Real Numbers

Correct answer: A. √10 − 3

Explanation: The governing concept is rationalisation of a denominator containing a surd. First, √9=3, so the expression becomes 1/(√10+3). Multiply numerator and denominator by the conjugate √10−3. The result is [1(√10−3)]/[(√10+3)(√10−3)]. Using the difference-of-squares identity, the denominator becomes (√10)²−3²=10−9=1. Therefore the fraction simplifies to √10−3, so option A is correct. Option B is only the original denominator, not the rationalised value. Option C introduces the incorrect denominator 19. Option D is an equivalent reciprocal-style expression before rationalisation, but its denominator still contains a surd and it is not the required simplified rationalised form. The calculation also confirms the sign and denominator exactly.

Correct answer: A. \(\frac{\sqrt{11}+\sqrt{2}}{9}\)

Explanation: To rationalise the denominator, multiply the numerator and denominator by its conjugate, \(\sqrt{11}+\sqrt{2}\). The denominator becomes \((\sqrt{11}-\sqrt{2})(\sqrt{11}+\sqrt{2})=11-2=9\), while the numerator becomes \(\sqrt{11}+\sqrt{2}\). Hence, the rationalised form is \(\frac{\sqrt{11}+\sqrt{2}}{9}\). Exam tip: use the identity \((a-b)(a+b)=a^2-b^2\).

Correct answer: A. (11\sqrt{2})

Explanation: ( \sqrt{98}=7\sqrt{2} ), ( \sqrt{162}=9\sqrt{2} ), and ( \sqrt{50}=5\sqrt{2} ). So the result is (11\sqrt{2}).

Correct answer: B. 13\sqrt{11}

Explanation: Since \(44=4\times11\) and \(99=9\times11\), we have \(\sqrt{44}=2\sqrt{11}\) and \(\sqrt{99}=3\sqrt{11}\). Therefore, \(2\sqrt{44}+3\sqrt{99}=2(2\sqrt{11})+3(3\sqrt{11})=4\sqrt{11}+9\sqrt{11}=13\sqrt{11}\). Hence, option B is correct. Options A and C use incorrect coefficients, while option D changes the radicand incorrectly. Exam tip: simplify each radical first, then combine like radicals.

Correct answer: B. \(14\sqrt{7}\)

Explanation: Since \(112=16\times7\), \(\sqrt{112}=4\sqrt{7}\). Also, \(175=25\times7\), so \(2\sqrt{175}=2\times5\sqrt{7}=10\sqrt{7}\). Therefore, \(4\sqrt{7}+10\sqrt{7}=14\sqrt{7}\), making option B correct. Exam tip: after reducing surds to the same radical, add or subtract their coefficients only.

Correct answer: A. \(5\sqrt{3}\)

Explanation: \(\sqrt{48}=\sqrt{16\times3}=4\sqrt{3}\), so \(5\sqrt{48}=20\sqrt{3}\). Similarly, \(\sqrt{75}=\sqrt{25\times3}=5\sqrt{3}\), so \(3\sqrt{75}=15\sqrt{3}\). Therefore, \(20\sqrt{3}-15\sqrt{3}=5\sqrt{3}\), making option A correct. In such questions, first simplify each surd and then combine only like surd terms.

Correct answer: A. 40

Explanation: The governing concept is the product rule for square roots: √a × √b = √(ab) for non-negative a and b. Applying it gives √32 × √50 = √(32 × 50) = √1600. Since 1600 = 40², its principal square root is 40. A second verification is possible by simplifying each radical: √32 = √(16×2) = 4√2 and √50 = √(25×2) = 5√2. Their product is (4√2)(5√2) = 20×2 = 40. Thus option A is correct. Option B doubles the answer, option C is an incomplete or incorrect simplification because the two √2 factors must also be multiplied, and option D adds radicands instead of multiplying them.

Correct answer: B. (3)

Explanation: Direct answer: Option B, 3. Rewrite the expression as \(\frac{\sqrt{180}}{\sqrt{20}}\). Since both radicands are positive, \(\frac{\sqrt a}{\sqrt b}=\sqrt{\frac ab}\). Therefore \(\frac{\sqrt{180}}{\sqrt{20}}=\sqrt{\frac{180}{20}}=\sqrt9=3\). Another check is to simplify first: \(180=9\times20\), so \(\sqrt{180}=\sqrt{9\times20}=3\sqrt{20}efore dividing by \(\sqrt{20}\), giving 3. Option A, 9, is the radicand quotient before taking the square root; it stops one step too early. Option B is correct because \(\sqrt9=3\). Option C, \(\sqrt{160}\), incorrectly subtracts or combines the radicands instead of dividing them. Option D, 6, has no valid calculation and is not the result. The denominator is non-zero, so the division is defined. Remember: when dividing square roots of positive numbers, divide the radicands and then take the square root.

Correct answer: A. 14

Explanation: \(\frac{\sqrt{200}}{\sqrt{2}}=\sqrt{\frac{200}{2}}=\sqrt{100}=10\), and \(\sqrt{16}=4\). Therefore, the value is \(10+4=14\). Option B incorrectly reflects a subtraction-based calculation rather than evaluating the quotient and then adding \(\sqrt{16}\). In the exam, remember that for positive radicands, \(\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}\).

