JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD
JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD B.Tech I MATHEMATICS-I Previous Question Paper (May/June – 2019 - R16)
Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 Units. Answer any one full question from each unit. Each question carries 10 marks and may have a, b, c as sub questions.
PART-A
Solve the following differential equation x(dy/dx) - y = x2.
Answer
The equation is a linear differential equation of the form dy/dx + P(x)y = Q(x). Rewriting it as dy/dx - (1/x)y = x, we identify P(x) = -1/x and Q(x) = x. The integrating factor is IF = e∫(-1/x)dx = e-ln|x| = 1/x. Multiplying by the IF gives d/dx(y/x) = 1. Integrating both sides, y/x = x + C, which yields the solution y = x2 + Cx.
Find the complimentary function for the equation (d4y/dx4) + 4y = cos 2x + cos 4x.
Answer
The characteristic equation is m4 + 4 = 0. Adding and subtracting 4m2, we get (m2 + 2)2 - 4m2 = 0, which factors as (m2 - 2m + 2)(m2 + 2m + 2) = 0. Solving these quadratic equations gives m = 1 ± i and m = -1 ± i. The complimentary function is yc = ex(C1 cos x + C2 sin x) + e-x(C3 cos x + C4 sin x).
Find the value of k such that the rank of the matrix [[1,2,3],[2,k,7],[3,6,10]] is 2.
Answer
Sign in to view the complete answer paper.
Find the LU decomposition of A = [[1,3],[-4,1]].
Answer
Sign in to view the complete answer paper.
If a square matrix A has an Eigen value λ then what is the Eigen value of the matrix kAT where k ≠ 0 is a scalar.
Answer
Sign in to view the complete answer paper.
If a matrix A = [[-1,0,0],[2,-3,0],[1,4,2]] then what are the Eigen values of A2?
Answer
Sign in to view the complete answer paper.
If u = exyz find the value of ∂3u/∂x∂y∂z.
Answer
Sign in to view the complete answer paper.
If v = x3y/(x3+y3) find the value of x(∂v/∂x) + y(∂u/∂y).
Answer
Sign in to view the complete answer paper.
Form the partial differential equation by eliminating the arbitrary constants a, b: z = (x + a)(y + b).
Answer
Sign in to view the complete answer paper.
Solve zp + yq = x.
Answer
Sign in to view the complete answer paper.
PART-B
Solve (D2 - 4)y = x sin x
Answer
Sign in to view the complete answer paper.
Find the current at any time t > 0 in a circuit having in series a constant electromotive force 40V, a resistor 10 Ω, and an inductor 0.2 H given that the initial current is zero. Find the current when emf is 150 cos 200 t.
Answer
Sign in to view the complete answer paper.
Solve (D2 + 2D + 1)y = x2 cos x
Answer
Sign in to view the complete answer paper.
Solve by the method of variation of parameters: (d2y/dx2) - y = 2/(1+ex).
Answer
Sign in to view the complete answer paper.
Factorize the matrix by the LU decomposition method [[2,-3,1],[3,4,2],[2,-3,4]]
Answer
Sign in to view the complete answer paper.
For what values of λ and μ do the system of equations x+y+z=6, x+2y+3z=10, x+2y+λz = μ have i) no solution, ii) unique solution iii) more than one solution?
Answer
Sign in to view the complete answer paper.
Find the value of k for which the system of equations: (k+1)x + 8y = 4k, kx + (k+3)y = 3k - 1 has infinitely many solutions.
Answer
Sign in to view the complete answer paper.
Verify Cayley Hamilton theorem for the matrix A = [[1,2,0],[-1,1,2],[1,2,1]] and obtain A-1 and A3.
Answer
Sign in to view the complete answer paper.
Reduce the quadratic form 3x2 + 3y2 + 3z2 - 2yz + 2zx + 2xy to its canonical form.
Answer
Sign in to view the complete answer paper.
Determine the functional dependence and find the relation between u = (x-y)/(x+y), v = xy/(x-y)2.
Answer
Sign in to view the complete answer paper.
If y1 = x2x3/x1, y2 = x3x1/x2, y3 = x1x2/x3, Find the Jacobian of y1, y2, y3 with respect to x1, x2, x3.
Answer
Sign in to view the complete answer paper.
Obtain the Taylor’s expansion of tan-1(y/x) about (1,1) upto and including second degree terms.
Answer
Sign in to view the complete answer paper.
Find a point within a triangle such that the sum of the squares of its distances from the three angular points is a minimum.
Answer
Sign in to view the complete answer paper.
Solve the partial differential equations:
(x2 - yz)p + (y2 - zx)q = (z2 - xy)
Answer
Sign in to view the complete answer paper.
x2(∂z/∂x) + y2(∂z/∂y) = (x + y)z.
Answer
Sign in to view the complete answer paper.
Solve the partial differential equations
p2 + q2 = z2(x + y)
Answer
Sign in to view the complete answer paper.
x2p2 + y2q2 = z.
Answer
Sign in to view the complete answer paper.