Thapar University 2006 M.C.A Mathematics-1 - Question Paper
Thapar Institute of Engineering & Technology
MCA (1st Year)
Final Term exam
MA101 (Mathematics-1)
Thapar Institute of Engineering and Technology School of Mathematics and Computer Applications End Semester Examination- December 15,2006 Mathematics-1 (MA-101)
Time: 3 hours
Max. Marks. 45
Note: Attempt any FIVE questions. Attempt all parts of each question in a sequence at one place. Do write your tutorial group on the top of your answer sheet.
Q1 .(a) If y - sin(a cos"1 -Jx), then prove that
An2-a2 4m+ 2
y**\
. yn )
lim
jt*o
(b) For what values of a, m and b does the function
3, x = 0
-x2 + 3 x + a, mx + 5,
/(*) =
OcjccI
lx2
satisfy the hypothesis of the Mean Value Theorem on the interval [0,2] ?
lies on the curve r = -sin(0/3) and hence find the slope of the curve at this point.
[1 3 7t
(3.S+3.5+2)
Q2. (a) State and prove the integral test.
P 9>
(b) Show that if an converges, then
'l + sin(a)'
converges.
n=l
d z d z
(c) For the function z- /(x, y), transform the equation y+j = 0 into polar
ox dy
coordinate system.
Q3. (a) Find the absolute maximum and minimum of the function
f(x>y) ~ (4x-x2)cosy on the regioni 5 x< 3 and - <y< .
4 4
(3+2+4)
(b) Obtain the relation r = --N, for the torsion x of a curve at any point,
ds
where symbols have their usual meanings.
(c) At time t = 0, a particle is located at the point (1,2,3)- It travels in a straight line to the point (4,1,4), has speed 2 at (1,2,3) and constant acceleration 3i-j+k. Find an equation for the position vector r(t) of the particle at time t.
(2.5+2.5+4)
Q4. (a) Evaluate the following integral by making use of proper substitution
R
where R is the square with vertices (1,0), (2,1),(1,2) and (0,1).
(b) Evaluate the double integral by changing the order of integration
2_ JFx
I
J ydydx.
(4+5)
Q5. (a) Evaluate the line integral J(sin x y)dx COS x dy where C is the triangle
whose vertices are (0,0), -,ojand 1
(b) State tangential form of the Greens theorem in the plane and verify it for the
integral J(x2 - 2xy) dx + (x2y + 3)dy along the boundary of the region bounded c
by y2 =8*and* = 2.
(4+5)
Q6. (a) Solve the differential equation xy(l + xy2) = 1.
dx
(b) Find the solution of the differential equation ydx - xdy + 3x2y2e*1 dx = 0.
(c) Use Taylors formula to find the quadratic approximation of e1 siny at the origin. Estimate the error in the approximation if |x| < 0.1 and \y\ <0.1.
(3.5+2.5+3)
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| Earning: Approval pending. |
