1
Zero is rational, because —
💡 Hint:
q cannot be zero.
Explanation:
0 = 0/1. Both numerator and denominator are integers and the denominator is 1.
Class 9 · Maths · Chapter 1 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Number Systems
Number Systems
How to use this page:
1. Read — 1.1 rationals · 1.2 irrationals and the square-root spiral · 1.3 decimals · 1.4 number line · 1.5 operations · 1.6 exponents, diagram, worked example, board tip
2. Check — each lesson has its own questions; the number follows the lesson
3. Mastery ★ — all of that lesson correct. Redo the wrong ones
4. The memory figure is a number line with the rational number 1/2 and the irrational number √2 both marked.
In the 15-chapter Bihar book this is chapter 1. Ganita Manjari 2026-27 uses a different order. Progress stays in this browser.
🏅 Your achievements — 0/6 badges
🔥 0day streak · longest 0
⭐ 0total XP · today 0
🎓 1level
Next level: 150 XP to goright answer +10 · after a hint +5 · review +6 · badge +25
Last 7 days:
Everything is saved in this phone/browser — no login.
- 1 Rational numbers — Exercise 1.1
- 2 Irrational numbers and the square-root spiral — Exercise 1.2
- 3 Decimal expansions — Exercise 1.3
- 4 Real numbers on the number line — Exercise 1.4
- 5 Operations and rationalising the denominator — Exercise 1.5
- 6 Laws of exponents for real numbers — Exercise 1.6
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
Rational numbers — Exercise 1.1
परिमेय संख्याएँ — प्रश्नावली 1.1 · NCERT 1.1 · Exercise 1.1
N, W, Z and QNotes
The natural numbers are 1, 2, 3, …. The set is N. Add zero and you have the whole numbers. The set is W. Add the negative integers and you have the integers. The set is Z. Z comes from the German word zahlen, which means to count.
A number is rational when it can be written as p/q. Here p and q are integers and q ≠ 0. Since 4 = 4/1, every integer is rational. Since 0 = 0/1, zero is rational too. So are 1/2 and −3/5.
Between two rational numbers there are infinitely many rational numbers. One way is the average (a + b)/2. Repeat it. Another way: use a common denominator and fill the numerators in between.
| Set | Symbol | Example |
|---|---|---|
| Natural | N | 1, 2, 3, … |
| Whole | W | 0, 1, 2, 3, … |
| Integers | Z | …, −2, −1, 0, 1, 2, … |
| Rational | Q | 0, 4/1, 1/2, −3/5 |
Worked exampleExample
Question: Write three rational numbers between 1 and 2.
Step 1. Use the common denominator 4. Then 1 = 4/4 and 2 = 8/4.
Step 2. The numerators between 4 and 8 are 5, 6 and 7.
Step 3. The numbers are 5/4, 6/4 and 7/4. Also 6/4 = 3/2.
Check: 1 < 5/4 < 3/2 < 7/4 < 2. Each is in the form p/q and the denominator is not zero.
10-second revision
- Q = p/q, q ≠ 0
- Every integer is rational, because n = n/1
- Infinitely many rationals lie between two rationals
Board tip · BSEBBoard tip
Write zero as 0/1. Never write q = 0, because division by zero is not allowed.
Board tip · CBSEBoard tip
Give a reason with true or false. Every natural number is whole, but every integer is not whole.
Check your understandingall correct = mastery ★
2
Every integer is a whole number.
Explanation:
False. −3 is an integer but not a whole number. The whole numbers are 0, 1, 2, ….
3
The symbol for the set of rational numbers is ______.
💡 Hint:
The letter linked with quotient.
Explanation:
The symbol is Q.
4
One rational number between 3 and 4 is —
💡 Hint:
Take the average (3+4)/2.
Explanation:
(3+4)/2 = 7/2 = 3.5, which lies between 3 and 4.
5
Find one rational number between 1/2 and 3/4 by the average. Show the steps.
Model answer (board format)
The average is (1/2 + 3/4)/2. First the sum: 1/2 = 2/4, so 2/4 + 3/4 = 5/4. Then (5/4)/2 = 5/8. And 1/2 = 4/8 < 5/8 < 6/8 = 3/4.
