If (\sqrt{a}=5\sqrt{2}), what is the value of (a)?
Correct answer: C
Step 1: Square both sides. Step 2: (a=(5\sqrt{2})^2=25\times2=50). Step 3: Apply ((k\sqrt{m})^2=k^2m) correctly.
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.
TOPIC PRACTICE
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Up to 25 questions from this page. Select your focus, then start.
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Hard · Level 4
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- (10)
- (25)
- (50)
- (100)
Hard · Level 4
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- (\sqrt{2}) and (-\sqrt{2})
- (\sqrt{3}) and (2\sqrt{3})
- (\sqrt{5}) and (-\sqrt{5})
- (\sqrt{7}) and (1-\sqrt{7})
Hard · Level 4
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- (2+\sqrt{3})
- (2-\sqrt{3})
- (\sqrt{3}-2)
- (\frac{2+\sqrt{3}}{7})
Hard · Level 4
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- (a=49)
- (a=81)
- (a=90)
- (a=121)
Hard · Level 4
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- (2\sqrt{3}+3)
- (5\sqrt{3})
- (6)
- (2+3\sqrt{3})
Hard · Level 4
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- (16)
- (25)
- (36)
- (40)
Hard · Level 4
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- (8\sqrt{3})
- (6\sqrt{3})
- (4\sqrt{3})
- (\sqrt{102})
Hard · Level 4
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- Rational
- Irrational
- Integer
- Zero
Hard · Level 4
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- (\sqrt{17})
- (\sqrt{20})
- (\sqrt{24})
- (\sqrt{26})
Hard · Level 4
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- Rational
- Irrational
- Integer
- Natural number
Hard · Level 4
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- (3)
- (\sqrt{9}) and hence (3)
- (9\sqrt{3})
- (\sqrt{24})
Hard · Level 4
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- (\sqrt{8}-2\sqrt{2})
- (\sqrt{18}-2\sqrt{2})
- (\sqrt{20}-2\sqrt{5})
- (\sqrt{27}-2\sqrt{3})
Hard · Level 4
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- (1)
- (-1)
- (2\sqrt{2})
- (\sqrt{2})
Hard · Level 4
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- (1)
- (2)
- (4)
- (5)
Hard · Level 4
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- ((\sqrt{6})^2)
- (\sqrt{12}\times\sqrt{3})
- (\sqrt{6}+\sqrt{24})
- ((\sqrt{5}+2)(\sqrt{5}-2))
Hard · Level 4
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- It is always rational
- It is always irrational
- It is always an integer
- It is always zero
Hard · Level 4
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- (a=2,b=7)
- (a=3,b=12)
- (a=5,b=6)
- (a=7,b=8)
Hard · Level 4
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- (2\sqrt{2}-2)
- (2\sqrt{2}+2)
- (\sqrt{2}-1)
- (\sqrt{2}+1)
Hard · Level 4
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- (7-2\sqrt{10})
- (3)
- (7+2\sqrt{10})
- (\sqrt{3})
Hard · Level 4
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- (x=\sqrt{8},y=-2\sqrt{2})
- (x=\sqrt{3},y=\sqrt{12})
- (x=\sqrt{5},y=\sqrt{20})
- (x=\sqrt{6},y=\sqrt{24})
Hard · Level 4
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- (\frac{\sqrt{2}}{2})
- (\frac{1}{2})
- (0.75)
- (\sqrt{4})
Hard · Level 4
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- (6\sqrt{3})
- (5\sqrt{3})
- (3\sqrt{6})
- (\sqrt{42})
Hard · Level 4
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- If (p) is not a perfect square, (\sqrt{p}) is irrational
- If (p) is prime, (\sqrt{p}) is irrational
- If (p=36), (\sqrt{p}) is irrational
- If (p=50), (\sqrt{p}) is irrational
Hard · Level 4
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- (1)
- (5)
- (\sqrt{6})
- (2\sqrt{6})
Hard · Level 4
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- (12\sqrt{2}), irrational
- (18), rational
- (6\sqrt{2}), irrational
- (22), rational
In which option are both numbers irrational and their sum is also irrational?
