Engineering Mathematics Test 2

Question 1
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The value of the integral \(\displaystyle\int_0^{2\pi}\left(\dfrac{3}{9+\sin^2 \theta }\right)d\theta \) is

A
\(\dfrac{2\pi}{\sqrt{10}}\)

B
\(2\sqrt{10} \pi\)

C
\(\sqrt{10} \pi\)

D
π

Question 2
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A student is solving the linear algebra problem. He has calculated the eigenvalues of matrix M as 4, 16, and 64. If he also wants to calculate the 100 × det(M-1)T where det(M-1)Tis the determinant of det(M-1)T then the value of 2000 × det (M-1)T is______.(up to 2 decimal places)

A
0.48-0.49

B
0.484-0.491

C
0.487-0.496

D
0.489-0.492

Question 3
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A class of 30 students occupy a classroom containing 5 rows of seats, with 8 seats in each row. If the student seat themselves at random, the probability that the sixth seat in the fifth row will be empty is

A
1/5

B
1/3

C
1/4

D
2/5

Question 4
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Consider a sentence consist of eight words: "Do not count your chickens before they hatch" placed on the floor written in different pieces of paper. These eight pieces of paper are kept in a bag. One of the pieces is drawn at random from the bag. What is the expected length of the word drawn?
NOTE:
Answer upto 2 decimal places

A
4.62-4.63

B
4.624-4.631

C
4.627-4.636

D
4.629-4.632

Question 5
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Consider the function f(x) = x + ln x and f is differentiable on (1, e) and f(x) is continuous on [1, e]. Determine the c value using Lagrange’s Mean value theorem.

A
e

B
e – 1

C
\(\frac{{\rm{e}}}{{{\rm{e}} - 1{\rm{\;}}}}\)

D
\(\frac{{{\rm{e}} - 1}}{{{\rm{e\;}}}}\)

Question 6
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Each of the nine words in the sentence”. The quick brown fox jumps over the lazy dog” is written on a separate piece of paper. These nine pieces of paper are kept in a box. One of the pieces is drawn at random from the box. The expected length of the word drawn is ______. (The answer should be rounded to one decimal place.)

A
3.8-3.9

B
3.84-3.91

C
3.87-3.96

D
3.89-3.92

Question 7
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Given simultaneous equations have infinitely many solution.
2x + 3y + z = 0
4x + 6y + 6z = 0
8x + 12ay + 2z = 0
Which of the following is true for a?

A
a = 1

B
a ≠ 1

C
a = -1

D
a = 0

Question 8
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The probability density function of a random variable function y has the following probability function:
y
0
1
2
3
4
5
6
7
p(y)
0
m
m
3m
3m
m²
4m²
5m² + m

find p(y ≥ 5)?

A
\(\frac{1}{5}\)

B
\(\frac{2}{5}\)

C
\(\frac{3}{5}\)

D
5

Question 9
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Two events A and B are such that P(A) = 0.5, P(B) = 0.3 and P(A ∩ B) = 0.1. Which of the following is/are true?

A
\(P\left( {A{\rm{|}}B} \right) = \frac{1}{5}\)

B
\(P\left( {A{\rm{|}}A \cup B} \right) = \frac{5}{7}\)

C
P(A|A ∩ B) = 1

D
\(P\left( {A \cap B{\rm{|}}A \cup B} \right) = \frac{1}{7}\)

Question 10
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If a random variable X has a Poisson distribution with mean 5, then the expectation E[(X + 2)²] equals ______.

A
54-54

B
544-541

C
547-546

D
549-542

Question 11
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Consider the matrix \(M =\left[ {\begin{array}{*{20}{c}} 5\\ 4 \end{array}\begin{array}{*{20}{c}} { - 1}\\ 1 \end{array}} \right]\). Which one of the following statements is not TRUE for the eigen values and eigen vectors of the matrix M?

A
Only one independent eigen vector exists

B
Both the eigen value are different

C
Two independent eigen vector exists

D
Product of all the eigen values is 9

Question 12
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Let X₁, X₂ be two independent normal random variables with means μ₁, μ₂ and standard deviations σ₁, σ₂ respectively. Consider Y =X₁ – X₂ ; μ₁ = μ₂ =1, σ₁ = 1, σ₂ = 2. Then,

A
Y is normal distributed with mean 0 and variance 1

B
Y is normally distributed with mean 0 and variance 5

C
Y has mean 0 and variance 5, but is NOT normally distributed

D
Y has mean 0 and variance 1, but is NOT normally distributed

Question 13
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Find the value of \(\mathop {{\rm{lim}}}\limits_{x \to 0\;} \frac{{{{({p^x} - 1)}^3}}}{{({q^x} - 1).sin2x.\;{\rm{log}}\left( {1\; + \;x} \right)}}\)