Check your answer:
Irrational numbers and the square-root spiral — Exercise 1.2
अपरिमेय संख्या और वर्गमूल सर्पिल — प्रश्नावली 1.2 · NCERT 1.2 · Exercise 1.2 · classroom activity
What cannot be written as p/qNotes
A number is irrational when it cannot be written as p/q, where p and q are integers and q ≠ 0. √2, √3, √5 and π are irrational. So is a decimal such as 0.10110111011110…, in which the run of 1s keeps growing.
Every irrational number is real. Not every real number is irrational, because 1/2 is both real and rational. Not every point of the number line is of the form √m. If m is a natural number, √m stays positive or zero. Negative points and 1/2 do not fit that form.
√4 = 2 and √9 = 3 are rational. So it is wrong to say that the square root of every positive integer is irrational.
Exercise 1.2 classroom activity — the square-root spiralActivity
Take a large sheet. From a point O draw OP₁ of unit length. Draw P₁P₂ perpendicular to OP₁, again of length 1 unit. Then OP₂ is the hypotenuse. By Pythagoras, OP₂ = √(1² + 1²) = √2.
Now draw P₂P₃ perpendicular to OP₂, of length 1. Then OP₃ = √((√2)² + 1²) = √3. The next perpendicular gives OP₄ = √4 = 2. In general OPₙ = √n. Join the points and you get a spiral of √2, √3, √4, ….
Worked exampleExample
Question: Prove that √2 is irrational.
Step 1. Suppose √2 = p/q, where p and q are integers, q ≠ 0, and they have no common factor other than 1.
Step 2. Square both sides: p² = 2q². So p² is even, and therefore p is even. Write p = 2m.
Step 3. (2m)² = 2q² gives 4m² = 2q², so q² = 2m². Then q² is even, and therefore q is even.
Step 4. Both p and q are even, so 2 is a common factor. This clashes with their being coprime. The supposition is false. √2 is irrational.
10-second revision
- Irrational = not of the form p/q
- √2 is irrational, √4 = 2 is rational
- In the spiral, OPₙ = √n
Board tip · BSEBBoard tip
In the proof, suppose it is rational, then show that both being even is a contradiction.
Board tip · CBSEBoard tip
The √m statement is only the positive root for a natural m. Negative points are a different case.
Check your understandingall correct = mastery ★
1
Every irrational number is a real number.
Explanation:
True. The real numbers are the rationals together with the irrationals.
2
Which of these is rational?
💡 Hint:
Which one is a perfect square?
Explanation:
√9 = 3 = 3/1, which is rational. √2, √3 and π are irrational.
3
In the square-root spiral the length of OP₃ is ______ unit.
💡 Hint:
The new perpendicular side is 1 and OP₂ = √2.
Explanation:
√((√2)² + 1²) = √3.
4
Is the square root of every positive integer irrational? Answer with one example.
Model answer (board format)
No. √4 = 2 = 2/1, which is rational. √9 = 3 is rational too. A square root such as √2 is irrational, but the square root of a perfect square is an integer.
Check your answer:
Decimal expansions — Exercise 1.3
दशमलव प्रसार — प्रश्नावली 1.3 · NCERT 1.3 · Exercise 1.3
Terminating, recurring, or neitherNotes
Divide p by q. The remainder either becomes 0 or starts to repeat. If the remainder becomes 0, the decimal terminates. Examples: 7/8 = 0.875 and 1/2 = 0.5. If the remainders repeat, the decimal is non-terminating recurring. Examples: 1/3 = 0.333… and 1/7 = 0.142857142857….
The count of repeating remainders is less than the divisor. For 1/3 there is one remainder and the divisor is 3. For 1/7 there are six remainders and the divisor is 7. So the repeating block of 1/17 has at most 16 digits.
The decimal of a rational number terminates or recurs. The converse is also true. A decimal that neither ends nor repeats is irrational.
Worked exampleExample
Question: Write 0.999… in the form p/q.
Step 1. Put x = 0.999….
Step 2. One digit repeats, so multiply by 10: 10x = 9.999….
Step 3. Subtract: 10x − x = 9.999… − 0.999… = 9. So 9x = 9.
Step 4. x = 9/9 = 1 = 1/1. Thus 0.999… and 1 are the same rational number.
10-second revision
- Terminating or recurring decimal = rational
- Neither terminating nor recurring = irrational
- 0.999… = 1
Board tip · BSEBBoard tip
Put the bar on the repeating block. The block of 1/7 is 142857, not only the digit 1.
