Paper I PYQs-2010
Section A
1.(a) Show that the set
P[t]={at2+bt+c/a,b,c∈R}forms a vector space over the field R. Find a basis for this vector space. What is the dimension of this vector space?
[8M]
1.(b) Determine whether the quadratic form
q=x2+y2+2xz+4yz+3z2is positive definite.
[8M]
1.(c) Prove that. between any two real roots of exsinx=1, there is at least one real root of excosx+1=0.
[8M]
1.(d) Let f be a function defined on R such that
f(x+y)=f(x)+f(y),x,y∈RIf f is differentiable-at one point of R, then prove that f is differentiable on R
[8M]
1.(e) If a plane cuts the axes in A,B,C and (a,b,c) are the coordinates of the centroid of the triangle ABC, then show that the equation of the plane is xa+yb+zc=3.
[8M]
1.(f) Find the equations of the spheres passing through the circle
x2+y2+z2−6x−2z+5=0,y=0and touching the plane 3y+4z+5=0.
[8M]
2.(a) Show that the vectors
α1=(1,0,−1),α2=(1,2,1),α3=(0,−3,2)form a basis for R3. Find the components of (1,0,0) w.r.t. the basis {α1,α2,α3}.
[10M]
2.(b) Find the characteristic polynomial of (001102013). Verify Cayley-Hamilton theorem for this matrix and hence find its inverse.
[10M]
2.(c) Let A=(5−6−6−1.423−6−4). Find an invertible matrix P such that P−1 AP is a diagonal matrix.
[10M]
2.(d) Find the rank of the matrix
(1211224347−1−22533621348689)
[10M]
3.(a) Discuss the convergence of the integral ∫∞0dx1+x4sin2x
[10M]
3.(b) Find the extreme value of xyz if x+y+z=a.
[10M]
3.(c) Let
f(x,y)={xy(x2−y2)x2+y2, if (x,y)≠(0,0)0, if (x,y)=(0,0)
Show that:
(i) fxy(0,0)≠fyx(0,0)
(ii) f is differentiable at (0,0)
[10M]
3.(d) Evaluate ∬, where D is the region bounded by the parabolas y=2 x^{2} and y=1+x^{2}.
[10M]
4.(a) Prove that the second degree equation
x^{2}-2 y^{2}+3 z^{2}+5 y z-6 z x-4 x y+ 8 x-19 y-2 z-20=0represents a cone whose vertex is (1,-2,3).
[10M]
4.(b) If the feet of three normals drawn from a point \mathrm{P} to the ellipsoid \dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1 lie in the plane \dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1, prove that the feet of the other three normals lie in the plane \dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}+1=0.
[10M]
4.(c) If \dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{3} represents one of the three mutually perpendicular generators of the cone 5 \mathrm{yz}-8 \mathrm{zx}-3 \mathrm{xy}=0, find the equations of the other two.
[10M]
4.(d) Prove that the locus of the point of intersection of three tangent planes to the ellipsoid \dfrac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\dfrac{\mathrm{y}^{2}}{\mathrm{b}^{2}}+\dfrac{\mathrm{z}^{2}}{\mathrm{c}^{2}}=1, which are parallel to the conjugate diametral planes of the ellipsoid \dfrac{x^{2}}{\alpha^{2}}+\dfrac{y^{2}}{\beta^{2}}+\dfrac{z^{2}}{\gamma^{2}}=1 is \dfrac{x^{2}}{\alpha^{2}}+\dfrac{y^{2}}{\beta^{2}}+\dfrac{z^{2}}{\gamma^{2}}=\dfrac{a^{2}}{\alpha^{2}}+\dfrac{b^{2}}{\beta^{2}}+\dfrac{c^{2}}{\gamma^{2}}.
[10M]
Section B
5.(a) Show that \cos (\mathrm{x}+\mathrm{y}) is an integrating factor of
y d x+[y+\tan (x+y)] d y=0Hence solve it.
[8M]
5.(b) Solve
\dfrac{d^{2} y}{d x^{2}}-2 \dfrac{d y}{d x}+y=x e^{x} \sin x
[8M]
5.(c) A uniform rod AB rests with one end on a smooth vertical wall and the other on a smooth inclined plane, making an angle \alpha with the horizon. Find the positions of equilibrium and discuss stability.
[8M]
5.(d) A particle is thrown over a triangle from one end of a horizontal base and grazing the vertex falls on che other end of the base. If \theta_{1} and \theta_{2} be the base angles and \theta be the angle of projection, prove that,
\tan \theta=\tan \theta_{1}+\tan \theta_{2}[8M]
5.(e) Prove that the horizontal line through the centre of pressure of a rectangle immersed in a liquid with one side in the surface, divides the rectangle in two parts, the fluid pressure on which, are in the ratio, 4:5.
