UNIT-I: Calculus-I

1. Answer any three questions: \( \pmb{[2\times 3]} \)
   1. What do you mean by asymptote? Does asymptote exist for every curve?
   2. Your answer: Find the value of \( \lim\limits_{x \to 1} x^{\frac{1}{x – 1} }\)
   3. Define the point of inflection of a curve.
   4. Find the envelope of the straight line \( \frac{x}{a}+\frac{y}{b}=1 \), where the parameters \( a \) and \( b \) are connected by the relation \( ab = c^2 \).
   5. Write the Leibnitz theorem for successive derivatives up to the 4th order.
2. Answer any one question \( \pmb{[10\times 1]} \):
   1. 1. If \( s \) be the length of the arc \( 3ay^{2}=x(x-a)^{2} \), measured from the origin to any point \( (x, y) \), show that \( 3s^{2} = 4x^2 + 3y^2 \). \( \pmb{[5] } \)
      2. Show that the curve \( y = 3x^5 – 40x^3 + 3x – 20 \) is concave upwards for \( -2 \lt x \lt 0 \) and \( 2 \lt x \lt \infty \), but convex upwards for \( -\infty \lt x \lt -2 \) and \( 0 \lt x \lt 2 \). Also, show that \( x = -2, 0, 2 \) are its points of inflection. \( \pmb{[2+2+1] } \)
   2. 1. Trace the curve \( x^{2}y^{2}=a(y^{2}-x^{2}) \). \( \pmb{[5] } \)
      2. If \( \alpha, \beta \) be the roots of the equation \( ax^2 + bx + c = 0 \), show that \( \lim\limits_{x \to a} \frac{1-\cos \left(ax^2 + bx + c \right)}{\left(x-\alpha \right)^{2}}=\frac{1}{2}a^{2}\left(\alpha^{2}-\beta^{2} \right) \). \( \pmb{[5] } \)

   UNIT-II: Calculus-II

   1. Answer any two questions: \( \pmb{[2\times 2] } \)
      1. If \( I_{n}=\int^{\frac{\pi}{4}}_{0} \tan^n x \, dx\), where \( n \) is a positive integer greater than 1, then prove that \( I_{n}=\frac{1}{n-1}-I_{n-2}\).
      2. The volume of the solid generated by the revolution of the curve \( y = \frac{1}{x} \), bounded by \( y = 0, x = 2, x=b~(~0\lt b \lt 2 ) \) about the x-axis is 3. Find the value of \( b \).
      3. What is the formula for finding the area of the curve \( y = \psi(t), x = \phi(t) \), where \( t \) is the parameter?
   2. Answer any two questions: \( \pmb{[5\times 2] } \)
      1. If \( I_{m,n}=\int^{\frac{\pi}{2}}_{0} \cos^m x \sin nx \, dx\) then show that \( I_{m,n}= \frac{1}{m+n}+\frac{1}{m+n}I_{m-1,n-1}\). And also deduce that \( I_{m,n}= \frac{1}{2^{m+1}}\left[2+\frac{2^2}{2}+\frac{2^3}{3}+…+\frac{2^m}{m} \right]\).
      2. Find the area of the surface generated by revolving the curve \( x = \cos t, y = 2 + \sin t \), where \( 0 \leq t \leq 2\pi \), about the x-axis.
      3. Find the arc length parameter along the curve \(C:\”{r}(t)=\left(1+2t \right)\hat{i}+\left(1+3t \right)\hat{j}+6\left(1-t \right)\hat{k} \) from the point \(t=0\).