A
\(\frac{{{{(\log p)}^3}\;}}{{logq}}\)

B
\(\frac{{{{(\log p)}^3}\;}}{{2logq}}\)

C
\(\frac{{3\log p\;}}{{2logq}}\)

D
\(\frac{{3\log p\;}}{{logq}}\)

Question 14
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What is the minimum value on the interval [4, 5].of the below given function?
f(x) = 3x3 – 40.5x2 + 180x + 7

A
269.5-269.5

B
269.54-269.51

C
269.57-269.56

D
269.59-269.52

Question 15
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If \(A = \;\left[ {\begin{array}{*{20}{c}}1&0&0\\1&0&1\\0&1&0\end{array}} \right]\) then A¹⁰² is _____.

A
\(\left[ {\begin{array}{*{20}{c}}1&0&0\\{51}&1&0\\{51}&0&1\end{array}} \right]\)

B
\(\left[ {\begin{array}{*{20}{c}}1&0&0\\{50}&1&0\\{50}&0&1\end{array}} \right]\)

C
\(\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\)

D
\(\;\left[ {\begin{array}{*{20}{c}}1&{50}&{50}\\0&1&0\\0&0&1\end{array}} \right]\)

Question 16
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The rank of the matrix
\(\left[ {\begin{array}{*{20}{c}} 1&{ - 1}&0&0&0\\ 0&0&1&{ - 1}&0\\ 0&1&{ - 1}&0&0\\ { - 1}&0&0&0&1\\ 0&0&0&1&{ - 1} \end{array}} \right]\)

A
4-4

B
44-41

C
47-46

D
49-42

Question 17
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I have in my pocket ten coins. Nine of them are ordinary coins with equal chances of coming up head and tail when tossed and the tenth has two heads.
If I toss the coin and it comes up heads, what is the probability that it is the coin with two heads?

A
\(\frac{1}{{22}}\)

B
\(\frac{2}{{11}}\)

C
\(\frac{{11}}{{20}}\)

D
\(\frac{1}{{20}}\)

Question 18
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Let the function f satisfies \(f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = {x^2},\;x \ne 0\).
The value of the integral \(\mathop \smallint \limits_1^2 {x^2}f\left( x \right)dx\) is

A
\(\frac{{ - 7}}{5}\)

B
\(\frac{{ - 7}}{3}\)

C
\(\frac{7}{5}\)

D
\(\frac{7}{3}\)

Question 19
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For real constants a and b, let \(M = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{\frac{1}{{\sqrt 2 }}}\\ a&b \end{array}} \right]\) be an orthogonal matrix. Then which of the following statements is/are always TRUE?

A
a + b = 0

B
\(b = \sqrt {1 - {a^2}}\)

C
\(ab = - \frac{1}{2}\)

D
M² = I₂

Question 20
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Consider the following system of linear equations,
\({x_1} + 2{x_2} = {b_1}\)
\(2{x_1} + 4{x_2} = {b_2}\)
\(3{x_1} + 7{x_2} = {b_3}\)
\(3{x_1} + 9{x_2} = {b_4}\)
Which one of the following conditions ensures that a solution exists for the above system?

A
b₂ = 2b₁ and 6b₁ – 3b₃ + b₄ = 0

B
b₃ = 2b₁ and 6b₁ – 3b₃ + b₄ = 0

C
b₂ = 2b₁ and 3b₁ – 6b₃ + b₄ = 0

D
b₃ = 2b₁ and 3b₁ – 6b₃ + b₄ = 0

Question 21
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Find the value of \(\mathop {{\rm{lim}}}\limits_{x \to 0} \;{\frac{{\left( {{a^x} + {b^x} + {c^x} + {d^x}} \right)}}{4}^{\frac{1}{x}}}\)?

A
1

B
\(\sqrt[2]{{abcd}}\)

C
0

D
\(\sqrt[4]{{abcd}}\)

Question 22
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Let \(A = {\rm{\;}}\left( {\begin{array}{*{20}{c}} 3&0&0\\ 0&2&{ - 5}\\ 0&{\rm{\alpha }}&{ - 2} \end{array}} \right)\) for some a ϵ R. Suppose there exists a 3 × 3 matrix P such that
\({{\rm{P}}^{ - 1}}{\rm{AP}} = \left( {\begin{array}{*{20}{c}} 3&0&0\\ 0&3&0\\ 0&0&{ - 3} \end{array}} \right).\) Then the value of α is

A
-1

B
0

C
1

D
3

Question 23
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If \(\mathop \smallint \nolimits_0^{2\pi } \left| {x\;sin\;x} \right|dx = k\pi ,\) then the value of k is equal to ______.

