RGU Question Paper Mathematics Semester II 2021: Calculus & Differential Equations (New and Old)

June-2021

B.A/B. Sc. (II Semester) Examination

MATHEMATICS

Paper: MATH-121 (OLD & NEW)

(Calculus and Differential Equations)

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Notes: (i) Answer all questions.

(ii) The figures in the margin indicate full marks for the questions.

I. Answer any five parts: 𝟐 × 𝟓 = 𝟏𝟎

(a) Evaluate the limit $\underset{x\to 0}{\lim }(1+x{)}^{1/x}$ .

(b)Discuss the continuity of the function

at 𝑥 = 1 and at 𝑥 = 2.

(c) Discuss the differentiability of the function

(d) What do you mean by order and degree of a differential equation?

(e) State Lagrange’s Mean Value theorem. Give its geometrical interpretation.

(f) Define Tangents and Normal; write the formula for finding their length.

(g) Obtain the reduction formula for the integral

(h) Examine the extreme values of $y=e^{-x}$.

II. Answer any four parts: 𝟓 × 𝟒 = 𝟐𝟎

(a) State Leibnitz theorem. If $y={\tan }^{-1}\left(x\right)$, then show that

(b) Show that the sum of intercepts made on the axes by the tangents at any point on the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is constant.

(c) Find the rectilinear asymptotes of the curve

(d) Trace the curve $x=aco{s}^3\theta , y=asi{n}^3\theta$.

(e) Find the angle of intersection of the curves 

(f) If $y^2=4ax$, prove that

(g) The part of the parabola $y^2=4ax$ bounded by the latus rectum revolves about the tangent at the vertex. Find the surface area of the curved surface of the solid thus generated.

III. Answer any five parts: 𝟏𝟎 × 𝟓 = 𝟓𝟎

(a)(i) Find the infinite expansion of the function

(5)

(ii) Evaluate the integral

using reduction formula. (5)

(b)(i) If $y=cos\left(m{\sin }^{-1}\left(x\right)\right)$, show that

(5)

(ii) Find the radius of curvature at any point on the curve

 (5)

(c)(i) Find the asymptote of the curve $x^3+y^3-3axy=0$. (5)

(ii) Find the radius of curvature of the polar curve 𝑟 = 𝑎(1 − cos 𝜃) at 𝜃 = 𝜋/2. (5)

(d)(i) Find the area bounded by the cardiode 𝑟 = 𝑎(1 − cos 𝜃). (5)

(ii) Find the length of the cycloid 𝑥 = 𝑎(1 + sin 𝜃), 𝑦 = 𝑎(1 − cos 𝜃), measured from the origin. (5)

(e)(i) Describe the method of formation of a differential equation. Hence, form the differential equation of a circle of radius 𝑎. (5)

(ii) Show that 𝑦 = 𝐴 cos 𝑥 + 𝐵 sin 𝑥 is a solution of (5)

Find the value of 𝐴 and 𝐵 if 𝑦 = 0 and 𝑥 = 0 and $\frac{dy}{dx}=1$ at 𝑥 = 0.

(f) (i) Solve the following: (5)

(ii) For the equation 𝑀𝑑𝑥 + 𝑁𝑑𝑦 = 0, show that $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$ is thecondition of exactness. (5)

(g)(i) Describe completely the method of reduction to normal form of differential equations. (5)

(ii) Solve by changing the independent variable (5)

(h)(i) Find the integral factor of the differential equation

where 𝑃 and 𝑄 are functions of 𝑥 alone. (5)

(ii) Solve the following: (5)