B.Sc. Mathematics Honours GE-1 Question Paper 2021 CBCS)

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

Group A

Answer any four questions.

1. (a) Find the asymptotes of $y^{3}-x^{2}y-2xy^{2}+2x^{3}-7xy+3y^{2}+2x^{2}+2x+2y+1=0$
(b) Find the envelope of the family of lines $x \cos \alpha + y \sin \alpha = a \sin \alpha \cos \alpha$ where $\alpha$ is a parameter.
5+7

2. (a) Show that $\int_{0}^{\pi/2} \sin^{5}x \cos^{6}x dx = \frac{8}{693}$
(b) Find the entire area of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$.
(c) Find the length of the curve $x=a(\cos \theta+\theta \sin \theta)$, $y=a(\sin \theta-\theta \cos \theta)$ between $\theta=0$ and $\theta=\theta_{1}$.
4+4+4

3. (a) If $\lim_{x\rightarrow0}\frac{a \sin x-\sin 2x}{\tan^{3}x}$ is finite, find the value of $a$ and the limit.
(b) Evaluate $\int \tan^{5}x dx$.
(c) Show that the area bounded by one arc of the cycloid $x=a(\theta-\sin \theta)$, $y=a(1-\cos \theta)$ and the X axis is $3\pi a^{2}$ sq.units.
4+4+4

4. (a) Reduce the equation $8x^{2}+8xy+2y^{2}+26x+13y+15=0$ to its canonical form.
(b) If $r_{1}$ and $r_{2}$ are two mutually perpendicular radius vectors of the ellipse $r^{2}=\frac{b^{2}}{1-e^{2}\cos^{2}\theta}$ Prove that $\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}$ Where $a$ and $b$ are semi-major and semi-minor axes of the ellipse.
6+6

5. (a) Find the equation of the sphere containing the circle $x^{2} + y^{2} + z^{2} + 7x - 2z + 2 = 0, 2x + 3y + 4z - 8 = 0$ as one of its great circles.
(b) Find the equation of the cone whose vertex is the origin and which has $lx+my+nz=p, ax^{2}+by^{2}+cz^{2}=1$ as its guiding curve.
6+6

6. (a) If $y^{1/m}+y^{-1/m}=2x$ then prove that $(x^{2}-1)y_{n+2}+(2n-1)xy_{n+1}+(n^{2}-m^{2})y_{n}=0$
(b) If $\alpha, \beta$ be the roots of the equation $ax^{2}+ bx +c=0$ then show that $\lim_{x\rightarrow \alpha}\frac{1-\cos(ax^{2}+bx+c)}{(x-\alpha)^{2}}=\frac{1}{2}a^{2}(\alpha-\beta)^{2}$.
(c) Find the points of inflection of the curve $x=a \tan \theta$, $y=a \sin \theta \cos \theta$
4+4+4

7. (a) A plane passing through a fixed point $(a, b, c)$ cuts the axes at A, B, C. Show that the locus of the centre of the sphere OABC is $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2$.
(b) Find the equation of the cylinder whose generators are parallel to the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and whose guiding curve is $x^{2}+y^{2}=4, z=1$.
(c) Reduce the following equation to its canonical form: $2x^{2}-4xy-y^{2}+20x-2y+17=0$
3+4+5

8. (a) If $y=\sin(m \sin^{-1}x)$ then show that $(1-x^{2})y_{n+2}-(2n+1)xy_{n+1}+(m^{2}-n^{2})y_{n}=0$
(b) Find the point of intersection of the tangents at $\theta=\alpha$ and $\theta=\beta$ on the conic $\frac{l}{r}=1+e \cos \theta$
6+6

Group B

Answer any six questions.

9. Examine if the ODE $e^{y}dx+(1+xe^{y})dy=0$ is exact.
2

10. If the expression $ax + by$ changes to $a'x' + b'y'$ by a rotation of the rectangular axes about the origin, prove that $a^{2}+b^{2}=a'^{2}+b'^{2}$.
2

11. Write down the equation of the sphere one of whose diameter has end points $(2, 1, 3)$ and $(0, 4, 5)$. Find its radius.
2

12. Find $\lim_{x\rightarrow0}(\frac{1}{x}-\frac{1}{\sin x})$
2

13. Find the polar equation of the straight line joining the two points $(r_{1}, \theta_{1})$ and $(r_{2}, \theta_{2})$.
2

14. Find the perimeter of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$.
2

15. Show that $\frac{1}{3x^{3}y^{3}}$ is an integrating factor of $y(xy+2x^{2}y^{2})dx+x(xy-x^{2}y^{2})dy=0$.
2

16. Find the angle through which the axes are to be rotated so that the equation $17x^{2}-18xy-7y^{2}=1$ may be reduced to the form $Ax^{2}+By^{2}=1, A>0$. Find also A, B.
2

17. Find the points on the conic $\frac{12}{r}=1-4 \cos \theta$ whose radius vector is 4.
2

18. Show that the curve $y=e^{-x^{2}}$ has points of inflection at $x=\pm\frac{1}{\sqrt{2}}$
2

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

Group A

Answer any four questions.

