Which option correctly describes (\frac{\sqrt{45}}{3})?
Correct answer: B
Step 1: (\sqrt{45}=3\sqrt{5}). Step 2: (\frac{\sqrt{45}}{3}=\sqrt{5}) which is irrational. Step 3: Even after division check the remaining radical.
Mathematics
Irrational numbers
अपरिमेय संख्याएँ
In Class 10 Mathematics, this topic from the Real Numbers chapter introduces irrational numbers as numbers that cannot be expressed as a ratio of two integers. Students learn to identify them through non-terminating, non-repeating decimal expansions, distinguish them from rational numbers, and locate them on the number line. The topic also develops understanding of examples such as √2 and how irrational numbers fit into the wider system of real numbers.
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Expert · Level 6
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- Rational because denominator is (3)
- Irrational because it equals (\sqrt{5})
- Rational because (45) is divisible by (3)
- Integer because radical disappears
Expert · Level 6
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- (\frac{3\sqrt{2}}{2}) and irrational
- (2) and rational
- (\sqrt{2}) and irrational
- (\frac{1}{2}) and rational
Expert · Level 6
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- (\sqrt{2})
- (\sqrt{4})
- (\frac{3}{2})
- (1.25)
Expert · Level 6
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- (\sqrt{2}=-\frac{a}{b}) would be rational which is impossible
- (a) would be irrational
- Since (b\neq 0) the answer is (1)
- It is true in every case
Expert · Level 6
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- It is (6\sqrt{2}) and irrational
- It is (64) and rational
- It is (\sqrt{64}) and rational
- It is (0) and rational
Expert · Level 6
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- (\sqrt{3}+\sqrt{3}=2\sqrt{3})
- (\sqrt{5}+(2-\sqrt{5})=2)
- (\sqrt{2}+\sqrt{8}=3\sqrt{2})
- (\sqrt{7}+\sqrt{11}) is irrational
Expert · Level 6
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- (\frac{7}{3})
- (\sqrt{11})
- (\sqrt{12}+\sqrt{3})
- (0.25)
Expert · Level 6
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- (10)
- (\sqrt{10})
- (4\sqrt{2})
- (2\sqrt{6})
Expert · Level 6
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- (x) is rational
- (x) is irrational
- (x=1)
- (x=0)
Expert · Level 6
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- (\frac{\sqrt{5}-1}{2})
- (\frac{\sqrt{5}+1}{2})
- (\sqrt{5}-1)
- (\frac{2}{\sqrt{5}-1})
Expert · Level 6
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- Irrational because it is decimal
- Rational because (\sqrt{0.04}=0.2)
- Irrational because it is a square root
- Natural number
Expert · Level 6
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- Rational because denominator is a perfect square
- Irrational because (\sqrt{\frac{2}{9}}=\frac{\sqrt{2}}{3})
- Integer because (9) is a perfect square
- Zero because numerator is small
Expert · Level 6
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- (0.3333\ldots)
- (2.12112111211112\ldots) with no repeating block
- (5.75)
- (1.272727\ldots)
Expert · Level 6
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- (x) is not a perfect square
- (x) is negative
- (x) is necessarily prime
- (x=0)
Expert · Level 6
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- Make a right triangle with legs (1) and (1) and use the hypotenuse
- Mark any point anywhere
- Take (\sqrt{2}) as (2)
- Place it only between (0) and (1)
Expert · Level 6
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- It is (5) and rational
- It is (\sqrt{6}) and irrational
- It is (\sqrt{5}) and irrational
- It is (6) and rational
Expert · Level 6
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- Irrational because it has a square root
- Rational because (4+\sqrt{9}=7)
- Irrational because (9) is odd
- Not rational because (4) is added
Expert · Level 6
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- When (r=0)
- When (s=0)
- When (r=1)
- Never
Expert · Level 6
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- (2\sqrt{2}) and irrational
- (\sqrt{48}) and irrational
- (48) and rational
- (0) and rational
Expert · Level 6
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- The first is greater
- The second is greater
- Both are equal
- Both are rational
Expert · Level 6
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- Whether it becomes the square root of a perfect square
- Whether it has a large digit
- Whether it is written in the middle of the page
- Whether it is only positive
Expert · Level 6
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- (3\sqrt{7}) and irrational
- (\sqrt{35}) and irrational
- (35) and rational
- (5\sqrt{7}) and irrational
Expert · Level 6
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- Because (\frac{3}{2}) is irrational
- Because adding an irrational number to a rational number gives an irrational result
- Because (\sqrt{5}) is an integer
- Because every fraction is irrational
Expert · Level 6
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- ((\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2}))
- (\sqrt{6}+\sqrt{2})
- (\sqrt{6}-\sqrt{2})
- (\sqrt{12}+\sqrt{2})
Expert · Level 6
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- When (a) can be written as a ratio of two perfect squares
- When (a) is only written in decimal form
- When (a) is less than (1)
- When (a) is positive
Which option gives the correct form and type of (\sqrt{2}+\frac{1}{\sqrt{2}})?
