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

Candidates are required to give their answers in their own words as far as practicable. Illustrate the answers wherever necessary.

Unit - I

1. Answer any three questions: $3 \times 2 = 6$
(a) If $y=x^{n-1} \log x$ prove that $y_{n}=\frac{(n-1)!}{x}$.
(b) Evaluate $\lim_{x \rightarrow 0}(\frac{\sin x}{x})^{\frac{1}{x^{2}}}$.
(c) Find the envelope of $y=mx+a\sqrt{1+m^{2}}$ parameter $m$.
(d) State Leibnitz's rule for successive differentiation. Find the $n$-th order derivative of $\cos(ax+b)$, where $a, b$ are constants.
(e) Find the points of inflexion of the curve $y=(x+1) \tan^{-1} x$.

2. Answer any one question: $1 \times 10 = 10$
(a) (i) Trace the curve $(x^{2}+y^{2})x-a(x^{2}-y^{2})=0, (a>0)$.
(ii) If $y^{\frac{1}{m}}+y^{-\frac{1}{m}}=2x$, then prove that $(x^{2}-1)y_{n+2}+(2n+1)xy_{n+1}+(n^{2}-m^{2})y_{n}=0$.
(b) (i) Find the asymptotes of the curve $y^{3}+x^{2}y+2xy^{2}-y+1=0$.
(ii) Find the ranges of values of $x$ in which the curve $y=3x^{5}-40x^{3}+3x-20$ is concave upwards or downwards. Also find the points of inflexion.

Unit - II

3. Answer any two of the following: $2 \times 2 = 4$
(a) Find the perimeter of the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$.
(b) Find the length of the arc of the parabola $x^{2}=4ay$ measured from the vertex to one extremity of the latus rectum.
(c) Calculate the area bounded by the curves $y=x^{2}$ and $x=y^{2}$.

4. Answer any two questions: $2 \times 5 = 10$
(a) Show that the volume of the solid generated by the revolution of the curve $y(x^{2}+a^{2})=a^{3}$ about the asymptote of the curve is $2\pi^{2}a^{3}$.
(b) Find the reduction formula of $\int \sin^{p}x \cos^{q}x dx$ where $q$ is positive integer and $p$ is negative integers. Hence prove that $\int \frac{\cos^{5}x}{\sin^{4}x} dx = -\frac{\cos^{4}x}{3 \sin^{3}x} + \frac{4 \cos^{2}x}{3 \sin x} + \frac{8 \sin x}{3}$.
(c) Show that the arc of the upper half of the cardioide $r=a(1-\cos \theta)$ is bisected at $\theta = \frac{2\pi}{3}$. Show also that the perimeter of the curve is $8a$.

Unit - III

5. Answer any three questions: $3 \times 2 = 6$
(a) Find the values of $a$ for which the plane $x+y+z=a\sqrt{3}$ touches the sphere $x^{2}+y^{2}+z^{2}-2x-2y-6=0$.
(b) Write down reflection properties of parabola and hyperbola.
(c) Find the equation of the cylinder generated by straight lines parallel to $z$-axis and passing through the curve of intersection of the plane $4x+3y-2z=5$ and the surface $3x^{2}-y^{2}+2z^{2}=1$.
(d) Find the equation of the sphere which passes through the circle $x^{2}+y^{2}=4, z=0$ and is cut by the plane $x+2y+2z=0$, in a circle of radius 3.
(e) Show that the equation $4xy-3x^{2}=1$ is transformed to $X^{2}-4Y^{2}=1$ by rotating the axes through an angle $\tan^{-1}2$.

6. Answer any one question: $1 \times 5 = 5$
(a) The expression $ax^{2}+2hxy+by^{2}+2gx+2fy+c$ changed to $a'x'^{2}+2h'x'y'+b'y'^{2}+2g'x'+2f'y'+c'$ when the axes are rotated through an angle $\theta$. Show that $a+b=a'+b'$.
(b) Show that the sum of the reciprocals of two perpendicular focal chords of a conic is constant.

7. Answer any one question: $1 \times 10 = 10$
(a) (i) Prove that the two conics $\frac{l_{1}}{r}=1+e_{1}\cos \theta$ and $\frac{l_{2}}{r}=1+e_{2}\cos(\theta-\gamma)$ will touch one another if $l_{1}^{2}(1-e_{2}^{2})+l_{2}^{2}(1-e_{1}^{2})=2l_{1}l_{2}(1-e_{1}e_{2}\cos \gamma)$.
(ii) A sphere of constant radius $r$ passes through origin $O$ and meets the axes in $A, B, C$. Prove that the locus of the foot of perpendicular from $O$ to the plane $ABC$ is given by $(x^{2}+y^{2}+z^{2})^{2}(x^{-2}+y^{-2}+z^{-2})=4r^{2}$.
(b) (i) Reduce the equation $x^{2}-5xy+y^{2}+8x-20y+15=0$ to its canonical form and determine the nature of the conic.
(ii) Show that the section of the surface $yz+zx+xy=a^{2}$ by the plane $lx+my+nz=p$ will be a parabola if $\sqrt{l}+\sqrt{m}+\sqrt{n}=0$.

Unit - IV

8. Answer any two questions: $2 \times 2 = 4$
(a) Find the differential equation of all parabolas having their axes parallel to $y$ axis.
(b) Find an integrating factor of the differential equation $(x^{2}+y^{2}+2x)dx+2ydy=0$.
(c) Reduce the equation $\sin y \cdot \frac{dy}{dx} = \cos x (2 \cos y - \sin^{2}x)$ into a linear equation.

9. Answer any one question: $1 \times 5 = 5$
(a) Show that $y = \frac{Q}{P} - e^{-\int P dx} [\int e^{\int P dx} d(\frac{Q}{P}) + C]$ is a solution of the differential equation $\frac{dy}{dx}+Py=Q$, where $P$ and $Q$ are functions of $x$ only.
(b) If $\frac{1}{N}(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x})$ is a function of $x$ alone, say $f(x)$, then prove that $e^{\int f(x)dx}$ is an integrating factor of $Mdx+Ndy=0$.