B.Sc. Mathematics Honours GE-1 Question Paper 2022 CBCS)

Group - A

1. Answer any ten questions : $2 \times 10 = 20$

(a) If $y = x^{n-1} \log x$; show that $y_n = \frac{(n-1)!}{x}$
(b) Show that origin is a point of inflexion on the curve $y = x \cos 2x$.
(c) Prove that the curve $y = e^x$ is convex to the x-axis at every point.
(d) Find the envelope of the family of straight lines $y = mx + \sqrt{a^2 m^2 + b^2}$, m being parameter.
(e) If $I_n = \int_{0}^{\pi/4} \tan^n x dx$, show that $I_{n+1} - I_{n-1} = \frac{1}{n}$
(f) Find the length of the perimeter of the astroid $x^{2/3} + y^{2/3} = a^{2/3}$.
(g) Determine the nature of the conic presented by $9x^2 + 24xy + 16y^2 - 126x + 82y - 59 = 0$
(h) Find the polar co-ordinate of the point whose Cartesian co-ordinates are $(\sqrt{3}, 1)$.
(i) Find the equation to a sphere whose centre is $(2, 5, 4)$ and which passes through the point $(-1, 3, 2)$.
(j) Define non-linear ODE of first order.
(k) Examine if the ODE $(1 + xy)y dx + (1 - xy)x dy = 0$ is exact.
(l) Find the area of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.
(m) Find the equation of the cylinder whose generators are parallel to the line $-3x = 6y = 2z$ and whose guiding curve is the ellipse $2x^2 + y^2 = 1, z = 0$.
(n) Define asymptote of a curve.
(o) Through an example, explain singular solution of an ODE.

Group - B

2. Answer any four questions: $5 \times 4 = 20$

(a) Find the asymptotes of the curve $x(x - y)^2 - 3(x^2 - y^2) + 8y = 0$.
(b) Show that the section of the hyperbolic paraboloid $\frac{x^2}{2} - \frac{z^2}{3} = y$ by the plane $3x - 3y + 4z + 2 = 0$ is a hyperbola.
(c) Solve the ODE: $(x^2 y^3 + 2xy) dy = dx$.
(d) Find a reduction formula for $\int \sin^m x \cos^n x dx$, where m and n are positive integers. Use it to evaluate $\int_{0}^{\pi/2} \sin^{18} x \cos^6 x dx$.
(e) Find a and b such that $\lim_{x \to 0} \frac{x(1 + a \cos x) - b \sin x}{x^3} = 1$
(f) If $r_1$ and $r_2$ be two mutually perpendicular radius vectors of the ellipse $r^2 = \frac{b^2}{1 - e^2 \cos^2 \theta}$, prove that $\frac{1}{r_1^2} + \frac{1}{r_2^2} = \frac{1}{a^2} + \frac{1}{b^2} \cdot [b^2 = a^2(1 - e^2)]$.

Group - C

3. Answer any two questions : $10 \times 2 = 20$

(a) (i) Use suitable integrating factor to solve the ODE $x dy - y dx - \cos \frac{1}{x} dx = 0$.
(ii) Show that the area bounded by one arch of the cycloid $x = a(\theta - \sin \theta)$, $y = a(1 - \cos \theta)$ and the x-axis is $3\pi a^2$ sq units.
(b) (i) Find the equation of a cone whose vertex is the point $P(\alpha, \beta, \gamma)$ and whose generating lines pass through the conic $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, z = 0$. If the section of this cone by the plane $x = 0$ is a rectangular hyperbola, show that the locus of P is $\frac{x^2}{a^2} + \frac{y^2 + z^2}{b^2} = 1$.
(ii) Find the equation of a sphere that passes through the points $(1, 0, 0), (0, 1, 0), (0, 0, 1)$ and touches the plane $2x + 2y - z = 15$.
(c) (i) State Leibnitz's theorem for $n^{th}$ derivative of the product of two functions. Use this to solve the following problem: If $y = e^{a \sin^{-1} x}$, then prove that $(1 - x^2) y_{n+2} - (2n + 1)x y_{n+1} - (n^2 + a^2) y_n = 0$
(ii) If PM, PN be perpendiculars drawn from any point P on the curve $y = ax^3$ upon the coordinate axes, show that the envelope of MN is $27y + 4ax^3 = 0$.
(d) (i) Find the surface area and the volume of the ellipsoid formed by the revolution of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ round its major axis.
(ii) If $J_n = \int_{0}^{\pi/2} \cos^n x dx$, show that $J_n = \frac{n-1}{n} J_{n-2} (n > 2)$. Hence find $J_8$.