Correct answer: A
Step 1: (\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}). Step 2: (\sqrt{2}+\frac{\sqrt{2}}{2}=\frac{3\sqrt{2}}{2}) which is irrational. Step 3: Rationalize the denominator before combining terms.
Which number is irrational and lies between (1) and (2)?
Correct answer: A
Step 1: (1<2<4). Step 2: Hence (1<\sqrt{2}<2) and (\sqrt{2}) is irrational. Step 3: To locate an irrational number compare squares.
If (a+b\sqrt{2}=0) where (a) and (b) are rational and (b\neq 0) then what conclusion follows?
Correct answer: A
Step 1: The equation gives (\sqrt{2}=-\frac{a}{b}). Step 2: (-\frac{a}{b}) is rational but (\sqrt{2}) is irrational. Step 3: Such questions use irrationality to create a contradiction.
Which option correctly describes (\sqrt{32}+\sqrt{50}-\sqrt{18})?
Correct answer: A
Step 1: (\sqrt{32}=4\sqrt{2}), (\sqrt{50}=5\sqrt{2}), and (\sqrt{18}=3\sqrt{2}). Step 2: The result is (6\sqrt{2}) which is irrational. Step 3: Add and subtract coefficients of like radicals.
Which option shows that adding an irrational number to an irrational number can give a rational result?
Correct answer: B
Step 1: (\sqrt{5}) is irrational and (2-\sqrt{5}) is also irrational. Step 2: Their sum is (2) which is rational. Step 3: There is no single always rule for the sum of two irrational numbers.
Which number is irrational but its square is rational?
Correct answer: B
Step 1: (\sqrt{11}) is irrational because (11) is not a perfect square. Step 2: ((\sqrt{11})^2=11) which is rational. Step 3: Squaring may remove the radical.
Which option gives the correct value of (\sqrt{2}\sqrt{8}+\sqrt{3}\sqrt{12})?
Correct answer: A
Step 1: (\sqrt{2}\sqrt{8}=\sqrt{16}=4). Step 2: (\sqrt{3}\sqrt{12}=\sqrt{36}=6), so the sum is (10). Step 3: In products combine radicals and check for perfect squares.
If (x=\sqrt{3}-\sqrt{2}) then which statement about (x) is correct?
Correct answer: B
Step 1: Suppose (\sqrt{3}-\sqrt{2}) is rational. Step 2: Squaring gives (5-2\sqrt{6}), forcing (\sqrt{6}) to be rational which is false. Step 3: The difference of unlike radicals is not directly an integer.
Which option correctly writes (\frac{2}{\sqrt{5}+1}) with a rational denominator?
Correct answer: A
Step 1: Multiply by the conjugate (\sqrt{5}-1). Step 2: (\frac{2(\sqrt{5}-1)}{5-1}=\frac{\sqrt{5}-1}{2}). Step 3: Multiplying by the conjugate makes the denominator rational.
Which option correctly describes (\sqrt{0.04})?
Correct answer: B
Step 1: (0.04=\frac{4}{100}). Step 2: (\sqrt{0.04}=\frac{2}{10}=0.2) which is rational. Step 3: Every square root is not irrational.
Which option correctly describes (\sqrt{\frac{2}{9}})?
Correct answer: B
Step 1: (\sqrt{\frac{2}{9}}=\frac{\sqrt{2}}{3}). Step 2: (\sqrt{2}) is irrational and remains irrational after division by (3). Step 3: Check both numerator and denominator of the fraction.
