RGU Question Paper Mathematics Semester I 2021: Calculus

 

DECEMBER—2021

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 instructions given.

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

Section—A

1. Answer any four of the following questions : 5×4=20

(a) (i) Obtain the reduction formula for

(ii) What kind of discontinuity the function f (x ) = [x] has at x = 0, where [×] denotes greatest integer function? 2

(b) (i) Show that the function

is discontinuous at x = 0. 3

(ii) Let f (x) and g (x) be continuous functions in [a, b] and f'(x) = g'(x) in [a, b]. Show that in [a, b], f (x ) - g (x) is a constant. 2

(c) (i) Define curvature and radius of curvature of a plane curve. 2

(ii) Show that x = 0 is an asymptote to the curve $y=e^{1/x}$. 3

(d) Find the whole length of the curve

5

(e) If lx + my = 1 is normal to the parabola $y^2=4ax$, then prove that

 5

(f) (i) If m and n are positive integers greater than 1, and

then show that $J_{m,n}=J_{n,m}$. 3

(ii) Define tangent and normal to a curve y = f (x). 2

Section—B

2. Answer any three of the following questions : 10×3=30

(a) (i) Find the entire area of the cardioid $r=a(1+cos\theta )$. 5

(ii) Show that the function f (x ) =|x| is continuous at x = 0 but is not derivable at x = 0. 5

(b) Find the volume and the surface area of the solid generated by revolving the cycloid $x=a(\theta +sin\theta ), y=a(1+cos\theta )$ about its base. 10

(c) Obtain the reduction formula for

Hence, evaluate $\underset{0}{\overset{\mathrm{\pi }/2}{\int }}{\cos }^{6}xdx$. 10

(d) (i) Find the nth derivative of $y=e^{ax}\sin bx$. 5

(ii) If $y=a\cos \left(\log x\right)+b\sin \left(\log x\right)$, prove that

5

(e) State and prove Taylor’s theorem for Lagrange’s form of remainder. 10

Section—C

3. Answer any two of the following questions : 15×2=30

(a) (i) Using $\mathrm{ϵ}-\mathrm{\delta }$ definition, show that the function f (x) = cos x is continuous at any real number. 5

(ii) Show that if a function is derivable at a point, then it must be continuous at that point. 5

(iii) In the mean value theorem

Find the value of $\theta$ if $x=1, h=3$ and $f\left(x\right)=\sqrt{x}$. 5

(b) (i) Find the area of the whole ellipse

10

(ii) Find the radius of curvature at any point $(r,\theta )$ of the cardioid $r=a(1-cos\theta )$ and show that it varies as $r^{1/2}$. 5

(c) (i) State Rolle’s theorem. Verify Rolle’s theorem for the function

in the interval [1, 3]. 5

(ii) Using Maclaurin’s theorem expand $f\left(x\right)=e^x$ in an infinite power series of x. Write the Lagrange’s form of remainder after n terms. 5

(iii) Find the equations of tangent and normal at the point $(\alpha ,\beta )$ on the curve y = alog(sin x). 5

(d) (i) Show that

touches the curve $y=b{e}^{-x/a}$ at the point where the curve crosses the y-axis. 5

(ii) Let f(x) = (x - 1)(x - 2)(x - 3). If a = 0 and b = 4, find the value of c such that

5

(iii) Using the method of integration, find the perimeter of the circle

5