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#### Model Question Paper

11th Standard

Reg.No. :
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Maths

Time : 03:00:00 Hrs
Total Marks : 100

Part  A

Choose the Correct Answer from  given four Alternatives

20 x 1 = 20
1. The function f:[0,2π]➝[-1,1] defined by f(x)=sin x is

(a)

one-to-one

(b)

on to

(c)

bijection

(d)

cannot be defined

2. If the function f:[-3,3]➝S defined by f(x)=x2 is onto, then S is

(a)

[-9,9]

(b)

R

(c)

[-3,3]

(d)

[0,9]

3. Let X={1,2,3,4}, Y={a,b,c,d} and f={f(1,a),(4,b),(2,c),(3,d),(2,d)}. Then f is

(a)

an one-to-one function

(b)

an onto function

(c)

a function which is not one-to-one

(d)

not a function

4. If A = {(x,y) : y = ex, x∈R} and B = {(x,y) : y=e-x, x ∈ R} then n(A∩B) is

(a)

Infinity

(b)

0

(c)

1

(d)

2

5. If A = {(x,y) : y = sin x, x ∈ R} and B = {(x,y) : y = cos x, x ∈ R} then A∩B contains

(a)

no element

(b)

infinitely many elements

(c)

only one element

(d)

cannot be determined

6. Let R be the set of all real numbers. Consider the following subsets of the plane R x R: S = {(x, y) : y =x + 1 and 0 < x < 2} and T = {(x,y) : x - y is an integer} Then which of the following is true?

(a)

T is an equivalence relation but S is not an equivalence relation

(b)

Neither S nor T is an equivalence relation

(c)

Both S and T are equivalence relation

(d)

S is an equivalence relation but T is not an equivalence relation.

7. Let A and B be subsets of the universal set N, the set of natural numbers. Then A'∪[(A⋂B)∪B'] is

(a)

A

(b)

A'

(c)

B

(d)

N

8. For non-empty sets A and B, if A ⊂ B then (A x B) ⋂ (B x A) is equal to

(a)

A ⋂ B

(b)

A x A

(c)

B x B

(d)

none of these.

9. The number of relations on a set containing 3 elements is

(a)

9

(b)

81

(c)

512

(d)

1024

10. Let R be the universal relation on a set X with more than one element. Then R is

(a)

not reflexive

(b)

not symmetric

(c)

transitive

(d)

none of the above

11. The value of loga b logb c logc a is

(a)

2

(b)

1

(c)

3

(d)

4

12. If 8 and 2 are the roots of x2+ax+c=0 and 3,3 are the roots of x2+dx+b=0;then the roots of the equation x2+ax+b = 0 are

(a)

1,2

(b)

-1,1

(c)

9,1

(d)

-1,2

13. If a and b are the roots of the equation x2-kx+c = 0 then the distance between the points (a, 0) and (b, 0)

(a)

$\sqrt { { 4k }^{ 2 }-c }$

(b)

$\sqrt { { k }^{ 2 }-4c }$

(c)

$\sqrt { 4c-{ k }^{ 2 } }$

(d)

$\sqrt { k-8c }$

14. If  $\frac { kx }{ (x+2)(x-1) } =\frac { 2 }{ x+2 } +\frac { 1 }{ x-2 }$ ,then the value of k is

(a)

1

(b)

2

(c)

3

(d)

4

15. If $\alpha$ and $\beta$ are the roots of 2x2 - 3x - 4 = 0 find the value of $\alpha^2+\beta^2$

(a)

$\frac{41}{4}$

(b)

$\frac{\sqrt{14}}{2}$

(c)

0

(d)

none of these

16. If $\alpha$ and $\beta$ are the roots of 2x2+4x+5=0 the equation where roots are 2$\alpha$ and 2$\beta$ is:

(a)

4x2+ 4x + 5 = 0

(b)

2x2 + 4x + 50 = 0

(c)

x2+4x+5=0

(d)

x2+4x+10=0

17. Solve 3x2 + 5x - 2≤0

(a)

(2,$\frac{1}{3}$)

(b)

[2,$\frac{1}{3}$]

(c)

(-2,$\frac{1}{3}$)

(d)

(-2,$\frac{-1}{3}$)

