PAKISTAN NAVY RECRUITMENT
MATHEMATICS TEST (PART 1) • 100 MCQS
Pakistan Navy Sailor Mathematics Test Preparation (Part 1)
100 Most Repeated Past Paper Mathematics Questions for Technical Sailor, Marine Sailor, Medical Sailor/FMT, Marine Engineering (ME) and Weapon Engineering (WE). Master sets, algebra, logarithms, matrices, quadratic equations, geometry, mensuration, trigonometry and statistics with instant interactive feedback.
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1
If universal set U has 50 elements, set A has 28 elements, and set B has 24 elements with 10 elements in their intersection, how many elements are in (A union B)?
Explanation: n(A U B) = n(A) + n(B) - n(A n B) = 28 + 24 - 10 = 42.
2
Which of the following numbers belongs to the set of irrational numbers?
Explanation: sqrt(7) is non-terminating and non-repeating decimal, hence irrational. sqrt(16)=4 and 22/7 are rational.
3
If set P = {2, 3, 5, 7}, how many proper subsets can be formed from set P?
Explanation: The total number of subsets is 2^n = 2^4 = 16. Number of proper subsets = 2^n - 1 = 16 - 1 = 15.
4
What is the characteristic of the common logarithm log10(0.00358)?
Explanation: In scientific notation, 0.00358 = 3.58 x 10^-3. The exponent of 10 gives the characteristic: -3.
5
What is the value of log to the base 2 of 128 [log_2(128)]?
Explanation: Since 2^7 = 128, log_2(128) = 7.
6
According to De Morgan's laws, the complement of (A union B) is equal to:
Explanation: De Morgan's first law states: (A U B)' = A' n B'.
7
Evaluate the logarithmic expression: log_a(1) for any positive base a > 0 (a != 1).
Explanation: Since a^0 = 1, the logarithm of 1 to any valid base is always 0.
8
If log10(x) + log10(5) = 2, what is the value of x?
Explanation: log10(5x) = 2 => 5x = 10^2 = 100 => x = 100 / 5 = 20.
9
What is the multiplicative identity element in the set of real numbers?
Explanation: For any real number a, a * 1 = 1 * a = a. Hence, 1 is the multiplicative identity (while 0 is the additive identity).
10
If set X has 3 elements and set Y has 4 elements, what is the number of elements in the Cartesian product (X x Y)?
Explanation: n(X x Y) = n(X) * n(Y) = 3 * 4 = 12 ordered pairs.
11
If x + 1/x = 5, what is the value of x^2 + 1/x^2?
Explanation: Squaring both sides: (x + 1/x)^2 = x^2 + 1/x^2 + 2 = 25 => x^2 + 1/x^2 = 25 - 2 = 23.
12
Factorize completely the algebraic expression: 16x^2 - 49y^2.
Explanation: Using difference of two squares identity a^2 - b^2 = (a - b)(a + b): (4x)^2 - (7y)^2 = (4x - 7y)(4x + 7y).
13
If x + y = 8 and xy = 15, what is the numerical value of x^3 + y^3?
Explanation: x^3 + y^3 = (x + y)^3 - 3xy(x + y) = 8^3 - 3(15)(8) = 512 - 360 = 152.
14
What is the remainder when the polynomial P(x) = 2x^3 - 3x^2 + 4x - 5 is divided by (x - 2)?
Explanation: By Remainder Theorem, R = P(2) = 2(2)^3 - 3(2)^2 + 4(2) - 5 = 2(8) - 3(4) + 8 - 5 = 16 - 12 + 8 - 5 = 7.
15
Find the Highest Common Factor (HCF) of the expressions: 12x^3 y^2 and 18x^2 y^4.
Explanation: HCF of 12 and 18 is 6. Lower power of x is x^2, lower power of y is y^2 => 6x^2 y^2.
16
If a + b = 6 and a - b = 4, what is the value of 4ab?
Explanation: 4ab = (a + b)^2 - (a - b)^2 = 6^2 - 4^2 = 36 - 16 = 20.
17
Factorize the quadratic trinomial: x^2 - 9x + 20.
Explanation: Find factors of +20 adding to -9: (-4) + (-5) = -9 and (-4)*(-5) = 20 => (x - 4)(x - 5).
18
What is the formula for expanding (a + b + c)^2?