Board tip · CBSEBoard tip
For a question such as 1/17, do not finish the division. Write the maximum digits as one less than the divisor.
Check your understandingall correct = mastery ★
1
The decimal of 7/8 is —
💡 Hint:
Division by 8 gives remainder 0.
Explanation:
7/8 = 0.875, which terminates. So it is rational.
2
0.10110111011110… is irrational, because it neither ends nor repeats a block.
Explanation:
True. The run of 1s keeps growing, so one fixed block does not repeat.
3
1/3 = 0.______… where one digit repeats.
💡 Hint:
On division by 3 the remainder stays 1.
Explanation:
1/3 = 0.333….
4
0.999… is equal to —
💡 Hint:
Put x = 0.999… and use 10x − x.
Explanation:
9x = 9, so x = 1.
5
The repeating block of 1/17 can have at most ______ digits.
💡 Hint:
The count is less than the divisor.
Explanation:
The divisor is 17, so at most 16 digits.
6
Convert 0.454545… into p/q. Write each step.
Model answer (board format)
x = 0.454545…. Two digits repeat, so 100x = 45.454545…. Then 100x − x = 45, so 99x = 45. Thus x = 45/99 = 5/11.
Check your answer:
Real numbers on the number line — Exercise 1.4
संख्या रेखा पर वास्तविक संख्या — प्रश्नावली 1.4 · NCERT 1.4 · Exercise 1.4 · √2
One point, one numberNotes
The rationals and the irrationals together are the real numbers. Every real number has exactly one point on the number line. Every point is exactly one real number.
Looking closer at a decimal is called successive magnification. For 3.765, look first between 3 and 4, then between 3.7 and 3.8, then between 3.76 and 3.77. The point 3.765 falls in that last interval. 4.2626… is magnified the same way up to four decimal places.
1/2 is rational and √2 is irrational. Each has its own point on the line. 1/2 lies between 0 and 1. √2 lies between 1 and 2, because 1² = 1 < 2 < 4 = 2².
Worked exampleExample
Question: Mark √2 on the number line. Use the same unit as the segment from 0 to 1.
Step 1. On the line draw OA = 1 unit. O is the origin.
Step 2. At A draw a perpendicular AB = 1 unit.
Formula: Pythagoras, OB² = OA² + AB².
Substitution: OB² = 1² + 1² = 2. So OB = √2 units, because length is positive.
Step 3. Take the compass with centre O and radius OB. Cut the line on the positive side at P. P is √2.
10-second revision
- Real = rational + irrational
- √2 lies between 1 and 2
- See 3.765 in 3–4, then 3.7–3.8, then 3.76–3.77
Board tip · BSEBBoard tip
In magnification, write the integer interval first. The first home of 3.765 is 3 and 4.
Board tip · CBSEBoard tip
For √5, go first to √4 = 2, then add a unit perpendicular and take the hypotenuse √5.
Check your understandingall correct = mastery ★
1
Where is √2 on the number line?
💡 Hint:
1² = 1 and 2² = 4. The number 2 lies between them.
Explanation:
1 < √2 < 2, because 1 < 2 < 4.
2
Every point on the number line is of the form √m, where m is a natural number.
Explanation:
False. 1/2 and −1 cannot be written that way. √m is not negative.
3
The hypotenuse of a right triangle with unit legs is ______ unit.
💡 Hint:
1² + 1².
Explanation:
√(1+1) = √2 units.
4
The first interval in the successive magnification of 3.765 is —
💡 Hint:
The integer part is 3.
Explanation:
3.765 lies between 3 and 4. Next it lies between 3.7 and 3.8.
5
Write the three magnifications, in order, for seeing 3.765 on the number line.
Model answer (board format)
First between 3 and 4. Then between 3.7 and 3.8. Then between 3.76 and 3.77. The point 3.765 lies in this third interval.
Check your answer:
Operations and rationalising the denominator — Exercise 1.5
संक्रियाएँ और हर का परिमेयकरण — प्रश्नावली 1.5 · NCERT 1.5 · Exercise 1.5
When the result stays irrationalNotes
The sum, difference, product and non-zero quotient of two rationals is rational again. Irrationals obey the commutative, associative and distributive laws for addition and multiplication. But two irrationals do not always give an irrational. √2 + (−√2) = 0 is rational. √2 × √2 = 2 is rational. √18 / √2 = √9 = 3 is rational.
