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Mathematics MCQs

MCQs on Algebra, Calculus, Statistics and Geometry

44 Questions
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Prepare Mathematics with confidence

Develop speed and accuracy in arithmetic, algebra, geometry, percentages, ratios and other quantitative concepts.

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1
Hard css pms fpsc

Differentiate f(x) = sin(x) + cos(x) with respect to x

  • A cos(x) - sin(x)
  • B sin(x) - cos(x)
  • C -sin(x) + cos(x)
  • D cos(x) + sin(x)
Permalink
Correct Answer: Option A — cos(x) - sin(x)
Explanation: d/dx[sin(x)] = cos(x) and d/dx[cos(x)] = -sin(x). Therefore the derivative of sin(x)+cos(x) = cos(x)-sin(x). Standard trigonometric derivatives must be memorized for calculus-based competitive exams. These are applied in finding maxima minima rates of change tangent line slopes and in solving differential equations.
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2
Hard css fpsc ppsc

A bag has 5 red and 3 blue balls. What is the probability of drawing 2 red balls?

  • A 5/14
  • B 10/28
  • C 10/56
  • D 5/28
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Correct Answer: Option A — 5/14
Explanation: P = C(5,2)/C(8,2) = 10/28 = 5/14. C(5,2)=10 favorable outcomes from C(8,2)=28 total. This is a combination-based probability problem. When drawing without replacement order does not matter so combinations are used. Such problems appear in probability sections of CSS FPSC and higher competitive examinations.
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3
Hard css pms fpsc ppsc

What is the value of log base 2 of 32 plus log base 2 of 4?

  • A 7
  • B 8
  • C 9
  • D 6
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Correct Answer: Option A — 7
Explanation: log₂(32) + log₂(4) = 5 + 2 = 7 using the property logb(x)+logb(y)=logb(xy) = log₂(128) = 7. Alternatively compute directly: 2⁵=32 so log₂(32)=5 and 2²=4 so log₂(4)=2. Logarithm addition and the change-of-base formula are frequently tested in competitive exam algebra sections.
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4
Hard css pms fpsc

The function f(x) = x² is an example of what type of function?

  • A Linear
  • B Even
  • C Odd
  • D Inverse
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Correct Answer: Option B — Even
Explanation: f(x) = x² is an even function because f(-x) = (-x)² = x² = f(x). Even functions are symmetric about the y-axis. Odd functions satisfy f(-x) = -f(x) and are symmetric about the origin. Classifying functions as even odd or neither is a standard algebra and calculus topic in higher competitive exam mathematics papers.
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5
Hard css pms fpsc ppsc

Solve: x² + 4x + 4 = 0 and identify the nature of roots

  • A Two distinct real roots
  • B One repeated real root
  • C Two complex roots
  • D No roots
Permalink
Correct Answer: Option B — One repeated real root
Explanation: Discriminant = b²-4ac = 16-16 = 0. When D=0 there is exactly one repeated root: x = -b/2a = -4/2 = -2. This is a perfect square trinomial (x+2)² = 0. Recognizing perfect square quadratics allows instant factoring and reveals repeated roots without computing the full quadratic formula.
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6
Hard css pms fpsc

What is the inverse of the 2x2 matrix with rows [1 2] and [3 4]?

  • A [-2 1] and [1.5 -0.5]
  • B [4 -2] and [-3 1]
  • C [1 -2] and [-3 4]
  • D Does not exist
Permalink
Correct Answer: Option A — [-2 1] and [1.5 -0.5]
Explanation: For matrix [[1 2][3 4]] det = 4-6 = -2. Inverse = (1/det)×[[d -b][-c a]] = (-1/2)×[[4 -2][-3 1]] = [[-2 1][1.5 -0.5]]. Matrix inversion is used in solving linear systems and transformations. It exists only when the determinant is non-zero and is tested in CSS PMS mathematical topics.
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7
Hard css pms fpsc

What is the value of (2+3i)(2-3i)?

  • A 4+9i
  • B 4-9i
  • C 13
  • D 7
Permalink
Correct Answer: Option C — 13
Explanation: (2+3i)(2-3i) = 2²-(3i)² = 4-9i² = 4-9(-1) = 4+9 = 13. The product of a complex number and its conjugate always gives a real number equal to sum of squares of real and imaginary parts. This property is used to rationalize complex denominators when dividing complex numbers.
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8
Hard css pms fpsc ppsc

In a class 60% passed math 70% passed English and 20% failed both. What % passed both?

  • A 50%
  • B 40%
  • C 30%
  • D 60%
Permalink
Correct Answer: Option A — 50%
Explanation: Eccentricity of a circle is 0. Eccentricity measures how much a conic deviates from being circular: circle e=0 ellipse 0<e<1 parabola e=1 and hyperbola e>1. Understanding all four conic sections and their properties including eccentricity foci directrix and standard equations is required for CSS PMS and engineering entrance examinations.
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9
Hard css pms fpsc ppsc

A mixture of 40 liters has milk:water ratio 3:1. How much water to add for ratio 3:2?

  • A 5 liters
  • B 8 liters
  • C 10 liters
  • D 12 liters
Permalink
Correct Answer: Option C — 10 liters
Explanation: Milk = 30L water = 10L. New ratio 3:2 means water = (2/3)×30 = 20L. Add 20-10 = 10 liters. Mixture problems require identifying the invariant component (milk) and solving for the new amount of the variable component. Such alligation problems are extremely common in competitive exam quantitative aptitude.
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10
Hard css pms fpsc

What is tan(A+B) equal to?