Explanation: Standard algebraic identity: (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca.
19
Simplify the algebraic fraction: (x^2 - 16) / (x - 4) for x != 4.
Explanation: (x^2 - 16) / (x - 4) = (x - 4)(x + 4) / (x - 4) = x + 4.
20
If a^3 - b^3 = 56 and a - b = 2, what is the value of a^2 + ab + b^2?
Explanation: Since a^3 - b^3 = (a - b)(a^2 + ab + b^2) => 56 = 2 * (a^2 + ab + b^2) => a^2 + ab + b^2 = 28.
21
Solve for x in the linear equation: 5(2x - 3) - 3(x + 2) = 17.
Explanation: 10x - 15 - 3x - 6 = 17 => 7x - 21 = 17 => 7x = 38 => x = 38/7.
22
Find the solution set of the linear equation: 4x - 7 = 2x + 9.
Explanation: 4x - 2x = 9 + 7 => 2x = 16 => x = 8.
23
Solve the system of simultaneous linear equations: 2x + y = 11 and x - y = 1.
Explanation: Adding the two equations: (2x + y) + (x - y) = 11 + 1 => 3x = 12 => x = 4. Then y = x - 1 = 3.
24
What is the solution set of the absolute value equation: |3x - 5| = 7?
Explanation: Case 1: 3x - 5 = 7 => 3x = 12 => x = 4. Case 2: 3x - 5 = -7 => 3x = -2 => x = -2/3.
25
Solve the linear inequality: 3 - 2x < 11 for real numbers x.
Explanation: -2x < 11 - 3 => -2x < 8. Dividing by -2 reverses inequality sign: x > -4.
26
If 3x + 2y = 13 and 2x + 3y = 12, what is the value of (x + y)?
Explanation: Adding both equations: 5x + 5y = 25 => 5(x + y) = 25 => x + y = 5.
27
At what point do the lines represented by 3x - 2y = 6 cross the y-axis (y-intercept)?
Explanation: On the y-axis, x = 0: 3(0) - 2y = 6 => -2y = 6 => y = -3. The point is (0, -3).
28
The sum of ages of a naval officer and his son is 54 years. Four years ago, the officer was 46 years old. What is the son's present age?
Explanation: Officer's age 4 years ago was 46, so his present age is 46 + 4 = 50. Son's present age = 54 - 50 = 4 years.
29
Find the value of k for which the system of equations 2x + ky = 7 and 4x + 6y = 14 has infinitely many solutions.
Explanation: For coincident lines: a1/a2 = b1/b2 = c1/c2 => 2/4 = k/6 = 7/14 => 1/2 = k/6 => k = 3.
30
Which inequality represents all numbers x whose distance from 0 is at most 6?
Explanation: |x| <= 6 represents -6 <= x <= 6 on the real number line.
31
What is the standard general form of a quadratic equation in one variable x?
Explanation: The standard quadratic equation is ax^2 + bx + c = 0 with real coefficients and leading coefficient a != 0.
32
Solve the quadratic equation by factoring: 2x^2 + 5x - 3 = 0.
Explanation: 2x^2 + 6x - x - 3 = 0 => 2x(x + 3) - 1(x + 3) = 0 => (2x - 1)(x + 3) = 0 => x = 1/2 or x = -3.
33
What is the discriminant formula of the quadratic equation ax^2 + bx + c = 0?
Explanation: The discriminant Delta = b^2 - 4ac determines the nature of the roots of a quadratic equation.
34
If the discriminant of a quadratic equation is negative (b^2 - 4ac < 0), what is the nature of its roots?
Explanation: When Delta < 0, the square root of a negative quantity yields imaginary/complex conjugate roots.
35
If alpha and beta are roots of the quadratic equation 3x^2 - 5x + 7 = 0, what is the sum of the roots (alpha + beta)?
Explanation: By Vieta's formulas, sum of roots = -b/a = -(-5)/3 = 5/3.
36
If alpha and beta are roots of the equation 2x^2 + 4x - 9 = 0, what is the product of the roots (alpha * beta)?
Explanation: Product of roots = c/a = -9/2.
37
For what value of m will the equation x^2 - 6x + m = 0 have equal roots?
