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CSS MCQs

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44 Questions
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CSS is Pakistan’s premier federal competitive examination. Strong preparation requires conceptual clarity, broad general knowledge and consistent practice across compulsory and optional areas.

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1
Mathematics Hard

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

  • Acos(x) - sin(x)
  • Bsin(x) - cos(x)
  • C-sin(x) + cos(x)
  • Dcos(x) + sin(x)
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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
Mathematics Hard

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

  • A5/14
  • B10/28
  • C10/56
  • D5/28
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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
Mathematics Hard

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

  • A7
  • B8
  • C9
  • D6
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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
Mathematics Hard

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

  • ALinear
  • BEven
  • COdd
  • DInverse
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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
Mathematics Hard

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

  • ATwo distinct real roots
  • BOne repeated real root
  • CTwo complex roots
  • DNo roots
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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
Mathematics Hard

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]
  • DDoes not exist
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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
Mathematics Hard

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

  • A4+9i
  • B4-9i
  • C13
  • D7
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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
Mathematics Hard

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

  • A50%
  • B40%
  • C30%
  • D60%
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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
Mathematics Hard

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

  • A5 liters
  • B8 liters
  • C10 liters
  • D12 liters
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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
Mathematics Hard

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)
  • Ctan A × tan B
  • Dtan A + tan B
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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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