B.Sc. Mathematics Honours C-1 Question Paper 2021 (CBCS)

Candidates are required to give their answers in their own words as far as practicable.
The figures in the right-hand margin indicate full marks.

Group A

Answer any four questions: 12 x 4 = 48

1. (a) Find the equation of the asymptotes of the curve
$$r^{n}f_{n}(\theta)+r^{n-1}f_{n-1}(\theta)+....+f_{0}(\theta)=0$$
(b) If $I_{n}=\int_{0}^{\pi/2}cos^{n-2}x \sin nx dx$ show that $2(n-1)I_{n}=1+(n-2)I_{n-1}$ and hence deduce $I_{n}=\frac{1}{n-1}$.

2. (a) Circles are described on the double ordinates of the parabola $y^{2}=4ax$ as diameters. Prove that the envelope is the parabola $y^{2}=4a(x+a)$.
(b) If $y=\sin(m \cos^{-1}\sqrt{x})$ then prove that $\lim_{x\rightarrow0}\frac{y_{n+1}}{y_{n}}=\frac{4n^{2}-m^{2}}{4n+2}$.
(c) Find a, b, c such that $\frac{ae^{x}-b \cos x+ce^{-x}}{x \sin x}\rightarrow2$ as $x\rightarrow0$.

3. (a) Show that the arc of the upper half of the cardiode $r=a(1-\cos \theta)$ is bisected at $\theta=\frac{2}{3}\pi$. Find also the perimeter of the curve.
(b) Show that the curve $re^{\theta}=a(1+\theta)$ has no point of inflexion.
(c) Find the asymptotes of the parametric curve $x=\frac{t^{2}+1}{t^{2}-1}$ and $y=\frac{t^{2}}{t-1}$.

4. (a) Show that feet of the normals from the point $(\alpha, \beta, \gamma)$ to the ellipsoid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1$ lie on the intersection of the ellipsoid and the cone $\frac{\alpha a^{2}(b^{2}-c^{2})}{x}+\frac{\beta b^{2}(c^{2}-a^{2})}{y}+\frac{\gamma c^{2}(a^{2}-b^{2})}{z}=0$.
(b) Find the equation of the right circular cylinder whose axis is $\frac{x}{1}=\frac{y}{-2}=\frac{z}{2}$ and radius is 2.

5. (a) Prove that $\cosh(x + y) = \cosh x \cosh y + \sinh x \sinh y$.
(b) Two spheres of radii $r_{1}$ and $r_{2}$ cut orthogonally. Prove that the radius of their common circle is $\frac{r_{1}r_{2}}{\sqrt{{r_{1}}^{2}+{r_{2}}^{2}}}$.
(c) Find the polar equation of the normal to the conic $\frac{l}{r}=1+e \cos \theta$, $e>0$.

6. (a) Find the equation of the generator of $x^{2}+y^{2}=z^{2}$ through the point (3, 4, 5).
(b) Given that the asteroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=c^{\frac{2}{3}}$ is the envelope of the family of ellipses $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, show that $a+b=c$.
(c) State the existence and uniqueness theorem for the solution of ordinary differential equation.

7. (a) Solve: $x\frac{dy}{dx}-y=x\sqrt{x^{2}+y^{2}}$.
(b) If m and n are positive integers, show that $\int_{a}^{b}(x-a)^{m}(b-x)^{n}dx=\frac{m!n!}{(m+n+1)!}(b-a)^{m+n+1}$.
(c) Solve $y=2px+y^{2}p^{3}$ and find the general and singular solutions.

8. (a) Compute the length of the curve $x=2 \cos \theta$, $y=\sin 2\theta$, $0\le\theta\le\pi$.
(b) Find the points of inflection on the curve $r(\theta^{2}-1)=a\theta^{2}$.
(c) If $I_{n}=\int_{0}^{1}x^{n}\tan^{-1}xdx$, n being positive integer greater than 2, prove that $(n+1)I_{n}+(n-1)I_{n-2}=\frac{\pi}{2}-\frac{1}{n}$.

Group B

Answer any six questions: 2 x 6 = 12

9. Find the value of $\lim_{x\rightarrow\infty}[a_{0}x^{m}+a_{1}x^{m-1}+...+a_{m}]^{1/x}$ where m is a positive integer and $a_{0}\ne0$.

10. Let $I_{n}=\int_{0}^{1}(\ln x)^{n}dx$. Prove that $I_{n}=(-1)^{n}n!$, n being positive integer.

11. The curves $y=x^{n}$ and $y^{m}=x$ ($m,n>0$) meet at (0, 0) and (1, 1). Find the area between these two curves.

12. Find $\alpha$ if $x^{\alpha}$ be an integrating factor of $(x-y^{2})dx+2xy dy=0$.

13. Find the curve for which the curvature is zero at every point and which passes through the point (0, 0) where $\frac{dy}{dx}=3/2$.

14. Solve the differential equation: $4x^{3}ydx+(x^{4}+y^{4})dy=0$.

15. Generate a reduction formula for $\int \tan^{n}x dx$, $n\in Z^{+}$ and $n>1$.

16. Find the equations of the straight lines in which the plane $2x+y-z=0$ cuts the cone $4x^{2}-y^{2}+3z^{2}=0$.

17. Find the asymptote (if any) of the curve $y=a \log[\sec(\frac{x}{a})]$.

18. On the ellipse $r(5-2 \cos \theta)=21$, find the point with the greatest radius vector.