# Punjabi University 2008 B.Sc Mathematics - Question Paper

Tuesday, 05 February 2013 07:10Web

B.Sc Mathematics Mathematics (Calculus) Paper-I

**Time :**three Hours] [Ma

**x.**

**Marks :**105

Instructions :

**(**Attempt all ques..

**1)****(**every ques. carries equal marks.

**2)****1.**

**(**State and Prove Leibnitz’s theorem.

**a)**OR

dn

**(**If In = dxn (xn.log

**a)****x)**then Prove that

In = n. In–1 + (n –

**1)**! and Hence deduce

? one 1 1?

In = n ! ? logx + one + two + three + .... + n?

? ?

o :

x then obtain y5 (0).

m

ec2?, y = tan

? then Prove that

n + two + (2n +

**1)**x.yn + one + (n2 – m

**2)**yn = 0.

dn ?logx?

n! ?

1 1?

dxn ? x

? = (–1)n

?

xn + one ? logx – one – 2

– ..... – ?

?

eries ? 1

nP

is divergent for P = one and convergent for P > 1.

**(**State and Prove cauchy root test for series.

**a)****(**explain the convergence (any two) :

**b)**(i)

1 1.3

2 + 2.5 +

1.3.5

2.5.8 +

1.3.5.7

2.5.8.11 + ......

(ii)

x

2.3 +

x2

3.4 +

x3

4.5 + ...........

8 ? one ?

(iii)

?

n = 1

?(n3 + 1)3– n?

**(**obtain he radius of convergence of the power series (any on

**c)****e)**:

(i)

8

?

n = 0

n3 n

3n x

x2 x4 x6

**(**one –

**i****i)**2! +

4! – 6! + ….

**3.**

**(**State and prove Langrange’s mean value theorem.

**a)****(**Attempt any 3 :

**b)****(**If 3a – 4b + 6c – 12 d = 0 then prove that 1 root of the formula

**i)**ax3 + bx2 + cx + d = 0, a ? 0 lies ranging from –1 and 0.

**(**Verify Mean value theorem for f

**i****i)****(**= log x, g

**x)****(**= tan–1x, ? x ? [1, 3]

**x)**4 log 3

Hence prove that three < cot–1 two < 4

**(**obtain the coefficient of x4 in the expansion of log (cos x).

**i****i****i)****(**Expand log x in the increasing powers of (x – 1). Where 0 < x = 2.

**i****v)**lim ?

**(**Evaluate x? 0 (Sec

**v)**x)cot x.

?/2

( n

???n – 1

? ? ?

**4.**

**(**If In = )

**a)**0

x sin x dx then prove that In + n · (n –

**1)**In – two = n · ?2?

**(**Attempt any 2 :

**b)****(**find the limit :

**i)**lim ??

1? ?

2? ? 3?

1

? n?? n

??1 +

? ?1 +

? ?1 +

? …….?1 + ??

n? eight ??

n? ?

n? ? n?

? n??

**(**obtain the length of the arc of the curve x = a (? + sin ?), y = a (1 – cos ?),

**i****i)**0 = ? = 2?.

**(**find the curved surface area of the sphere with radius a.

**i****i****i)****(**obtain the quantity of the solids generated by rotating of the astroid

**i****v)**2

?x? 3

2

?y? 3

? ? + ? ?

=

**1.**about x-axis.

?a?

?b?

**5.**

**(**Show that

**a)**?M

?y =

?N

?x is necessary and sufficient condition for the differential

formula M(x,y) dx + N (x, y) dy = 0 to be exact.

**(**Attempt any 3 :

**b)****(**Solve : xdx + y dy + x dy – y dx = 0

**i)****(**obtain the orthogonal intersecting curves to the curves r2 = c2 cos 2?

**i****i)**? dx?

?dx

? dx

**(**Solve : ?x – y dy?

**i****i****i)**?dy – 1?

= dy

? ? ? ?

**(**Solve : P2 – four P + three = 0 where P = dy dx

**i****v)****(**Solve : (y – px)2 = 4p2 + 1.

**v)**1

1

**6.**

**(**In usual notation, Prove that f

**a)****(**· eax · V = eax · f(D +

**D)****a)**· V

**(**Solve (any three)

**b)****(**(D2 –

**i)****1)**y = x2 · cos x

**(**(D2 + D + 1)2 y = 0

**i****i)****(**(D2 – 5D +

**i****i****i)****6)**y = two · e3x + three · e2x + e6x

**(**x2y'' – 3xy' + 4y = x2 · log x

**i****v)**d2y dy

**(**2x dx2 + dx + 2y = 24x

**v)****7.**

**(**obtain the Radial and Transverse Components of velocity and acceleration of a

**a)**Particle moving in a plane.

OR

**(**State and prove the legal regulations of conservation of energy.

**a)****(**Attempt any 2 :

**b)****(**A gun is mounted on a hill of height h above a level plain. If the greatest horizontal range for provided muzzle velocity V is found by firing at an

**i)**angle of elevation ? then prove that cosec2 ? = two ?1 +

?

gh?

v2 ? .

**(**describe simple harmonic motion and find its formula in the form

**i****i)**x = a cos (pt + ?). Also find its periodic time.

**(**A Particle moves on the curve r = a· e? in such a way that the radial d?

**i****i****i)**component of its acceleration is always zero. Prove that

dt = constant and

the magnitudes of its velocity and acceleration are directly proportional to r.

Earning: Approval pending. |