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Section-A

1. a) Solve the inequality |x| + |x – 1| > 1 5

b)

x

x

Evaluate l im 8x(

1+ x

X:((

 2. a) b) Find the values of a and b so that function differentiable at x = 1 If y = (Tan x)Cot x + (Cot x)Tan x find dy dx . f (x) = x3 ax + b if if x < 1 x = 1 is continuous and 5 5 3. a) A 6 feet tall man is walking towards a lamp post 16 feet high at a speed of 5 ft/sec. At what rate is the tip of shadow moving, at what rate is the length of shadow moving? 5

dn l nx (-1)n

= dxn (

x x

X (:

n! -1 -1 -1

b) Show

[l nx -1

]

n+1

3 …….

2

n

3

x+y3

if (x,y) ? (0, 0)

4. a) If f (x, y) = x2+y2 The evaluate f x(0, 0), f y(0, 0) if possible.

(x,y) =(0,0)

0 b) Using mean value theorem to show that |Tan x + Tan y| = |x + y| for all real x and y in the interval

-2 ,2[

ð ð

]

5. a) Find by Maclaurin’s formula the 1st four terms of expansion of f (x) = e ax Cos bx and write the remainder after n terms.

X

:

1/x 1/x x

a +b

l im

b) Use L’Hospital’s rule to prove that = ab,a>0, b>0

x(

2

5

8

ð

Section-B

22

x-a

6. a) Evaluate ? 4 dx. 5

xb) Evaluate ?sec3x dx. 5

7. a) Integrate the following w.r.t. x

5

3

1 + x

b) Cosx

1 -Cosx 5

ð/2

-

8. a) Show that

ln Sinx dx= 2ln2.

1

pq

b) IfIn=? x(1 –x)ndx whereÞ,q,n arepositive.Show(qn + Þ+ 1)In= qn In-1.Wherenis+ve integer.

*** B.A/B.Sc-I (09/A) xxiii ***

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