Engineering Mathematics Test 1

Question 1
2 / -0.33

The solutions of the equation 3yy’ + 4x = 0 represents a:

A
Family of circles

B
Family of ellipses

C
Family of Parabolas

D
Family of hyperbolas

Question 2
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Match the list

List – X

List – Y
(Order in Accuracy Quadrature)
A
Simpson’s 1/3 rule
I
O(n⁴)
B
Trapeziodal rule
II
O(n⁵)
C
Simpson’s 3/8 rule
III
O(n²)

A
A – I, B – III, C - II

B
A – I, B – II, C – III

C
A – II, B – I, C – III

D
A – III, B – II, C – I

Question 3
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[1, 1, 2] is an Eigen vector of the matrix, \(A = \left[ {\begin{array}{*{20}{c}}3&1&{ - 1}\\2&2&{ - 1}\\2&2&0\end{array}} \right]\) corresponding to the Eigen value x. Then value of x is _______

A
1

B
2

C
3

D
4

Question 4
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The type of partial differential equation \(\frac{{{\partial ^2}P}}{{\partial {x^2}}} + \frac{1}{2}\frac{{{\partial ^2}P}}{{\partial x\partial y}} - \frac{{5\partial P}}{{\partial x}} + \frac{{2\partial P}}{{\partial y}} = 0\) is

A
Elliptical

B
Parabolic

C
Hyperbolic

D
None

Question 5
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If \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {4 - x}&{x \le 2}\\ {kx - 4}&{x > 2} \end{array}} \right.\) is a continuous function for all real values of x, then f(8) is equal to ________.

A
20-20

B
204-201

C
207-206

D
209-202

Question 6
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A box contains 2 red and 3 black balls. Three balls are randomly chosen from the box and are placed in a bag. Then the probability that there are 1 red and 2 black balls in the bag, is ________.

A
0.59-0.61

B
0.594-0.611

C
0.597-0.616

D
0.599-0.612

Question 7
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Consider a function f(x, y, z) given by
f(x, y, z) = (x² + y² – 2z²)(y² + z²)
The partial derivative of this function with respect to x at the point x = 2, y = 1 and z = 3 is _______

A
40-40

B
404-401

C
407-406

D
409-402

Question 8
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The rank of the matrix \(A=\left[ \begin{matrix} -1 & 2 & -1 & 0 \\ 2 & 4 & 4 & 2 \\ 0 & 0 & 1 & 5 \\ 1 & 6 & 3 & 2 \\\end{matrix} \right]\) is _________.

A
3-3

B
34-31

C
37-36

D
39-32

Question 9
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The order and degree of the differential equation representing the family of curves \({y^2} = 2C\left( {x + \sqrt C } \right)\), where C is an arbitrary constant, are

A
The order is 1

B
The Degree is 3

C
The order is 3

D
The Degree is 1

Question 10
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Values of the function \(y = \frac{1}{{1 + {x^2}}}\) are y(0) = 1; y(1) = 0.5; y(2) = 0.2; y(3) = 0.1; y(4) = 0.0588; y(5) = 0.0385; y(6) = 0.027 using Simpson’s one-third rule, the value of the integral \(\mathop \smallint \limits_0^6 \frac{{dx}}{{1 + {x^2}}}\) is ______

A
1.364-1.368

B
1.3644-1.3681

C
1.3647-1.3686

D
1.3649-1.3682

Question 11
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In Eight throw of a die, 5 or 6 is considered as success, then the standard deviation is ______

A
0.33

B
1.33

C
2.33

D
0.67

Question 12
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Consider a function f(y) = y³ - 7y² + 5 given on interval [p, q]. If f(y) satisfies hypothesis of Rolle’s theorem and p = 0 then what is the value of q?

A
7-7

B
74-71

C
77-76

D
79-72

Question 13
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Let y(x) be the solution to the differential equation \(4\frac{{{{\rm{d}}^2}{\rm{y}}}}{{{\rm{d}}{{\rm{x}}^2}}} + 12\frac{{{\rm{dy}}}}{{{\rm{dx}}}} + 9{\rm{y}} = 0,{\rm{\;y}}\left( 0 \right) = 1,{\rm{y'}}\left( 0 \right) = - 4\). Then y(1) equals:

A
\( - \frac{1}{2}{{\rm{e}}^{ - 3/2}}\)

B
\( - \frac{3}{2}{{\rm{e}}^{ - 3/2}}\)

C
\( - \frac{5}{2}{{\rm{e}}^{ - 3/2}}\)

D
\( - \frac{7}{2}{{\rm{e}}^{ - 3/2}}\)

Question 14
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Evaluate \(\mathop {\lim }\limits_{x \to 1} \frac{{{x^x} - x}}{{x - 1 - \log x}}\).

