(i) 2- o 0 & 1 )--7

( Roll Number:

Thapar University, Patiala School of Mathematics and Computer Applications

BE (Semester 1st) UMA001: Mathematics- 1 Mid Semester Test September, 2014

Time: 2hrs; MM: 30 Course Coordinator: Dr S S Bhatia

Note: All Questions are compulsory. Attempt all parts of a question in sequence

I. (a)Prove that

(x n (1 - x)n )= n!((1 -x)" -("C1)2 (l xyl-1 x +("C 2)2 (1 x)"-2 x2 + (Ll)nxn)

(b) If y = tan1+ x , then prove that 1-x

x2)1,7+2 + 2(n + pxyn+1 + n(n + 1) y n = 0 . [3+3.5]

X 3 + X 2 2. (a) Sketch the graph of function y = by using

x2 +x

(i) asymptotes and dominant terms (if any) (ii) rise and fall, and concavity. Also indicate the points of maxima, minima, inflection and cusp ( if any).

(b) Iff(x) and g(x) are differentiable on (a, b) and continuous in [a, , then prove that there exists a point c E (a, b ) such that

(b) _- f (a))(r) (g(b) - g(a))1.1(c), g' (x) 0, VX E (a ,b) . [5+2.5]

1 ) tan x

3.(a) Use L' Hospital rule to determine Lim -

as x -> 0. lx

(b) Sketch the region 0 0 < 7r, r = -1 .

(c) Show that x2 > (1 + x)(log(1 + x))2 for all x > 0 . [3+1+3]

4(a) If f is a twice differentiable function of x, then show that curvature of a curve f m(x)

Y- = f(x) at any point is given by K = and hence show that, in an \3/ + [ f (x)1` r 2

x2 u 2 ellipse -F + = I , (a>b) the radius of curvature at an end of the major axis is equal to

a b

( 2 semi-latus rectum b of the ellipse. a

(b) Graph the following polar curves and find all the points of intersections of (i) r = (1 -:cos0) (ii) r = 2cos0 [5+4]

d n dx"