Correct answer: B
Step 1: (\sqrt{3}) and (2\sqrt{3}) are both irrational. Step 2: Their sum is (3\sqrt{3}), which is irrational. Step 3: In sum questions, identify whether like surds cancel or combine.
If (x=2-\sqrt{3}), what is the rationalized form of (\frac{1}{x})?
Correct answer: A
Step 1: The conjugate of the denominator in (\frac{1}{2-\sqrt{3}}) is (2+\sqrt{3}). Step 2: (\frac{1}{2-\sqrt{3}}\times\frac{2+\sqrt{3}}{2+\sqrt{3}}=\frac{2+\sqrt{3}}{4-3}=2+\sqrt{3}). Step 3: Multiplying by the conjugate removes the radical from the denominator.
In which option is (\sqrt{a}) irrational?
Correct answer: C
Step 1: (49), (81), and (121) are perfect squares. Step 2: (90) is not a perfect square, so (\sqrt{90}) is irrational. Step 3: To decide the nature of a square root, first check perfect squares.
Which option gives the correct value of (\sqrt{3}(2+\sqrt{3}))?
Correct answer: A
Step 1: Use the distributive law. Step 2: (\sqrt{3}(2+\sqrt{3})=2\sqrt{3}+(\sqrt{3})^2=2\sqrt{3}+3). Step 3: Remember that (\sqrt{3}\times\sqrt{3}=3).
If (2+\sqrt{n}) is irrational and (n) is a positive integer, which (n) is suitable?
Correct answer: D
Step 1: (2) is rational. So (2+\sqrt{n}) is irrational when (\sqrt{n}) is irrational. Step 2: (40) is not a perfect square, so (\sqrt{40}) is irrational. Step 3: First eliminate perfect squares from the given integers.
Which option correctly gives the sum of (\sqrt{75}) and (\sqrt{27})?
Correct answer: A
Step 1: (\sqrt{75}=5\sqrt{3}) and (\sqrt{27}=3\sqrt{3}). Step 2: The sum is (5\sqrt{3}+3\sqrt{3}=8\sqrt{3}). Step 3: Do not combine separate square roots directly into one root.
If (a=\sqrt{2}+\sqrt{5}), what will be the nature of (a^2)?
Correct answer: B
Step 1: ((\sqrt{2}+\sqrt{5})^2=2+5+2\sqrt{10}). Step 2: This is (7+2\sqrt{10}), which has an irrational part. Step 3: When squaring a sum of two different surds, pay attention to the middle term.
Which of the following numbers is not between (4) and (5)?
Correct answer: D
Step 1: (4=\sqrt{16}) and (5=\sqrt{25}). Step 2: (17), (20), and (24) lie between (16) and (25), but (26) is greater than (25). Step 3: For positive square roots, comparing squares is easier.
If (x=\sqrt{7}), what is the nature of (3x-2)?
Correct answer: B
Step 1: (3x-2=3\sqrt{7}-2). Step 2: (3\sqrt{7}) is irrational, and subtracting a rational number keeps it irrational. Step 3: A non-zero rational multiple of a surd remains irrational.
Which option gives the correct value of (\frac{\sqrt{27}}{\sqrt{3}})?
Correct answer: B
Step 1: (\frac{\sqrt{27}}{\sqrt{3}}=\sqrt{\frac{27}{3}}). Step 2: This is (\sqrt{9}=3), which is rational. Step 3: In division, simplifying the radicals together is a quick method.
Which option is equal to (0)?
Correct answer: A
Step 1: (\sqrt{8}=2\sqrt{2}). Step 2: Therefore (\sqrt{8}-2\sqrt{2}=0), which is rational. Step 3: Sometimes terms that look irrational cancel completely.
If (x=1+\sqrt{2}), what is the value of (x^2-2x)?
Correct answer: A
Step 1: (x^2-2x=x(x-2)). Step 2: With (x=1+\sqrt{2}), (x-2=\sqrt{2}-1), so the product ((1+\sqrt{2})(\sqrt{2}-1)=1). Step 3: Recognizing conjugate-like forms makes calculation shorter.
Which number lies between (\sqrt{3}) and (2\sqrt{3})?
