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JNTU B.Tech I Year MATHEMATICS I Previous Question Paper (September - 2023 - R13)

B.Tech | I Year | I-I | R13 | September - 2023

Answer PaperPaper Code: 111ABMaximum Marks: 75Time: 3 Hours

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

(25 Marks)

125 marks
a

Using Cayley - Hamilton theorem find inverse for

A = [[1, 4], [2, 3]]
.

2 marks

Answer

The characteristic equation of A is det(A - λI) = 0, which gives λ2 - 4λ - 5 = 0. By the Cayley-Hamilton theorem, A2 - 4A - 5I = 0. Multiplying by A-1, we get A - 4I - 5A-1 = 0, so A-1 = (1/5)(A - 4I). Substituting A = [[1, 4], [2, 3]] and I = [[1, 0], [0, 1]], we find A-1 = (1/5)[[-3, 4], [2, -1]] = [[-0.6, 0.8], [0.4, -0.2]].

b

Define Index and Signature of a canonical form.

3 marks

Answer

The Index (p) is the number of positive terms in the canonical form of a quadratic form. The Signature (s) is the difference between the number of positive terms (p) and the number of negative terms (n) in the canonical form, defined as s = p - n.

c

State Lagrange’s mean value theorem.

2 marks

Answer

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d

In polar coordinates, if

x = r cos θ, y = r sin θ
, then show that
∂(x, y) / ∂(r, θ) = r
.

3 marks

Answer

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e

Write the relation between beta and gamma functions.

2 marks

Answer

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f

Express

∫01 xm-1 (1-x)n-1 dx
in terms of gamma function.

3 marks

Answer

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g

Write the laws of natural growth and decay in mathematical form.

2 marks

Answer

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h

Find the particular integral of

(D2 + 2)y = x
.

3 marks

Answer

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i

State any two properties of Laplace transform.

2 marks

Answer

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j

Find the Laplace transform of

t2 u(t-1)
.

3 marks

Answer

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Part B

(50 Marks)

210 marks
a

Prove that

A = [(1/2)(1+i), (1/2)(1+i); (1/2)(1-i), (1/2)(1-i)]
is a unitary and find A-1.

5 marks

Answer

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b

Find the rank of the matrix

[[1, 3, 4, 3], [3, 9, 2, 3], [1, 3, 4, 1]]
by reducing it to the normal form.

5 marks

Answer

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OR
310 marks
a

Test for consistency and solve: 5x + 3y + 7z = 4, 3x + 26y + 2z = 9, 7x + 2y + 10z = 5.

5 marks

Answer

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b

Reduce the quadratic form

x12 + 2x22 - x32 + 2x1x2 - x2x3 + 2x3x1
into a sum of squares by an orthogonal transformation and also discuss its nature.

5 marks

Answer

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410 marks
a

Verify Rolle’s theorem for

f(x) = x2 - 6x + 8
in [2,4].

5 marks

Answer

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b

If

u = x + 3y + 2z, v = x - 2y + z, w = 2x - y + 3z
, show that they are functionally related, and find the relation.

5 marks

Answer

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OR
510 marks
a

Find the maximum and minimum values of

x3 + 3xy2 - 15x2 - 15y2 + 72x
.

5 marks

Answer

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b

A rectangular box open at top is to have volume of 32 cubic ft. Find the dimensions of the box requiring least material for its construction.

5 marks

Answer

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610 marks
a

Prove that

∫0π/2 tan θ dθ = π/2
.

5 marks

Answer

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b

Express

∫01 xm-1 (1-x)n-1 / (a+bx)m+n dx
as a beta function.

5 marks

Answer

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OR
710 marks
a

Find by the double integration, the area lying inside the circle

r = a sin θ
and outside the Cardioid
r = a(1 - cos θ)
.

5 marks

Answer

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b

Find the volume of the ellipsoid

x2/a2 + y2/b2 + z2/c2 = 1
.

5 marks

Answer

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810 marks
a

A body kept in air with temperature 25 °C cools from 140 °C to 80 °C in 20 minutes. Find when the body cools down to 30 °C.

5 marks

Answer

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b

Find the orthogonal trajectories of the family of curves

rn = an cos(nθ)
.

5 marks

Answer

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OR
910 marks
a

Solve

(D2 + 1)y = x sin 2x
.

5 marks

Answer

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b

If a voltage of 20 cos 5t is applied to a series circuit consisting of 10 ohm resistor and 2 henry inductor, determine the current at any time t.

5 marks

Answer

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1010 marks
a

Find the Laplace transform of

∫0t (e-t sin t / t) dt
.

5 marks

Answer

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b

Find the L{f(t)}, where

f(t) = cos t, if t > 3π; 0, if t < 3π
.

5 marks

Answer

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OR
1110 marks
a

Apply convolution theorem to find the inverse Laplace transforms of

1 / (s2(s+4))
.

5 marks

Answer

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b

Solve

(D2 + 2D + 5)y = et sin t
, given that y(0) = 0, y'(0) = 1 using Laplace transforms.

5 marks

Answer

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This page contains the solved question paper with answers of MATHEMATICS I from JNTU for B.Tech CHEM I Year I-I, September - 2023, R13.

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