1. (a) Find the asymptotes of $y^{3}-x^{2}y-2xy^{2}+2x^{3}-7xy+3y^{2}+2x^{2}+2x+2y+1=0$
(b) Find the envelope of the family of lines $x \cos \alpha + y \sin \alpha = a \sin \alpha \cos \alpha$ where $\alpha$ is a parameter.
5+7

2. (a) Show that $\int_{0}^{\pi/2} \sin^{5}x \cos^{6}x dx = \frac{8}{693}$
(b) Find the entire area of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$.
(c) Find the length of the curve $x=a(\cos \theta+\theta \sin \theta)$, $y=a(\sin \theta-\theta \cos \theta)$ between $\theta=0$ and $\theta=\theta_{1}$.
4+4+4

3. (a) If $\lim_{x\rightarrow0}\frac{a \sin x-\sin 2x}{\tan^{3}x}$ is finite, find the value of $a$ and the limit.
(b) Evaluate $\int \tan^{5}x dx$.
(c) Show that the area bounded by one arc of the cycloid $x=a(\theta-\sin \theta)$, $y=a(1-\cos \theta)$ and the X axis is $3\pi a^{2}$ sq.units.
4+4+4

4. (a) Reduce the equation $8x^{2}+8xy+2y^{2}+26x+13y+15=0$ to its canonical form.
(b) If $r_{1}$ and $r_{2}$ are two mutually perpendicular radius vectors of the ellipse $r^{2}=\frac{b^{2}}{1-e^{2}\cos^{2}\theta}$ Prove that $\frac{1}{r_{1}^{2}}+\frac{1}{r_{2}^{2}}=\frac{1}{a^{2}}+\frac{1}{b^{2}}$ Where $a$ and $b$ are semi-major and semi-minor axes of the ellipse.
6+6

5. (a) Find the equation of the sphere containing the circle $x^{2} + y^{2} + z^{2} + 7x - 2z + 2 = 0, 2x + 3y + 4z - 8 = 0$ as one of its great circles.
(b) Find the equation of the cone whose vertex is the origin and which has $lx+my+nz=p, ax^{2}+by^{2}+cz^{2}=1$ as its guiding curve.
6+6

6. (a) If $y^{1/m}+y^{-1/m}=2x$ then prove that $(x^{2}-1)y_{n+2}+(2n-1)xy_{n+1}+(n^{2}-m^{2})y_{n}=0$
(b) If $\alpha, \beta$ be the roots of the equation $ax^{2}+ bx +c=0$ then show that $\lim_{x\rightarrow \alpha}\frac{1-\cos(ax^{2}+bx+c)}{(x-\alpha)^{2}}=\frac{1}{2}a^{2}(\alpha-\beta)^{2}$.
(c) Find the points of inflection of the curve $x=a \tan \theta$, $y=a \sin \theta \cos \theta$
4+4+4

7. (a) A plane passing through a fixed point $(a, b, c)$ cuts the axes at A, B, C. Show that the locus of the centre of the sphere OABC is $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}=2$.
(b) Find the equation of the cylinder whose generators are parallel to the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and whose guiding curve is $x^{2}+y^{2}=4, z=1$.
(c) Reduce the following equation to its canonical form: $2x^{2}-4xy-y^{2}+20x-2y+17=0$
3+4+5

8. (a) If $y=\sin(m \sin^{-1}x)$ then show that $(1-x^{2})y_{n+2}-(2n+1)xy_{n+1}+(m^{2}-n^{2})y_{n}=0$
(b) Find the point of intersection of the tangents at $\theta=\alpha$ and $\theta=\beta$ on the conic $\frac{l}{r}=1+e \cos \theta$
6+6

Group B

Answer any six questions.

9. Examine if the ODE $e^{y}dx+(1+xe^{y})dy=0$ is exact.
2

10. If the expression $ax + by$ changes to $a'x' + b'y'$ by a rotation of the rectangular axes about the origin, prove that $a^{2}+b^{2}=a'^{2}+b'^{2}$.
2

11. Write down the equation of the sphere one of whose diameter has end points $(2, 1, 3)$ and $(0, 4, 5)$. Find its radius.
2

12. Find $\lim_{x\rightarrow0}(\frac{1}{x}-\frac{1}{\sin x})$
2

13. Find the polar equation of the straight line joining the two points $(r_{1}, \theta_{1})$ and $(r_{2}, \theta_{2})$.
2

14. Find the perimeter of the astroid $x^{2/3}+y^{2/3}=a^{2/3}$.
2

15. Show that $\frac{1}{3x^{3}y^{3}}$ is an integrating factor of $y(xy+2x^{2}y^{2})dx+x(xy-x^{2}y^{2})dy=0$.
2

16. Find the angle through which the axes are to be rotated so that the equation $17x^{2}-18xy-7y^{2}=1$ may be reduced to the form $Ax^{2}+By^{2}=1, A>0$. Find also A, B.
2

17. Find the points on the conic $\frac{12}{r}=1-4 \cos \theta$ whose radius vector is 4.
2

18. Show that the curve $y=e^{-x^{2}}$ has points of inflection at $x=\pm\frac{1}{\sqrt{2}}$
2