[8M]
5.(f) Find the directional derivation of \overrightarrow{\mathrm{V}}^{2}, where, \overrightarrow{\mathrm{V}}=\mathrm{xy}^{2} \overrightarrow{\mathrm{i}}+\mathrm{zy}^{2} \overrightarrow{\mathrm{j}}+\mathrm{xz}^{2} \overrightarrow{\mathrm{k}} at the point (2,0,3) in the direction of the outward normal to the surface x^{2}+y^{2}+z^{2}=14 at the point (3,2,1).
[8M]
6.(a) Solve the following differential equation
\dfrac{d y}{d x}=\sin ^{2}(x-y+6)[8M]
6.(b) Find the general solution of
\dfrac{d^{2} y}{d x^{2}}+2 x \dfrac{d y}{d x}+\left(x^{2}+1\right) y=0[12M]
6.(c) Solve
\left(\dfrac{d}{d x}-1\right)^{2}\left(\dfrac{d^{2}}{d x^{2}}+1\right)^{2} \quad y=x+e^{x}[10M]
6.(d) Solve by the method of variation of parameters the following equation
\left(x^{2}-1\right) \dfrac{d^{2} y}{d x^{2}}-2 x \dfrac{d y}{d x}+2 y=\left(x^{2}-1\right)^{2}[10M]
7.(a) A aniform chain of length 2 l and weight W, is suspended from two points A and B in the same horizontal line. A load P is now hung from the middle point \mathrm{D} of the chain and the depth of this point below \mathrm{AB} is found to be \mathrm{h}. Show that each terminal tension is,
\dfrac{1}{2}\left[\mathbf{P} \cdot \dfrac{l}{\mathbf{h}}+\mathbf{W} \cdot \dfrac{\mathbf{h}^{2}+i^{2}}{2 \mathrm{h} l}\right][14M]
7.(b) A particle moves with a central acceleration \dfrac{\mu}{(\text { distance })^{2}}, it is projected with velocity \mathrm{V} at \mathrm{a} distance \mathrm{R}. Show that its path is a rectangular hyperbola if the angle of projection is,
\sin ^{-1}\left[\dfrac{\mu}{\mathrm{VR}\left(\mathrm{V}^{2}-\dfrac{2 \mu}{\mathrm{R}}\right)^{1 / 2}}\right][13M]
7.(c) A smooth wedge of mass M is placed on a smooth horizontal plane and a particle of mass \mathrm{m} slides down its slant face which is inclined at an angle \alpha to the horizontal plane, Prove that the acceleration of the wedge is,
\dfrac{m g \sin \alpha \cos \alpha}{M+m \sin ^{2} \alpha}[13M]
8.(a)(i) Show that \overrightarrow{\mathrm{F}}=\left(2 \mathrm{x} y+\mathrm{z}^{3}\right) \overrightarrow{\mathrm{i}}+\mathrm{x}^{2} \overrightarrow{\mathrm{j}}+3 \mathrm{z}^{2} \mathrm{x} \overrightarrow{\mathrm{k}} is a conservative field. Find its scalar potential and also the work done in moving a particle from (1,-2,1) to (3,1,4).
[5M]
8.(a)(ii) Show that, \nabla^{2} f(r)=\left(\dfrac{2}{r}\right) f^{\prime}(r)+f^{\prime \prime}(r), where
r=\sqrt{x^{2}+y^{2}+z^{2}}[5M]
8.(b) Use divergence theorem to evaluate,
\iint_{\mathrm{s}}\left(\mathrm{x}^{3} \mathrm{dy} \mathrm{dz}+\mathrm{x}^{2} \mathrm{y} \mathrm{dz} \mathrm{dx}+\mathrm{x}^{2} \mathrm{z} \mathrm{dy} \mathrm{d} \mathrm{x}\right)where S is the sphere, x^{2}+y^{2}+z^{2}=1.
[10M]
8.(c) If \vec{A}=2 y \vec{i}-z \vec{j}-x^{2} \vec{k} and S is the surface of the parabolic cylinder y^{2}=8 x in the first octant bounded by the planes y=4, z=6, evaluate the surface integral,
\iint_{\mathrm{S}} \overrightarrow{\mathrm{A}} \cdot \hat{\mathrm{n}} \overrightarrow{\mathrm{dS}}[10M]
8.(d) Use Green’s theorem in a plane to evaluate the integral, \int_{\mathrm{C}}\left[\left(2 \mathrm{x}^{2}-\mathrm{y}^{2}\right) \mathrm{d} \mathrm{x}+\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right) \mathrm{dy}\right], where \mathrm{C} is the boundary of the surface in the xy-plane enclosed by y=0 and the semi-circle, y=\sqrt{1-x^{2}}.
[10M]