B.Sc. Mathematics Honours C-1 Question Paper 2023 (CBCS)

Full Marks: 60 Time: Three Hours

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

Group A

Answer any ten questions: $2\times10=20$

1. Solve: $4x^3y\,dx+(x^4+y^4)\,dy=0$.
2. If $y=x^{n-1}\log x$, prove that $y_n=\frac{(n-1)!}{x}$.
3. Find the area of the region bounded by $x=y^3$, the $y$-axis, $y=3$ and $y=6$.
4. What does the equation $11x^2+16xy-y^2=0$ become on turning the axes through an angle $\tan^{-1}\frac12$?
5. Find the equations of the straight lines in which the plane $2x+y-z=0$ cuts the cone $4x^2-y^2+3z^2=0$.
6. If $y=\frac{x^2-6}{x^3-x^2-2x}$, find $y_n$.
7. Show that the abscissa of the point of inflection on the curve $y^2=f(x)$ satisfies $[f(x)]^2=2f(x)f'(x)$.
8. Find the oblique asymptotes of the curve $y=\frac{3x}{2}\log\!\left(e-\frac{1}{3x}\right)$.
9. On the ellipse $r(5-2\cos\theta)=21$, find the point with the greatest radius vector.
10. Solve: $(xy^2-e^{1/x^2})\,dx-x^2y\,dy=0$.
11. Write Leibnitz’s theorem of successive derivatives up to 4th order.
12. Find $\lim_{x\to1}x^{\frac{1}{x-1}}$.
13. The volume generated by $y=\frac1x$ about the $x$-axis, bounded by $y=0$, $x=2$, $x=b$ $(0<b<2)$, is $3$. Find $b$.
14. Determine the type of the conic $8x^2+10xy+3y^2+22x+14y+15=0$.
15. Show that $M(x,y)\,dx+N(x,y)\,dy=0$ is exact if $\frac{\partial N}{\partial x}=\frac{\partial M}{\partial y}$.

Group B

Answer any four questions: $5\times4=20$

16. If $y=\sin(m\cos^{-1}\sqrt{x})$, prove that $\lim_{x\to0}\frac{y_{n+1}}{y_n}=\frac{4n^2-m^2}{4n+2}$.
17. If $I_{m,n}=\int_0^{\pi/2}\sin^m x\cos^n x\,dx$, prove that $I_{m,n}=\frac{n-1}{m+n}I_{m,n-2}$. Hence find $\int_0^1 x^m\sqrt{1-x^2}\,dx$.
18. Find the area of the segment of the parabola $y=x^2-7x+9$ cut off by the line $y=3-2x$.
19. Solve $y=2px+y^2p^3$ and find the general and singular solutions.
20. Find $a,b$ such that $\lim_{x\to0}\frac{a\sin2x-b\sin x}{x^3}=1$.
21. Find the asymptotes of $x^3-x^2y-y^2x+y^3+2x^2-4y^2+2xy+x+y+1=0$.

Group C

Answer any two questions: $10\times2=20$

22. (a) A sphere of radius $r$ passes through the origin and cuts the axes in $A,B,C$. Prove that the locus of the foot of the perpendicular from the origin to the plane $ABC$ is $$(x^2+y^2+z^2)^2(x^{-2}+y^{-2}+z^{-2})=4r^2.$$ (b) Find the polar equation of the normal to the conic $\frac{l}{r}=1+e\cos\theta$.

23. (a) Find the equation of the cylinder whose generators are parallel to $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$ and which passes through $z=0$, $3x^2+7y^2=12$.
(b) Show that $lx+my+nz=p$ is a tangent plane to $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ if $a^2l^2+b^2m^2+c^2n^2=p^2$.

24. (a) Reduce $(px-y)(x-py)=2p$ by the substitutions $x^2=u$, $y^2=v$. Hence solve it and show that the singular solution is $$(2-x^2-y^2)^2=4x^2y^2.$$ (b) Given $x^{2/3}+y^{2/3}=c^{2/3}$ as the envelope of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, show that $a+b=c$.

25. (a) Reduce $x^2+3y^2+3z^2-2yz-2xy-2zx+1=0$ to canonical form and determine its type.
(b) Show that $$\int_a^b (x-a)^m(b-x)^n\,dx=\frac{m!n!}{(m+n+1)!}(b-a)^{m+n+1}.$$