JNTU
JNTU B.Tech I MATHEMATICS-I Previous Question Paper (September - 2023 - R16)
Note: i) Question paper consists of Part A, Part B.
ii) Part A is compulsory, which carries 25 marks. In Part A, answer all questions.
iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
Part A
Solve π¦ππ₯ β π₯ππ¦ = π(π₯2 + π¦2)ππ₯.
Answer
The given equation is y dx - x dy = a(x2 + y2) dx. Dividing by x2, we get (y dx - x dy)/x2 = a(1 + (y/x)2) dx. This simplifies to -d(y/x) = a(1 + (y/x)2) dx. Integrating both sides, -tan-1(y/x) = ax + c, or y = x tan(c - ax).
Find the complementary function for the equation π4π¦/ππ₯4 + 4π¦ = πππ 2π₯ + πππ 4π₯.
Answer
The characteristic equation is m4 + 4 = 0. We can write this as (m2)2 + 4 + 4m2 - 4m2 = 0, which is (m2 + 2)2 - (2m)2 = 0. This factors to (m2 - 2m + 2)(m2 + 2m + 2) = 0. The roots are m = 1 Β± i and m = -1 Β± i. The complementary function is yc = ex(C1 cos x + C2 sin x) + e-x(C3 cos x + C4 sin x).
Find the LU decomposition of π΄ = [[1, 3], [4, -1]].
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Reduce the following matrix to upper triangular form (Echelon form) by elementary row transformations [[1, 2, 3], [2, 5, 7], [3, 1, 2]].
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Find the sum and product of the Eigen values of the matrix π΄ = [[2, 5, 7], [1, 4, 6], [2, -2, 3]].
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Find the nature, index and signature of the Quadratic form: π = 2π₯1π₯2 + 2π₯1π₯3 + 2π₯2π₯3
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If π’ = π₯π¦, show that π3π’/ππ₯2ππ¦ = π3π’/ππ₯ππ¦2.
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If π’ = πππ -1((π₯+π¦)/(βπ₯+βπ¦)), then show that π₯(ππ’/ππ₯) + π¦(ππ’/ππ¦) = -1/2 πππ‘π’.
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Form the partial differential equation by eliminating the arbitrary constants π and π from π§ = ππ₯π + ππ¦π.
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Solve π§π + π¦π = π₯.
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Part B
Solve (π·2 + 2π· + 1)π¦ = π₯2πππ π₯.
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According to Newtonβs law of cooling, the rate at which a substance cools in moving air is proportional to the difference between the temperature of the substance and that of the air. If the temperature of the air is 30Β°πΆ and the substance cools from 100Β°πΆ to 70Β°πΆ in 15 minutes, find when the temperature will be 40Β°πΆ?
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Apply the method of variation of parameters to solve π2π¦/ππ₯2 + π¦ = π‘πππ₯.
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Find the rank of the matrix π΄ = [[2, 3, -1, -1], [1, -1, -2, -4], [3, 6, 1, 3], [3, 0, -2, -7]].
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Solve the system of equations 3π₯ + 3π¦ + 2π§ = 1; π₯ + 2π¦ = 4; 10π¦ + 3π§ = -2; 2π₯ - 3π¦ - π§ = 5.
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Solve the system of equations 4π₯ + π¦ + π§ = 4, π₯ + 4π¦ - 2π§ = 4, 3π₯ + 2π¦ - 4π§ = 6 by LU- Decomposition method.
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Verify Cayley-Hamilton theorem. Also find π΄4 and π΄-1 when π΄ = [[2, -1, 2], [-1, 2, -1], [1, -1, 2]].
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Reduce the quadratic form 8π₯2 + 7π¦2 + 3π§2 - 12π₯π¦ + 4π₯π§ - 8π¦π§ into canonical form by orthogonal reduction. Also find its nature?
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If π πππ’ = (π₯2π¦2)/(π₯+π¦), show that π₯(ππ’/ππ₯) + π¦(ππ’/ππ¦) = 3 π‘πππ’.
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If π¦1 = (π₯2π₯3)/π₯1, π¦2 = (π₯3π₯1)/π₯2, π¦3 = (π₯1π₯2)/π₯3, Find the Jacobian of π¦1, π¦2 ,π¦3 with respect to π₯1, π₯2 ,π₯3.
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Obtain the Taylorβs expansion of π‘ππ-1(π¦/π₯) about (1,1) up to and including second degree terms.
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Find a point within a triangle such that the sum of the squares of its distances from the three vertices is a minimum.
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Solve π¦2π§/π₯ π + π₯π§π = π¦2.
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Solve π(1 + π) = ππ§.
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Form the partial differential equation from π§ = ππ₯3 + ππ¦3 by eliminating π and π.
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Solve π2π(π₯2 + π¦2) = π2 + π.
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