Punjabi University 2008 B.Sc Mathematics - Question Paper

B.Sc Mathematics Mathematics (Calculus) Paper-I

Time : three Hours] [Max. Marks : 105


Instructions : (1) Attempt all ques..

(2) every ques. carries equal marks.



1. (a) State and Prove Leibnitz’s theorem.
OR
dn
(a) If In = dxn (xn.log 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 – m2) 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.




(a) State and Prove cauchy root test for series.


(b) explain the convergence (any two) :



(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?


(c) obtain he radius of convergence of the power series (any one) :



(i)

8
?
n = 0

n3 n
3n x

x2 x4 x6

(ii) one –

2! +

4! – 6! + ….




3. (a) State and prove Langrange’s mean value theorem.
(b) Attempt any 3 :

(i) If 3a – 4b + 6c – 12 d = 0 then prove that 1 root of the formula
ax3 + bx2 + cx + d = 0, a ? 0 lies ranging from –1 and 0.

(ii) Verify Mean value theorem for f(x) = log x, g(x) = tan–1x, ? x ? [1, 3]
4 log 3
Hence prove that three < cot–1 two < 4

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

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


lim ?
(v) Evaluate x? 0 (Sec


x)cot x.






?/2
( n


???n – 1

? ? ?

4. (a) If In = )
0

x sin x dx then prove that In + n · (n – 1) In – two = n · ?2?


(b) Attempt any 2 :

(i) find the limit :



lim ??


1? ?


2? ? 3?

1
? n?? n

??1 +

? ?1 +

? ?1 +

? …….?1 + ??

n? eight ??

n? ?

n? ? n?

? n??



(ii) obtain the length of the arc of the curve x = a (? + sin ?), y = a (1 – cos ?),
0 = ? = 2?.

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

(iv) obtain the quantity of the solids generated by rotating of the astroid


2
?x? 3

2
?y? 3

? ? + ? ?

= 1. about x-axis.

?a?

?b?







5. (a) Show that

?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. (b) Attempt any 3 :
(i) Solve : xdx + y dy + x dy – y dx = 0

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


? dx?

?dx

? dx

(iii) Solve : ?x – y dy?

?dy – 1?

= dy

? ? ? ?



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

(v) Solve : (y – px)2 = 4p2 + 1.





1

1

6. (a) In usual notation, Prove that f(D) · eax · V = eax · f(D + a) · V

(b) Solve (any three)

(i) (D2 – 1) y = x2 · cos x

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

(iii) (D2 – 5D + 6) y = two · e3x + three · e2x + e6x

(iv) x2y'' – 3xy' + 4y = x2 · log x

d2y dy
(v) 2x dx2 + dx + 2y = 24x



7. (a) obtain the Radial and Transverse Components of velocity and acceleration of a
Particle moving in a plane.

OR

(a) State and prove the legal regulations of conservation of energy. (b) Attempt any 2 :
(i) 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

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

gh?
v2 ? .


(ii) describe simple harmonic motion and find its formula in the form
x = a cos (pt + ?). Also find its periodic time.

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

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.