Correct answer: A. 1

Explanation: Since \(4.84=(2.2)^2\), we have \(\sqrt{4.84}=2.2\). Similarly, \(1.44=(1.2)^2\), so \(\sqrt{1.44}=1.2\). Therefore, the required value is \(2.2-1.2=1\). Options B and D are only the values of the second and first square roots respectively, while option C is not the correct difference. Exam tip: Recognise decimal numbers as squares of terminating decimals before subtracting.

Correct answer: A. ( \frac{47}{40} )

Explanation: The first term is ( \frac{4}{5} ) and the second is ( \frac{3}{8} ). The sum is ( \frac{32}{40}+\frac{15}{40}=\frac{47}{40} ).

Correct answer: B. ( \frac{41}{77} )

Explanation: ( \sqrt{\frac{81}{121}}=\frac{9}{11} ) and ( \sqrt{\frac{4}{49}}=\frac{2}{7} ). The difference is ( \frac{63}{77}-\frac{22}{77}=\frac{41}{77} ).

Correct answer: B. ( \sqrt{31} )

Explanation: Since (5^2<31<6^2), ( \sqrt{31} ) lies between (5) and (6). Since (31) is not a perfect square, it is irrational.

Correct answer: A. \(\sqrt{45}\)

Explanation: Compare the squares of the two positive numbers: \((\sqrt{45})^2=45\), whereas \(\left(\frac{13}{2}\right)^2=\frac{169}{4}=42.25\). Since \(45>42.25\), we have \(\sqrt{45}>\frac{13}{2}\). Therefore, option A is correct. Exam tip: For positive square roots or positive numbers, comparing their squares is often quicker than using decimal approximations.

Correct answer: C. (8) and (9)

Explanation: (8^2=64) and (9^2=81), so ( \sqrt{68} ) lies between (8) and (9). Compare nearby perfect squares.

Correct answer: C. 9 and 10

Explanation: The governing concept is locating a principal square root by comparing the number with consecutive perfect squares. The nearby perfect squares are 9² = 81 and 10² = 100. Because 81 < 86 < 100, taking the positive square root preserves the order and gives 9 < √86 < 10. Therefore √86 lies between 9 and 10, making option C correct. Its approximate value is about 9.27, which provides an additional check. The interval 7 to 8 is too low because 8² is only 64. The interval 8 to 9 is also too low because 9² is 81. The interval 10 to 11 is too high because √86 is less than 10.

Correct answer: A. ( -\sqrt{50} )

Explanation: ( \sqrt{50}\approx7.07 ), so ( -\sqrt{50}\approx-7.07 ). It is greater than ( -7.2 ) because it is closer to zero.

Correct answer: C. Irrational real number

Explanation: A non-terminating decimal without regular repetition is irrational. Only fixed repeating decimals are rational.

Correct answer: A. It is terminating

Explanation: A rational number has a terminating decimal when, after reducing the fraction, the denominator has no prime factors other than 2 and 5. This happens because powers of 2 and 5 can be multiplied to make a power of 10. Therefore the decimal form in this question ends after a finite number of digits, so option A is correct. It is not irrational or undefined.

Here, the denominator is 80, and the factorisation is \(80=2^4\times5\). The numerator 21 has no common factor with 80, so the fraction is already in simplest form. Since its denominator contains only 2 and 5, \(21/80=0.2625\), which terminates. Thus the correct statement is that it is a terminating decimal.

Correct answer: B. It is an irrational number.

Explanation: The diagonal of the square is \(\sqrt{1^2+1^2}=\sqrt{2}\) cm. Since 2 is not a perfect square, \(\sqrt{2}\) cannot be expressed exactly as a ratio of two integers; therefore, it is irrational. It is neither an integer nor a natural number, so options C and D are incorrect. Exam tip: the square root of a non-perfect square is generally irrational.

Correct answer: B. अपरिमेय संख्या

Explanation: The diagonal of a square is given by side × \(\sqrt{2}\), so its length is \(5\sqrt{2}\) cm. Since \(\sqrt{2}\) is irrational and 5 is a non-zero rational number, \(5\sqrt{2}\) is also irrational. Therefore, option B is correct. Exam tip: multiplying an irrational number by a non-zero rational number keeps the result irrational.

Correct answer: C. It is a non-terminating non-recurring decimal, so it is irrational.

Explanation: In the decimal expansion \(0.101001000100001\ldots\), the number of zeros between successive 1s keeps increasing, so no fixed block repeats regularly. Hence it is a non-terminating, non-recurring decimal and therefore irrational. The use of only the digits 0 and 1 does not make a number rational. Exam tip: a decimal is rational only if it terminates or eventually repeats a fixed pattern.

Correct answer: C. It is irrational because its decimal expansion is non-terminating and non-repeating.

Explanation: The decimal expansion \(0.101001000100001\ldots\) is non-terminating, and the number of zeros between successive 1s keeps increasing. Hence, no fixed block of digits repeats regularly. A rational number has either a terminating or an infinitely repeating decimal expansion, so this number is irrational. Exam tip: an infinite decimal is not automatically irrational; an infinite recurring decimal, such as \(0.333\ldots\), is rational.

Correct answer: B. -5

Explanation: The two expressions are conjugates, so use \((a+b)(a-b)=a^2-b^2\): \((3+\sqrt{14})(3-\sqrt{14})=3^2-(\sqrt{14})^2=9-14=-5\). Option A results from missing the negative sign. Exam tip: whenever conjugate pairs appear, apply the difference-of-squares identity directly.

Correct answer: A. (3)

Explanation: Multiplying by the conjugate ( \sqrt{7}+2 ) makes the denominator (7-4=3). Use difference of squares in the conjugate method.