Set Theory, Integer Bounds & Arithmetic Past Papers

Concept: Arithmetic Order of Operations (BODMAS/PEMDAS).

Formula: Perform multiplication before subtraction.

Solution:

• First multiply: \(3 \times 2 = 6\)



• Then subtract: \(10 - 6 = 4\)

Why other options are incorrect:

• Option C: Incorrectly performs subtraction first: \((10 - 3) \times 2 = 14\).

Concept: Anagram Numerical Equivalence.

Formula: \(\sum \text{Letter values} = \text{Constant}\)

Solution: "EORGNSU" is an anagram containing the exact same set of letters as "SURGEON". Since each letter's numerical value is fixed, the total sum remains unchanged at 99.

Why other options are incorrect:

• Options A, B, and D: Arbitrary numbers that ignore the principle of algebraic addition of identical variables.

Concept: Commutative Property of Variable Sums.

Formula: \(A+B+C = C+B+A\)

Solution: Reordering letter variables leaves their numerical total invariant.

Why other options are incorrect:

• Options A, B, and D: Disregard the basic algebraic principle that changing variable order does not alter sum.

Concept: Order of Operations.

Formula: Multiplication before subtraction.

Solution:

• \(10 - (3 \times 2) = 10 - 6 = 4\).

Why other options are incorrect:

• Option C: Evaluates left-to-right: \((10 - 3) \times 2 = 14\).

Concept: Anagram Sum Equivalence.

Formula: N/A

Solution: Identical letters maintain an identical total sum of 99.

Why other options are incorrect:

• Options A, B, and D: Violate fixed variable algebraic sum rules.

Concept: Divisibility & Digit Product Rules.

Formula: \(N \pmod{d_1 \times d_2} = 0\)

Solution: Digits of 24 are 2 and 4. Product = \(2 \times 4 = 8\). Since \(24 / 8 = 3\), 24 is a multiple of its digits' product.

Why other options are incorrect:

• Option A: Digits product = 2; 21 is odd (not divisible by 2).

Concept: Number Property Evaluation.

Formula: \(P = 2, Q = 9, R = 5\)

Solution:

• Smallest positive single-digit even number \(P = 2\).



• Largest single-digit odd number \(Q = 9\).



• \(P + Q + R = 2 + 9 + 5 = 16\).

Why other options are incorrect:

• Option C: Calculates \((9 + 5) \times 2 = 28\) (not 30).

Concept: Number Bounds & Factors.

Formula: Bounds = \(12 \le N \le 23\).

Solution:

• Smallest prime in range = 13 (\(X\)).



• Largest multiple of 3 in range = 21 (\(Y\)).



• Smallest multiple of 11 in range = 22 (\(Z\)).

Why other options are incorrect:

• Option A: Includes 24 and 11 which violate strictly 'less than 24 and greater than 11'.

Concept: Bounded Interval Whole Numbers.

Formula: Interval = \((23, 31)\).

Solution: Prime = 29 (X), Max odd < 35 in range = 29 (Y), Min even = 24 (Z).

Why other options are incorrect:

• Option A: Includes interval boundary 31.

Concept: Whole Number Sets.

Formula: Interval = \((23, 31)\).

Solution: Prime = 29, Max odd = 29, Min even = 24.

Why other options are incorrect:

• Option A: Includes upper boundary 31.

Concept: Integer Interval Bounds.

Formula: Interval = \((11, 24)\).

Solution: Smallest prime = 13 (X), Largest multiple of 3 = 21 (Y), Smallest multiple of 11 = 22 (Z).

Why other options are incorrect:

• Option A: Includes boundary values 24 and 11.