If r is rational, r ≠ 0, and s is irrational, then r + s, r − s, rs and r/s are irrational. So 2 + √3 and 2√3 are irrational.
For positive a and b, √(ab) = √a √b, √(a/b) = √a / √b, (√a + √b)(√a − √b) = a − b, (a + √b)(a − √b) = a² − b, and (√a + √b)² = a + 2√(ab) + b. To rationalise a denominator, multiply by the conjugate. π = c/d in looks, but c and d are not both integers, so π does not become rational.
Worked exampleExample
Question: Rationalise the denominator of 1/(√5 − √2).
Formula: (√a − √b)(√a + √b) = a − b. Here a = 5 and b = 2.
Step 1. Multiply and divide by the conjugate √5 + √2. This multiplies by 1.
Step 2. Numerator = √5 + √2.
Step 3. Denominator = (√5)² − (√2)² = 5 − 2 = 3.
Answer: (√5 + √2)/3. The denominator 3 is rational.
10-second revision
- Non-zero rational × irrational = irrational
- The sum of two irrationals can be rational
- The conjugate of 1/(√a − √b) is √a + √b
Board tip · BSEBBoard tip
When you write π = circumference/diameter, add that the two lengths are not both integers.
Board tip · CBSEBoard tip
If the denominator has a root, the answer should have a rational denominator. Flipping the fraction is not enough.
Check your understandingall correct = mastery ★
1
2 + √3 is —
💡 Hint:
Rational plus irrational.
Explanation:
2 is rational and √3 is irrational, so the sum is irrational.
2
√2 × √8 is a rational number.
Explanation:
True. √2 × √8 = √16 = 4 = 4/1. The product of two irrationals can be rational.
3
The form of 1/√2 with a rational denominator is ______.
💡 Hint:
Multiply by √2/√2.
Explanation:
1/√2 × √2/√2 = √2/2.
4
π stays irrational even though π = c/d, because —
💡 Hint:
p/q needs p and q to be integers.
Explanation:
The circumference and the diameter are real lengths. Their ratio is not a ratio of integers.
5
The sum of two irrational numbers is always irrational.
Explanation:
False. √2 + (−√2) = 0, which is rational.
6
Rationalise the denominator of 1/(√7 − √3). Write the formula, the substitution and the answer.
Model answer (board format)
Formula (√a − √b)(√a + √b) = a − b. Here a = 7 and b = 3. Multiply by √7 + √3. Numerator √7 + √3. Denominator 7 − 3 = 4. Answer (√7 + √3)/4.
Check your answer:
Laws of exponents for real numbers — Exercise 1.6
वास्तविक संख्याओं के घातांक — प्रश्नावली 1.6 · NCERT 1.6 · Exercise 1.6
a > 0 and rational exponentsNotes
Let a > 0 be real and n a positive integer. Then ⁿ√a is the positive b for which bⁿ = a. In exponents, ⁿ√a = a1/n. If m and n have no common factor other than 1 and n > 0, then am/n = (ⁿ√a)m = ⁿ√(am).
Let a > 0 and let p, q be rational. Then ap · aq = ap+q, (ap)q = apq, ap / aq = ap−q and ap bp = (ab)p. Also a0 = 1 and a−n = 1/an.
The two paths for 43/2 give the same answer. (√4)3 = 23 = 8 and √(43) = √64 = 8. A fractional exponent on a negative base is not used in this chapter.
Worked exampleExample
Question: Find 322/5.
Step 1. Write 32 = 25.
Formula: am/n = (a1/n)m.
Substitution: 321/5 = (25)1/5 = 2.
Step 2. 322/5 = (321/5)2 = 22 = 4.
Check: 45 = (22)5 = 210 = 1024 and 322 = 1024. The two paths agree.
10-second revision
- Need a > 0
- a^p · a^q = a^(p+q)
- 32^(2/5) = 4
Board tip · BSEBBoard tip
Add or subtract the exponents first, then find the simple number. 2^(2/3) · 2^(1/3) = 2^1 = 2.
Board tip · CBSEBoard tip
Write a negative exponent as 1/a^n. 125^(−1/3) = 1/5.
Check your understandingall correct = mastery ★
1
641/2 equals —
💡 Hint:
The positive square root.
