Irrational numbers

Correct answer: C. (50)

Explanation: 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.

Correct answer: B. (\sqrt{3}) and (2\sqrt{3})

Explanation: 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.

Correct answer: A. (2+\sqrt{3})

Explanation: 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.

Correct answer: C. (a=90)

Explanation: 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.

Correct answer: A. (2\sqrt{3}+3)

Explanation: 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).

Correct answer: D. (40)

Explanation: 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.

Correct answer: A. (8\sqrt{3})

Explanation: 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.

Correct answer: B. Irrational

Explanation: 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.

Correct answer: D. (\sqrt{26})

Explanation: 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.

Correct answer: B. Irrational

Explanation: 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.

Correct answer: B. (\sqrt{9}) and hence (3)

Explanation: 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.

Correct answer: A. (\sqrt{8}-2\sqrt{2})

Explanation: 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.

Correct answer: A. (1)

Explanation: 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.

Correct answer: B. (2)

Explanation: 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.

Correct answer: C. (\sqrt{6}+\sqrt{24})

Explanation: 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.

Correct answer: B. It is always irrational

Explanation: 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.

Correct answer: B. (a=3,b=12)

Explanation: 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.

Correct answer: A. (2\sqrt{2}-2)

Explanation: 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.

Correct answer: A. (7-2\sqrt{10})

Explanation: 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.

Correct answer: A. (x=\sqrt{8},y=-2\sqrt{2})

Explanation: 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.

Correct answer: A. (\frac{\sqrt{2}}{2})

Explanation: 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.

Correct answer: A. (6\sqrt{3})

Explanation: 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.

Correct answer: C. If (p=36), (\sqrt{p}) is irrational

Explanation: 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.

Correct answer: A. (1)

Explanation: 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.

Correct answer: A. (12\sqrt{2}), irrational

Explanation: 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.