions

11th Standard

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Maths

Use Blue Pen only
Time : 00:50:00 Hrs
Total Marks : 50

Part-A

25 x 2 = 50
1. Identify the Quadrant in which a given measure lies; -550

2. Identify the Quadrant in which a given measure lies; 3280

3. 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?

4. If $\triangle ABC$ is a right triangle and if $\angle A=\frac{\pi}{2}$, then prove that
$\sin^2B+\sin^2C=1$

5. If  $\triangle$ABC is a right triangle and if $\angle A$= $\pi/{2}$ , then prove that cos B-cosC =-1+2$\sqrt { 2 } cos\frac { B }{ 2 } sin\frac { C }{ 2 }$

6. Expand: sin(A + B + C).

7. Expand:tan(A + B + C).

8. Find the values of sin 18°.

9. Solve: sinx-cosx+$\sqrt{2}$=0

10. Evaluate: sin-1 $(cos^{-1}{3\over5})$

11. How many numbers are there between 100 and 500 with the digits 0, 1, 2, 3, 4, 5 ? if
(i) repetition of digits allowed
(ii) the repetition of digits is not allowed.

12. How many three-digit odd numbers can be formed using the digits 0, 1, 2, 3, 4, 5? if The repetition of digits is allowed

13. To travel from a place A to place B, there are two different bus routes B1, B2, two different train routes T1  and one air route A1. From place B to place C there is one bus route say B1, two different train routes T'1: T'2 and one air route A1 .Find the number of routes of commuting from place A to place C via place without using similar mode of transportation.

14. By the principle of mathematical induction, prove that for n > 1
$1·2 + 2·3 + .. +n(n + 1)={n(n+1)(n+2)\over 3}$

15. Using the mathematical induction, show that for any natural number n
${1\over 2.5}+{1\over 5.8}+{1\over 8.11}+...+{1\over (3n-1)(3n+2)}={n\over 6n+4}$

16. By the principle of mathematical induction, prove that for n > 1,
$1^2+2^2+3^2+L+n^2>{n^3\over 3}$

17. Prove that using the Mathematical induction $sin (α) + sin\left(\alpha+{\pi\over 6}\right)+ sin\left(\alpha+{2\pi\over 6}\right)+...+ sin\left(\alpha+{(n-1)\pi\over 6}\right)={sin\left[\alpha+{(n-1)^\pi\over 12}\right]\times sin\left(n\pi\over12\right)\over sin\left(\pi\over 12\right)}$

18. A School library has 75 books on Mathematics, 35 books on Physics. A student can choose only one book In how many ways a student can choose a book on Mathematics or Physics?

19. A person wants to buy a car. There are two brands of car available in the market and each brand has 3 variant models and each model comes in five different colours as in figure. In how many ways she can choose a car to buy?

20. Evaluate $\frac { n! }{ r!(n-r)! }$ when n=50, r=47.

21. Let N denote the number of days. If the value of N! is equal to the total number of hours in N days then find the value of N?

22. If $\frac { 6! }{ n! }$ = 6, then find the value of n.

23. What is the unit digit of the sum 2! + 3! + 4! + ...+ 22! ?