RGU Question Paper Mathematics Semester I 2022: Calculus

December-2022

BA/B.Sc (I Semester) Examination (CBCS)

MATHEMATICS

Paper: MAT-CC-111

( Calculus)

Full Marks :80

Pass Marks :35%

Time :Three Hours

Note: 1. Answer the questions according to the instruction given in each Section.
2. The figures in the margin indicate full marks for the questions.

Section-A

1. Answer any four of the following questions: 5x4=20

(a) Determine

(b) Define tangent and normal of a curve with suitable diagram. Also write the equation to find the tangent and normal of a curve.

(c) Find the nth differential coefficient of cos(ax+b).

(d) Show that

does not exist.

(e) If $y=(x^2-1{)}^n$, then prove that

(f) Find the length of the curve

from x = 1 to x=2.

Section-B

2. Answer any three of the following questions : 10x3=30

(a) State and prove Taylor's theorem for Lagrange's form of remainder. 10

(b) Trace the cycloid x = a(t + sint), y = a(1 + cost). 10

(c) (i) Find the values of a and b, if

5

(ii) Find the volume obtained by revolving the ellipse

about the x-axis. 5

(d) Find the asymptotes of the cubic curve

10

(e) (i) Find the equation of tangent and normal at any point (x, y) of the curve

5

(ii) Prove that the curvature of the circle $x^2+y^2=a^2$ is constant. 5

Section - C 

3. Answer any two of the following questions: 15x2=30

(a) Expand the following by Maclaurin's series : 4+4+3+4=15

(b) (i) State and prove Leibnitz's theorem. 8

(ii) If $y={\tan }^{-1}\left(x\right)$, then show that

7 

(c) (i) If 

discuss the continuity of f(x) at x=-1. If f(x) is not continuous, then redefine it, if possible. 8

(ii) Trace the curve

7

(d) (i) Find the volume of generated by the curve solid

7

(ii) Find the volume obtained by revolving the ellipse

about the x-axis. 8