Paper 4: Fundamentals of Business Mathematics & Statistic

3.84 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Calculus

SELF EXAMINATION QUESTIONS

Evaluate :

1. (i)

2 1

0

x

2

(^) (ii)^ x 4x^2 + 12 (iii)
2 1
0
x 2x
3
(^) +
(^) [Ans. (i)^12 , (ii) 7, (iii)^73 ]

2. (i)
   1
   0 dx (ii)

3

2 xdx (iii)

(^14)
0 2x dx (iv)
(^42)
0 x dx (v)
6
1
xdx.
(^6)
[Ans. (i) 1, (ii)^52 , (iii)^25 , (iv)^643 , (v)^3512 ]

3. (i) ( )
   1
   0 2x 3 dx+ (ii) ( )

(^12)
(^) − 1 x 1 dx+ (iii) ( )
(^23)
(^) − 2 x 2 dx+
(iv) ( )
(^22)
0 x x 1 dx.− + [Ans.^ (i) 4, (ii)
(^8) ,
3 (iii) 64, (iv)
(^8) ,
3 ]

4. (i)
   7
   6

dx

(^) x 4− (ii)
8
2
dx
(^) 2x 3+ (iii)
2
1
dx
(^) ax b+ (iv) ( )
1
0 2
(^1) dx.
(^) 2x 1+
[Ans. (i) log ,^32 (ii) 2 71 19log (iii) a^1 log 2a b log a b( + )− ( + )^ (iv) 31 ]

5. (i)

9

0

x^1 dx

x

(^) −
(ii)
4 2
0
5 x 3x 7dx.
x

− +

[Ans.^ (i) 12, (ii) 76]

(iii) ( )

2

1 3 /2

x x 2 5 dx

+x− x [Ans. (^) ( )
(^1) 58 42 2
5 − ]
6. (i) 20 e dxx (ii) (^) abe dxmx (iii) 02 e dx.−x [Ans. (i) e^2 – 1, (ii) m^1 (e emb− ma) (iii) 1 – e– 2 ]

7. (i)
   2
   1 logx dx (ii)

2

1 xlogx dx. (iii) ( )

1

0

x log 1 2x dx.+

[Ans. (i) 2 log 2 – 1, (ii) 2 log 2− 43 (iii)^38 log 3]

8. (i)
   e
   1 x logx dx (ii) ( )

1

0 x log x 2 dx.+ [Ans.^ (i)

e 1^2

4 4+ , (ii) 2 log 2−^32 log3+ 43 ]

9. (i)

(^2) x
1 xe dx (ii)
(^1) x
0 xe dx
(iii) 01 x e dx2 x (iv) 12 x e dx.2 x (v) 10 x e dx.2 3x
[Ans. (i) e^2 , (ii) 1, (iii) e – 2, (iv) 2e^2 – e (v) 271 (5e 2^3 − ) ]