Explanation:
641/2 = √64 = 8.
2
93/2 = ______.
💡 Hint:
(√9)³.
Explanation:
(√9)³ = 3³ = 27.
3
If a > 0, then a0 = 1.
Explanation:
True. A zero exponent has the value 1 when the base is positive.
4
Simplify 22/3 · 21/3. Write the law and the substitution.
Model answer (board format)
Law a^p · a^q = a^(p+q). Here a = 2, p = 2/3, q = 1/3. The exponent is 2/3 + 1/3 = 1. So 2^1 = 2.
Check your answer:
❓ Full question bank — with answers and explanations — 59 questions
Pick a type. The 30 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false60 tried · 0 correctFill in the blanksFill60 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason40 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA110 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option. A wrong try brings a hint.
1
Which of these is a natural number?
💡 Hint:
Naturals start at 1.
Explanation:
1 is natural. 0 is whole, −2 is an integer, and 1/2 is a rational fraction.
2
The correct p/q form of 0 is —
💡 Hint:
The denominator is not zero.
Explanation:
0 = 0/1. Neither 1/0 nor 0/0 is allowed.
3
√16 is —
💡 Hint:
16 is a perfect square.
Explanation:
√16 = 4, which is rational.
4
The recurring block of 1/7 is —
💡 Hint:
Six remainders, divisor 7.
Explanation:
1/7 = 0.142857142857….
5
A terminating decimal is —
💡 Hint:
1/2 = 0.5.
Explanation:
1/2 = 0.5 terminates. 1/3 and 1/7 recur. √2 does not repeat.
6
√3 lies between which pair?
💡 Hint:
1² = 1, 2² = 4, and 1 < 3 < 4.
Explanation:
1 < √3 < 2.
7
√12 / √3 equals —
💡 Hint:
Use √(a/b) = √a / √b the other way.
Explanation:
√12 / √3 = √(12/3) = √4 = 2.
8
125−1/3 equals —
💡 Hint:
A negative exponent takes the reciprocal.
Explanation:
1251/3 = 5, so 125−1/3 = 1/5.
9
In the spiral the length of OP₄ is —
💡 Hint:
OPₙ = √n.
Explanation:
OP₄ = √4 = 2 units.
10
Between two rationals the rational numbers are —
💡 Hint:
Keep taking averages.
Explanation:
Infinitely many. (a+b)/2 keeps giving a new rational.
11
For a > 0, (a3)2 equals —
💡 Hint:
The exponents multiply.
Explanation:
(a3)2 = a6.
12
Writing 1/√2 as √2/2 is called —
💡 Hint:
The denominator becomes rational.
Explanation:
This is called rationalising the denominator.
True or false
0/61
Every natural number is a whole number.
Explanation:
True. 1, 2, 3, … are among the whole numbers.
2
Every rational number is a whole number.
Explanation:
False. 1/2 is rational but not a whole number.
3
Every real number is irrational.
Explanation:
False. Rational numbers are real too.
4
π is rational because it is written as c/d.
Explanation:
False. c and d are not both integers. π is irrational.
5
2√5 is irrational.
Explanation:
True. 2 is a non-zero rational and √5 is irrational.
6
If a > 0, then a5 / a2 = a3.
Explanation:
True. The exponent is 5 − 2 = 3.
Fill in the blanks
0/61
The symbol for the whole numbers is ______.
💡 Hint:
Zero is included.
Explanation:
0, 1, 2, 3, ….
2
√9 = ______.
💡 Hint:
The positive root.
Explanation:
3² = 9.
3
The decimal of 1/2 is ______.
💡 Hint:
One half.
Explanation:
A terminating decimal.
4
√2 × √2 = ______.
💡 Hint:
√(2×2).
Explanation:
The product of two irrationals became rational.
5
641/2 = ______.
💡 Hint:
8 × 8 = 64.
Explanation:
√64 = 8.
6
For a > 0, a0 = ______.
💡 Hint:
Any positive base.
Explanation:
The zero exponent.
Match
0/21
Match the number with its type.
Column B: A. Terminating rational · B. Recurring rational · C. Irrational root · D. Irrational, circle ratio
1. √2
2. 0.875
3. 0.333…
4. π
Explanation:
√2 is a root, 0.875 terminates, 0.333… recurs, and π is an irrational ratio.