A
1

B
2

C
3

D
4

Question 15
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If f(x) = 12x^{4 / 3} – 6x^{1 / 3}, xϵ [-1, 1] then absolute minimum value of the function is

A
-2.25

B
2.25

C
6

D
18

Question 16
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The matrix ‘A’ is defined as \(A = \left[ {\begin{array}{*{20}{c}} 1&2&{ - 3}\\ 0&3&2\\ 0&0&{ - 2} \end{array}} \right]\). The determinant of the matrix B = 3A³ + 5A² – 6A + 2J is

A
110

B
1100

C
4400

D
440

Question 17
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The Wronskian of the functions x², 3x + 2, 2x + 3 is _______

A
-5

B
-8

C
-10

D
10

Question 18
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A lot has 20% defective items and 20 items are chosen randomly from this lot. The probability that exactly 3 of the chosen items are defective is _____

A
0.205

B
0.939

C
0.501

D
0.0036

Question 19
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The value of α for which the system has more than one solution is _______
x + y + z = 0
y + 2z = 0
αx + z = 0

A
-1

B
0

C
1 / 2

D
1

Question 20
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The value of the integral \(\underset{a}{\overset{\text{ }\!\!b\!\!\text{ }}{\mathop \int }}\,\text{x}{{\cos }^{2}}\text{xdx}\) is\(\frac{{{\pi }^{2}}}{4}. \)
What is the correct value of a and b?

A
b = π ÷ 2

B
a = 0

C
a = π ÷ 4

D
b = π

Question 21
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Find the positive root of x⁴ – x = 10 after 1^st iteration and 2^nd iteration (x₂) with initial value x₀ = 2. Using Newton-Raphson method

A
After the 1^st iteration the value is 1.8709

B
After the 1^st iteration the value is 1.9132

C
After the 2^nd iteration the value is 1.9172

D
After the 2^nd iteration the value is 1.8577

Question 22
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A continuous random variable has a probability density function as
f(x) = 3x², 0 ≤ x ≤ 1, Find a and b such that
(i) P(X ≤ a) = P(X > a) and
(ii) P(X > b) = 0.05

A
a = 0.68

B
b = 0.53

C
a = 0.79

D
b = 0.98

Question 23
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Which of the following statements is/are true?

A
The product of Eigen values of an upper triangular matrix equals the product of its diagonal elements

B
There exist n × n matrices A and B such that AB = I_n and such that BA ≠ I_n

C
If A and B are similar matrices (i.e. there exists an invertible matrix Q such that A = QBQ^-1), then det (A) = det (B)

D
Let A be an n × n invertible matrix. Then rows of A are linearly independnet

Question 24
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The vector \(\vec V = \left( {x + y + az} \right)i + \left( {bx + 2y - z} \right)j + + \left( { - x + cy + 2z} \right)k\) is irrotational. Where a, b and c are constants. Find the divergence of the vector \(\vec V\).

A
5-5

B
54-51

C
57-56

D
59-52

Submit Test

	List – X		List – Y (Order in Accuracy Quadrature)
A	Simpson’s 1/3 rule	I	O(n⁴)
B	Trapeziodal rule	II	O(n⁵)
C	Simpson’s 3/8 rule	III	O(n²)

Engineering Mat...

Question 1

Family of circles

Family of ellipses

Family of Parabolas

Family of hyperbolas

Question 2

A – I, B – III, C - II

A – I, B – II, C – III

A – II, B – I, C – III

A – III, B – II, C – I

Question 3

1

2

3

4

Question 4

Elliptical

Parabolic

Hyperbolic

None

Question 5

20-20

204-201

207-206

209-202

Question 6

0.59-0.61

0.594-0.611

0.597-0.616

0.599-0.612

Question 7

40-40

404-401

407-406

409-402

Question 8

3-3

34-31

37-36

39-32

Question 9

The order is 1

The Degree is 3

The order is 3

The Degree is 1

Question 10

1.364-1.368

1.3644-1.3681

1.3647-1.3686

1.3649-1.3682

Question 11

0.33

1.33

2.33

0.67

Question 12

7-7

74-71

77-76

79-72

Question 13

\( - \frac{1}{2}{{\rm{e}}^{ - 3/2}}\)

\( - \frac{3}{2}{{\rm{e}}^{ - 3/2}}\)

\( - \frac{5}{2}{{\rm{e}}^{ - 3/2}}\)

\( - \frac{7}{2}{{\rm{e}}^{ - 3/2}}\)

Question 14

1

2

3

4

Question 15

-2.25

2.25

6

18

Question 16

110

1100

4400

After the 1^st iteration the value is 1.8709

After the 1^st iteration the value is 1.9132

After the 2^nd iteration the value is 1.9172

After the 2^nd iteration the value is 1.8577

There exist n × n matrices A and B such that AB = I_n and such that BA ≠ I_n

If A and B are similar matrices (i.e. there exists an invertible matrix Q such that A = QBQ^-1), then det (A) = det (B)