   UNIT-III: Geometry

   1. Answer any three questions: \( \pmb{[2\times 3] } \)
      1. Determine the type of the conic \( 8x^2 + 10xy + 3y^2 + 22x + 14y + 15 = 0 \).
      2. Show the plane \( z -1=0 \) which intersects the ellipsoid \( \frac{x^2}{48}+\frac{y^2}{12}+\frac{z^2}{4}=1 \) is an ellipse. Determine its semi-axes.
      3. Find the equation of the sphere through the circle \( x^2 + y^2 + z^2 = 25 \), \( x + 2y -z+ 2 = 0 \) and the point \( (1, 1, 1) \).
      4. Find the equation of the cylinder where generators are parallel to the line \(x=-\frac{y}{2}=\frac{Z}{3} \) and whose guiding curve is the ellipse \(x^2 +y^2 =1, z=3 \).
      5. Find the equation of the right circular cone where axis is \(\frac{x}{1}=\frac{y}{0}=\frac{Z}{-2} \) and radius is equal to \(7 \).
   2. Answer any one question: \( \pmb{[5\times 1] } \)
      1. Find the polar equation of the tangent to the circle \( r = 2d \cos \theta \) at the point whose vectorial angle is \( \theta_0 \).
      2. Find the equations of the generating lines of the hyperboloid of one sheet \(\frac{x^2}{4}+\frac{y^2}{9}-\frac{z^2}{16}=1 \) which passes through the point \((2,3,-4)\).
   3. Answer any one question: \( \pmb{[10\times 1] } \)
      1. 1. Find the locus of a luminous point, if the ellipsoid \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \) casts a circular shadow on the plane \(z=0\). \( \pmb{[5] } \)
         2. Reduce the equation \(4x^2 -4xy +y^2-8x-6y+5=0 \) to canonical form. \( \pmb{[5] } \)
      2. 1. Prove that the locus of the foot of the perpendicular from the a focus of the conic \(\frac{l}{r}=1-e\cos \theta \) on a tangent to it, is given by \(r^2 (1-e^2)-2ler \cos \theta – l^2=0 \). \( \pmb{[5] } \)
         2. Prove that the axes of sections of the conicoid \(ax^2 +by^2+cz^2=1 \) which pass through the line \(\frac{x}{l}=\frac{y}{m}=\frac{z}{n} \) lie on the cone \(\frac{(b-c)}{x}\left(mz-ny \right)+\frac{(c-a)}{y}\left(nx-lz \right)+\frac{(a-b)}{z}\left(ly-mx \right)=0 \). \( \pmb{[5] } \)

   UNIT-IV: Differential Equations

   1. Answer any two questions: \( \pmb{[2\times 2] } \)
      1. Find the integrating factor of the following differential equation \( \frac{dy}{dx}+ \frac{xy}{1-y^2} -y\sqrt{x} =0 \).
      2. Reduce the equation \( \sin y \frac{dy}{dx} = \cos x (2 \cos y – \sin^2 x) \) into a linear equation.
      3. Show that the equation \( M(x, y) \, dx + N(x, y) \, dy = 0 \) will be exact if \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).
   2. Answer any one question: \( \pmb{[5\times 1] } \)
      1. 1. Show that the substitution \( z = ax + by + c \) changes \( y^{\prime} = f(ax + by + c) \) into an equation with separable variables, and apply this method to solve the equation \( y^{\prime} = \sin^2 (x-y+1) \).
         2. Reduce the equation \(\left( 2x^2 -1\right)\left( \frac{dy}{dx}\right)^2+\left( x^2 + y^2 +2xy+2\right)\frac{dy}{dx}+2y^2+1=0 \) to clairaut’s form by the substitution \(x+y=u \) and \(xy-1=v \), hence solve the equation.
      2. 1. Show that if the substitution \(y_1 \) and \(y_2 \) be the solutions of the equation \(\frac{dy}{dx}+Py=Q \) where \(P\) and \(Q\) are functions of \(x\) alone, and \(y_2 =y_1 z\) then \( z=1+ae^{-\int \frac{Q}{y_1} dx}\).
         2. Show that the solution \(\frac{dy}{dx}+Py=Q \) can also be written in the form \(y=\frac{P}{Q}-e^{\int P \, dx}\left[c+\int e^{\int P \, dx} \, \left(\frac{P}{Q} \right) \right] \).