Explanation: For equal roots, Delta = 0: b^2 - 4ac = 0 => (-6)^2 - 4(1)(m) = 0 => 36 - 4m = 0 => 4m = 36 => m = 9.
38
Find the roots of the equation: x^2 - 49 = 0.
Explanation: x^2 = 49 => x = +-sqrt(49) = +-7.
39
What is the quadratic equation whose roots are 3 and -5?
Explanation: Sum of roots S = 3 + (-5) = -2. Product P = 3 * (-5) = -15. Equation: x^2 - Sx + P = 0 => x^2 - (-2)x + (-15) = x^2 + 2x - 15 = 0.
40
How many solutions does a quadratic equation in one variable have over the complex numbers?
Explanation: By the Fundamental Theorem of Algebra, a degree-2 polynomial equation has exactly two roots (counted with multiplicity).
41
What is the order of a matrix having 3 rows and 2 columns?
Explanation: The order of a matrix is denoted as (rows x columns) => 3 x 2.
42
Calculate the determinant of the 2x2 matrix A = [[4, 2], [3, 5]].
Explanation: det(A) = ad - bc = (4 * 5) - (2 * 3) = 20 - 6 = 14.
43
A square matrix A is called a singular matrix if:
Explanation: A square matrix is singular if its determinant is zero, meaning its multiplicative inverse does not exist.
44
What is the transpose of a row matrix [1, 2, 3]?
Explanation: The transpose of a 1x3 row matrix interchanges rows and columns, producing a 3x1 column matrix.
45
If matrix A is symmetric, which condition must hold true?
Explanation: A matrix is symmetric if it equals its transpose (A^T = A). If A^T = -A, it is skew-symmetric.
46
What is the multiplicative identity matrix I of order 2x2?
Explanation: The 2x2 identity matrix has 1s along the principal diagonal and 0s elsewhere: [[1, 0], [0, 1]].
47
If A = [[2, 1], [0, 3]] and B = [[1, 4], [2, 1]], what is the element in row 1, column 1 of product (A * B)?
Explanation: Element c11 = (row 1 of A) * (column 1 of B) = (2 * 1) + (1 * 2) = 2 + 2 = 4.
48
What is the formula for the multiplicative inverse A^-1 of a non-singular square matrix A?
Explanation: The inverse of matrix A is given by A^-1 = (1 / |A|) * Adj(A), provided |A| != 0.
49
In solving a system of two linear equations by Cramer's rule, the variable x is evaluated as:
Explanation: According to Cramer's rule, x = det(Ax) / det(A) = Dx / D, where D != 0.
50
If matrix M has order 2 x 3 and matrix N has order 3 x 4, what is the order of product matrix (M * N)?
Explanation: Since the number of columns in M (3) matches rows in N (3), product exists with order 2 x 4.
51
If 12 technical sailors can overhaul a naval generator in 6 days, how many days will 8 sailors take at the same work rate?
Explanation: Inverse proportion: Men1 * Days1 = Men2 * Days2 => 12 * 6 = 8 * D2 => 72 = 8 * D2 => D2 = 9 days.
52
Find the fourth proportional to the numbers: 4, 7, and 12.
Explanation: 4 / 7 = 12 / x => 4x = 7 * 12 = 84 => x = 84 / 4 = 21.
53
What is the mean proportional between 9 and 25?
Explanation: Mean proportional x = sqrt(a * b) = sqrt(9 * 25) = sqrt(225) = 15.
54
A warship's radar detected 360 targets during an exercise. If 15% were friendly vessels, how many targets were friendly?
Explanation: 15% of 360 = (15 / 100) * 360 = 0.15 * 360 = 54.
55
If a : b = 2 : 3 and b : c = 4 : 5, find the compound ratio a : b : c.
Explanation: Make b equal (LCM of 3 and 4 is 12): a:b = 8:12 and b:c = 12:15 => a : b : c = 8 : 12 : 15.
56
If y varies directly as x and y = 24 when x = 6, what is y when x = 10?
Explanation: y = kx => k = 24 / 6 = 4. When x = 10, y = 4 * 10 = 40.
57
A medical sailor purchased surgical equipment for Rs. 4,000 and sold it for Rs. 4,600. What was the percentage profit?
Explanation: Profit = 4600 - 4000 = 600. Profit % = (600 / 4000) * 100 = 15%.
