Mathematics Problem Set for Self As-

sessment

This set contains 44 problems spanning the following topics:

1 – Set Theory

2 – Functions

3 – Differentiation

4 – Maxima and Minima

5 – Integration

6 – Progressions

7 – Vectors and Matrices

8 – System of Linear Equations

9 – Theory of Quadratic Equations

10. – Permutations & Combinations

Try to solve all the problems in 3 hours. Check how many of your solutions match the answer set at the

end. If that number is 32 or less, it is strongly recommended that you should attend the remedial Math-

ematics course offered by the institute before the start of the PGP. Please report yourself for the same.

1. What is the number of elements in the set {1, 1, 2, 3, 3.5, 4, 4, 44}?

2. What is the number of elements in the power set of the set {1,2,{1,2}}? Write down that power set.

3. True or false: if A is a subset of B, and B is a proper subset of the sample space, then AC cannot be a subset of B. (Here AC is the complement of the set A.)

4. Simplify to the simplest form: (AUB)C∩(AUC)C

5. If f(x) = x1/2, x>0 then what is f({f-1(f-1(x))}2) where f-1(.) is the inverse function of f(.)?

6. Is f(x) =sin-1(x), 0<x<2 a function? Why or why not?

7. If f(x) = sin(x) and g(x) = cos(x), 0<x<π/2, then what is the relation between f2(x) and g2(x) over the same range?

8. Obtain f(g(x))-g(f(x)) where g(x) = 0.5x and f(x) = x+2.

9. If y=x3, x is a real number, then obtain dxdy

.

10. Obtain ( )−sidx

xnxd e sin(e ) .

11. Obtain xdlog(e ).

dx−

12.If y = e2x+1, x a real number, then obtain ( )2

2d dy

dx

2

dx

y3 2 5 .

− +

13. True or false: if the derivative of a function f(x) does not exist at x=x0, then f(x) cannot be continuous at x0.

14. Obtain all the maxima and minima (whichever exists) of the function 2 12xe +

where x is a real number.

15. True or false: if a function has only one minimum and only one maximum, then the minimum of the function is always less than the maximum.

16. True or false: a function must have either a maximum or a minimum at a point where its derivative is 0.

17. Is the function2x /2f(x) e−= , (x is a real number,) symmetric?

18. Evaluate1

x

0

e dx.−∫ .

19. Evaluate 1

2 x

0

(x e sinx)dx.+ +∫

20. Evaluateπ

∫/2

2

0

xdxsin .

21. Evaluate2

21

xdx.

2 x+∫

22. Obtain the constant c such that 10x0

ce dx 1.∞ − =∫

23. Obtain the sum 2 5 8+ + +Lup to 100 terms.

24. Obtain the infinite sum 2 3(0.3) (0.3) (0.3) .+ + +L

25. The first 12 terms in arithmetic progression has an average of 6. When the 13th term is considered as well, the average becomes 7. Write down the first two terms of the progression.

26. Suppose in the market the word of mouth spreads as follows. If a customer (generation 1) finds a product unsatisfactory he shares it with 10 of his friends (gen-eration 2) and each one of them in turn shares with 10 of their friends (generation 3) and so on. Assuming that at each stage sharing takes place with people who did not

hear about it before how many customers in total would know about it when it reach-es to the 10th generation?

27. Obtain the rank of the following matrix:

1 2 1

2 3 5 .

4 7 7

28. Obtain the determinant of the following matrix:

5 4 3

0 5 4 .

0 0 5

29. Obtain the product of the two matrices given in the problems 27 and 28 respec-

tively, written in that order.

30. Is the matrix obtained in problem 29 invertible? Why or why not?

31. Is the following system of linear equations solvable? Why or why not? 2x+3y = 5 4x+6y = 8

32. How many solution sets does the following system of equations have? Obtain all the solutions. x+y = 5 x2-y2 = 25

33. Solve the following set of linear equations: 5x+4y+3z = 1 5y+4z = 2 2z = 9

34. How many sets of points (x,y), such that x> 0 and y>0, will solve the following: 5x+4y>1, 5y+4x <2?

35. How many sets of points (x,y), such that x> 0 and y > 0, will solve the following: 5x+4y = 1, 5y+4x >2?

36. Find the roots of the quadratic equation 23x 7x 4 0.+ + =

37. Find the sum and the product of the roots of the equation given in problem 36.

38. Find the minimum value of 23x 7x 4+ + over real x.

39. Let c be a real number such that the quadratic equation 3x2+7x+4+c = 0 has two equal roots. What is the value of c?

40. Out of 5 tennis players, how many teams of 4 players could be formed?

41. A coin is tossed (2 outcomes: H and T) and a die is thrown (6 outcomes: 1,2,..,6). How many joint outcomes (like H1, T3 etc.) are possible?

42. A coin is tossed thrice and a die is thrown twice. How many joint outcomes (like HHT14, THT32 etc.) are possible?

43. A bag contains 5 white and 4 black balls. If two balls are drawn one by one not replacing the ball drawn in first drawing find the possible number of groups (i) having 2 white balls and (ii) having 1 white and 1 black ball.

44. In how many different ways can 5 boys be arranged in a straight line so that two specific boys always stand together?

Answers

1. 6.

2.8; {{}, {1},{2},{1,2},{{1,2}},{1,{1,2}},{2,{1,2}},{1,2,{1,2}}}.

3.True.

4. (AUBUC)c or AC∩BC

∩CC.

5.x4.

6. No it is not a function as sin(x) = sin(π-x) for any x such that π/2 < x<2.

7.f2(x)+g2(x) = 1.

8.1.

9.2/31

3yor

21

3x .

10. cosx esinx – excos (ex)

11. -1.

12. ( ) ( ) ( )22 2x 1 2x 1(2x 1) e 1 e 5 logy y 1 y 5.+ ++ − + = − +

13. False: |x| does not have a derivative at 0 but is continuous there.

14. Neither maximum nor minimum exists.

15. False: Consider the function f(x) = ( )( )

( )( )2

2

1 if x 0

1 if x<0

x

x 1 .

1−

− +

+ ≥ +

It is maximum at -1 with value -1 and minimum at 1 with value 1.

16. False: x3 does not have an optimum point at 0.

17. Yes, as f(x) = f(-x), it is symmetric around 0.

18. 1-e-1.

19. 1/3 + e – cos1.

20. π/4.

21. ½log3.

22. 10.

23. 15050.

24. 3/7.

25. -5, -3.

26. 1010

1,111,111, 19

111 .=−

27. 2.

28. 125.

29.

5 14 16

10 23 43 .

20 51 75

30. No, as its rank cannot be more than 2.

31. No. The system is inconsistent as we get 0 = 2 by 2x(i) – (ii).

32. One solution set: x=5, y= 0.

33. x = 0.2, y = 0.24, z = 3.72.

34. Infinitely many.

35. Not a single point.

36. -1,-4/3.

37. Sum: - 7/3, product 4/3.

38. -1/12 (at x=-7/6).

39. 1/12.

40. 5.

41. 12.

42. 288.

43.(i) 10, (ii) 20.

44. 48.