18. If $\frac{1}{\sqrt{3}\times\sqrt{2}}=\sqrt{3}+a$ then a is

(a)

$\sqrt{2}$

(b)

-$\sqrt{2}$

(c)

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

(d)

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

19. If $\pi <2\theta <\frac { 3\pi }{ 2 }$, then $\sqrt { 2+\sqrt { 2+2cos4\theta } }$ equals to

(a)

-2 cosፀ

(b)

-2 sinፀ

(c)

2 cosፀ

(d)

2 sinፀ

20. Which of the following is not true?

(a)

​​​​​​sinፀ = $\frac { -3 }{ 4 }$

(b)

cosፀ = -1

(c)

tanፀ = 25

(d)

secፀ = $\frac { 1 }{ 4 }$

21. Part B

Answer all the Two Mark Questions

10 x 2 = 20
22. If A = { 0, 1, 2, 3, 4, 5, 6, 7 } is a set. Then,

23. Try to write the following intervals in symbolic form:
(i) $\{x:x\in R,-2\le x \le 0 \},$
(ii) $\{ x:x\in R, 0<x< 8 \},$
(iii) $\{ x:x \in R, -8\le -2 \}$
(iv) $\{x:x\in R, -5\le x \le 9 \}$

24. Find two irrational numbers such that their sum is a rational number. Can you find two irrational numbers whose product is a rational number?

25. Find a positive number small than $\frac { 1 }{ { 2 }^{ 1000 } }$ . Justify

26. 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?

27. A plane is 1 km from one landmark and 2 km from another. From the planes point of view the land between them subtends an angle of 450. How far apart are the land marks?

28. Show that cos2 A + cos2 B - 2 cos A cos B cos (A + B) =sin2 (A + B)

29. If $\cos { \left( \alpha -\beta \right) } +\cos { \left( \beta -\gamma \right) } +\cos { \left( \gamma -\alpha \right) } =\frac { -3 }{ 2 }$ then prove that $\cos { \alpha } +\cos { \beta } +\cos { \gamma } =\sin { \alpha } +\sin { \beta } +\sin { \gamma } =0$

30. Part C

Answer all the Three  Mark Questions

10 x 3 = 30
31. Find the pairs of equal sets from the following sets. A = {0}, B = {x: x > 15 and x < 5}, C = {x: x - 5 = 0}, D = {x: x2 = 25}, E = {x: x is an integral positive root of the equation x2 - 2x - 15 = 0}.

32. Which of the following sets are finite and which are infinite?
Set of concentric circles in a plane.

33. If one root of the equation 3x2+kx-81=0 is the square of the other then find k

34. If one root of the equation 2x2-ax+64 = 0 is twice that of the other then find the value of a

35. Prove that sin (270° - $\theta$) sin (90° - $\theta$) - cos (270° - $\theta$) cos (90° + $\theta$) + 1 = 0.

36. Prove that: cos 24° + cos 55° + cos 125° + cos 204° + cos 300°$=\frac{1}{2}$

37. Prove that n!(n+2)=n!+(n+1)!

38. If (n+2)!=60(n-1)! find n.

39. Prove that in the expansion of (1+x)n ,the Co-efficient of terms equidistant from the beginning and from the end are equal

40. Show that the sequence where log a,$log\frac { { a }^{ 2 } }{ b^{ 1 } } log\frac { { a }^{ 2 } }{ { b }^{ 2 } }$  ..is an A.P

41. Part D

Answer all the Five  Mark Questions

6 x 5 = 30
42. Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
Show that A ∩ (B ∩ C) = (A ∩ B) ∩ C

43. A polygon has 90 diagonals. Find the number of its sides?

44. There are 5 teachers and 20 students. Out of them a committee of 2 teachers and 3 students is to be formed. Find the number of ways in which this can be done. Further find in how many of these committees
(i) a particular teacher is included?
(ii) a particular student is excluded?

45. Show that the sum of (m + n)th and (m - n)th term of an A.P is equal to twice the mth term.

46. A man repays an amount of Rs.3250 by paying Rs.20 in the first month and then increases the payment by Rs.15 per month. How long will it take him to clear the amount?

47. Find the equation of a straight line parallel to 2x + 3y = 10 and which is such that the sum of its intercepts on the axes is 15.