B.Sc. Mathematics Honours C-1 Question Paper 2019 (CBCS)

Full Marks: 60 Time: 3 Hours

Unit - I

1. Answer any three of the following questions: $3 \times 2 = 6$
(a) If $y = e^{ax} \cos^{2} bx$, find $y_{n} (a, b > 0)$.
(b) Find the oblique asymptotes of the curve $y = \frac{3x}{2} \log (e - \frac{1}{3x})$.
(c) If $y = x^{n-1} \log x$ then prove that $y_{n} = \frac{(n-1)!}{x}$.
(d) What is reciprocal spiral? Sketch it.
(e) The parabolic path is given by $y = x \tan \theta - \frac{x^{2}}{4h \cos^{2} \theta}$, what will be the asymptote of parabolic paths? 2. Answer any one question: $1 \times 10 = 10$
(a) (i) Find the evolute of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$.
(ii) Let $P_{n} = D^{n}(x^{n} \log x)$. Prove that $P_{n} = nP_{n-1} + (n-1)!$. Hence show that $P_{n} = n! (\log x + 1 + \frac{1}{2} + \dots + \frac{1}{n})$.
(b) (i) Prove that the envelope of circles whose centres lie on the rectangular hyperbola $xy = c^{2}$ and which pass through its centre is $(x^{2} + y^{2})^{2} = 16c^{2}xy$.
(ii) Find the point of inflexion on the curve $(\theta^{2} - 1)r = a\theta^{2}$.

Unit - II

3. Answer any two questions: $2 \times 2 = 4$
(a) If $I_{n} = \int_{0}^{\pi/2} \cos^{n-2} x \sin x dx, n > 2$ Prove that $2(n-1)I_{n} = 1 + (n-2)I_{n-1}$.
(b) Find the length of the curve $x = e^{\theta} \sin \theta$ and $y = e^{\theta} \cos \theta$ between $\theta = 0$ to $\theta = \frac{\pi}{2}$.
(c) Find the reduction formula for $\int \cos^{n} x \sin (nx) dx$. 4. Answer any two questions: $2 \times 5 = 10$
(a) Prove that the volume of the solid obtained by revolving the lemniscate $r^{2} = a^{2} \cos 2\theta$ about the initial line is $\frac{1}{2} \pi a^{3} \{ \frac{1}{\sqrt{2}} \log (\sqrt{2} + 1) - \frac{1}{3} \}$.
(b) If $I_{m,n} = \int_{0}^{1} x^{m} (1 - x)^{n} dx$, where $m$ and $n$ are positive integers, then prove that $(m + n + 1)I_{m,n} = nI_{m,n-1}$ and deduce that $I_{m,n} = \frac{m!n!}{(m + n + 1)!}$.
(c) Evaluate the surface area of the solid generated by revolving the cycloid $x = a(\theta - \sin \theta)$, $y = a(1 - \cos \theta)$ about the line $y = 0$.

Unit - III

5. Answer any three questions: $3 \times 2 = 6$
(a) Find the centre and foci of the conic $x^{2} - 2y^{2} - 2x + 8y - 1 = 0$.
(b) Find the equation of the sphere of which the circle $x^{2} + y^{2} + z^{2} + 2x - 2y + 4z - 3 = 0, 2x + y + z = 4$ is a great circle.
(c) Find the condition that the line $\frac{l}{r} = A \cos \theta + B \sin \theta$ may touch the conic $\frac{l}{r} = 1 - e \cos \theta$.
(d) For what angle must the axes be turned to remove the term $xy$ from $7x^{2} + 4xy + 3y^{2} = 0$.
(e) Find the equation of cone whose vertex is origin and the base curve is $x^{2} + y^{2} = 4, z = 2$. 6. Answer any one question: $1 \times 5 = 5$
(a) If $r$ be the radius of the circle $x^{2} + y^{2} + z^{2} + 2ux + 2vy + 2wz + d = 0, lx + my + nz = 0$ then prove that $(r^{2} + d)(l^{2} + m^{2} + n^{2}) = (mw - nv)^{2} + (nu - lw)^{2} + (lv - mu)^{2}$.
(b) Show that the 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 cone $\frac{\alpha a^{2} (b^{2} - c^{2})}{x} + \frac{\beta b^{2} (c^{2} - a^{2})}{y} + \frac{\gamma c^{2} (a^{2} - b^{2})}{z} = 0$. 7. Answer any one question: $1 \times 10 = 10$
(a) (i) Show that the plane $3x - 2y - z = 0$ cuts the cones $21x^{2} - 4y^{2} - 5z^{2} = 0$ and $3yz - 2zx + 2xy = 0$ in the same pair of perpendicular lines.
(ii) Find the equation of the cylinder, whose generators are parallel to the straight line $\frac{x}{2} = \frac{y}{3} = \frac{z}{5}$ and which passes through the conic $z = 0, 3x^{2} + 7y^{2} = 12$.
(b) (i) Find the locus of the point of intersection of the perpendicular generators of the hyperboloid $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1$.
(ii) Reduce the equation $x^{2} + 3y^{2} + 3z^{2} - 2xy - 2yz - 2zx + 1 = 0$ to its canonical form and determine the type of quadratic represented by it.

Unit - IV

8. Answer any two questions: $2 \times 2 = 4$
(a) Find the integrating factor of the differential equation $(2xy + 3x^{2}y + 6y^{3})dx + (x^{2} + 6y^{2})dy = 0$.
(b) Show that the general solution of the equation $\frac{dy}{dx} + Py = Q$ can be written in the form $y = k(u - v) + v$, where $k$ is a constant and $u$ and $v$ are its two particular solutions.
(c) Solve: $\frac{dy}{dx} + y \cos x = xy^{n}$. 9. Answer any one question: $1 \times 5 = 5$
(a) The population of a country increases at the rate proportional to the number of inhabitants. If the population doubles in 30 years, in how many years will it triple?
(b) Solve: $(px^{2} + y^{2})(px + y) = (p + 1)^{2}$ by using the transformation $u = xy, v = x + y$, where $p = \frac{dy}{dx}$.