Basic Mathematics R22 Januaryfebruary 2024 Question Paper & Answers - B.Tech BT I-I Sem

1

Part-A is a compulsory question which consists of ten sub-questions from all units carrying equal marks.

10 marks

a

If A and B are two matrices, what is condition that A+B=B+A to be true.

1 marks

Answer

The condition is that matrices A and B must have the same order (dimensions), specifically both must be m x n matrices, for the addition to be defined.

b

If

A = [[1, 5, 7], [0, 7, 9], [11, 8, 9]]

, then find the trace of A.

1 marks

Answer

The trace of a matrix is the sum of its diagonal elements. For matrix A, the trace is 1 + 7 + 9 = 17.

c

Find the modulus and principal argument of 1-i.

1 marks

Answer

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d

If

a = 3i-j+k, b = 2i+4j-3k and c = i-2j+k

, then find

a+b-c

.

1 marks

Answer

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e

Express sinh(x) in terms of exponential functions.

1 marks

Answer

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f

Find the sum

tan^-1(1/3) + tan^-1(1/7) + tan^-1(1/13) + tan^-1(1/21)

.

1 marks

Answer

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g

Find the domain and range of

f(x) = (x^2-9)/(x-3)

.

1 marks

Answer

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h

Find the period of

sin^2(x)

.

1 marks

Answer

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i

Form a differential equation for y if

y = Ax^2 + B/x

.

1 marks

Answer

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j

Four dice are thrown independently. Find the probability that exactly two dice show up a three or five.

1 marks

Answer

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210 marks

a

If α and β are roots of equation

x^2+x+1=0

, then find the value of

(α^3+β^3)/(α^2+β^2)

.

5 marks

Answer

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b

Compute the inverse of the matrix

A = [[1, 2, 1], [-1, 1, 2], [2, -1, 1]]

.

5 marks

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310 marks

a

If α and β are roots of equation

x^2+ax+1=0

. If the equation whose roots are

(α-1/β)

and

(β-1/α)

is

x^2-ax+k=0

, then find k.

5 marks

Answer

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b

Evaluate the determinant:

[[b^2-ab, b-c, bc-ac], [ab-a^2, a-b, b^2-ab], [bc-ac, c-a, ab-a^2]]

.

5 marks

Answer

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410 marks

a

Prove that

(a+ib)^(m/n) + (a-ib)^(m/n) = 2(a^2+b^2)^(m/2n) * cos(m/n * tan^-1(b/a))

.

5 marks

Answer

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b

Show that the points (1, 5), (2, 4) and (3, 3) are collinear.

5 marks

Answer

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510 marks

a

Prove that

(cos(α+iβ))/(sin(α+iβ)) = (sin(2α)+i*sinh(2β))/(cos(2α)+cosh(2β))

.

5 marks

Answer

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b

If a, b, and c are mutually perpendicular vectors of equal magnitudes, then find the angle between the vectors a and a+b+c.

5 marks

Answer

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610 marks

a

If

y = (1-3cos x + sin^2 x)/(cos x - sin^2 x)

, then find dy/dx.

5 marks

Answer

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b

Find the partial fractions of

1/(x^2(x^2+5))

.

5 marks

Answer

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710 marks

a

Find the value of

sin^-1(12/13) - sin^-1(3/5)

.

5 marks

Answer

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b

Find the partial fractions of

(3x+1)/((x-3)(x+1))

.

5 marks

Answer

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810 marks

a

Let the function

f:[0,1]->R

be defined by

f(x) = 4^x / (4^x+2)

. Find the value of

f(1/40) + f(2/40) + ... + f(39/40)

.

5 marks

Answer

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b

If

x = a(θ+sin θ), y = a(1-cos θ)

, then find

dy/dx

at

θ=π/4

.

5 marks

Answer

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910 marks

a

If the function

f(x) = k1*x^2 - 1, (x <= π)

and

f(x) = cos(x) + k2, (x > π)

is twice differentiable, then find the ordered pair

(k1, k2)

.

5 marks

Answer

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b

If

f(x) = (2^x - 2^-x)/(2^x + 2^-x)

and

g(x) = log_e(x)

, then find

integral of g(f(x)) dx from -π/4 to π/4

.

5 marks

Answer

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1010 marks

a

Solve

dy/dx = (2log x + 1)/(sin y + y cos y)

.

5 marks

Answer

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b

Solve

dy/dx = 1/(x+y+2)^2

.

5 marks

Answer

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1110 marks

a

If A and B are any two events such that

P(A)=2/5, P(A∩B)=3/20

, then find the conditional probability

P(A'|A'∪B')

.

5 marks

Answer

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b

A computer producing factory has two plants T1 and T2. T1 produces 20% and T2 produces 80%. 7% total are defective. Given P(defective|T1) = 10 * P(defective|T2). A computer is selected and is not defective. Find probability it was produced in plant T2.

5 marks

Answer

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JNTU B.Tech I Year BASIC MATHEMATICS Previous Question Paper (January/February - 2024 - R22)

Part A

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