Correct answer: B
Step 1: (\sqrt{3}) is about (1.732), and (2\sqrt{3}) is about (3.464). Step 2: (2) lies between these two values. Step 3: For comparison, you may use estimation or squaring.
Which of the following expressions is not rational?
Correct answer: C
Step 1: Simplify each option first. Step 2: (\sqrt{6}+\sqrt{24}=\sqrt{6}+2\sqrt{6}=3\sqrt{6}), which is irrational. Step 3: Radicals may cancel in multiplication, but not always in addition.
If (m) is a positive integer and (m) is not a perfect square, which statement about (\sqrt{m}+4) is correct?
Correct answer: B
Step 1: Since (m) is not a perfect square, (\sqrt{m}) is irrational. Step 2: (4) is rational, and adding it to an irrational number gives an irrational number. Step 3: Connect the non-perfect-square condition directly with the nature of the square root.
Which option is a correct example that makes (\sqrt{a}\times\sqrt{b}) rational?
Correct answer: B
Step 1: (\sqrt{a}\times\sqrt{b}=\sqrt{ab}). Step 2: For (a=3,b=12), (ab=36), so (\sqrt{36}=6), which is rational. Step 3: Check whether the product inside the radical becomes a perfect square.
Which number is equal to (\frac{2}{\sqrt{2}+1})?
Correct answer: A
Step 1: The conjugate of the denominator is (\sqrt{2}-1). Step 2: (\frac{2}{\sqrt{2}+1}\times\frac{\sqrt{2}-1}{\sqrt{2}-1}=\frac{2(\sqrt{2}-1)}{2-1}=2\sqrt{2}-2). Step 3: Choosing the correct conjugate sign is very important.
If (x=\sqrt{5}-\sqrt{2}), what is (x^2)?
Correct answer: A
Step 1: Use ((a-b)^2=a^2-2ab+b^2). Step 2: (x^2=5-2\sqrt{10}+2=7-2\sqrt{10}). Step 3: Do not forget the negative sign in the middle term when squaring a difference.
In which option is (x+y) rational while both (x) and (y) are irrational?
Correct answer: A
Step 1: (\sqrt{8}=2\sqrt{2}), so (x) and (y=-2\sqrt{2}) are both irrational. Step 2: Their sum is (2\sqrt{2}-2\sqrt{2}=0), which is rational. Step 3: Opposite irrational terms can give a rational sum.
If (0<x<1) and (x) is irrational, which example is suitable?
Correct answer: A
Step 1: (\sqrt{2}) is irrational, and dividing by non-zero rational (2) keeps it irrational. Step 2: (\frac{\sqrt{2}}{2}) is about (0.707), so it lies between (0) and (1). Step 3: Check both the interval condition and the nature of the number.
Which option is the correct simplified form of (\sqrt{3}+\sqrt{12}+\sqrt{27})?
Correct answer: A
Step 1: (\sqrt{12}=2\sqrt{3}) and (\sqrt{27}=3\sqrt{3}). Step 2: The total sum is (\sqrt{3}+2\sqrt{3}+3\sqrt{3}=6\sqrt{3}). Step 3: Converting all terms into like surds makes addition easy.
Which option explains the irrationality of (\sqrt{p}) incorrectly?
Correct answer: C
Step 1: (36) is a perfect square. Step 2: (\sqrt{36}=6), which is rational, so the statement for (p=36) is incorrect. Step 3: Checking perfect squares is the safest way to decide the nature of a square root.
If (a=\sqrt{2}+\sqrt{3}) and (b=\sqrt{3}-\sqrt{2}), what is the value of (ab)?
Correct answer: A
Step 1: View (ab) as ((\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})). Step 2: This equals ((\sqrt{3})^2-(\sqrt{2})^2=3-2=1). Step 3: Since addition order does not change the sum, recognize the conjugate form.
If (A=(3+\sqrt{2})^2-(3-\sqrt{2})^2), what is the correct value and nature of (A)?
Correct answer: A
Step 1: Use ((a+b)^2-(a-b)^2=4ab). Step 2: Here (a=3) and (b=\sqrt{2}), so (A=4\times3\times\sqrt{2}=12\sqrt{2}), which is irrational. Step 3: In such questions, use the identity instead of expanding both squares fully.
QUIZ COMPLETE