Business Mathematics 1

ASSIGNMENT – A

1. (A) Simplify:

[(a+x)^1/2 + (a-x)^1/2] / [(a+x)^1/2 – (a-x)^1/2] = a / x 1^1/2

 

B. Prove by way of Mathematical Induction :

1³+2³+3³+4³+ – – – – – – – – – – – – – – – – – + n³ = [n²(n+1)²] / 4 1^1/2

 

2. Prove the following Trigonometric Identity: 3

2(Sin⁶A + Cos⁶A) – 3(Sin⁴A + Cos⁴A) + 1 = 0

 

3. Draw the Graph of the following In-Equations and find the Common Feasible s Solution along with the Vertices :

x+y <= 4 ; 3x+y >= 3 ; x+4y >= 4 ; x <= 3 ; y >= 2 3

4. (A) Show whether the following is a ‘Tautology’ or a ‘Contradiction’ ?

((p ® q) Ù (q ® r)) ® (p ® r) 1^1/2

 

B. For what values of ‘k’ the terms :

1 + k , 5/6 + k , 13/18 + k are in Geometric Progression ? 1^1/2

 

5. Find dy/dx if : 3

(tan^-1x)^y + (y)^cotx = 1

 

6. Prove by applying the properties of Determinants : 3

 

x y z 1 1 1

x² y²z² = x² y² z² = (y – z)(z – x)(x – y)(yz+zx+xy)

yz zx xy x³ y³ z³

<strong>ASSIGNMENT – B</strong>

Answer any Three questions

&nbsp;

&nbsp;

1. (A) If a , b , c are in Arithmetic Progression prove : 2^1/2

1 / (b^1/2+c^1/2) , 1 / (c^1/2+a^1/2) , 1 / (a^1/2+b^1/2) are also in A.P.

&nbsp;

(B) Integrate the following function : 2^1/2

(x) / [(x+1)(x-1)³]

&nbsp;

2. The angle of elevation of a jet plane from a point A on the ground is 60^· . After a f flight of 15 seconds, the angle of elevation changes to 30^·. If the jet plane is f l flying at a constant height of 1500Ö3m, find the Speed of the jet plane. 5

&nbsp;

3. (A) Kamla borrowed from Ratan a certain sum at certain rate for two years

Simple Interest. She lent this sum at the same rate to Hari for two 5

years Compound Interest. At the end of two years, she received

Rs. 210 as compound Interest, But paid Rs. 200 only as a Simple

Interest. Find the Sum and the Rate of Interest.

&nbsp;

4. (A) Solve the following System of Equations in Matrix Form : 2^1/2

2/x + 3/y + 10/z = 4

4/x – 6/y + 5/z = 1

6/x + 9/y – 20/z = 2

&nbsp;

B. A survey has been taken on methods of computer travel. Each respondent was

Asked to check BUS, TRAIN, or AUTOMOBILE as a major method of traveling to work. More than one answer was permitted. The results reported was as follows : BUS – 30 people; TRAIN – 35 people; AUTOMOBILE – 100 people; BUS and TRAIN – 15 people; BUS and AUTOMOBILE – 15 people; TRAIN and AUTOMOBILE – 20 people; and all three methods – 5 people. How many people completed the survey form. 2^1/2

<strong>ASSIGNMENT – C</strong>

OBJECTIVE TYPE QUESTIONS

&nbsp;

1. If in a G.P. a = 4 , r = 5 then S₆ = ?

a. 15265 b. 15625

c. 15562 d. 15225

&nbsp;

2. The Disjunction of ‘p’ and ‘q’ is true if :

a. Both p and q are true

b. When p is true and q is false

c. When p is false and q is true

d. When p or q is true

&nbsp;

&nbsp;

3. Value of the Determinant Cosx Sinx is :

-Sinx Cosx

a. Cos²x – Sin²x b. 0

b. 1 d. Cosx + Sinx

&nbsp;

4. Derivative of x + (1/x) w.r.t x is equal to :

a. 1 – x b. 1 – 1/x²

b. 1 + (1/x) d. 1 + 1/x²

&nbsp;

5. Sum of money will double itself at 10% per year Simple Interest in :

a. 5 yrs b. 2 yrs

c. 20 yrs d. 10 yrs

&nbsp;

6. In an In-Equation x &lt;= 2; solution for x is = ?

a. All real values less than 2.

b. Value equal to 2.

c. All real values less than and equal to 2.

d. All real values other than 2.

