B.Sc. Mathematics Honours GE-1 Question Paper 2018 CBCS)

Unit-I

1. Answer any three questions :
(a) If $\lim_{x\rightarrow0}\frac{ae^{x}+be^{-x}+2 \sin x}{\sin x+x \cos x}=2,$ find the values of a and b.
(b) Draw a rough sketch of $y=\cosh x.$
(c) Find the nth derivative of $\frac{1}{x^{2}-a^{2}}$.
(d) Find the range of values of x for which $y=x^{4}-6x^{3}+12x^{2}+5x+7$ is concave downwards.
(e) From any point P on the parabola $y^{2}=4ax,$ perpendiculars PM and PN are drawn to the co-ordinate axes. Find the envelope of the line MN.

2. Answer any one questions :
(a) i) Trace the curve $xy^{2}=a^{2}(a-x)$
ii) If $y=(\sin^{-1}x)^{2}$ prove that $(1-x^{2})y_{n+2}-(2n+1)xy_{n+1}-n^{2}y_{n}=0$
(b) i) Find the asymptotes of the curve $y^{3}-yx^{2}+y^{2}+x^{2}-4=0$
ii) Find if there is any point of inflexion on the curve $y-3=6(x-2)^{5}$

Unit-II

3. Answer any two of the following:
(a) Find the differential of arc length for the curve $x=a(1-\cos \theta)$, $y=a(\theta+\sin \theta)$.
(b) Find the area of the circle $r=2a \sin \theta$.
(c) Find the reduction formula for $\int \sec^{n}x dx$.

4. Answer any two questions :
(a) Establish the reduction formula for $\int_{0}^{\pi/2} \sin^{m}x \cos^{n}x dx$, m, n being positive integers, greater than 1. Hence Calculate $\int_{0}^{\pi/2} \sin^{5}x \cos^{6}x dx$.
(b) Find the area bounded by the parabola $4y=3x^{2}$ and the straight line $3x-2y+12=0$.
(c) Find the volume and surface area generated by the revolution of the cardioid $r=a(1+\cos \theta)$ about initial line.

Unit-III

5. Answer any three questions :
(a) Find the angle through which the axes are to be rotated so that the equation $x\sqrt{3}+y+6=0$ may be reduced to the form $x=c$.
(b) If the pair of straight lines $x^{2}-2pxy-y^{2}=0$ and $x^{2}-2qxy-y^{2}=0$ be such that each pair bisects the angles between the other pair, then prove that $pq=-1$.
(c) Find the the equation of the sphere for which the circle $x^{2}+y^{2}+z^{2}+2x-4y+2z+5=0, x-2y+3z+1=0$ is a great circle.
(d) Find the point of intersection of the lines $r \cos(\theta-\alpha)=p$ and $r \cos(\theta-\beta)=p$.
(e) Write down the reflection property of ellipse.

6. Answer any one question :
(a) Show that the distance between two fixed points is unaltered by a rotation of axes.
(b) Find the equation of the cylinder whose generators are parallel to the straight line $2x=y=3z$ and which passes through the circle $y=0$, $x^{2}+z^{2}=6$.

7. Answer any one question :
(a) i) Prove that the plane $ax+by+cz=0 (a,b,c \neq 0)$ cuts the cone $yz+zx+xy=0$ in perpendicular straight lines, if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$.
ii) Reduce the equation $x^{2}+4xy+4y^{2}+4x+y-15=0$ to its standard form.
(b) i) Show that the equation of the circle which passes through the focus of the curve $\frac{l}{r}=1-e \cos \theta$ and touches it at the point $\theta=\alpha$ is $r(1-e \cos \alpha)^{2}=l \cos(\theta-\alpha)-el \cos(\theta-2\alpha)$.
ii) Prove that the five normals from a given point to a paraboloid lie on a cone.

Unit-IV

8. Answer any two questions :
(a) Determine the order and the degree of the differential equation $\sqrt{y+(\frac{dy}{dx})^{2}}=1+x$.
(b) Find an integrating factor of the differential equation $x^{2}ydx-(x^{3}+y^{3})dy=0$.
(c) Define singular solution of a differential equation.

9. Answer any one question :
(a) Find a solution of the differential equation $\frac{dy}{dx}-y \tan x=0$ in the form $y=y_{1}(x)$. Hence solve $\frac{dy}{dx}-y \tan x=\cos x$ by the substitution $y=y_{1}(x).v(x)$.
(b) By the substitution $x^{2}=u$ and $y^{2}=v$ reduce the equation $(px-y)(x-py)=2p$ to clairaut's form and find general solution.