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

Full Marks: 60 Time: Three Hours

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

(i) Find $y_{n}$ for the function $y = \frac{x^{n}}{x-1}$
(ii) Show that the curve $y^{3} = 8x^{2}$ is concave to the foot of the ordinate everywhere except at the origin.
(iii) If the axes are rotated through an angle $45^{\circ}$ without changing the origin, then find the new form of the equation $x^{2} - y^{2} = a^{2}$.
(iv) Find the equation of the circle lying on the sphere $x^{2} + y^{2} + z^{2} - 2y - 4z = 11$ and having its centre at (1, 3, 4).
(v) Find the total area of the circle $x^{2} + y^{2} + 2x = 9$.
(vi) If $I_{n} = \int_{0}^{\pi/4} \tan^{n} x dx$, for $n \ge 2$, find the value of $I_{n} + I_{n-2}$.
(vii) Find the asymptotes of the curve $x^{3} + y^{3} = 3axy$.
(viii) Find the integrating factor of $(1 + x^{2}) y_{1} + y = e^{\tan^{-1} x}$.
(ix) Find the singular solution of $y = x \frac{dy}{dx} - (\frac{dy}{dx})^{2}$.
(x) Find the nature of the conic $3x^{2} + 2xy + 3y^{2} - 16x + 20 = 0$.
(xi) Calculate the sum of the reciprocals of two perpendicular focal chord of the conic $l/r = 1 + e \cos \theta$.
(xii) Show that $\lim_{x \to \infty} (\frac{ax+1}{ax-1})^{x} = e^{2/a}, a > 0$.
(xiii) If $u = \sin ax + \cos ax$, show that $u_{n} = a^{n} \{ 1 + (-1)^{n} \sin 2ax \}^{\frac{1}{2}}$.
(xiv) Solve $p - \frac{1}{p} - \frac{x}{y} + \frac{y}{x} = 0$ where $p \equiv \frac{dy}{dx}$.
(xv) Evaluate $\lim_{x \to \infty} (\sqrt{x^{2} + 2x} - x)$.

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

(i) State and prove Leibnitz's theorem. If $y = \tan^{-1} x$, find $(y_{n})_{0}$ by using Leibnitz's theorem.
(ii) Prove that the locus of the middle points of focal chords of a conic is another conic.
(iii) If $J_{n} = \int \sin n\theta \sec \theta d\theta$, show that $J_{n} + J_{n-2} = -\frac{2}{n-1} \cos (n-1)\theta$. Hence deduce the value of $\int_{0}^{\pi/2} \frac{\sin 3\theta \cos 3\theta}{\cos \theta} d\theta$.
(iv) If $S$ be the length of the arc of $3ay^{2} = x(x-a)^{2}$, measured from the origin to the point (x, y), show that $3s^{2} = 4x^{2} + 3y^{2}$.
(v) Find the equation to the right circular cylinder of radius a, whose axis passes through the origins and makes equal angles with the co-ordinates axes.
(vi) Solve: $16x^{2} + 2(\frac{dy}{dx})^{2} y - (\frac{dy}{dx})^{3} x = 0$.

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

(i) (a) Explain L'Hospital Rule. Using L'Hospital Rule prove that $$\lim_{x \to 0} \left[ \frac{a_{1}^{\frac{1}{x}} + a_{2}^{\frac{1}{x}} + \dots + a_{n}^{\frac{1}{x}}}{n} \right]^{nx} = a_{1} a_{2} \dots a_{n}.$$ (b) Find the envelope of the straight line $\frac{x}{a} + \frac{y}{b} = 1$ where a and b are variable parameters connected by the relation $a+b=c$.

(ii) (a) What is a great circle? Obtain the equation of the sphere having the circle $x^{2} + y^{2} + z^{2} + 10y - 4z - 8 = 0$, $x + y + z = 3$ as the great circle.
(b) Reduce the equation $3x^{2} + 5y^{2} + 3z^{2} + 2yz + 2zx + 2xy - 4x - 8z + 5 = 0$ to the standard form and find the nature of the conic.

(iii) (a) Find the volume of ellipsoid generated by the revolution of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ about major axis and minor axis.
(b) Define singular and general solution of the differential equation. Find both solutions of the following differential equation: $p^{3}x - p^{2}y - 1 = 0$.

(iv) (a) Find the rectilinear asymptotes of the following curve: $$x^{3} + x^{2}y - xy^{2} - y^{3} + 2xy + 2y^{2} - 3x + y = 0.$$ (b) If $f(m,n) = \int_{0}^{\pi/2} \cos^{m} x \sin nx dx$, prove that $$f(m,n) = \frac{1}{m+n} + \frac{m}{m+n} f(m-1, n-1), m,n > 0.$$ Hence deduce that $$f(m,n) = \frac{1}{2^{m+1}} \left( \frac{2}{1} + \frac{2^{2}}{2} + \frac{2^{3}}{3} + \dots + \frac{2^{m}}{m} \right).$$