2
Match the law with its form. a > 0.
Column B: A. a^(p−q) · B. (ab)^p · C. a^(p+q) · D. a^(pq)
1. Product
2. Power of a power
3. Quotient
4. Same exponent
Explanation:
Product adds, a power multiplies, quotient subtracts, and a^p b^p = (ab)^p.
Assertion–reason
0/4Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): √2 is irrational.
Reason (R): √2 cannot be written as p/q with integers p and q.
Explanation:
Both are true and R explains A.
2
Assertion (A): 0.333… is rational.
Reason (R): Every integer is rational.
Explanation:
Both are true, but R does not explain this decimal. The explanation is that the decimal recurs.
3
Assertion (A): There are infinitely many rational numbers between 1 and 2.
Reason (R): √4 is irrational.
Explanation:
A is true. R is false, because √4 = 2 is rational.
4
Assertion (A): Every real number is irrational.
Reason (R): A rational decimal terminates or recurs.
Explanation:
A is false. R is true.
Coefficient practice
0/4These equations are not results of this maths chapter. This is only practice in filling coefficients. A blank coefficient means 1.
1
This is coefficient practice, not a result of the number-systems chapter. H2 + O2 makes water. Balance by filling coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice, not a result of the number-systems chapter. N2 + H2 makes NH3. Balance by filling coefficients (blank = 1).
N2 + H2 → NH3
Explanation: N2 + 3H2 → 2NH3
3
This is coefficient practice, not a result of the number-systems chapter. C + O2 makes CO2. Balance by filling coefficients (blank = 1).
C + O2 → CO2
Explanation: C + O2 → CO2
4
This is coefficient practice, not a result of the number-systems chapter. H2 + Cl2 makes HCl. Balance by filling coefficients (blank = 1).
H2 + Cl2 → HCl
Explanation: H2 + Cl2 → 2HCl
Classify
0/21
Place each number as rational or irrational.
√4
√2
0.25
π
Explanation:
√4 = 2 and 0.25 = 1/4 are rational. √2 and π are irrational.
2
Choose the kind of decimal.
7/8
1/3
√2
1/2
Explanation:
7/8 and 1/2 terminate. 1/3 recurs. √2 does not recur.
Very short answer
0/41
Define a rational number in one line.
Model answer (board format)
A number that can be written as p/q, with p and q integers and q ≠ 0, is rational.
Check your answer:
2
Write two rational numbers between 3 and 4.
Model answer (board format)
7/2 and 13/4. Here 7/2 = 3.5 and 13/4 = 3.25. Both lie between 3 and 4.
Check your answer:
3
What is an irrational number? Give one example.
Model answer (board format)
A number that cannot be written as p/q is irrational. An example is √2.
Check your answer:
4
Write the value of a0 for a > 0.
Model answer (board format)
a0 = 1.
Check your answer:
Short answer
0/41
Convert 0.272727… into p/q.
Model answer (board format)
x = 0.272727…. Then 100x = 27.272727…. So 99x = 27 and x = 27/99 = 3/11.
Check your answer:
2
Rationalise the denominator of 1/√7. Write the steps.
Model answer (board format)
1/√7 × √7/√7 = √7/7. The denominator 7 is rational.
Check your answer:
3
Find 163/4. Show one full path.
Model answer (board format)
16 = 24. Then 161/4 = 2, so 163/4 = 23 = 8.
Check your answer:
4
Show that √8 × √2 is rational and that 3 + √2 is irrational.
Model answer (board format)
√8 × √2 = √16 = 4, which is rational. 3 is rational and √2 is irrational, so the sum is irrational.
Check your answer:
Long answer
0/31
Write the contradiction proof that √2 is irrational. At the end say why √4 does not fall into this proof.
Model answer (board format)
Suppose √2 = p/q in lowest terms. Then p² = 2q², so p is even. From p = 2m we get q² = 2m², so q is even too. This clashes with being coprime. √4 = 2 = 2/1 is already in the form p/q, so we do not start by calling it irrational.
Check your answer:
2
Find the first three hypotenuses of the square-root spiral. Then write the three magnifications of 3.765.
Model answer (board format)
OP₁ = 1. OP₂ = √(1²+1²) = √2. OP₃ = √(2+1) = √3. OP₄ = √(3+1) = √4 = 2. The number 3.765 lies first between 3 and 4, then between 3.7 and 3.8, then between 3.76 and 3.77.
