MCQs
1. The factorization of \( 12x + 36 \) is:
(a) \( 12(x + 3) \)
(b) \( 12(3x) \)
(c) \( 12(3x + 1) \)
(d) \( x(12 + 36x) \)
Correct Answer: (a) \( 12(x + 3) \)
Explanation: Take 12 as the common factor, leaving \( (x + 3) \) as the factorized expression.
2. The factors of \( 4x^2 – 12y + 9 \) are:
(a) \( (2x + 3)^2 \)
(b) \( (2x – 3)^2 \)
(c) \( (2x – 3)(2x + 3) \)
(d) \( (2 + 3x)(2 – 3x)^2 \)
Correct Answer: (b) \( (2x – 3)^2 \)
Explanation: The trinomial is a perfect square: \( (2x – 3)^2 \), since \( (2x – 3)(2x – 3) = 4x^2 – 12x + 9 \).
3. The HCF of \( a^3b^3 \) and \( ab^2 \) is:
(a) \( a^3b^3 \)
(b) \( ab^2 \)
(c) \( a^4b^5 \)
(d) \( a^2b \)
Correct Answer: (b) \( ab^2 \)
Explanation: The HCF is the product of the lowest powers of the common factors \( a \) and \( b \): \( ab^2 \).
4. The LCM of \( 16x^2, 4x, \) and \( 30xy \) is:
(a) \( 480x^3y \)
(b) \( 240xy \)
(c) \( 240x^2y \)
(d) \( 120x^4y \)
Correct Answer: (a) \( 480x^3y \)
Explanation: The LCM is the product of the highest powers of all the factors in the terms.
5. Product of LCM and HCF = ________ of two polynomials.
(a) sum
(b) difference
(c) product
(d) quotient
Correct Answer: (c) product
Explanation: The product of LCM and HCF of two polynomials equals the product of the polynomials themselves.
6. The square root of \( x^2 – 6x + 9 \) is:
(a) \( \pm(x – 3) \)
(b) \( \pm(x + 3) \)
(c) \( x – 3 \)
(d) \( x + 3 \)
Correct Answer: (a) \( \pm(x – 3) \)
Explanation: \( x^2 – 6x + 9 = (x – 3)^2 \), so its square root is \( \pm(x – 3) \).
7. The LCM of \( (a – b)^2 \) and \( (a – b)^4 \) is:
(a) \( (a – b)^2 \)
(b) \( (a – b)^3 \)
(c) \( (a – b)^4 \)
(d) \( (a – b)^6 \)
Correct Answer: (c) \( (a – b)^4 \)
Explanation: The LCM is the highest power of \( (a – b) \), which is \( (a – b)^4 \).
8. Factorization of \( x^3 + 3x^2 + 3x + 1 \) is:
(a) \( (x + 1)^3 \)
(b) \( (x – 1)^3 \)
(c) \( (x + 1)(x^2 + x + 1) \)
(d) \( (x – 1)(x^2 – x + 1) \)
Correct Answer: (a) \( (x + 1)^3 \)
Explanation: The given expression is a perfect cube: \( (x + 1)^3 \).
9. Cubic polynomial has degree:
(a) 1
(b) 2
(c) 3
(d) 4
Correct Answer: (c) 3
Explanation: The degree of a polynomial is the highest power of the variable, which is 3 in this case.
10. One of the factors of \( x^3 – 27 \) is:
(a) \( x – 3 \)
(b) \( x + 3 \)
(c) \( x^2 – 3x + 9 \)
(d) Both a and c
Correct Answer: (a) \(x-3\)
Explanation: \( x^3 – 27 = (x – 3)(x^2 + 3x + 9) \), so \( x – 3 \) is factors.
2. Factorize the following expressions
Please factorize the expressions by following the pervious exercise.
3. Find LCM and HCF by prime factorization method
(i) \( 4x^3 + 12x^2, 8x^2 + 16x \)
\[4x^3 + 12x^2 = 4x^2(x + 3), \quad 8x^2 + 16x = 8x(x + 2)\]
\[\text{HCF} = 4x, \quad \text{LCM} = 8x^2(x + 3)(x + 2)\]
(ii) \( x^3 + 3x^2 – 4x, x^2 – x – 6 \)
\[x^3 + 3x^2 – 4x = x(x^2 + 3x – 4) = x(x + 4)(x – 1)\]
\[x^2 – x – 6 = (x – 3)(x + 2)\]
\[\text{HCF} = x, \quad \text{LCM} = x(x + 4)(x – 1)(x – 3)(x + 2)\]
(iii) \( x^2 + 8x + 16, x^2 – 16 \)
\[x^2 + 8x + 16 = (x + 4)^2, \quad x^2 – 16 = (x + 4)(x – 4)\]
\[\text{HCF} = (x + 4), \quad \text{LCM} = (x + 4)^2(x – 4)\]
(iv) \( x^3 – 9x, x^2 – 4x + 3 \)
\[x^3 – 9x = x(x^2 – 9) = x(x – 3)(x + 3)\]
\[x^2 – 4x + 3 = (x – 3)(x – 1)\]
\[\text{HCF} = x(x – 3), \quad \text{LCM} = x(x – 3)(x + 3)(x – 1)\]
4. Find square root by factorization and division method of the expression \( 16x^4 + 8x^2 + 1 \).
\[16x^4 + 8x^2 + 1 = (4x^2 + 1)^2\]
\[\sqrt{16x^4 + 8x^2 + 1} = 4x^2 + 1\]
5. Huria is analyzing the total cost of her loan, modeled by \( C(x) = x^2 – 8x + 15 \). Find the optimal repayment period.
\[C(x) = x^2 – 8x + 15 = (x – 5)(x – 3)\]
\[x = 5, x = 3\]
The optimal repayment periods are 3 years or 5 years.
