B.Sc. Mathematics Honours C-1 Question Paper 2018 (CBCS)

Full Marks: 60 Time: 3 Hours

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

(a) Find the range of values of $x$ for which $y=x^{4}-6x^{3}+12x^{2}+5x+7$ is concave upward.
(b) If $n$ be any positive integer, find the value of $lim_{x\rightarrow n}\frac{x-n}{sin~\pi x}$.
(c) If $y=2cos~x(sin~x-cos~x)$ then find the value of $(y_{20})_{0}$.
(d) Find the asymptotes, if any of the curve $y=log~sec(x/a)$.
(e) Show that abscissa of the points of inflexion on the curve $y^{2}=f(x)$ satisfying $[f(x)]^{2}=2f(x)f''(x)$.

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

(a) (i) If $y=sin(m~cos^{-1}\sqrt{x})$ then prove that $lim_{x\rightarrow0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2}$.
(ii) Find all the asymptotes of the curve $x^{3}-2x^{2}y+xy^{2}+x^{2}-xy+2=0$.
(iii) If $f(x)=ax^{3}+3bx^{2}$. Find $a$ and $b$ so that $(1, -2)$ is a point of inflexion of $f$.
(b) (i) Trace the curve $x=a(\theta+sin~\theta)$, $y=a(1-cos~\theta)$.
(ii) Find the values of $a$ and $b$ so that $lim_{x\rightarrow0}\frac{a~sin2x-b~sin~x}{x^{3}}=1$.
(iii) Obtain the envelope of the circle drawn upon the radii vectors of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ as diameter.

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

(a) Find the entire area enclosed by the curve $r=a~cos~2\theta$.
(b) Obtain reduction formula for $\int cosec^{n}x dx$.
(c) Show that in the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$, $s \propto x^{\frac{2}{3}}$, $s$ being measured from the point for which $x = 0$.

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

(a) Prove that the surface of the solid obtained by revolving the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ round its minor axis is $2\pi a^{2}[1+\frac{1-e^{2}}{2e}log(\frac{1+e}{1-e})]$ where $b^{2}=a^{2}(1-e^{2})$.
(b) If $I_{m,n}=\int_{0}^{\pi/2}sin^{m}x~cos^{n}x dx$, $m, n$ being positive integers greater than 1, prove that $I_{m,n}=\frac{n-1}{m+n}I_{m,n-2}$. Hence find the value of $\int_{0}^{1}x^{6}\sqrt{1-x^{2}}dx$.
(c) Show that the arcs of the curves $x=f(t)-\varphi'(t), y=\varphi(t)+f'(t)$ and $x=f'(t) sin~t - \varphi'(t) cos~t, y=f'(t) cos~t + \varphi'(t) sin~t$ corresponding to same interval of variation of $t$ have equal lengths.

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

(a) Find the angle of rotation about the origin which will transform the equation $x^{2}-y^{2}=4$ into $x'y+2=0$.
(b) Prove that the equations $x=1+\lambda, y=-1+\frac{2z}{\lambda}$ represents a generator of $x^{2}-2yz=1$. Find also other system of generators which lie on $x^{2}-2yz=1$.
(c) Find the equation of the cylinder whose generating line is parallel to x-axis and guiding curve is $3x+2y-5=0, 5x^{2}-2y^{2}+7z^{2}=1$.
(d) Find the point of intersection of the two tangents at $\alpha$ and $\beta$ to the Conic $\frac{l}{r}=1+e~cos~\theta$.
(e) Find the nature of the conicoid $3x^{2}-2y^{2}-12x-12y-6z=0$.

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

(a) Prove that the discriminant of the Conic $Ax^{2}+By^{2}+Cxy+Dx+Ey+F=0$ is invariant under rotation of axes.
(b) The section of a cone whose guiding curve is the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, z=0$ by the plane $x=0$ is a rectangular hyperbola. Show that locus of the vertex is the surface $\frac{x^{2}}{a^{2}}+\frac{y^{2}+z^{2}}{b^{2}}=1$.

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

(a) (i) Show that the Centre of the sphere which always touch the lines $y=mx, z=c$ and $y=-mx, z=-c$ lie on the surface $mxy+cz(1+m^{2})=0$.
(ii) Find the equation of the right circular cylinder whose guiding curve is $x^{2}+y^{2}+z^{2}=9, x-y+z=3$.
(b) (i) Find the locus of the point of intersection of the perpendicular generators of the hyperbolic paraboloid $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=2z$.
(ii) If the normal be drawn at one extremity $L$ of the latus rectum $PSP'$ on the conic $\frac{l}{r}=1+e~cos~\theta$ where $S$ is the pole, then show that the distance from focus $S$ of the other point in which the normal meets the conic is $\frac{l(1+3e^{2}+e^{4})}{1+e^{2}-e^{4}}$.

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

(a) For which value of $m$, $y=x^{m}$ is a solution of the equation $3x^{2}\frac{d^{2}y}{dx^{2}}+11x\frac{dy}{dx}-3y=0$.
(b) Let the differential equation be $a\frac{dy}{dx}+by=ke^{-\lambda x}$ where $a, b, k$ are positive constants and $\lambda$ is non-negative constant. Find the solution of differential for $\lambda=0$. Show that $y\rightarrow k/b$ as $x\rightarrow\infty$ ($\lambda=0$).
(c) Find an integrating factor of the equation $(x^{2}y-2xy^{2})dx-(x^{3}-3x^{2}y)dy=0$.

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

(a) Reduce the equation $x^{2}p^{2}+yp(2x+y)+y^{2}=0$ to Clairaut's form and obtain complete primitive.
(b) (i) In a certain culture of bacteria, the rate of increase is proportional to the number present. If it is found that their number doubles in 4 hours, what should be their number at the end of 12 hours?
(ii) Find the solution of $\frac{dy}{dx}-y~tan~x=cos~x$ by substitution $y=y_{1}(x)v(x)$ where $y_{1}=sec~x$.