  • A (tan A + tan B)/(1 - tan A·tan B)
  • B (tan A - tan B)/(1 + tan A·tan B)
  • C tan A × tan B
  • D tan A + tan B
Permalink
Correct Answer: Option A — (tan A + tan B)/(1 - tan A·tan B)
Explanation: tan(A+B) = (tan A+tan B)/(1-tan A·tan B). This compound angle formula is derived from sin(A+B)/cos(A+B). Compound angle formulas for sin cos and tan are essential for solving equations and proving identities. They appear in CSS PMS and engineering entrance exams requiring higher trigonometric manipulation.
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11
Hard css pms fpsc ppsc

What is the sum of squares formula: a² + b² equals?

  • A (a+b)² - 2ab
  • B (a-b)² + 2ab
  • C (a+b)(a-b)
  • D Both A and B
Permalink
Correct Answer: Option D — Both A and B
Explanation: a² + b² = (a+b)² - 2ab = (a-b)² + 2ab. Both identities are correct and equivalent. This means if sum (a+b) and product (ab) are known we can find sum of squares without knowing individual values. Such algebraic manipulation problems are frequently tested in competitive exams to assess algebraic fluency.
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12
Medium css fpsc ppsc nts

What is the expansion of (a-b)²?

  • A a² + b²
  • B a² - b²
  • C a² + 2ab + b²
  • D a² - 2ab + b²
Permalink
Correct Answer: Option D — a² - 2ab + b²
Explanation: (a-b)² = a² - 2ab + b². Along with (a+b)² = a²+2ab+b² and (a+b)(a-b) = a²-b² these form the three core algebraic identities. Recognizing which identity applies enables rapid expansion and factoring of algebraic expressions in competitive exam algebra sections saving significant time.
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13
Hard css pms fpsc

What does a radian measure?

  • A π/2 degrees
  • B 57.3 degrees
  • C 90 degrees
  • D 180 degrees
Permalink
Correct Answer: Option B — 57.3 degrees
Explanation: One radian is defined as the angle subtended at the center of a circle by an arc equal in length to the radius. 1 radian ≈ 57.3°. There are 2π radians in a full circle. Radian measure simplifies calculus and physics formulas and is essential for differentiation of trigonometric functions.
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14
Medium css fpsc ppsc nts

What is the slope of a line perpendicular to y = 2x + 3?

  • A 2
  • B -2
  • C 1/2
  • D -1/2
Permalink
Correct Answer: Option D — -1/2
Explanation: Slope of y=2x+3 is 2. Perpendicular slope = negative reciprocal = -1/2. Two lines are perpendicular when product of slopes = -1: 2×(-1/2)=-1. This condition is used in problems involving altitudes normals to curves and orthogonal intersections in coordinate geometry questions.
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15
Medium css fpsc ppsc

The sum of an infinite geometric series with first term 1 and ratio 1/2 is?

  • A 1
  • B 2
  • C 3
  • D 4
Permalink
Correct Answer: Option B — 2
Explanation: Sum = a/(1-r) = 1/(1-0.5) = 1/0.5 = 2. The series 1+1/2+1/4+1/8+... converges to 2. The condition |r|<1 must hold for convergence. Infinite geometric series appear in competitive exam algebra and also model real-world phenomena like bouncing balls and repeating decimal conversions.
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16
Medium css fpsc ppsc

If f(x) = x² - 4 what are the zeros of f(x)?

  • A x = ±1
  • B x = ±2
  • C x = ±3
  • D x = ±4
Permalink
Correct Answer: Option B — x = ±2
Explanation: Set f(x) = 0: x²-4 = 0 → x² = 4 → x = ±2. Zeros are x-values where the function crosses the x-axis. For quadratics zeros can be found by factoring completing the square or the quadratic formula. Finding zeros is fundamental for graphing analyzing and applying polynomial functions in competitive mathematics.
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17
Hard css pms fpsc

What is the determinant of the 2x2 matrix with rows [2 3] and [1 4]?

  • A 5
  • B 8
  • C 11
  • D 14
Permalink
Correct Answer: Option A — 5
Explanation: Det = ad-bc = (2×4)-(3×1) = 8-3 = 5. Determinants are used to solve linear systems find matrix inverses and identify singular matrices. A matrix with zero determinant is singular and uninvertible. Matrix algebra including determinants inverses and systems of equations is tested in CSS PMS and higher competitive exams.
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18
Medium css pms fpsc

What is the value of i² where i is the imaginary unit?

  • A 1
  • B -1
  • C i
  • D -i
Permalink
Correct Answer: Option B — -1
Explanation: The imaginary unit i is defined as √(-1) so i² = -1. Powers of i cycle every four: i¹=i i²=-1 i³=-i i⁴=1. Complex numbers extend the real number system to solve equations like x²+1=0. They appear in higher mathematics physics and engineering topics tested in advanced competitive exams.
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19
Hard css pms fpsc ppsc

What is the number of diagonals in a polygon with n sides?

  • A n(n-1)/2
  • B n(n-3)/2
  • C n(n+1)/2
  • D n²/2
Permalink
Correct Answer: Option B — n(n-3)/2
Explanation: Diagonals = n(n-3)/2. A pentagon (n=5) has 5×2/2 = 5 diagonals. This formula comes from selecting 2 vertices from n vertices (C(n 2)) then subtracting n sides. Polygon formulas for diagonals interior angles and exterior angles are standard geometry topics tested in higher competitive exam papers.
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20
Medium css fpsc ppsc

If variance of a data set is 25 what is the standard deviation?

  • A 5
  • B 10
  • C 15
  • D 25
Permalink
Correct Answer: Option A — 5
Explanation: Standard deviation = √variance = √25 = 5. Variance is the average of squared deviations from the mean. Standard deviation being the square root is expressed in the same units as the original data making it more interpretable. Both quantify data spread and appear in statistics-based competitive exam questions.
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