58
If 20% of an unknown quantity N is 90, what is the value of N?
Explanation: 0.20 * N = 90 => N = 90 / 0.20 = 450.
59
Divide Rs. 1,500 among three recruits in the ratio 2 : 3 : 5. What is the largest share?
Explanation: Sum of ratios = 2 + 3 + 5 = 10. Largest share = (5 / 10) * 1500 = Rs. 750.
60
If fuel consumption of a patrol craft increases by 25%, by what percentage must speed be decreased to keep expenditure unchanged?
Explanation: Reduction % = [r / (100 + r)] * 100 = [25 / 125] * 100 = 1/5 * 100 = 20%.
61
What is the complement of an angle measuring 38 degrees?
Explanation: Complementary angles sum to 90 degrees: 90 - 38 = 52 degrees.
62
What is the supplement of an angle measuring 115 degrees?
Explanation: Supplementary angles sum to 180 degrees: 180 - 115 = 65 degrees.
63
What is the sum of the interior angles of a regular hexagon (6-sided polygon)?
Explanation: Sum of interior angles = (n - 2) * 180 = (6 - 2) * 180 = 4 * 180 = 720 degrees.
64
According to the angle subtended by a semicircle, the angle in a semicircle is always:
Explanation: Thales' theorem states that any angle inscribed in a semicircle subtended by a diameter is 90 degrees.
65
In a right-angled triangle, if two legs measure 9 cm and 12 cm, what is the length of the hypotenuse?
Explanation: By Pythagoras theorem: c = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 cm.
66
How many degrees are in each exterior angle of a regular octagon (8-sided polygon)?
Explanation: Sum of exterior angles is always 360 deg. Each exterior angle = 360 / 8 = 45 degrees.
67
The line segment connecting the center of a circle to any point on its circumference is called:
Explanation: The radius is the segment from the center to the circumference (diameter is twice the radius).
68
Two angles of a navigation triangle measure 65 degrees and 45 degrees. What is the measure of the third angle?
Explanation: Sum of angles in a triangle is 180 degrees: 180 - (65 + 45) = 180 - 110 = 70 degrees.
69
A tangent line to a circle intersects the radius at the point of tangency at an angle of:
Explanation: A tangent to a circle is always perpendicular (forms 90 degrees) to the radius drawn to the point of contact.
70
A triangle having all three sides of unequal lengths is classified as a:
Explanation: A scalene triangle has all three sides of different lengths and all three angles of different measures.
71
What is the area of a circular helipad on a naval ship having a diameter of 28 meters? (Take pi = 22/7)
Explanation: Radius r = 28 / 2 = 14 m. Area = pi * r^2 = (22/7) * 14 * 14 = 22 * 2 * 14 = 616 m^2.
72
Find the circumference of a circle whose radius is 21 cm. (Take pi = 22/7)
Explanation: Circumference C = 2 * pi * r = 2 * (22/7) * 21 = 2 * 22 * 3 = 132 cm.
73
A rectangular ammunition storage room measures 15 m long and 8 m wide. What is its perimeter?
Explanation: Perimeter P = 2 * (length + width) = 2 * (15 + 8) = 2 * 23 = 46 meters.
74
What is the volume of a cylindrical freshwater tank of radius 7 m and height 10 m aboard a naval vessel? (pi = 22/7)
Explanation: Volume = pi * r^2 * h = (22/7) * 7 * 7 * 10 = 22 * 7 * 10 = 1540 m^3.
75
Find the total surface area of a solid cube whose edge length is 5 cm.
Explanation: Total surface area of cube = 6 * a^2 = 6 * (5^2) = 6 * 25 = 150 cm^2.
76
What is the area of a triangular navigation flag with base 16 cm and altitude (height) 12 cm?
Explanation: Area of triangle = 0.5 * base * height = 0.5 * 16 * 12 = 8 * 12 = 96 cm^2.
77
Find the volume of a sphere of radius 3 cm in terms of pi.
Explanation: Volume of sphere = (4/3) * pi * r^3 = (4/3) * pi * (27) = 4 * 9 * pi = 36*pi cm^3.
78
A square naval parade ground has an area of 625 square meters. What is the length of one side?
Explanation: Side = sqrt(Area) = sqrt(625) = 25 meters.
