B.Sc. Mathematics Honours C-1 Question Paper 2017 (CBCS)

Full Marks: 60 Time: 3 hours

UNIT - I

(Calculus - I)

1. Answer any three questions: $2 \times 3$

(a) What do you mean by asymptote? Does asymptote exist for every curve?
(b) Find the value of $lim_{x\rightarrow1}x^{\frac{1}{x-1}}$.
(c) Define point of inflection of a curve.
(d) Find the envelope of the straight line $\frac{x}{a}+\frac{y}{b}=1$ where the parameters $a$ and $b$ are connected by the relation $ab=c^{2}$.
(e) Write the Leibnitz theorem successive derivatives up to $4^{th}$ order.

2. Answer any one question: $10 \times 1$

(a) (i) If $s$ be the length of the arc $3ay^{2}=x(x-a)^{2}$ measured from the origin to any point $(x, y)$, show that $3s^{2}=4x^{2}+3y^{2}$.
(ii) Show that the curve $y=3x^{5}-40x^{3}+3x-20$ is concave upwards for $-2 < x < 0$ and $2 < x < \infty$ but convex upwards for $-\infty < x < -2$ and $0 < x < 2$. Also show that $x=-2,0,2$ are its points of inflexion.
(b) (i) Trace the curve: $x^{2}y^{2}=a(y^{2}-x^{2})$.
(ii) If $\alpha, \beta$ be the roots of the equation $ax^{2}+bx+c=0$ then show that $lim_{x\rightarrow \alpha}\frac{1-cos(ax^{2}+bx+c)}{(x-\alpha)^{2}}=\frac{1}{2}a^{2}(\alpha-\beta)^{2}$.

UNIT - II

(Calculus - II)

3. Answer any two questions: $2 \times 2$

(a) If $I_{n}=\int_{0}^{\pi/4}tan^{n}xdx,$ $n$ being a positive integer greater than 1, then prove that $I_{n}=\frac{1}{n-1}-I_{n-2}$.
(b) The volume of the solid generated by the revolution of the curve $y=\frac{1}{x}$, bounded by $y=0, x=2, x=b (0 < b < 2)$ about x-axis is 3. Find the value of $b$.
(c) What is the formula for finding area of the curve $y=\psi(t), x=\Phi(t)$, where $t$ is the parameter?

4. Answer any two questions: $5 \times 2$

(a) If $I_{m,n}=\int_{0}^{\pi/2}cos^{m}x~sin~mx~dx,$ then show that $I_{m,n}=\frac{1}{m+n}+\frac{1}{m+n}I_{m-1,n-1}$. Also, deduce that $I_{m,n}=\frac{1}{2^{m+1}}[2+\frac{2^{2}}{2}+\frac{2^{3}}{5}+\cdot\cdot\cdot+\frac{2^{m}}{m}]$.
(b) Find the area of the surface generated by revolving the curves $x=cos~t, y=2+sin~t, 0 \le t \le 2\pi$ about x-axis.
(c) Find the arc length parameter along the curve $C:\vec{r}(t)=(1+2t)\hat{i}+(1+3t)\hat{j}+6(1-t)\hat{k}$ from the point, where $t=0$.

UNIT - III

(Geometry)

5. Answer any three questions: $2 \times 3$

(a) Determine the type of the conic $8x^{2}+10xy+3y^{2}+22x+14y+15=0$.
(b) Show the plane $z-1=0$ which intersects the ellipsoid $\frac{x^{2}}{48}+\frac{y^{2}}{12}+\frac{z^{2}}{4}=1$ is an ellipse. Determine semi-axes.
(c) Find the equation of the sphere through the circle $x^{2}+y^{2}+z^{2}=25, x+2y-z+2=0$ and the point $(1, 1, 1)$.
(d) Find the equation of the cylinder where generators are parallel to the line $x=-\frac{y}{2}=\frac{z}{3}$ and whose guiding curve is the ellipse $x^{2}+y^{2}=1, z=3$.
(e) Find the equation of a right circular cone whose axis is $\frac{x}{1}=\frac{y}{0}=\frac{z}{-2}$ and radius equal to 7.

6. Answer any one question: $5 \times 1$

(a) Find the polar equation of the tangent to the circle $r=2d~cos~\theta$ at the point whose vectorial angle is $\theta_{1}$.
(b) Find the equations of the generating lines of the hyperboloid of one sheet $\frac{x^{2}}{4}+\frac{y^{2}}{9}-\frac{z^{2}}{16}=1$ which passes through the point $(2, 3, 4)$.

7. Answer any one question: $10 \times 1$

(a) (i) Find the locus of a luminous point, if the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ casts a circular shadow on the plane $z=0$.
(ii) Reduce the equation $4x^{2}-4xy+y^{2}-8x-6y+5=0$ to canonical form.
(b) (i) Prove that the locus of the foot of the perpendicular from a focus of the conic $\frac{l}{r}=1-e~cos~\theta$ on a tangent to it, is given by $r^{2}(1-e^{2})-2ler~cos~\theta-l^{2}=0$.
(ii) Prove that the axes of sections of the conicoid $ax^{2}+by^{2}+cz^{2}=1$ which pass through the line $\frac{x}{l}=\frac{y}{m}=\frac{z}{n}$ lie on the cone $\frac{(b-c)}{x}(mz-ny)+\frac{(c-a)}{y}(nx-lz)+\frac{(a-b)}{z}(ly-mx)=0$.

UNIT - IV

(Differential Equation)

8. Answer any two questions: $2 \times 2$

(a) Find the integrating factor of the following differential equation $\frac{dx}{dy}+\frac{xy}{1-y^{2}}-y\sqrt{x}=0$.
(b) Reduce the equation $sin~y\frac{dy}{dx}=cos~x(2~cos~y-sin^{2}x)$ into a linear equation.
(c) Show that the equation $M(x,y)dx+N(x,y)dy=0$ will be exact if $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$.

9. Answer any one question: $5 \times 1$

(a) (i) Show that the substitution $z=ax+by+c$ changes $y^{\prime}=f(ax+by+c)$ into an equation with separable variables, and apply this method to solve the equation $y^{\prime}=sin^{2}(x-y+1)$.
(ii) Reduce the equation $(2x^{2}-1)(\frac{dy}{dx})^{2}+(x^{2}+y^{2}+2xy+2)\frac{dy}{dx}+2y^{2}+1=0$ to Clairaut's form by the substitution $x+y=u$ and $xy-1=v$, hence solve the equation.
(b) (i) Show that if $y_{1}$ and $y_{2}$ be solutions of the equation $\frac{dy}{dx}+Py=Q$ where $P$ and $Q$ are functions of $x$ alone, and $y_{2}=y_{1}z$ then $z=1+ae^{-\int\frac{Q}{y_{1}}dx}$.
(ii) Show that the solution of $\frac{dy}{dx}+Py=Q$ can also be written in the form $y=\frac{Q}{P}-e^{-\int Pdx}[c+\int e^{\int Pdx}d(\frac{Q}{P})]$.