Case Study 1: Exponents and Divisibility
Himanshu prepared a project on real numbers, where he explained exponential laws and divisibility rules.
Questions:
- For what value of n, 4ⁿ ends in 0?
(a) 10 (b) when n is even (c) when n is odd (d) no value of n
Answer: (d) - If a is rational and n > 1, then aⁿ is:
(a) irrational (b) rational (c) always integer (d) only when n=0
Answer: (b) - If x, y are odd integers, then x² + y² is:
(a) even (b) not divisible by 4 (c) odd (d) both (a) and (b)
Answer: (d) - “One of every three consecutive integers is divisible by 3.”
(a) always true (b) always false (c) sometimes true (d) none
Answer: (a) - If n is odd, n² – 1 is divisible by:
(a) 22 (b) 55 (c) 8 (d) 88
Answer: (c)
Case Study 2: Representation of Irrational Numbers
Ravi used the Pythagoras theorem to represent √2 and √3 on a number line.
Questions:
- The identity used is:
(a) a²+b²=c² (b) (a+b)² (c) (a−b)² (d) a²−b²
Answer: (a) - √2 belongs to:
(a) integers (b) irrationals (c) rationals (d) whole numbers
Answer: (b) - Which pair are both irrational?
(a) √2, √3 (b) √4, √3 (c) √9, √16 (d) √25, √49
Answer: (a) - Approx value of √2 is:
(a) 1.2 (b) 1.3 (c) 1.4 (d) 1.5
Answer: (c) - √3 lies between:
(a) 1 and 2 (b) 2 and 3 (c) 3 and 4 (d) none
Answer: (b)
Case Study 3: Decimal Expansions
Students got decimals: 3.142857142857… and 0.1010010001…
Questions:
- 3.142857… is:
(a) terminating (b) non-terminating repeating (c) non-terminating non-repeating (d) none
Answer: (b) - 0.1010010001… is:
(a) rational (b) irrational (c) integer (d) terminating
Answer: (b) - Which fraction gives terminating decimal?
(a) 7/25 (b) 1/3 (c) 1/11 (d) 1/7
Answer: (a) - Rational decimals are:
(a) terminating (b) repeating (c) either terminating or repeating (d) non-repeating
Answer: (c) - Which is irrational?
(a) 22/7 (b) π (c) 3.5 (d) 7/2
Answer: (b)
Case Study 4: Rationalisation
Vimal simplified √3/(√5+√3) by multiplying with (√5−√3).
Questions:
- Rationalisation makes denominator:
(a) integer (b) irrational (c) decimal (d) fraction
Answer: (a) - (√5+√3)(√5−√3) = ?
(a) 5−3 (b) 25−9 (c) 8 (d) √2
Answer: (a) - Identity used:
(a) a²−b² (b) (a+b)² (c) (a−b)² (d) (a+b)(a+b)
Answer: (a) - Rationalisation of √7/(√7−√2) = ?
(a) (√7+√2)/5 (b) (√7+√2)/7 (c) √7+√2 (d) none
Answer: (a) - Rationalisation is used for:
(a) removing roots from denominator (b) making numerator rational (c) converting to decimal (d) none
Answer: (a)
Case Study 5: Square Roots in Real Life
A builder constructed a square garden of 50 m² area.
Questions:
- Side length = ?
(a) √50 (b) √25 (c) √100 (d) √7
Answer: (a) - √50 simplified = ?
(a) 5√2 (b) 10√2 (c) 25√2 (d) none
Answer: (a) - √50 is:
(a) rational (b) irrational (c) integer (d) natural
Answer: (b) - √50 lies between:
(a) 7 and 8 (b) 6 and 7 (c) 8 and 9 (d) 9 and 10
Answer: (a) - Approximate √50 = ?
(a) 6.9 (b) 7.07 (c) 7.5 (d) 8
Answer: (b)
Case Study 6: Laws of Exponents
A student solved (2³×2⁴)÷2² = 2⁵.
Questions:
- Law used:
(a) aᵐ×aⁿ (b) aᵐ/aⁿ (c) (aᵐ)ⁿ (d) both (a)&(b)
Answer: (d) - Simplify: (x²)³.
(a) x⁵ (b) x⁶ (c) x⁷ (d) x⁸
Answer: (b) - 9^(3/2) = ?
(a) 9 (b) 27 (c) 81 (d) none
Answer: (b) - If 2ˣ = 32, x=?
(a) 4 (b) 5 (c) 6 (d) 3
Answer: (b) - 8^(2/3) = ?
(a) 2 (b) 4 (c) 8 (d) 16
Answer: (b)
Case Study 7: Divisibility
Students tested 735 for divisibility by 3, 5, and 11.
Questions:
- Divisible by 3 because:
(a) last digit 0 (b) last digit 5 (c) digit sum divisible by 3 (d) none
Answer: (c) - Divisible by 5 because:
(a) last digit 0 or 5 (b) sum divisible by 5 (c) alt sum divisible by 11 (d) none
Answer: (a) - For 11, check:
(a) sum of digits (b) alternating sum (c) last 3 digits (d) last digit
Answer: (b) - 735 ÷ 11 = ?
(a) integer (b) not divisible (c) remainder 2 (d) none
Answer: (b) - Divisible by both 2 & 5?
(a) 15 (b) 20 (c) 25 (d) 35
Answer: (b)
Case Study 8: Terminating & Non-Terminating Fractions
Teacher wrote 1/2, 1/3, 1/8, 1/11.
Questions:
- 1/8 = ?
(a) 0.25 (b) 0.125 (c) 0.333… (d) 0.5
Answer: (b) - 1/3 = ?
(a) 0.25 (b) 0.5 (c) 0.333… (d) 0.125
Answer: (c) - Which is terminating?
(a) 1/8 (b) 1/3 (c) 1/11 (d) 1/7
Answer: (a) - Which is repeating decimal?
(a) 1/11 (b) 1/8 (c) 1/2 (d) 1/4
Answer: (a) - Which is irrational?
(a) 1/3 (b) √5 (c) 1/8 (d) 1/2
Answer: (b)
Case Study 9: Surds & Irrationals
Surveyor used √7 and √11 in measurement.
Questions:
- √7 and √11 are:
(a) rationals (b) irrationals (c) integers (d) whole numbers
Answer: (b) - √11 lies between:
(a) 2 and 3 (b) 3 and 4 (c) 4 and 5 (d) 5 and 6
Answer: (c) - Approx √7 = ?
(a) 2.6 (b) 2.65 (c) 2.7 (d) 2.8
Answer: (c) - √16 is:
(a) rational (b) irrational (c) prime (d) none
Answer: (a) - Which is irrational?
(a) √49 (b) √50 (c) √64 (d) √81
Answer: (b)
Case Study 10: Properties of Real Numbers
Students checked closure, commutative, associative properties.
Questions:
- Rational + Rational = ?
(a) rational (b) irrational (c) integer (d) natural
Answer: (a) - Rational × Rational = ?
(a) rational (b) irrational (c) prime (d) none
Answer: (a) - Addition is commutative for:
(a) natural (b) integers (c) rationals (d) all reals
Answer: (d) - Subtraction is:
(a) commutative (b) not commutative (c) associative (d) distributive
Answer: (b) - Multiplication is associative for:
(a) natural (b) integers (c) rationals (d) all of these
Answer: (d)