Which of the following numbers CANNOT be expressed as a sum of consecutive natural numbers?
(a) 14
(b) 18
(c) 16
(d) 20
Correct Answer: (c) 16
Explanation: 16 is a power of 2 (2⁴). Powers of 2 cannot be expressed as sums of consecutive natural numbers because they have only one odd divisor (1).
Logical Reasoning
Pattern Recognition
2
For any 4 consecutive numbers, all expressions formed with '+' and '–' signs yield:
(a) Odd numbers
(b) Even numbers
(c) Prime numbers
(d) Multiples of 3
Correct Answer: (b) Even numbers
Explanation: Changing a sign changes value by an even number. All 8 expressions have same parity, starting with even result.
Analytical Thinking
Mathematical Proof
3
The sum of two even numbers is divisible by 4 if:
(a) Both are multiples of 4
(b) Both leave remainder 2 when divided by 4
(c) Either (a) or (b)
(d) None of these
Correct Answer: (c) Either (a) or (b)
Explanation: Even numbers are either 4k or 4k+2. Sum of two same type: 4k+4m=4(k+m) or (4k+2)+(4m+2)=4(k+m+1).
Algebraic Thinking
Pattern Recognition
4
If a number is divisible by 12, it must be divisible by:
(a) 24
(b) 36
(c) 8
(d) 4
Correct Answer: (d) 4
Explanation: Since 12 = 4 × 3, any multiple of 12 contains factor 4. Not necessarily divisible by 24, 36, or 8.
Divisibility Rules
5
The remainder when 427 is divided by 9 is:
(a) 4
(b) 5
(c) 6
(d) 7
Correct Answer: (a) 4
Explanation: Digital root: 4+2+7=13 → 1+3=4. So remainder is 4.
Mental Calculation
Divisibility Rules
6
Which number is divisible by 11?
(a) 158
(b) 841
(c) 462
(d) 5529
Correct Answer: (c) 462
Explanation: Alternating sum: (4+2)-6=0, divisible by 11.
Applying Rules
7
The digital root of 489710 is:
(a) 2
(b) 3
(c) 4
(d) 5
Correct Answer: (a) 2
Explanation: 4+8+9+7+1+0=29 → 2+9=11 → 1+1=2.
Computational Skills
8
In cryptarithm AB × 5 = BC, what must A be?
(a) 1
(b) 2
(c) 3
(d) 4
Correct Answer: (a) 1
Explanation: If A ≥ 2, product would be 3-digit. BC is 2-digit, so A must be 1.
Logical Reasoning
9
If a number leaves remainder 3 when divided by 5, it can be represented as:
Explanation: 6 = 2 × 3. Since 2 and 3 are co-prime, divisible by 6 iff divisible by both.
Understanding Rules
Assertion & Reasoning (20 Questions)
20 Qs × 1 mark = 20 marks
1
Assertion (A): All odd numbers can be expressed as sums of two consecutive numbers. Reason (R): Odd numbers are of the form 2n+1, which equals n + (n+1).
(a) Both A and R are true and R explains A
(b) Both A and R are true but R does not explain A
(c) A is true but R is false
(d) A is false but R is true
Correct Answer: (a) Both A and R are true and R explains A
Explanation: A is true (e.g., 7=3+4, 9=4+5). R provides algebraic proof: 2n+1 = n + (n+1).
Logical Reasoning
Mathematical Proof
2
Assertion (A): For 4 consecutive numbers, all '+'/'–' expressions give even results. Reason (R): Changing a sign changes value by an even number.
(a) Both A and R are true and R explains A
(b) Both A and R are true but R does not explain A
(c) A is true but R is false
(d) A is false but R is true
Correct Answer: (a) Both A and R are true and R explains A
Explanation: A is verified. R explains why: switching +b to -b changes total by 2b (even), so parity remains same.
Mathematical Reasoning
True/False (10 Questions)
10 Qs × 1 mark = 10 marks
1
Every even number can be expressed as a sum of consecutive numbers.
True
False
Correct Answer: False
Explanation: Powers of 2 (like 2, 4, 8, 16) cannot be expressed as sums of consecutive natural numbers.
Counterexample Reasoning
2
Digital root of a multiple of 9 is always 9.
True
False
Correct Answer: True
Explanation: Multiples of 9 have digit sums divisible by 9, repeated summing yields 9.
Understanding Rules
Short Answer I (15 Questions - 2 Marks Each)
15 Qs × 2 marks = 30 marks
1
Express 15 as sums of consecutive numbers in two ways. 2 marks
Answer: 15 = 7 + 8 OR 15 = 4 + 5 + 6
Explanation: 15 can be expressed as sum of 2 consecutive numbers (7+8) or 3 consecutive numbers (4+5+6).
Number Decomposition
2
Show that for 4 consecutive numbers, a ± b ± c ± d is always even. 2 marks
Answer: Changing signs changes value by even amounts. Starting expression gives even result, so all do.
Explanation: Let numbers be n, n+1, n+2, n+3. Changing +b to -b changes total by 2b (even).
Algebraic Proof
Short Answer II (10 Questions - 3 Marks Each)
10 Qs × 3 marks = 30 marks
1
Prove: If a number is divisible by both 9 and 4, it is divisible by 36. 3 marks
Answer: Since 9 and 4 are co-prime, their LCM is 36. Divisibility by both implies divisibility by LCM.
Explanation: Let N be divisible by 9 and 4. Then N contains factors 3² and 2², so contains factor 2²×3²=36.
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