A
4-4

B
44-41

C
47-46

D
49-42

Question 24
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A probability function X has probability density function f(x) as given below
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\alpha x + \beta {x^2} - 1\;\;\;\;\;0 \le x \le 3}\\ {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise} \end{array}} \right.\)
If the expected value E(X) =\(\frac{5}{4}\) what it the value α + β?

A
\( \frac{17}{9}\)

B
\(-\frac{17}{9}\)

C
\( - \frac{17}{3}\)

D
\(\frac{17}{3}\)

Question 25
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The matrix \(A = \left[ {\begin{array}{*{20}{c}} {\frac{3}{2}}&0&{\frac{1}{2}}\\ 0&{ - 1}&0\\ {\frac{1}{2}}&0&{\frac{3}{2}} \end{array}} \right]\) has three distinct Eigen values and one of its Eigen vectors is \(\left[ {\begin{array}{*{20}{c}} 1\\ 0\\ { 1} \end{array}} \right]\). Which one of the following can be another Eigen vector of A?

A
\(\left[ {\begin{array}{*{20}{c}} 0\\ 0\\ { - 1} \end{array}} \right]\)

B
\(\left[ {\begin{array}{*{20}{c}} { - 1}\\ 0\\ 0 \end{array}} \right]\)

C
\(\left[ {\begin{array}{*{20}{c}} 1\\ 0\\ { - 1} \end{array}} \right]\)

D
\(\left[ {\begin{array}{*{20}{c}} 1\\ { - 1}\\ 1 \end{array}} \right]\)

Submit Test

y	0	1	2	3	4	5	6	7
p(y)	0	m	m	3m	3m	m²	4m²	5m² + m

Engineering Mat...

Question 1

\(\dfrac{2\pi}{\sqrt{10}}\)

\(2\sqrt{10} \pi\)

\(\sqrt{10} \pi\)

π

Question 2

0.48-0.49

0.484-0.491

0.487-0.496

0.489-0.492

Question 3

1/5

1/3

1/4

2/5

Question 4

4.62-4.63

4.624-4.631

4.627-4.636

4.629-4.632

Question 5

e

e – 1

\(\frac{{\rm{e}}}{{{\rm{e}} - 1{\rm{\;}}}}\)

\(\frac{{{\rm{e}} - 1}}{{{\rm{e\;}}}}\)

Question 6

3.8-3.9

3.84-3.91

3.87-3.96

3.89-3.92

Question 7

a = 1

a ≠ 1

a = -1

a = 0

Question 8

\(\frac{1}{5}\)

\(\frac{2}{5}\)

\(\frac{3}{5}\)

5

Question 9

\(P\left( {A{\rm{|}}B} \right) = \frac{1}{5}\)

\(P\left( {A{\rm{|}}A \cup B} \right) = \frac{5}{7}\)

P(A|A ∩ B) = 1

\(P\left( {A \cap B{\rm{|}}A \cup B} \right) = \frac{1}{7}\)

Question 10

54-54

544-541

547-546

549-542

Question 11

Only one independent eigen vector exists

Both the eigen value are different

Two independent eigen vector exists

Product of all the eigen values is 9

Question 12

Y is normal distributed with mean 0 and variance 1

Y is normally distributed with mean 0 and variance 5

Y has mean 0 and variance 5, but is NOT normally distributed

Y has mean 0 and variance 1, but is NOT normally distributed

Question 13

\(\frac{{{{(\log p)}^3}\;}}{{logq}}\)

\(\frac{{{{(\log p)}^3}\;}}{{2logq}}\)

\(\frac{{3\log p\;}}{{2logq}}\)

\(\frac{{3\log p\;}}{{logq}}\)

Question 14

269.5-269.5

269.54-269.51

269.57-269.56

269.59-269.52

Question 15

\(\left[ {\begin{array}{*{20}{c}}1&0&0\\{51}&1&0\\{51}&0&1\end{array}} \right]\)

\(\left[ {\begin{array}{*{20}{c}}1&0&0\\{50}&1&0\\{50}&0&1\end{array}} \right]\)

\(\left[ {\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}} \right]\)

\(\;\left[ {\begin{array}{*{20}{c}}1&{50}&{50}\\0&1&0\\0&0&1\end{array}} \right]\)

Question 16

4-4

44-41

47-46

M² = I₂

b₂ = 2b₁ and 6b₁ – 3b₃ + b₄ = 0

b₃ = 2b₁ and 6b₁ – 3b₃ + b₄ = 0

b₂ = 2b₁ and 3b₁ – 6b₃ + b₄ = 0

b₃ = 2b₁ and 3b₁ – 6b₃ + b₄ = 0