POKHARA UNIVERSITY Exam Model Question 2079
FACULTY OF SCINCE AND Level B. E. FM 100
TECHNOLOGY Programme Bachelor PM 45
SCHOOL OF ENGINEERING Year/Part 1st year/1st semester Time 3 Hrs
Subject: Calculus I
Candidates are required to give answers in their own words as far as practicable.
The figure in the margin indicates full marks.
Attempt all the questions
x 2 + 1 for x 1
1. a. Show that: f(x)= is continuous at x = 1 but not differentiable at x = 1. 5
3x + 1 for x 1
b. If (
y = sin −1 x ) 2
, prove that (i) (1 − x 2 ) y2 − xy1 − 2 = 0
(ii )(1 − x 2 ) y n + 2 − (2n + 1) xyn +1 − n 2 y n = 0
5
c. State the Roll’s theorem .Verify the Roll’s Theorem y= f(x) =x 2 -4x+3 on [1,3] 4
x 2 x3 x 4
2. a. Show that: log(1+x)= x − + − + ....................... by expanding Maclaurin’s theorem . 5
2 3 4
b. Find the asymptotes of the curves, x2(x-y)2 - a2(x2+y2)=0 5
c. Find the radius of curvature of the curves y2 = 4x at the vertex (0,0). 4
dx
3. a. Integrate, 2 + cos x + sin x 5
1
cot
−1
b. Show that: (1 − x + x 2 )dx = − log 2 5
0
2
5
1
c. Show that: x 1 − x 2 dx = 4
6
0
256
4. a. Find the area of the region and the circle x2+y2=4 cut off by the line x-2y =-2 in the first two
quadrants. 5
b. Find the volume of the solid generated by revolving the region in the first quadrant bounded on
the left by the circle x2+y2 =3 and on the right by the line x = 3 and above the line y = 3
about y-axis. 5
c. Let U = f(x , y, z)be a homogeneous function of three independent variables x, y, and z of degree
u u u
n. then show that x +y +z = nu 4
x y z
dy y y
5. a. Solve: + log y = 2 (log y ) 2 5
dx x x
3
b. Solve the given initial value problem: x y"−2 xy'+2 y = 0, y (1) = y ' (1) = 1 .
2
, 5
2
c. Solve by using the method of variation of parameters, y”+4y=3cosec2x. 4
6. a. Use Lagrange’s multiplier to find the minimum value of x 2+y2+z2 subjects to constraint
a x+ by+ c z = p. 7
b. A tank initially contains 4lb of salt dissolved in 100 litre of water. Suppose that salt solution 2lb
of salt per litre is allowed to enter the tank at the rate of 5 lire /min and the uniform solution is
drained from the tank at the same rate .find the amount of salt in the tank after 10 minutes. 7
7 . Attempt all question. 8 2 = 16
a. Sketch the graph of parametric equations
x= t2 - 2t y = t+1 for 0t 4
u u u 3
b. If u = log (x3+y3+z3-3xyz) then prove that: + + =
x y z x + y + z .
c. Find the arc length of the curves y = x2 ; − 1 t 2 .
dy y y
d. Solve: = + tan
dx x x.
e. Find the particular integral of y”+4y =2sin2x.
f. Find the area enclosed by x-axis and the curve y = 3x - 5x2.
sin −1 x
1
g. Evaluate:
0 1 − x2
.
dx
h. What are the condition of saddle point and undecided information?
Best of Luck The End
