Real Numbers

Correct answer: A. \(2\sqrt5\)

Explanation: The governing concept is using a conjugate to rationalise the reciprocal of a surd expression. Since x = √5 + 2, multiply the numerator and denominator of 1/x by √5 − 2: 1/x = (√5 − 2)/[(√5 + 2)(√5 − 2)] = (√5 − 2)/(5 − 4) = √5 − 2. Now add this to x: x + 1/x = (√5 + 2) + (√5 − 2) = 2√5. Therefore option A is correct. Option B incorrectly combines constants and irrational terms, option C leaves out one √5 term, and option D is merely the given value of x rather than the requested sum. The cancellation of +2 and −2 is the key step.

Correct answer: A. \(\frac{4+\sqrt7}{9}\)

Explanation: The governing method is rationalisation using the conjugate of the denominator. Starting from 1/x = 1/(4 − √7), multiply numerator and denominator by 4 + √7. The denominator becomes (4 − √7)(4 + √7) = 4² − (√7)² = 16 − 7 = 9, while the numerator is 4 + √7. Thus 1/x = (4 + √7)/9, so option A is correct. Option B gives the numerator but omits division by 9. Option C incorrectly keeps the original sign instead of using the conjugate. Option D multiplies by 9 rather than dividing by the rationalised denominator. Multiplying the proposed answer by x gives [(4 + √7)(4 − √7)]/9 = 9/9 = 1.

Correct answer: A. \(\sqrt{4+9}\ne\sqrt4+\sqrt9\)

Explanation: The governing concept is the use of a counterexample to disprove a statement claimed to hold for all allowed numbers. In option A, the left side is √(4 + 9) = √13, whereas the right side is √4 + √9 = 2 + 3 = 5. Since √13 is not equal to 5, this example directly proves that square roots do not distribute over addition. Options B, C, and D are true only because one of the addends is zero: adding zero leaves the other number unchanged. These special cases do not establish the proposed rule for general positive numbers. Therefore option A is the valid counterexample and the unique correct answer.

Correct answer: D. √5 × √5

Explanation: The governing concept is the classification of rational and irrational numbers. A rational number can be written as p/q, where p and q are integers and q is nonzero. In option D, √5 × √5 = (√5)^2 = 5, and every integer is rational because it can be written as 5/1. The other expressions remain irrational: √2 + √3 is irrational, √7 + 2 is irrational because adding a rational number to an irrational number remains irrational, and π + 1 is irrational for the same reason. Therefore option D is the only rational expression. The multiplication sign matters: √5 + √5 would equal 2√5, which is irrational, but the given product simplifies exactly to 5.

Correct answer: B. 11√3

Explanation: The governing rule is to factor the radicand so that a perfect-square factor can be taken outside the square root. Since 363 = 121 × 3 and 121 = 11², we have √363 = √(11² × 3) = √(11²)√3 = 11√3. The principal square root is positive, so the outside factor is +11. Therefore option B is correct. A reliable check is to square the result: (11√3)² = 121 × 3 = 363. Option A does not preserve the correct factorization, while options C and D introduce incorrect extra factors. The expression is fully simplified because 3 has no perfect-square factor greater than 1, so no further extraction is possible.