Paper I PYQs-2016

1.(a) Let T:IB3→B4 be given by
T(x,y,z)=(2x−y,2x+z,(z)+2z,x+y+z).

Find the matrix of T with respect to standard basis of EB3 and R4 (i.e., (1,0,0), (0,1,0), etc. Examine if T is a linear map.

[8M]

1.(b) Show that x(1)+x)<log(1+x)<x$for$x>0.

[8M]

1.(c) Examine if the function f(x,y)=xyx2+y2,(x,y)≠(0,0) and f(0,0)=0 is continuous at (0,0). Find ∂f∂x and ∂f∂y at points other than origin.

[8M]

1.(d) If the point (2,3) is the mid-point of a chord of the parabola y2=4x, then obtain the equation of the chord.

[8M]

1.(e) For the matrix A=[−1222−1222−1], obtain the eigen value and get the value of A4+3A3−9A2.

[8M]

2.(a) After changing the order of integration of ∫∞0∫∞0e−xysinnxdxdy, show that ∫∞0sinnxxdx=π2.

[10M]

2.(b) A perpendicular is drawn from the centre of ellipse Qx2a2+y2b2=1 to any tangent. Prove that the locus of the foot of the perpendicular is given-by (x2+y2)2=a2x2+b2y2.

[10M]

2.(c) Using mean value theorem, find a point on the curve y=√x−2, defined on [2,3], where the tangent (is, parallel to the chord joining the end points of the curve.

[10M]

2.(d) Let T be a linear map such that T:v3→v2 defined by T(e1)=2f1−f2, T(e2)=f1+2f2TT(e3)=0f1+0f2, where e1,e2,e3 and f1,f2 are standard basis in v3 and V2. Find the matrix of T relative to these basis. Further take two other basis B1[(1,1,0)(1,0,1)(0,1,1)] and B2[(1,1)(1,−1)]. Obtain the matrix T1 relative to B1 and B2.

[10M]

3.(a) For the matrix A=[3−342−340−11], find two non-singular matrices P and Q such that PAQ=I. Hence find A−1.

[10M]

3.(b) Using Lagrange’s method of multipliers, find the point on the plane 2x+3y+4z=5 which is closest to the point (1,0,0).

[10M]

3.(c) Obtain the area between the curve x=3(secθ+cosθ) and its asymptote x=3.

[10M]

3.(d) Obtain the equation of the sphere on which the intersection of the plane 5x−2y+4z+7=0 with the sphere which has (0,1,0) and (3,−5,2) as the end points of its diameter is a great circle.

[10M]

4.(a) Examine whether the real quadratic form 4x2−y2+2z2>2xy−2yz−4xz is a positive definite or not. Reduce it to its Aiagonal form and determine its signature.

[10M]

4.(b) Show that the integral ∫∞0e−xxα−1dx,α>0 exists, by separately taking the cases for α≥1 and 0<α<1.

[10M]

4.(c) Prove that Γ(2z) = 22z−1√πΓ(z)Γ(z+12)

[10M]

4.(d) A plane xa+yb+zc=a2 cuts the coordinate plane at A, B, C. Find the equation of the cone with vertex at origin and guiding curve as the circle passing through A, B,\mathrm{C}$$.

[10M]

5.(a) Obtain the curve which passes through (1,2) and has a slope =−2xyx2+1 Dbtain one asymptote to the curve.

[8M]

5.(b) Solve the de to get the particular integral of d4ydx4+2d2ydx2+y=x2cosx.

[8M]

5.(c) A weight W is hanging with the help of two strings of length l and 2l in such a way that the other ends A and B of those strings lie on a horizontal line at a distance 2l. Obtain the tension in the two strings.

[8M]

5.(d) From a point in a smooth horizontal plane, a particle is projected with velocity u at angle α to the horizontal from the foot of a plane, inclined at an angle β with respect to the horizon. Show that if will strike the plane at right angles, if cot β=2tan(α−β).

[8M]

5.(e) If E be the solid bounded by the xy plane and the paraboloid the volume E and ˉF=(zxsinyz+x3)ˆi+ cosyzˆj + (3zy2−eλ2+y2)ˆk.

[8M]

6.(a) A stone is thrown vertically with the velocity which would just carry it to a height of 40 m. Two seconds later another stone is projected vertically from the same place with the same velocity. When and where will they meet?

[10M]

6.(b) Using the method of variation of parameters, solve

[10M]

6.(c) Water is flowing through a pipe of 80 mm diameter under a gauge pressure of 60kPa, with a mean velocity of 2 m/s. Find the total head, if the pipe is 7 m above the datum line.

[10M]

6.(d) Evaluate ∬S(∇×ˉf),ˆndS for ¯f=(2x−y)ˆi−yz2ˆj−y2zˆk where S is the upper half surface of the sphere x2+y2+z2=1 bounded by its projection on the xy plane.

[10M]

7.(a) State Stokes’ theorem. Verify the Stokes’ theorem for the function ¯f=xˆi+zˆj+2yˆk, where c is the curve obtained by the intersection of the plane z=x and the cylinder x2+y2=1 and S is the surface inside the interesected cone.

[15M]

7.(b) A uniform rod of weight W is resting against an equally rough horizon and a wall, at an angle α with the wall. At this condition, a horizontal force P is stopping them from sliding, implemented at the mid-point of the rod. Prove that P=Wtan(α−2λ), where λ is the angle of friction. Is there any condition on λ and α?

[15M]

7.(c) Obtain the singular solution of the differential equation

[10M]

8.(a) A body immersed in a liquid is balanced by a weight P to which it is attached by a thread passing over a fixed pulley and when half immersed, is balanced in the same manner by weight 2P. Prove that the density of the body and the liquid are in the ratio 3:2?

[10M]

8.(b) Solve the differential equation

[10M]

8.(c) Prove that ˉa×(ˉb×ˉc)=(ˉa×→b)×→c, if and only if either ˉb=¯0 or ˉc is collinear with ˉa or ˉb is perpendicular to both ˉa and ˉc.

[10M]

8.(d) A particle is acted on a force parallel to the axis of y whose acceleration is λy, initially projected with a velocity a√λ parallel to x -axis at the point where y=a. Prove that it will describe a catenary.

[10M]