Which option is a correct decimal example of an irrational number?
Correct answer: B
Step 1: Terminating or recurring decimals are rational. Step 2: The given decimal is non-terminating and has no fixed repeating block, so it is irrational. Step 3: Decide by checking repetition, not just by seeing many digits.
If (\sqrt{x}) is irrational then which conclusion is always correct?
Correct answer: A
Step 1: The square root of a perfect square is an integer and rational. Step 2: If (\sqrt{x}) is irrational then (x) cannot be a perfect square. Step 3: For irrational square roots first check perfect-square status.
Which option is related to a proper method of locating (\sqrt{2}) on the number line?
Correct answer: A
Step 1: (1^2+1^2=2). Step 2: So the hypotenuse of that right triangle is (\sqrt{2}). Step 3: The Pythagoras theorem helps locate irrational numbers on a number line.
Which statement is correct about (\sqrt{2}\cdot \sqrt{3})?
Correct answer: B
Step 1: The product of radicals is (\sqrt{2}\cdot \sqrt{3}=\sqrt{6}). Step 2: Since (6) is not a perfect square (\sqrt{6}) is irrational. Step 3: In multiplication the numbers inside radicals multiply, not add.
Which option correctly describes (4+\sqrt{9})?
Correct answer: B
Step 1: (\sqrt{9}=3). Step 2: (4+3=7), a rational number. Step 3: Do not call every expression with a square root irrational immediately.
If (r) is rational and (s) is irrational then when is (r+s=s) possible?
Correct answer: A
Step 1: From (r+s=s), subtract (s) from both sides to get (r=0). Step 2: (0) is rational, so the condition is possible. Step 3: Form a simple equation before judging number types.
Which option gives the correct form and type of (\sqrt{98}-\sqrt{50})?
Correct answer: A
Step 1: (\sqrt{98}=7\sqrt{2}) and (\sqrt{50}=5\sqrt{2}). Step 2: The difference is (2\sqrt{2}), which is irrational. Step 3: Directly subtracting numbers inside radicals is wrong.
Which option is correct for comparing (\sqrt{2}+\sqrt{18}) and (\sqrt{8}+\sqrt{12})?
Correct answer: A
Step 1: (\sqrt{2}+\sqrt{18}=4\sqrt{2}). Step 2: (\sqrt{8}+\sqrt{12}=2\sqrt{2}+2\sqrt{3}). Since (\sqrt{3}>\sqrt{2}), the second expression is greater. Step 3: Simplify first and compare carefully.
Which option is the best first check for identifying an irrational number in an exam?
Correct answer: A
Step 1: In square-root questions first check whether the number inside is a perfect square. Step 2: A perfect square may give a rational square root while a non-perfect square often gives an irrational value. Step 3: Simplifying is the safest first step in identification questions.
If (x=\sqrt{7}+\sqrt{28}), what is the correct simplified form and type of (x)?
Correct answer: A
Step 1: Since (28=4\cdot 7), (\sqrt{28}=2\sqrt{7}). Step 2: Now (\sqrt{7}+2\sqrt{7}=3\sqrt{7}), and (\sqrt{7}) is irrational. Step 3: In exams, combine like radicals by adding their coefficients.
Which option proves that (\frac{3}{2}+\sqrt{5}) is irrational?
Correct answer: B
Step 1: (\frac{3}{2}) is rational and (\sqrt{5}) is irrational. Step 2: If their sum were rational, then (\sqrt{5}) would become the difference of two rational numbers, which is impossible. Step 3: For rational-plus-irrational questions, contradiction is a very useful method.
Which number is rational even though irrational square roots appear in it?
Correct answer: A
Step 1: The first option is a product of conjugate terms. Step 2: ((\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})=6-2=4), which is rational. Step 3: Identifying conjugates helps remove radicals quickly.
If (0<a<1) and (a) is rational, when will (\sqrt{a}) definitely be rational?
Correct answer: A
Step 1: The square root of a rational fraction is rational when both numerator and denominator can be perfect squares. Step 2: For example, (\sqrt{\frac{4}{9}}=\frac{2}{3}), so a ratio of two perfect squares is a safe condition. Step 3: Being positive or less than (1) does not guarantee a rational square root.
QUIZ COMPLETE