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Probability, Mathematical Statistics & Numerical Methods (Pure & Applied Mathematics) Solved Questions & Notes (2026) - Apex Rankers

Natural & Physical Sciences > Pure & Applied Mathematics > Probability, Mathematical Statistics & Numerical Methods

100 Total Questions
~150 mins Read Time
1 Subject Areas
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Q. 1 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
For a standard normal distribution Z ~ N(0, 1), the mean and variance are:
A
Mean = 0, Variance = 0
B
Mean = 1, Variance = 0
C
Mean = 0, Variance = 1
✓ Correct
D
Mean = 1, Variance = 1
💡 Step-by-Step Explanation & Concept Rationale
Standardizing transforms any normal random variable X into Z with μ=0 and σ²=1.
Q. 2 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Newton-Raphson iterative formula for solving non-linear equation f(x) = 0 is:
A
x_(n+1) = x_n + f(x_n)
B
x_(n+1) = x_n - f(x_n) / f'(x_n)
✓ Correct
C
x_(n+1) = x_n / 2
D
x_(n+1) = f'(x_n) / f(x_n)
💡 Step-by-Step Explanation & Concept Rationale
Uses tangent line linear approximation to converge quadratically to the root.
Q. 3 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In hypothesis testing, a Type I error occurs when:
A
Sample size is too small
B
Failing to reject a false null hypothesis
C
Rejecting a true null hypothesis (False Positive)
✓ Correct
D
Calculation error
💡 Step-by-Step Explanation & Concept Rationale
Type I error probability is the significance level alpha (α), typically 0.05.
Q. 4 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Trapezoidal rule approximates the definite integral ∫ from a to b of f(x) dx as:
A
h/3 * [f(a) + f(b)]
B
(h / 2) * [f(a) + 2 f(x1) + ... + 2 f(x_(n-1)) + f(b)]
✓ Correct
C
h * [f(a) + f(b)]
D
(b - a) * f(a)
💡 Step-by-Step Explanation & Concept Rationale
Linear interpolation across sub-intervals yields composite trapezoidal quadrature.
Q. 5 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
The central limit theorem states that the sampling distribution of sample means approaches normal as sample size n grows:
A
Regardless of the underlying population distribution shape (for n ≥ 30)
✓ Correct
B
Never
C
Only for uniform distributions
D
Only if population is already normal
💡 Step-by-Step Explanation & Concept Rationale
Fundamental statistical theorem enabling inferential parametric confidence intervals.
Q. 6 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: For a standard normal distribution Z ~ N(0, 1), the mean and variance are:
A
Mean = 0, Variance = 1
✓ Correct
B
Mean = 0, Variance = 0
C
Mean = 1, Variance = 0
D
Mean = 1, Variance = 1
💡 Step-by-Step Explanation & Concept Rationale
Standardizing transforms any normal random variable X into Z with μ=0 and σ²=1.
Q. 7 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In professional practice: Newton-Raphson iterative formula for solving non-linear equation f(x) = 0 is:
A
x_(n+1) = f'(x_n) / f(x_n)
B
x_(n+1) = x_n - f(x_n) / f'(x_n)
✓ Correct
C
x_(n+1) = x_n / 2
D
x_(n+1) = x_n + f(x_n)
💡 Step-by-Step Explanation & Concept Rationale
Uses tangent line linear approximation to converge quadratically to the root.
Q. 8 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
In professional practice: In hypothesis testing, a Type I error occurs when:
A
Sample size is too small
B
Failing to reject a false null hypothesis
C
Rejecting a true null hypothesis (False Positive)
✓ Correct
D
Calculation error
💡 Step-by-Step Explanation & Concept Rationale
Type I error probability is the significance level alpha (α), typically 0.05.
Q. 9 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
In professional practice: Trapezoidal rule approximates the definite integral ∫ from a to b of f(x) dx as:
A
(b - a) * f(a)
B
h * [f(a) + f(b)]
C
h/3 * [f(a) + f(b)]
D
(h / 2) * [f(a) + 2 f(x1) + ... + 2 f(x_(n-1)) + f(b)]
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Linear interpolation across sub-intervals yields composite trapezoidal quadrature.
Q. 10 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
In professional practice: The central limit theorem states that the sampling distribution of sample means approaches normal as sample size n grows:
A
Never
B
Only if population is already normal
C
Only for uniform distributions
D
Regardless of the underlying population distribution shape (for n ≥ 30)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Fundamental statistical theorem enabling inferential parametric confidence intervals.
Q. 11 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: For a standard normal distribution Z ~ N(0, 1), the mean and variance are:
A
Mean = 1, Variance = 1
B
Mean = 1, Variance = 0
C
Mean = 0, Variance = 0
D
Mean = 0, Variance = 1
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Standardizing transforms any normal random variable X into Z with μ=0 and σ²=1.
Q. 12 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: Newton-Raphson iterative formula for solving non-linear equation f(x) = 0 is:
A
x_(n+1) = x_n - f(x_n) / f'(x_n)
✓ Correct
B
x_(n+1) = x_n / 2
C
x_(n+1) = x_n + f(x_n)
D
x_(n+1) = f'(x_n) / f(x_n)
💡 Step-by-Step Explanation & Concept Rationale
Uses tangent line linear approximation to converge quadratically to the root.
