Punjabi University 2008 B.Sc Mathematics - Question Paper
Tuesday, 05 February 2013 07:10Web
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. |