RGU Question Paper Mathematics Semester II 2020: Calculus & Differential Equations (New)

 

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BA/BSc (II Semester) Examination

MATHEMATICS

Paper : MATH–121 (New Course)

( Calculus and Differential Equation )

Full Marks : 80

Pass Marks : 35%

Time : Three Hours

Note : 1. Answer all questions.
2. The figures in the margin indicate full marks for the questions.

1. Answer any five questions of the following : 2×5=10 

(a) Find :
$\frac{dy}{dx}$
if $y=(sinx{)}^{\cos x}+(cosx{)}^{\sin x}$

(b) Evaluate :
$\underset{x\to \pi /2}{\lim }(sinx{)}^{\tan x}$

(c) Explain the nodes and cusps with suitable diagrams.

(d) Find the differential equation of the family of curves
$y=e^x(Acosx+Bsinx)$
where $A$ and $B$ are arbitrary constants.

(e) Find the order and the degree of the following differential equations :
(i) $\frac{d^{4}x}{d{t}^4}+\frac{d^{2}x}{d{t}^2}+{\left(\frac{dx}{dt}\right)}^5=e^t$
(ii) $y=\sqrt{x}\frac{dy}{dx}+\frac{k}{\left(\frac{dy}{dx}\right)}$

(f) Find the complementary function for the differential equation
$D^3-4D^2+5D-2=x$

(g) Find the length of the circumference of the curve $x^2+y^2=4$.

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

(a) State and prove Leibniz’s theorem for successive differentiation.

(b) If $y^{1/m}+y^{-1/m}=2x$, prove that
$(x^2-1)y_{n+2}+(2n+1)x{y}_{n+1}+(n^2-m^2)y_n=0$

(c) Find the equation of the tangent and the normal at the point t on the curve $x=a(t+sint)$ and $y=a(1-cost)$.

(d) Prove that the necessary and sufficient condition for the differential equation $Mdx+Ndy=0$, to be exact is
$\frac{\partial M}{\partial x}=\frac{\partial N}{\partial y}$
where $M$ and $N$ are functions of $x$ and $y$.

(e) Write the necessary and sufficient condition for any point $(x,y)$ to be a
multiple point. Examine the nature of the origin on the curve
$x^4-a{x}^2+ax{y}^2+a^2{y}^2=0$

(f) Show that the general solution of the Clairaut’s equation $y=px+f\left(p\right)$ is $y=cx+f\left(c\right)$, where $c$ is an arbitrary constant.

(g) Solve the Lagrange’s equation $y=2px+p^2$.

(h) Find the perimeter of the cardioid $r=a(1-cos\theta )$.

3. Answer any five questions of the following : 10×5=50

(a) Prove that the two solutions $y_1(x)$and $y_2\left(x\right)$ of the equation $a_0\left(x\right)y\text{'}\text{'}\left(x\right)+a_1\left(x\right)y\text{'}\left(x\right)+a_2\left(x\right)y\left(x\right)=0, a_0\left(x\right)\ne 0, x\in (a,b)$ are linearly dependent if and only if their Wronskian is identically zero. 10

(b) (i) If $I_n=\frac{d^n}{d{x}^n}\left(x^n\log \mathrm{x, }\right)$prove that $I_n=n{I}_{n-1}+(n-1)!$.
Hence show that
$I_n=n!\left(\log x+1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)$

(ii) Show that ${\sin }^{p}q+co{s}^{q}p$ attains a maximum when
$\theta ={\tan }^{-1}\left(\sqrt{p/q}\right)$. 5+5

(c) (i) Show that the semi-vertical angle of the cone of maximum volume and of given slant height is ${\tan }^{-1}\left(\sqrt{2}\right)$.

(ii) Evaluate :
$\underset{x\to 0}{\lim }\frac{(1+x{)}^{1/x-e}}{\mathrm{x. \; }}$5+5

(d) (i) Solve
$\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}=e^x\sin x$
by method of variation of parameters.

(ii) In the ellipse
$\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=\mathrm{1,}$
show that the radius of curvature at an end of the major axis is equal to the semi-latus rectum of the ellipse. 5+5

(e) (i) Find the asymptotes of the curve
$y^3-x^{2}y-2x{y}^2+2x^3-7xy+3y^2+2x^2+2x+2y+1=0$

(ii) Let
$I_n=\underset{0}{\overset{\pi /2}{\int }}{\sin }^{n}xdx$.
Then prove that $I_n=\frac{n-1}{n}{I}_{n-\mathrm{2, }}$where $n$ is a positive integer.
Hence deduce that
$I_n=\left\{\begin{array}{ll}\frac{n-1}{n}·\frac{n-3}{n-2}·\frac{n-5}{n-4}···\frac{2}{3}& when n is odd;\\ \frac{n-1}{n}·\frac{n-3}{n-2}·\frac{n-5}{n-4}···\frac{1}{2}·\frac{\pi }{2}& when n is even.\end{array}\right.$
5+5

(f) (i) Evaluate
$\underset{0}{\overset{1}{\int }}{x}^2{\left(1-x^2\right)}^{3/2}dx$
by using reduction formulae.
(ii) Find the length of the asteroid $x^{2/3}+y^{2/3}=a^{2/3}$. 5+5

(g) Solve the differential equations : 5+5
(i) $p^2+2pycotx=y^2, where p=\frac{dy}{dx}$
(ii) $(D^2+a^2)y=secax, where D=\frac{d}{dx}$

(h) (i) Solve :
$y\text{'}\text{'}-4xy\text{'}+(4x^2-1)y=-3e^{x^2}\sin 2x$
(ii) Test for exactness and solve the differential equation
$(1+x^2)y\text{'}\text{'}+4xy\text{'}+2y=se{c}^{2}x, y=0, y\text{'}=1$
when $x=0$. 5+5

(i) Evaluate
$\underset{0}{\overset{1}{\int }}{x}^2{\left(1-x^2\right)}^{3/2}dx$
using reduction formula. 10