RowQ
The Vault
RowQ
The Vault
CBSE Class 9 Maths · 11 questions · 26 marks
Up to Class 8 you mostly counted, added and divided; here you finally ask what a number really is. You will separate rationals from irrationals, place every one of them on a single unbroken number line, and learn to tidy up messy surds like 1/(5 - √2). The laws of exponents then stretch to cover fractional powers, which is what makes 8^(2/3) meaningful.
Which of the following numbers is irrational?
Answer
√11 is irrational. √64 = 8, which is an integer and hence rational. 0.4545454545... is non-terminating but recurring, so it is rational (it equals 45/99 = 5/11). 22/7 is a ratio of two integers, so it is rational — it is only an approximation of π, not π itself. Since 11 is not a perfect square, √11 has a non-terminating, non-recurring decimal expansion and is therefore irrational.
The value of (√7 + √3)(√7 - √3) is:
Answer
The value is 4. Using the identity (a + b)(a - b) = a² - b² with a = √7 and b = √3, we get (√7)² - (√3)² = 7 - 3 = 4. Notice that the irrational cross terms +√21 and -√21 cancel exactly, which is the whole reason conjugates are used to rationalise denominators.
Simplified, 32^(3/5) equals:
Answer
32^(3/5) = 8. Write 32 as a power of 2: 32 = 2⁵. Then 32^(3/5) = (2⁵)^(3/5) = 2^(5 × 3/5) = 2³ = 8. Equivalently, the fifth root of 32 is 2, and 2³ = 8.
The decimal expansion of a rational number whose denominator, in lowest terms, is 40 will be:
Answer
It will be terminating. A rational number in lowest terms has a terminating decimal expansion exactly when its denominator has only 2s and 5s in its prime factorisation. Here 40 = 2³ × 5, which contains no other prime factor, so the expansion terminates. For example, 9/40 = 0.225.
Assertion (A): The sum of the two irrational numbers 5 + √6 and 5 - √6 is a rational number. Reason (R): The sum of any two irrational numbers is always irrational.
Answer
A is true but R is false. Adding the two numbers gives (5 + √6) + (5 - √6) = 10, which is rational, so Assertion A is true. However, Reason R is false: the sum of two irrationals need not be irrational, and this very example is a counter-example. R would only hold in special cases, so it cannot be a general rule.
Express the recurring decimal 0.373737... (in which the block 37 repeats forever) in the form p/q, where p and q are integers and q ≠ 0.
Answer
Let x = 0.373737... Since two digits repeat, multiply by 100: 100x = 37.373737... Subtract the first equation from the second: 100x - x = 37.373737... - 0.373737..., so 99x = 37. Therefore x = 37/99, which is already in lowest terms since 37 is prime and does not divide 99.
Rationalise the denominator of 6/(4 - √10) and simplify your answer.
Answer
Multiply the numerator and denominator by the conjugate of the denominator, 4 + √10. 6/(4 - √10) = 6(4 + √10) / [(4 - √10)(4 + √10)] The denominator becomes 4² - (√10)² = 16 - 10 = 6. So the expression equals 6(4 + √10)/6 = 4 + √10. Hence 6/(4 - √10) = 4 + √10, which has no radical in the denominator.
Insert any two irrational numbers between 0.6 and 0.7.
Answer
Any non-terminating, non-recurring decimal lying between 0.6 and 0.7 works. Two such numbers are 0.6101001000100001... and 0.6787887888788887..., where the pattern of digits keeps changing so the expansion never repeats. Both lie strictly between 0.6 and 0.7 because each begins with 0.6 followed by digits that keep the value under 0.7. In fact infinitely many irrational numbers lie between any two distinct rationals.
If x = (√8 + √5)/(√8 - √5), find the value of x + 1/x, showing all steps.
Answer
Step 1 — Note that 1/x = (√8 - √5)/(√8 + √5), simply the reciprocal. Step 2 — Rationalise x: multiply numerator and denominator by (√8 + √5). x = (√8 + √5)² / [(√8 - √5)(√8 + √5)] = (8 + 5 + 2√40) / (8 - 5) = (13 + 2√40)/3. Since √40 = 2√10, x = (13 + 4√10)/3. Step 3 — Similarly, 1/x = (√8 - √5)² / (8 - 5) = (13 - 2√40)/3 = (13 - 4√10)/3. Step 4 — Add them: x + 1/x = (13 + 4√10)/3 + (13 - 4√10)/3 = (13 + 13)/3 = 26/3. Hence x + 1/x = 26/3, a rational number, because the irrational parts cancel.
(a) Simplify (81)^(-1/4) × (243)^(1/5). (b) Show that 1/(3 + √7) + 1/(3 - √7) is rational, and find its value.
Answer
(a) Write each base as a power of 3: 81 = 3⁴ and 243 = 3⁵. (81)^(-1/4) = (3⁴)^(-1/4) = 3^(-1) = 1/3. (243)^(1/5) = (3⁵)^(1/5) = 3¹ = 3. Multiplying, (1/3) × 3 = 1. So the value is 1. (b) Take the common denominator (3 + √7)(3 - √7) = 9 - 7 = 2. The numerator is (3 - √7) + (3 + √7) = 6. So the sum equals 6/2 = 3. Since 3 can be written as 3/1 with integer numerator and non-zero integer denominator, the sum is rational. The irrational parts -√7 and +√7 cancelled in the numerator, which is why the result is rational.
Read the following and answer the questions that follow: A school is laying a square tiled floor for a new reading room. The architect fixes the area of the square floor at 50 square metres and marks the side length on a number line drawn along the wall so the tiling team can measure it accurately. (a) Write the exact side length of the square floor in surd form, simplified. (b) Is this side length rational or irrational? Justify your answer. (c) The team wants the side length correct to two decimal places. Find it. (d) If a decorative border strip is priced per metre and the team needs the exact perimeter, express the perimeter in simplified surd form.
Answer
(a) Area = side², so side = √50 = √(25 × 2) = 5√2 metres. (b) It is irrational. If 5√2 were rational, then dividing by 5 would make √2 rational, but 2 is not a perfect square, so √2 has a non-terminating, non-recurring decimal expansion. Hence 5√2 is irrational. (c) Using √2 ≈ 1.41421, side = 5 × 1.41421 = 7.07105 metres, which is 7.07 metres correct to two decimal places. (d) Perimeter = 4 × side = 4 × 5√2 = 20√2 metres, which is approximately 28.28 metres.
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