RGU Question Paper Mathematics Semester I 2021: Differential & Integral Calculus

DECEMBER—2021

B.Sc (I Semester) Examination

MATHEMATICS

Paper : MAT–GE–001 (CBCS Course)

( Differential and Integral Calculus )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer the questions as per the instruction given in each Section.

2. The figures in the margin indicate full marks for the questions.

Section—A

I. Answer any four parts : 5×4=20

(a) Evaluate 

using reduction formula.

(b) Define discontinuity of a function and its types.

(c) Using ε-δ definition of limit , prove that 

(d) Evaluate 

using reduction formula.

(e) Evaluate 

(f) Find the nth derivative of $\log (x^2-a^2)$

Section—B

II. Answer any three parts : 10×3=30

(a) Evaluate 

where m, n are positive integers.

(b) State Taylor’s Theorem with Lagrange’s form of remainder

(c) (i) Find the Area of the Cardioid r = a(1 − cosθ). 5

(ii) Evaluate 

5

(d) (i) Define Tangent and normal. Find the equation Tangent and Normal. 5

(ii) Equation of Tangent at point (2,3) of the curve $y^3=a{x}^3+b$ is 𝑦 = 4𝑥 − 5. Find the value of a and b. 5

(e) Find the volume of the obtained by the revolution of the area enclosed by the curve $y^2(2a-x)=x^3$.

Section—C

III. Answer any two parts : 15×2=30

(a) (i) State and Prove Rolle’s and Lagrange Mean value Theorem. 10

(ii) Find ‘c’ of the the mean value theorem if f(x) = x(x-1)(x-2) ; a=0, b= ½. 5

(b) (i) Evaluate the continuity of the function

at x = 0, 1 and 2. 8

(ii) Find the Volume of the Solid obtained by revolving the ellipse

about X-axis.

(c) i. State and Prove Euler’s Theorem on Homogeneous function. 5

ii. Verify Euler’s theorem for the function $u=x^4-3x^{3}y+5x^2{y}^2-2y^{4.}$  5

iii. State and Prove Leibnitz’s Theorem. 5