&nbsp;

7. If a + b + c = 0, value of x^{(a.a) / bc} . x^{(b.b) / ac}. x^{(c.c) / ab} is = ?

a. x b. x³

b. x⁰d. x^abc

&nbsp;

8. If A={1,2,3,4} , B={2,4,6,8} , C={3,4,5,6} then A’ÈB is equal to :

a. AÇB’ b. A’ÈB’

c. (AÈB)’ d. (AÇB’)’

&nbsp;

&nbsp;

&nbsp;

9. If the two Linear Equations :

(k – 3)x + 3y – k = 0 and kx + ky = 12 has Infinite Many Solutions, then value of k will be = ?

a. k = 0 b. k = 6

c. k = -6 c. None of the Above

&nbsp;

10. If the value of (7+3Ö5) / (3+Ö5) is equal to (3+Ö5) / 2 then the value of

(7-3Ö5) / (3-Ö5) is equal to :

a. (3+Ö5) / 2 b. 3Ö5

c. (3-Ö5) / 2 d. 7Ö5

&nbsp;

11. What will be the value of A Å B if

A = 1 1 0 B = 1 0 0 0

0 1 0 0 1 1 0

1 1 0 1 0 1 1

0 0 1

&nbsp;

a. 1 1 1 0 b. 1 1 1 0

0 1 1 0 0 1 1 0

1 1 1 0 1 0 1 1

1 0 1 1 1 0 1 1

&nbsp;

&nbsp;

&nbsp;

1 0 1 1 1 0 1 1

c. 1 1 1 0 d. 0 1 0 1

0 0 1 0 1 1 1 0

1 0 1 1 1 0 1 1

&nbsp;

&nbsp;

&nbsp;

&nbsp;

12. A tower is 100Ö3m high. What will be the angle of elevation if its top from a point is 100m away from its foot.

a. 30 b. 0

c. 90 d. 60

&nbsp;

13. If U be the set of all triangles in a plane, and A is the set of all triangles with

at least one angle different from 60 then A’ is :-

a. All triangles in a plane

b. Set of all equilateral triangles

c. Set of all triangles except equilateral triangle

d. None of the above

&nbsp;

14 Integration of Cosec(ax+b)Cot(ax+b) is = ?

a. -Cosec(ax+b) b. -Cot(ax+b)

c. -[Cosec(ax+b)] / (ax+b) d. -[Cosec(ax+b] / a

&nbsp;

&nbsp;

15 Sum of the Sequence 15+24+33+42+51+60+69+78+87+96 is

a. 500 b. 505

c. 550 d. 555

&nbsp;

16 In a quadratic equation if b² &lt; 4ac the roots are :

a. Real b. Equal

c. Imaginary d. Unequal

&nbsp;

17. Value of Cos1.Cos2.Cos3.Cos4 . . . . . . . . . . . . . . . . . . . . Cos180 is

a. 0 b. 1

c. 1/Ö2 d. Ö3/2

&nbsp;

18. If any two rows or columns of a Determinant are identical, the Value of the D e Determinant becomes

a. 1 b. 0

c. No Change d. -1

&nbsp;

19. 101th term of an A.P. of the function [n(n+1)] / (2n-1) is :

a. 50.36 b. 50.24

c. 50.45 d. 50.13

&nbsp;

20. In a group of 65 people, 40 like Cricket, 10 like both Cricket &amp; Tennis. How many of them like only Tennis and not Cricket?

a. 25 b. 35

c. 65 d. 40

&nbsp;

21. Do you speak English?

a. Is a Statement.

b. Is a Declarative Sentence and Statement.

c. Is a Declarative Sentence but not a Statement.

d. None of the Above.