Q. 13 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
According to standard examination standards: In hypothesis testing, a Type I error occurs when:
A
Sample size is too small
B
Rejecting a true null hypothesis (False Positive)
✓ Correct
C
Calculation error
D
Failing to reject a false null hypothesis
💡 Step-by-Step Explanation & Concept Rationale
Type I error probability is the significance level alpha (α), typically 0.05.
Q. 14 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
According to standard examination standards: Trapezoidal rule approximates the definite integral ∫ from a to b of f(x) dx as:
A
h * [f(a) + f(b)]
B
h/3 * [f(a) + f(b)]
C
(b - a) * f(a)
D
(h / 2) * [f(a) + 2 f(x1) + ... + 2 f(x_(n-1)) + f(b)]
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Linear interpolation across sub-intervals yields composite trapezoidal quadrature.
Q. 15 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
According to standard examination standards: The central limit theorem states that the sampling distribution of sample means approaches normal as sample size n grows:
A
Never
B
Only if population is already normal
C
Regardless of the underlying population distribution shape (for n ≥ 30)
✓ Correct
D
Only for uniform distributions
💡 Step-by-Step Explanation & Concept Rationale
Fundamental statistical theorem enabling inferential parametric confidence intervals.
Q. 16 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: For a standard normal distribution Z ~ N(0, 1), the mean and variance are:
A
Mean = 0, Variance = 0
B
Mean = 1, Variance = 0
C
Mean = 1, Variance = 1
D
Mean = 0, Variance = 1
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Standardizing transforms any normal random variable X into Z with μ=0 and σ²=1.
Q. 17 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: Newton-Raphson iterative formula for solving non-linear equation f(x) = 0 is:
A
x_(n+1) = x_n - f(x_n) / f'(x_n)
✓ Correct
B
x_(n+1) = x_n / 2
C
x_(n+1) = f'(x_n) / f(x_n)
D
x_(n+1) = x_n + f(x_n)
💡 Step-by-Step Explanation & Concept Rationale
Uses tangent line linear approximation to converge quadratically to the root.
Q. 18 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
From an applied perspective: In hypothesis testing, a Type I error occurs when:
A
Sample size is too small
B
Calculation error
C
Failing to reject a false null hypothesis
D
Rejecting a true null hypothesis (False Positive)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Type I error probability is the significance level alpha (α), typically 0.05.
Q. 19 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
From an applied perspective: Trapezoidal rule approximates the definite integral ∫ from a to b of f(x) dx as:
A
(b - a) * f(a)
B
(h / 2) * [f(a) + 2 f(x1) + ... + 2 f(x_(n-1)) + f(b)]
✓ Correct
C
h/3 * [f(a) + f(b)]
D
h * [f(a) + f(b)]
💡 Step-by-Step Explanation & Concept Rationale
Linear interpolation across sub-intervals yields composite trapezoidal quadrature.
Q. 20 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
From an applied perspective: The central limit theorem states that the sampling distribution of sample means approaches normal as sample size n grows:
A
Only if population is already normal
B
Never
C
Regardless of the underlying population distribution shape (for n ≥ 30)
✓ Correct
D
Only for uniform distributions
💡 Step-by-Step Explanation & Concept Rationale
Fundamental statistical theorem enabling inferential parametric confidence intervals.
Q. 21 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: For a standard normal distribution Z ~ N(0, 1), the mean and variance are:
A
Mean = 0, Variance = 0
B
Mean = 1, Variance = 1
C
Mean = 1, Variance = 0
D
Mean = 0, Variance = 1
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Standardizing transforms any normal random variable X into Z with μ=0 and σ²=1.
Q. 22 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: Newton-Raphson iterative formula for solving non-linear equation f(x) = 0 is:
A
x_(n+1) = x_n - f(x_n) / f'(x_n)
✓ Correct
B
x_(n+1) = x_n / 2
C
x_(n+1) = f'(x_n) / f(x_n)
D
x_(n+1) = x_n + f(x_n)
💡 Step-by-Step Explanation & Concept Rationale
Uses tangent line linear approximation to converge quadratically to the root.
Q. 23 MULTIPLE_CHOICE
Difficulty: MEDIUM (1 Mark)
Under standard operational protocols: In hypothesis testing, a Type I error occurs when:
A
Sample size is too small
B
Calculation error
C
Failing to reject a false null hypothesis
D
Rejecting a true null hypothesis (False Positive)
✓ Correct
💡 Step-by-Step Explanation & Concept Rationale
Type I error probability is the significance level alpha (α), typically 0.05.
Q. 24 MULTIPLE_CHOICE
Difficulty: HARD (1 Mark)
Under standard operational protocols: Trapezoidal rule approximates the definite integral ∫ from a to b of f(x) dx as:
A
(h / 2) * [f(a) + 2 f(x1) + ... + 2 f(x_(n-1)) + f(b)]
✓ Correct
B
h/3 * [f(a) + f(b)]
C
h * [f(a) + f(b)]
D
(b - a) * f(a)
💡 Step-by-Step Explanation & Concept Rationale
Linear interpolation across sub-intervals yields composite trapezoidal quadrature.
Q. 25 MULTIPLE_CHOICE
Difficulty: EASY (1 Mark)
Under standard operational protocols: The central limit theorem states that the sampling distribution of sample means approaches normal as sample size n grows:
A
Regardless of the underlying population distribution shape (for n ≥ 30)
✓ Correct
B
Never
C
Only if population is already normal
D
Only for uniform distributions
💡 Step-by-Step Explanation & Concept Rationale
Fundamental statistical theorem enabling inferential parametric confidence intervals.
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