B.Sc. Mathematics Honours C-1 Question Paper 2020 (CBCS)

Candidates are required to give their answers in their own words as far as practicable.
The figures in the margin indicate full marks.

Group A

Answer any three from the following questions: 3 x 20 = 60

1. (a) Evaluate the following limits: $\lim_{x\rightarrow0}x \ln(\sin x)$ in $(0,\pi)$.
(b) Show that the four asymptotes of the curve $(x^{2}-y^{2})(y^{2}-4x^{2})+6x^{3}-5x^{2}y-3xy^{3}+2y^{3}-x^{2}+3xy-1=0$ cut the curve in eight points which lie on the circle $x^{2}+y^{2}=1$.
(c) Prove that the envelope of a variable circle whose centre lies on the parabola $y^{2}=4ax$ and which passes through its vertex is $2ay^{2}+x(x^{2}+y^{2})=0$.
(d) What are the points of inflection of the function $f(x)=3x^{4}-8x^{3}$.

2. (a) What do you mean by rectilinear asymptotes to a curve?
(b) Find the equation of the envelope of the family of curve represented by equation $x^{2} \sin \alpha + y^{2} \cos \alpha = a^{2}$.
(c) If $y=(\sin^{-1}x)^{2}$ show that $(1-x^{2})y_{n+2}-(2n+1)xy_{n+1}-n^{2}y_{n}=0$. Also find $y_{n}(0)$.
(d) Find the asymptotes of the curve $(x+y)(x-2y)(x-y)^{2}+3xy(x-y)+x^{2}+y^{2}=0$.

3. (a) If $I_{n}=\int_{0}^{1}x^{n} \tan^{-1}x dx$, $n>2$ then prove that $(n+1)I_{n}+(n-1)I_{n-2}+\frac{1}{n}=\frac{\pi}{2}$.
(b) Determine the length of one arc of the cycloid $x=a(\theta-\sin \theta)$, $y=a(1-\cos \theta)$.
(c) Find the reduction formula for $\int \sin^{m}x \cos^{n}x dx$ where either m or n or both are negative integers. And hence find $\int \frac{\cos^{4}x}{\sin^{2}x}dx$.
(d) Find the whole length of the loop of the curve $9ay^{2}=(x-2a)(x-5a)^{2}$.

4. (a) Find the eccentricity and the vertex of the conic $r=3 \sec^{2} \frac{\theta}{2}$.
(b) Find the polar equation of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{20}=1$.
(c) A sphere of radius k passes through the origin and meets the axes in A, B, C. Prove that the locus of the centroid of the triangle ABC is the sphere $9(x^{2}+y^{2}+z^{2})=4k^{2}$.
(d) Show that the plane $y+6=0$ intersects the hyperbolic paraboloid $\frac{x^{2}}{5}-\frac{y^{2}}{4}=6z$ in parabola.

5. (a) For what angle must the axes be turned to remove the term $x^{2}$ from $x^{2}-4xy+3y^{2}=0$.
(b) Find the centre and the radius of the circle $3x^{2}+3y^{2}+3z^{2}+x-5y-2=0$, $x+y=2$.
(c) P is a variable point such that its distance from the xy-plane is always equal to one fourth the square of its distance from the y-axis. Show that the locus of P is a cylinder.
(d) Reduce the equation $7x^{2}+y^{2}+z^{2}+16yz+8zx-8xy+2x+4y-40z-14=0$ to the canonical form and find the nature of the conicoid it represents.

6. (a) Solve: $(1+y^{2})dx-(\tan^{-1}y-x)dy=0$.
(b) Find the singular solution of $xp^{2}-(y-x)p-y=1$.
(c) Solve and find the singular solutions of $p^{4}=4y(xp-2y)^{2}$.
(d) Solve: $y(xy+2x^{2}y^{2})dx+x(xy-x^{2}y^{2})dy=0$.