Check your answer:
3
Write the four laws of exponents. Then simplify 22/3 · 21/3 and 71/2 / 71/4. Also write the condition on the base.
Model answer (board format)
Need a > 0 and rational p, q. Then a^p a^q = a^(p+q), (a^p)^q = a^(pq), a^p/a^q = a^(p−q), a^p b^p = (ab)^p. First: 2^(2/3+1/3) = 2^1 = 2. Second: 7^(1/2−1/4) = 7^(1/4).
Check your answer:
BSEB model questions · practice
0/6This model set is for practice. It is not a question from any year annual examination.
1
Which of these is not a whole number?
💡 Hint:
A negative is not a whole number.
Explanation:
−1 is an integer, not a whole number.
2
The p/q form of 0.999… is —
💡 Hint:
9x = 9.
Explanation:
0.999… = 1 = 1/1.
3
321/5 = ______.
💡 Hint:
2.
Explanation:
25 = 32.
4
In the square-root spiral each new perpendicular is 1 unit.
Explanation:
True. That is the classroom activity of Exercise 1.2. OPₙ = √n.
5
Write four rational numbers between 5 and 6 using denominator 6.
Model answer (board format)
5 = 30/6 and 6 = 36/6. Between them: 31/6, 32/6, 33/6 and 34/6.
Check your answer:
6
Write the decimal difference between rational and irrational. Classify 0.5, 1/3 and √2. Write one sentence on 0.999….
Model answer (board format)
A rational decimal terminates or recurs. An irrational decimal neither terminates nor recurs. 0.5 is a terminating rational, 1/3 is a recurring rational, and √2 is irrational. Since 0.999… = 1, it is rational.
Check your answer:
CBSE-style questions
0/6These are competency-based practice questions. They are not copies of a past paper.
1
A student says √9 is irrational because of the root sign. The right decision is —
💡 Hint:
Check the perfect square.
Explanation:
√9 = 3 = 3/1. The root sign does not by itself make a number irrational.
2
Point A on the number line lies between 1 and 2, and its decimal neither ends nor repeats. A is —
💡 Hint:
Look at the kind of decimal.
Explanation:
Such a decimal belongs to an irrational number. √2 lies between 1 and 2 in the same way.
3
Assertion (A): 2 + √5 is irrational.
Reason (R): The sum of a non-zero rational and an irrational is irrational.
Explanation:
Both are true and R explains A.
4
Assertion (A): √2 × √8 is rational.
Reason (R): The product of two irrational numbers is always irrational.
Explanation:
A is true because the product is 4. R is false.
5
Rita left 1/(√5 − √2) unchanged. What should she do? Write the final value.
Model answer (board format)
She should multiply and divide by the conjugate √5 + √2. The denominator is (√5)² − (√2)² = 3. The answer is (√5 + √2)/3.
Check your answer:
6
A question gives base −8 and exponent 1/3. What caution should a student write from the condition in this chapter?
Model answer (board format)
In this chapter the laws of exponents are for a positive real base. −8 is not positive, so the form a^(m/n) is not applied here.
Check your answer:
🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
🏛️ Model questions on one page →
The verified label will be used only when a source page for the question is available.
CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
🔁 Spaced review — today’s questions
Wrong questions return soon; correct ones return after a few days.
🧠 What you learned + equation sheet
What you learned
- A rational number is p/q, where p and q are integers and q ≠ 0. Zero is rational too.
- An irrational number cannot be written as p/q. √2 is irrational and √4 = 2 is rational.
- A rational decimal terminates or recurs. An irrational decimal neither ends nor repeats.
- Every real number has exactly one point on the number line, and every point is one real number.
- The sum, difference, product and quotient of a non-zero rational and an irrational are irrational. Two irrationals can give either kind.
- If a > 0 and p, q are rational, the four laws of exponents apply. A denominator is rationalised with the conjugate.
| What | Keep this |
|---|---|
| Rational | p/q, q ≠ 0 |
| Irrational | not p/q, such as √2 |
| Terminating | 7/8 = 0.875 |
| Recurring | 1/3 = 0.333… |
| Rationalise | 1/√2 = √2/2 |
| Exponents | a^p · a^q = a^(p+q), a > 0 |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.