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ANNA UNIVERSITY CHENNAI :: CHENNAI – 600 025 B.E / B.TECH. DEGREE EXAMINATIONS – I YEAR ANNUAL PATTERN
MODEL QUESTION PAPER MA 1X01 - ENGINEERING MATHEMATICS - I
(Common to all Branches of Engineering and Technology) Regulation 2004 Time : 3 Hrs Answer all Questions PART – A (10 x 2 = 20 Marks) Maximum: 100 Marks
⎡ 3 −1 1 ⎤ 1. Find the sum and product of the eigen values of the matrix ⎢− 1 5 − 1⎥ ⎢ ⎥ ⎢ 1 −1 3 ⎥ ⎣ ⎦
2. If x = r cosθ, y = r sinθ, find
∂ ( r ,θ ) ∂( x, y )
3. Solve (D3+D2+4D+4)y = 0. 4. The differential equation for a circuit in which self-inductance L and capacitance C neutralize each other is L
i + = 0. Find the current i as a function of t. dt 2 C
d 2i
5. Find, by double integration, the area of circle x2+y2 = a2. 6. Prove that curl grad φ = o . 7. State the sufficient conditions for a function f(z) to be analytic. 8. State Cauchy’s integral theorem.
9. Find the Laplace transform of unit step function at t = a.
10. Find L-1 [
s+3 ]. s + 4s + 13
2
1
PART – B (5 x 16 = 80 marks)
2 − 2⎞ ⎛ 7 ⎜ ⎟ 11.(a).(i). Verify Cayley-Hamilton theorem for the matrix A = ⎜ − 6 − 1 2 ⎟ . ⎜ 6 2 −1⎟ ⎝ ⎠ Hence find its inverse. (8)
(ii). Find the radius of curvature at any point ‘t’ on the curve x = a (cost + t sint), y = a(sint−t cost) (8)
(OR)
⎡ 8 −6 2 ⎤ (b).(i). Diagonalise the matrix ⎢− 6 7 − 4⎥ by orthogonal transformation. (8). ⎢ ⎥ ⎢ 2 −4 3 ⎥ ⎣ ⎦
(ii). A rectangular box open at the top is to have volume of 32 c.c. Find the dimensions of the box requiring least material for its construction, by Lagrange’s multiplier method. (8).
d2y dy + 3(3x+2) − 36 y = 3x2+4x+1 12(a). (i). Solve (3x+2) 2 dx dx
2
(8)
(ii). For the electric circuit gover ned by (LD2+RD+ D= q=
1 ) q = E where C
d if L = 1 henry, R = 100 Ohms, C = 10-4 farad and E = 100 volts, dt dq = 0 when t = 0, find the charge q and the current i. (8) dt
(OR)
(b).(i). Solve
dx + 2x + 3y = 0 , dt
3x+
dy + 2 y = 2e 2 t dt
(8)
2
MODEL QUESTION PAPER MA 1X01 - ENGINEERING MATHEMATICS - I
(Common to all Branches of Engineering and Technology) Regulation 2004 Time : 3 Hrs Answer all Questions PART – A (10 x 2 = 20 Marks) Maximum: 100 Marks
⎡ 3 −1 1 ⎤ 1. Find the sum and product of the eigen values of the matrix ⎢− 1 5 − 1⎥ ⎢ ⎥ ⎢ 1 −1 3 ⎥ ⎣ ⎦
2. If x = r cosθ, y = r sinθ, find
∂ ( r ,θ ) ∂( x, y )
3. Solve (D3+D2+4D+4)y = 0. 4. The differential equation for a circuit in which self-inductance L and capacitance C neutralize each other is L
i + = 0. Find the current i as a function of t. dt 2 C
d 2i
5. Find, by double integration, the area of circle x2+y2 = a2. 6. Prove that curl grad φ = o . 7. State the sufficient conditions for a function f(z) to be analytic. 8. State Cauchy’s integral theorem.
9. Find the Laplace transform of unit step function at t = a.
10. Find L-1 [
s+3 ]. s + 4s + 13
2
1
PART – B (5 x 16 = 80 marks)
2 − 2⎞ ⎛ 7 ⎜ ⎟ 11.(a).(i). Verify Cayley-Hamilton theorem for the matrix A = ⎜ − 6 − 1 2 ⎟ . ⎜ 6 2 −1⎟ ⎝ ⎠ Hence find its inverse. (8)
(ii). Find the radius of curvature at any point ‘t’ on the curve x = a (cost + t sint), y = a(sint−t cost) (8)
(OR)
⎡ 8 −6 2 ⎤ (b).(i). Diagonalise the matrix ⎢− 6 7 − 4⎥ by orthogonal transformation. (8). ⎢ ⎥ ⎢ 2 −4 3 ⎥ ⎣ ⎦
(ii). A rectangular box open at the top is to have volume of 32 c.c. Find the dimensions of the box requiring least material for its construction, by Lagrange’s multiplier method. (8).
d2y dy + 3(3x+2) − 36 y = 3x2+4x+1 12(a). (i). Solve (3x+2) 2 dx dx
2
(8)
(ii). For the electric circuit gover ned by (LD2+RD+ D= q=
1 ) q = E where C
d if L = 1 henry, R = 100 Ohms, C = 10-4 farad and E = 100 volts, dt dq = 0 when t = 0, find the charge q and the current i. (8) dt
(OR)
(b).(i). Solve
dx + 2x + 3y = 0 , dt
3x+
dy + 2 y = 2e 2 t dt
(8)
2