B.Sc. Mathematics Honours GE-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.

Answer any three from the following questions: 3 x 20

1. (a) Show that the curve $y^{3}=8x^{2}$ is Concave to the foot of the ordinate everywhere except at Origin.
(b) State some natures of Hyperbolic Sine.
(c) If $y=2 \cos x(\sin x-\cos x)$, show that $y_{10}(0)=2^{10}$.
(d) Find the envelopes of the straight line $\frac{x}{a}+\frac{y}{b}=1$ where the parameters a and b are connected by the relation $a^{2}+b^{2}=c^{2}$
4+4+6+6

2. (a) If $y=(ax+b)^{m}$ find $D^{n}(ax+b)^{m}$.
(b) Evaluate $Lt_{x\rightarrow0}(\cos mx)^{\frac{n}{x^{2}}}$.
(c) Find the length of a quadrant of the circle $r=2a \sin \theta$.
(d) Evaluate $\int_{0}^{\pi/2} \sin^{8}x \cos^{6}xdx$
(e) The circle $x^{2}+y^{2}=a^{2}$ revolves about the x-axis. Show that the surface area and the volume of the sphere thus generated are respectively $4\pi a^{2}$ and $\frac{4}{3}\pi a^{3}$.
4+4+4+4+4

3. (a) Evaluate $\int_{0}^{\pi/4} \tan^{5}xdx$.
(b) Find the volume of the solid generated by revolving the part of parabola $x^{2}=4ay$, $a>0$ between the ordinates $y=0$ and $y=a$ about its axis.
(c) Find the area of the smaller portion enclosed by the curves $x^{2}+y^{2}=9$ and $y^{2}=8x$.
(d) Trace out the curve cycloid $x=a(\theta-\sin \theta)$, $y=a(1-\cos \theta)$
4+4+6+6

4. (a) Through what angle must be the axis be turned to remove xy term from $7x^{2}+4xy+3y^{2}=0$.
(b) If pair of 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, prove that $pq+1=0$.
(c) 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 $x^{2}+y^{2}=9$, $z=1$.
(d) The plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ meets the co-ordinate axes A, B, C. Find the equation of the cone generated by the straight lines drawn from O to meet the circle ABC.
4+4+6+6

5. (a) Show that the semi-latus rectum of a conic is the harmonic mean between the segments of a focal chord.
(b) Find the equation of the circle on the sphere $x^{2}+y^{2}+z^{2}=49$ whose centre is at the point (2,-1,3).
(c) Show that the straight line $r \cos(\theta-\alpha)=p$ touches the conic $\frac{l}{r}=1+e \cos \theta$ if $(l \cos \alpha-ep)^{2}+l^{2} \sin^{2} \alpha = p^{2}$.
(d) Find the equation of the plane which passes through the point (2,1,-1) and is orthogonal to each of the planes $x-y+z=1$ and $3x+4y-2z=0$.
4+4+6+6

6. (a) Find the differential equation of all circles passing through the origin having centres on the x-axis.
(b) Find an integrating factor of the differential equation $(3x^{2}y^{4}+2xy)dx+(2x^{3}y^{3}-x^{2})dy=0$
(c) Find the general and the singular solutions of $y=px+\sqrt{a^{2}p^{2}+b^{2}}$.
(d) Reduce the differential equation $(px^{2}+y^{2})(px+y)=(p+1)^{2}$ to clairaut's form by the substitution $u=xy$, $v=x+y$ and then find the general solution. Where $p=\frac{dy}{dx}$
4+4+6+6