Paper I PYQs-2019

1.(a) Let T:R3→R3 be a linear operator on R3 defined by T(x,y,z)= (2y+z,x−4y,3x). Find the matrix of T in the basis {(1,1,1), (1,1,0), (1,0,0)}.

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1.(b) The eigenvalues of a real symmetric matrix A are -1, 1 and -2. The corresponding eigenvectors are 1√2(−1,1,0)T, (0,0,1)T and 1√2(−1,−1,0)T respectively. Find the matrix A4.

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1.(c) Find the volume lying inside the cylinder x2+y2−2x=0 and outside the paraboloid x2+y2=2z, while bounded by xy-plane.

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1.(d) Justify by using Rolle’s theorem or mean value theorem that there is no number k for which the equation x3−3x+k=0 has two distinct solutions in the interval [−1,1].

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1.(e) If the coordinates of the points A and B are respectively (bcosα,bsinα) and (acosβ,asinβ) and if the line joining A and B is produced to the point M(x,y) so that AM:MB=b:a, then show that xcosα+β2+ysinα+β2=0

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2.(a) Determine the extreme values of the function f(x,y)=3x2−6x+2y2−4y in the region {(x,y)∈R2:3x2+2y2≤20}

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2.(b) Consider the singular matrix

Given that one eigenvalue of A is 4 and one eigenvector that does not correspond to this eigenvalue 4 is (1100)T. Find all the eigenvalues of A other than 4 and hence also find the real numbers p, q, r that satisfy the matrix equation A4+pA3+qA2+rA=0

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2.(c) A line makes angles α,β,γ,δ with the four diagonals of a cube. Show that

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3.(a) Consider the vectors x1=(1,2,1,−1), x2=(2,4,1,1), x3=(−1,−2,0,−2) and x4=(3,6,2,0) in R4. Justify that the linear span of the set {x1,x2,x3,x4} is a subspace of R4, defined as

Can this subspace be written as {(α,2α,β,2α−3β):α,β∈R}? What is the dimension of this subspace?

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3.(b) The dimensions of a rectangular box are linear functions of time- l(t), w(t) and h(t). If the length and width are incréasing at the rate 2 cm/sec and the height is decreasing at the rate 3 cm/sec find the rates at which the volume V and with respect to time. If l(0)=10, w(0)=8 and the surface area S are changing h(0)=20, is V increasing or decreasing, when t=5 sec? What about S, when t=5 sec?

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3.(c) Show that the shortest distance between the straight lines

and

is 3√30. Find also the equation of the line of shortest distance.

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4.(a) Using elementary row operations, reduce the matrix A=[2130302511112113] to reduced echelon form and find the inverse of A and hence solve the system of linear equations AX=b, where X=(x,y,z,u)T and b=(2,1,0,4)T

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4.(b) Find the centroid of the solid generated by revolving the upper half of the cardioid r=a(1+cosθ) bounded by the line θ=0 about the initial line. Take the density of the solid as uniform.

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4.(c) A variable plane is parallel to the plane xa+yb+zc=0 and meets the axes at the points A,B and C. Prove that the circle ABC lies on the cone

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5.(a) Solve the differential equation (D2+1)y=x2sin2x;D≡ddx.

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5.(b) Solve the differential equation (px−y)(py+x)=h2p, where p=y′.

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5.(c) A 2 metres rod has a weight of 2N and has its centre of gravity at 120 cm from one end. At 20 cm,100 cm and 160 cm from the same end are hing loads of 3N, 7N and 10N respectively. Find the point at which the rod must be supported if it is to remain horizontal.

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5.(d) Let ˉr=ˉr(s) represent a space curve. Find d3ˉrds3 in terms of ˉT,ˉN and ˉB where ˉT, ˉN and ˉB represent tangent, principal normal and binormal respectively. Compute dˉrds⋅(d2ˉrds2×d3ˉrds3) in terms of radius of curvature and the torsion.

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5.(e) Evaluate ∫(2,1)(0,0)(10x4−2xy3)dx−3x2y2dy along the path x4−6xy3=4y2.

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6.(a) Solve by the method of variation of parameters the differential equation

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6.(b) Find the law of force for the orbit r2=a2cos2θ (the pole being the centre of the force).

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6.(c) Verify Stokes’ theorem for ˉV=(2x−y)ˆi−yz2ˆj−y2zˆk, where S is the upper half surface of the sphere x2+y2+z2=1 and C is its boundary.

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7.(a) Find the general solution of the differential equation

Hence find the particular solution satisfying the conditions

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7.(b) A vessel is in the shape of a hollow hemisphere surmounted by a cone held with the axis vertical and vertex uppermost. If it is filled with a liquid so as to submerge half the axis of the cone in the liquid and height of the cone be double the radius (r) of its base, find the resultant downward thrust of the liquid on the vessel in terms of the radius of the hemisphere and density (p) of the liquid.

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7.(c) Derive the Frenet-Serret formulae. Verify the same for the space curve x=3cost, y=3sint, z=4t

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8.(a) Find the general solution of the differential equation

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8.(b) A shot projected with a velocity u can just reach a certain point on the horizontal plane through the point of projection. So in order to hit a mark h metres above the ground at the same point, if the shot is projected at the same elevation, find increase in the velocity of projection.

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8.(c) Derive ∇2=∂2∂x2+∂2∂y2+∂2∂z2 in spherical coordinates and compute ∇2(x(x2+y2+z2)32) in spherical coordinates.

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