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#### Important question-chapter 1,2

11th Standard

Reg.No. :
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Maths

Use blue pen Only

Time : 00:50:00 Hrs
Total Marks : 100

Part A

Answer all the questions

10 x 1 = 10
1. Let f:R➝R be defined by f(x)=1-|x|. Then the range of f is

(a)

R

(b)

(1,∞)

(c)

(-1,∞)

(d)

(-∞,1]

2. For real numbers x and y, define xRy if x-y+√2 is an irrational number. Then the relation R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

none of these

3. 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.

4. If f:R➝R is given by f(x)=3x-5, then f-1(x) is

(a)

$\frac{1}{3x-5}$

(b)

$\frac{x+5}{3}$

(c)

does not exist since f is not one-one

(d)

does not exists since f is not onto

5. 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

6. Let X = {1, 2, 3, 4} and R = {(1, 1), (1, 2), (1, 3), (2, 2), (3, 3), (2, 1), (3, 1), (1, 4),(4, 1)}. Then R is

(a)

reflexive

(b)

symmetric

(c)

transitive

(d)

equivalence

7. The rule f(x) =x2 is a bijection if the domain and the co-domain are given by

(a)

R, R

(b)

R, (0, ∞)

(c)

(0, ∞); R

(d)

[0, ∞); [0,∞)

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

(a)

2

(b)

1

(c)

3

(d)

4

9. 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

10. If P(x) = x3 + 3x2 + 2x + 1, then the remainder on dividing p(x) by (x - 1) is

(a)

7

(b)

0

(c)

6

(d)

1

11. Part B

Answer any 10 of the questions

10 x 2 = 20
12. From the curve y=sin x, draw y=sin|x|. (Hint: sin(-x)=-sin x)

13. Show that the relation R on the set A={1,2,3} given by R={(1,1)(2,2)(3,3)(1,2)(2,3)} is reflexive but neither symmetric nor transitive.

14. Let A be the set of all 50 students of class XII. Let f:R➝N be a function defined by f(x)=Roll number of student x. Show that f is one-one but not onto.

15. Let f= {(1, 2), (3,4), (2, 2)} and s = {(2, 1), (3, 1), (4, 2)}. Find gof and fog.

16. If the equations x2 - ax + b = 0 and x2 - ex + f = 0 have one root in common and if the second equation has equal roots, then prove that ae = 2 (b + f).

17. If a3+b3=ab(8-3a-3b),show that log $\left( \frac { a+b }{ 2 } \right) =\frac { 1 }{ 3 }$  (log a +log b)

18. Solve 3|x- 2| + 7 = 19 for x.

19. Solve the following system of linear inequalities 3x-9≥0, 4x-10≤6;

20. Solve $\frac{6-x}{3}<\frac{x}{4}-1$

21. Resolve with partial fractions $\frac{1}{(x^2-1)(x+2)}$

22. Part C

Answer all the questions

10 x 3 = 30
23. Discuss the following relations for reflexivity, symmetricity and transitivity :
On the set of natural numbers, the relation R is defined by "xRy if x + 2y = 1".

24. On the set of natural number let R be the relation defined by aRb if a + b $\le$ 6. Write down the relation by listing all the pairs. Check whether it is transitive

25. Write a description of each shaded area. Use symbols U, A, B, C, U, ∩, ' and \ as necessary.

26. Write a description of each shaded area. Use symbols U, A, B, C, U, ∩, ' and \ as necessary.

27. Draw venn diagram of three sets A, B and C which illustrates the following:
A and B disjoint but both are subsets of C.

28. Find the domain and range of the function f(x)=$\frac { 1 }{ \sqrt { x-5 } }$.

29. If a2+b2=c2, show that a6+b6+3a2b2c2=c6.

30. Find a quadratic polynomial f(x) such that, f(0)=1; f(-2)=0 and f(1)=0.

31. Solve x=$\sqrt{x+20}$ for x∈R

32. Solve ${2x-1\over x}>-1$

33. Part D

Answer all the questions

10 x 5 = 50
34. Let A = {a, b, c, d}, B = {a, c, e}, C = {a, e}.
Show that A ∩ (B ∩ C) = (A ∩ B) ∩ C

35. Find the largest possible domain of the real valued function f(x)=$\frac { \sqrt { 4-{ x }^{ 2 } } }{ \sqrt { { x }^{ 2 }-9 } }$

36. Prove $log\frac { { a }^{ 2 } }{ bc } +log\frac { b^{ 2 } }{ ca } +log\frac { c^{ 2 } }{ ab } =0$

37. Solve the equation $8+9\sqrt{(3x-1)(x-2)}=3x^2-7x.$

38. Evaluate $\sqrt [ 3 ]{ {{{(45.4)}^{2}}\over{{(3.2)}^{2}}\times{(6.5)}^{2}} }$

39. Resolve the following rational expressions into partial fractions.
${{{(x-1)}^{2}}\over{x^3+x}}$

40. Resolve into partial fractions: $\frac{x+1}{x^2(x-1)}$

41. Solve the linear inequalities and exhibit the solution set graphically: x+y ≥ 3, 2x-y ≤ 5, -x+2y ≤ 3.

42. Forensic Scientists use h = 61.4+2.3F to predict the height h in centimeters for a female whose thigh bone (femur) measures F cm. If the height of the female lies between 160 to 170 cm find the range of values for the length of the thigh bone?

43. Determine the region in the Plane determined by the inequalities 3x+4y≤60,x+3y≤30,x≥0,y≥0