79
What is the volume of a rectangular metal container of dimensions 10 m x 5 m x 4 m?
Explanation: Volume = length * width * height = 10 * 5 * 4 = 200 m^3.
80
If the radius of a circular radar screen is doubled, by what factor does its surface area increase?
Explanation: Area is proportional to r^2. If r is doubled (2r), new area is proportional to (2)^2 = 4 times.
81
In a right-angled triangle, which trigonometric ratio is defined as Opposite side divided by Hypotenuse?
Explanation: sin(theta) = Opposite / Hypotenuse, cos(theta) = Adjacent / Hypotenuse, tan(theta) = Opposite / Adjacent.
82
What is the exact numerical value of tan(45 degrees)?
Explanation: tan(45 deg) = sin(45 deg) / cos(45 deg) = (1/sqrt(2)) / (1/sqrt(2)) = 1.
83
Which of the following is the fundamental Pythagorean trigonometric identity?
Explanation: The basic trigonometric identity is sin^2(theta) + cos^2(theta) = 1 for all angles theta.
84
What is the value of cos(60 degrees)?
Explanation: cos(60 deg) = 1/2 = 0.5 (which is also equal to sin(30 deg)).
85
From the top of a 50-meter-high coastal lighthouse, the angle of depression of a naval patrol boat is 45 degrees. How far is the boat from the base of the lighthouse?
Explanation: tan(45 deg) = height / distance => 1 = 50 / d => d = 50 meters.
86
What is the reciprocal trigonometric ratio of cos(theta)?
Explanation: sec(theta) = 1 / cos(theta), cosec(theta) = 1 / sin(theta), cot(theta) = 1 / tan(theta).
87
What is the value of the identity: 1 + tan^2(theta)?
Explanation: Standard trigonometric identity: 1 + tan^2(theta) = sec^2(theta).
88
Convert an angle of 180 degrees into circular radian measure.
Explanation: 180 degrees equals exactly pi radians (approx 3.14159 radians).
89
If sin(theta) = 3/5 in a right triangle, what is the value of cos(theta)?
Explanation: cos(theta) = sqrt(1 - sin^2(theta)) = sqrt(1 - 9/25) = sqrt(16/25) = 4/5.
90
What is the value of sin(90 degrees) + cos(0 degrees)?
Explanation: sin(90 deg) = 1 and cos(0 deg) = 1 => 1 + 1 = 2.
91
What is the arithmetic mean of the test scores: 14, 18, 22, 26, and 30?
Explanation: Mean = (14 + 18 + 22 + 26 + 30) / 5 = 110 / 5 = 22.
92
Find the median of the data set: 11, 7, 19, 5, 15, 13, 21.
Explanation: Arrange in ascending order: 5, 7, 11, 13, 15, 19, 21. The middle value (4th term) is 13.
93
In a statistical sample, the value that occurs with greatest frequency is called:
Explanation: The mode is the data value having the highest frequency of occurrence.
94
What is the range of the numbers: 45, 12, 88, 34, 76, 23?
Explanation: Range = Maximum value - Minimum value = 88 - 12 = 76.
95
When tossing an unbiased coin once, what is the probability of obtaining a Head?
Explanation: Total outcomes = {H, T} = 2. Favorable outcomes = {H} = 1. Probability = 1/2 = 0.5.
96
What is the probability of rolling an even number on a standard six-sided die?
Explanation: Even numbers on a die are {2, 4, 6} (3 outcomes). Probability = 3 / 6 = 1/2.
97
The sum of probabilities of all elementary mutually exclusive outcomes of an experiment is:
Explanation: The total probability of all mutually exclusive and exhaustive events in a sample space is always 1.
98
If the probability of an event happening is P(E) = 0.35, what is the probability of its complement P(not E)?
Explanation: P(not E) = 1 - P(E) = 1 - 0.35 = 0.65.
99
Find the arithmetic mean of the first five prime numbers (2, 3, 5, 7, 11).
Explanation: Sum = 2 + 3 + 5 + 7 + 11 = 28. Mean = 28 / 5 = 5.6.
100
A bag contains 5 red balls, 4 blue balls, and 3 green balls. What is the probability of drawing a blue ball at random?
Explanation: Total balls = 5 + 4 + 3 = 12. Favorable blue balls = 4. Probability = 4 / 12 = 1/3.
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