&nbsp;

22. Differentiation of Cos³x is

a. -3Cos²xSinx b. 3Cos²x

c. 3Cos²xSinx d. -3CosxSinx

&nbsp;

23. Simple Interest on Rs. 650 for 8 Months at the rate of 5 paise per rupee per month is

a. 65 / 3 b. 260

c. 280 d. 360

&nbsp;

&nbsp;

24. For an A.P., a1,a2,a3,a4,- – – – – – – – – -if a4 / a7 = 2/3 then a6 / a8 will be :

a. 3/2 b. 4/6

c. 5/4 d. 4/5

&nbsp;

&nbsp;

25. Value of -1⁵ . 1⁷ . -1^-3 . 1^-8 is :

a. 0 b. 1

c. -1 d. Infinity

&nbsp;

26. A software firm provides the following three characteristics for each program that it sells :

Language : FORTRAN (f) ; PASCAL(p) ; LISP(l)

Memory : 2meg(2) ; 4meg(4) ; 8meg(8)

Operating System: UNIX (u) ; DOS (d)

The number of Categories in this Classification is :

a. 18 b. 19

c. 7 d. None of the Above

&nbsp;

27. For Real and Equal Roots the value of the Determinant :

a. D &gt; 0 b. D&lt; 0

c. D != 0 d. D = 0

&nbsp;

28. For what values of ‘a’ does the Linear Equation :

ax + 8y + 6 = 0 &amp; 2x + (a-1)y – 7 = 0 have Unique Solution?

a. Values 4, 17, -4

b. Values between -4 and 17

c. Values other than 4, -4, 17

d. None of the Above

&nbsp;

29. If A = {a, b, c, d} and R be the Relation on A, whose Matrix is : 1 0 0 0

then R will be: 0 1 0 0

a. {(a,a),(a,b),(a,c)(a,d),(b,b),(c,c),(d,d)} 1 1 1 0

b. {(a,a),(a,c),(b,b),(b,c),(c,c),(d,d)} 0 1 0 1

c. {(a,a),(a,c),(b,b),(b,c),(b,d),(c,c),(d,d)}

d. {(a,a),(a,c),(b,b),(b,c),(b,d),(c,c),(c,d),(d,d)}

&nbsp;

30. Which term in the given Sequence is wrong?

10, 20, 40, 70, 120, 160

a. 40 b. 120

c. 160 d. 70

&nbsp;

31. A Sum of Rs. 2000 is invested at 16% rate of interest per annum. What will be the Amount after 2 years if the interest is Compounded Semi – annually?

a. Rs.2361.20 b. Rs.2163.20

c. Rs.2136.20 c. Rs.2613.20

&nbsp;

&nbsp;

32. If A = {a, e, i, o, u}

a. a Ë A b. a E A

c. a Í A c. None of the Above

&nbsp;

&nbsp;

33. Integration of e^5x

a. 5e^5x b. 5e^4x

c. e^5x / 5 d. e^5x

&nbsp;

34. If the Graph of two Linear Equations represents a Single Line then the Solution for the graph is

a. Consistent b. Inconsistent

c. Unique d. None of the Above

&nbsp;

35. Value of 2 Sin²q + Cos²q is = ?, where 45º

a. 3 / 2 b. 2

c. 2 / 3 d. 3 / Ö2

&nbsp;

36. If the difference between Simple and Compound Interest on a Sum of Money for 4 years at 5% per annum is Rs. 150 the Sum will be:

a. Rs.9637.52 b. Rs.9763.25

c. Rs.9367.25 d. Rs.9673.52

&nbsp;

37. If A = 5 4 A^-1 will be equal to

7 6

&nbsp;

&nbsp;

a. 5 7 b. -3 7/2

4 6 2 -5/2

&nbsp;

&nbsp;

c. 3 7/2 d. 3 -7/2

2 5/2 -2 5/2

&nbsp;

38. Differentiation of tan²x w.r.t. cos²x is:

a. tan²xcos²x b. sec⁴x

c. -sec⁴x d. tan2x/cos²x

&nbsp;

39. Value of (81 / 256)^-5/4is:

a. 3 / 2 b. -1024 / 243

c. 1024 / 243 d. -3/2

&nbsp;

40. Integration of (2x-3)²

a. 4/3x³-6x²+9x

b. 4/3x²-6x+9

c. 3/4x³-6x²+9x

d. 3/4x³-6x²+9