MCQ
Squares & roots, LCM & HCF; factoring, quadratic equations and logarithms MCQ - Practice Questions with Answers
Solve 15 Squares & roots, LCM & HCF; factoring, quadratic equations and logarithms questions for RAS/RPSC preparation.
Practice questions
Q1What is the cube root of 2197?
A cube root reverses cubing, so the number required is the one whose third power is 2197. The note gives the direct comparison: 12^3 = 1728 and 13^3 = 2197. Therefore the cube root is 13. This also shows why nearby values must be checked by cubing, not guessed from the last digit alone. Unit digits help in perfect-cube questions, but the final answer should be confirmed using known cubes or factorisation.
Q2Two bells ring every 12 minutes and 18 minutes. A student says the answer should be the HCF because both bells are common to the event. Which response is correct?
The wording decides whether LCM or HCF is needed. When events repeat and the question asks when they will happen together again, the required value is the least common multiple. The multiples of 12 are 12, 24, 36, and the multiples of 18 are 18, 36, so the first common time after the start is 36 minutes. HCF would answer a different kind of question, such as the greatest equal length or greatest equal divisor without a remainder.
Q3Match List I with List II. List I: 1. 784, 2. 1728, 3. 46, 4. 2197. List II: a. Not a perfect square despite ending in 6, b. Square root 28, c. Cube root 12, d. Cube root 13.
The study note gives these examples to separate reliable recognition from unit-digit guessing. Since 784 = 2^4 × 7^2, its square root is 28. Since 1728 = 2^6 × 3^3, its cube root is 12. The number 46 is a warning that ending in 6 does not prove a perfect square, although 36 is a square. Also, 13^3 = 2197, so the cube root of 2197 is 13.
Q4Which statement gives the safest test for deciding whether 784 is a perfect square with a whole-number square root?
Prime factorisation is the reliable recognition method for perfect squares. A whole-number perfect square has every prime factor raised to an even power. Here 784 = 2^4 × 7^2, so the square root is made by taking half of each exponent: 2^2 × 7 = 28. A valid unit digit can reject some numbers, but it cannot confirm a square by itself. Grouping factors in threes belongs to cube and cube-root questions.
Q5Assertion: In ax² + bx + c = 0, the coefficient a may be 0 and the equation will still be quadratic. Reason: The discriminant b² - 4ac decides the nature of roots.
The standard form of a quadratic equation is ax² + bx + c = 0 with a not equal to 0. If a becomes 0, the x² term disappears, so the equation is no longer quadratic. The reason statement is still correct by itself: the discriminant b² - 4ac is used to decide the nature of roots. A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative value gives no real roots in the basic real-number setting.
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More questions
6For 2x^2 + 3x - 2 = 0, what are the roots?
7In a quadratic equation ax^2 + bx + c = 0, what does a negative discriminant indicate in the basic real-number setting?
8Which factorisation is correct for x² - x - 12?
9Assertion: x² - 16 should be factorised as (x - 4)(x + 4). Reason: It is a difference of squares, so the middle-term splitting method is not needed.
10Two positive integers have HCF 6 and LCM 1260. If one integer is 84, what is the other integer?
11Which is the correct factorisation of 2x^2 + 7x + 3?
12Lengths of 48 cm and 60 cm are to be cut into the greatest equal pieces without any waste. Which value should be used?
13If log base a of x is 3, which power form is equivalent?
14For the quadratic equation 2x² + 3x - 2 = 0, which conclusion is correct?
15Which statement about logarithms is incorrect in the basic real-number setting used for CET?