Paper I PYQs-2016
Section A
1.(a)(i) Using elementary row operations, find the inverse of A=[121132101].
[6M]
1.(a)(ii) If A=[113526−2−1−3] then find A14+3A−2I.
[4M]
1.(b)(i) Using elementary row operation find the condition that the linear equations have a solution:
x−2y+z=a
2x+7y−3z=b
3x+5y−2z=c
[7M]
1.(b)(ii) If w1={(x,y,z)|x+y−z=0}, w2={(x,y,z)|3x+y−2z=0}, w3={(x,y,z)|x−7y+3z=0}, then find dim(w1∩w2∩w3) and dim(w1+w2).
[3M]
1.(c) Evaluate I=∫103√xlog(1x)dx
[10M]
1.(d) Find the equation of the sphere which passes though the circle x2+y2=4; z=0 and is cut by the plane x+2y+2z=0 in a circle of radius 3.
[10M]
1.(e) Find the shortest distance between the lines x−12=y−24=z−3 and y−mx=z=0. For what value of m will the two lines intersect?
[10M]
2.(a)(i) If M2(R) is space of real matrices of order 2×2 and P2(x) is the space of real polynomials of degree at most 2, then find the matrix representation of T:M2(R)→P2(x) such that T([abcd])=a+c+(a−d)x+(b+c)x2, with respect to the standard bases of M2(R) and P2(x), further find null space of T.
[10M]
2.(a)(ii) If T:P2(x)→P3(x) is such that T(f(x))=f(x)+5∫x0f(t)dt, then choosing {1,1+x,1−x2} and {1,x,x2,x3} as bases of P2(x)$and$P3(x) respectively, find the matrix of T.
[6M]
2.(b)(i) If A=[1−12−21−1123] is the matrix representation of a linear transformation T:P2(x)→P2(x) with respect to the bases {1−x,x(1−x),x(1+x)} and {1,1+x,1+x2} then find T.
[6M]
2.(b)(ii) Prove that eigen values of a Hermitian matrix are all real.
[8M]
2.(c) If A=[110110001], then find the eigen values and eigenvectors of A.
[6M]
3.(a) Find the maximum and minimum values of x2+y2+z2 subject to the conditions x24+y25+z225=1 and x+y−z=0.
[20M]
3.(b) Let f(x,y)={2x4y−5x3sy2+y5(x2+y2)2,(x,y)≠(0,0)0,(x,y)=(0,0)
Find a δ>0 such that |f(x,y)−f(0,0)|<0.01 whenever √x2+y2<δ.
[15M]
Remarks: Original question has printing mistake
3.(c) Find the surface area of the plane x+2y+2z=12 cut off by x=0, y=0 and x2+y2=16.
[15M]
4.(a) Find the surface generated by a line which intersects the line y=a=z,x+3z=a=y+z and parallel to the plane x+y=0.
[10M]
4.(b) Show that the cone 3yz−2zx−2xy=0 has an infinite set of three mutually perpendicular generators. If x1=y1=zz is a generator belonging to one such set, find the other two.
[10M]
4.(c) Evaluate ∬Rf(x,y)dxdy over the rectangle R=[0,1;0,1], where f(x,y)={x+y, if x2<y<2x20,
[15M]
4.(d) Find the locus of the point of intersection of three mutually perpendicular tangent planes to the conicoid ax2+by2+cz2=1.
[15M]
Section B
5.(a) Find a particular integral of d2ydx2+y=ex/2sinx√32.
[10M]
5.(b) Prove that the vectors →a=ˆiˆjˆk, →b=ˆiˆjˆk, →c=ˆiˆjˆk can form the sides of a triangle. Find the lengths of the medians of the triangle.
[10M]
5.(c) Solve:
dydx=11+x2(etan−1x−y)[10M]
5.(d) Show that the family of parabolas y2=4cx+4c2 is self orthogonal.
[10M]
5.(e) A particle moves with a central acceleration which varies inversely as the cube of the distance. If it is projected from an apse at a distance a from the origin with a velocity which is √2 times the velocity for a circle of radius a, then find the equation to the path.
[10M]
6.(a) Solve {y(1−xtanx)+x2cosx}dx−xdy=0.
[10M]
6.(b) Using the method of variation of parameters, solve the differential equation (D2+2D+1)y=e−xlog(x),[D≡ddx].
[15M]
6.(c) Find the general solution of the equation x2d3ydx3−4xd2ydx2+6dydx=4.
[15M]
6.(d) Using Laplace transformation, solve the following: y′′−2y′−8y=0, y(0)=3, y′(0)=6.
[10M]
7.(a) A uniform rod AB of length 2 a movable about a hinge at A rests with other end against a smooth vertical wall. If α is inclination of the rod to the vertical, prove that the magnitude of reaction of the hinge is 12W√4+tan2α, where W is the weight of the rod.
[15M]
7.(b) Two weights P and Q are suspended from a fixed point O by strings OA, OB and are kept apart by a light rod AB. If the strings OA and OB make angles α and β with the rod AB, show that the angle θ which the rod makes with the vertical is given by tanθ=P+QPcotα−Qcotβ.
[15M]
7.(c) A square ABCD, the length of whose sides is a, is fixed in a vertical plane with two of its sides horizontal. An endless string of length l(>4a) passes over four pegs at the angles of the board and through a ring of weight W which is hanging vertically. Show that the tension of the string is W(l−3a)2√l2−6la+8a2.
[20M]
8.(a) Find f(r) such that ∇f=→rr5 and f(1)=0.
[10M]
8.(b) Prove that the vector →a=3→i+ˆj−2ˆk, →b=−ˆi+3ˆj+4ˆk, →c=4ˆi−2ˆj−6ˆk can from the sides of a triangle. Find the length of the medians of the triangle.
[10M]
8.(c) A particle moves in a straight line. Its acceleration is directed towards a fixed point O in the line and is always equal to μ(a5x2)1/3 when it is at a distance x from O. If it starts from rest at a distance a from O, then find the time the particle will arrive at O.
[15M]
8.(d) For the cardioid r=a(1+cosθ), show that the square of the radius of curvature at any point (r,θ) is proportion to r. Also find the radius of curvature if θ=0, π4, π2.
[15M]