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Business Mathematics 1

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SKU: AMSEQ-030 Category:

ASSIGNMENT – A

1. (A) Simplify:

[(a+x)<sup>1/2</sup> + (a-x)<sup>1/2</sup>] / [(a+x)<sup>1/2</sup> – (a-x)<sup>1/2</sup>] = a / x 1<sup>1/2</sup>

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B. Prove by way of Mathematical Induction :

1<sup>3</sup>+2<sup>3</sup>+3<sup>3</sup>+4<sup>3</sup>+ – – – – – – – – – – – – – – – – – + n<sup>3</sup> = [n<sup>2</sup>(n+1)<sup>2</sup>] / 4 1<sup>1/2</sup>

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2. Prove the following Trigonometric Identity: 3

2(Sin<sup>6</sup>A + Cos<sup>6</sup>A) – 3(Sin<sup>4</sup>A + Cos<sup>4</sup>A) + 1 = 0

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

x+y &lt;= 4 ; 3x+y &gt;= 3 ; x+4y &gt;= 4 ; x &lt;= 3 ; y &gt;= 2 3

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

((p ® q) Ù (q ® r)) ® (p ® r) 1<sup>1/2</sup>

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B. For what values of ‘k’ the terms :

1 + k , 5/6 + k , 13/18 + k are in Geometric Progression ? 1<sup>1/2</sup>

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5. Find dy/dx if : 3

(tan<sup>-1</sup>x)<sup>y</sup> + (y)<sup>cotx</sup> = 1

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6. Prove by applying the properties of Determinants : 3

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x y z 1 1 1

x<sup>2</sup> y<sup>2 </sup>z<sup>2</sup> = x<sup>2</sup> y<sup>2</sup> z<sup>2</sup> = (y – z)(z – x)(x – y)(yz+zx+xy)

yz zx xy x<sup>3</sup> y<sup>3</sup> z<sup>3</sup>

<strong>ASSIGNMENT – B</strong>

Answer any Three questions

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

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

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(B) Integrate the following function : 2<sup>1/2</sup>

(x) / [(x+1)(x-1)<sup>3</sup>]

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

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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.

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

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

4/x – 6/y + 5/z = 1

6/x + 9/y – 20/z = 2

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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<sup>1/2</sup>

<strong>ASSIGNMENT – C</strong>

OBJECTIVE TYPE QUESTIONS

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1. If in a G.P. a = 4 , r = 5 then S<sub>6</sub> = ?

a. 15265 b. 15625

c. 15562 d. 15225

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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

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3. Value of the Determinant Cosx Sinx is :

-Sinx Cosx

a. Cos<sup>2</sup>x – Sin<sup>2</sup>x b. 0

b. 1 d. Cosx + Sinx

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

a. 1 – x b. 1 – 1/x<sup>2</sup>

b. 1 + (1/x) d. 1 + 1/x<sup>2</sup>

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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

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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.

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7. If a + b + c = 0, value of x<sup>(a.a) / bc</sup> . x<sup>(b.b) / ac</sup>. x<sup>(c.c) / ab</sup> is = ?

a. x b. x<sup>3</sup>

b. x<sup>0 </sup>d. x<sup>abc</sup>

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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’)’

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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

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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

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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

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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

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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

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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

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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

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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

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

a. 500 b. 505

c. 550 d. 555

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16 In a quadratic equation if b<sup>2</sup> &lt; 4ac the roots are :

a. Real b. Equal

c. Imaginary d. Unequal

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

a. 0 b. 1

c. 1/Ö2 d. Ö3/2

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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

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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

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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

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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.

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22. Differentiation of Cos<sup>3</sup>x is

a. -3Cos<sup>2</sup>xSinx b. 3Cos<sup>2</sup>x

c. 3Cos<sup>2</sup>xSinx d. -3CosxSinx

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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

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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

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25. Value of -1<sup>5</sup> . 1<sup>7</sup> . -1<sup>-3</sup> . 1<sup>-8</sup> is :

a. 0 b. 1

c. -1 d. Infinity

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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

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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

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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

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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)}

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30. Which term in the given Sequence is wrong?

10, 20, 40, 70, 120, 160

a. 40 b. 120

c. 160 d. 70

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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

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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

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33. Integration of e<sup>5x</sup>

a. 5e<sup>5x</sup> b. 5e<sup>4x</sup>

c. e<sup>5x</sup> / 5 d. e<sup>5x</sup>

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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

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35. Value of 2 Sin<sup>2</sup>q + Cos<sup>2</sup>q is = ?, where 45º

a. 3 / 2 b. 2

c. 2 / 3 d. 3 / Ö2

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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

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37. If A = 5 4 A<sup>-1</sup> will be equal to

7 6

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a. 5 7 b. -3 7/2

4 6 2 -5/2

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

2 5/2 -2 5/2

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38. Differentiation of tan<sup>2</sup>x w.r.t. cos<sup>2</sup>x is:

a. tan<sup>2</sup>xcos<sup>2</sup>x b. sec<sup>4</sup>x

c. -sec<sup>4</sup>x d. tan2x/cos<sup>2</sup>x

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39. Value of (81 / 256)<sup>-5/4 </sup>is:

a. 3 / 2 b. -1024 / 243

c. 1024 / 243 d. -3/2

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40. Integration of (2x-3)<sup>2</sup>

a. 4/3x<sup>3</sup>-6x<sup>2</sup>+9x

b. 4/3x<sup>2</sup>-6x+9

c. 3/4x<sup>3</sup>-6x<sup>2</sup>+9x

d. 3/4x<sup>3</sup>-6x<sup>2</sup>+9

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