MCQ test, Set 9

Answer: (C) $(4m + 5n)(4m - 5n)$

$16m^2 - 25n^2 = (4m)^2 - (5n)^2$

$= (4m + 5n)(4m - 5n)$

using $a^2 - b^2 = (a + b)(a - b)$

Answer: (A) $(2xz + 3)(2xz - 3)(4x^2z^2 + 9)$

$16x^4z^4 - 81 = (4x^2z^2)^2 - 9^2$

$= (4x^2z^2 + 9)(4x^2z^2 - 9)$

$= (4x^2z^2 + 9)[(2xz)^2 - 3^2]$

$= (4x^2z^2 + 9)(2xz + 3)(2xz - 3)$

Answer: (C) $(7z + 4)^2$

$49z^2 + 56z + 16 = (7z)^2 + 4^2 + 2(7z)(4)$

$= (7z + 4)^2$

$= (7z + 4)(7z + 4)$

using $a^2 + b^2 + 2ab = (a + b)^2$

Answer: (A) $(a - 3b)^2$

$a^2 - 6ab + 9b^2 = a^2 + (3b)^2 - 2(a)(3b)$

$= (a - 3b)^2$

$= (a - 3b)(a - 3b)$

using $a^2 + b^2 - 2ab = (a - b)^2$

Answer: (A) $(2x - 7 + 6y)(2x - 7 - 6y)$

$4x^2 - 28x + 49 - 36y^2 = (2x - 7)^2 - (6y)^2$

$= (2x - 7 + 6y)(2x - 7 - 6y)$

using $a^2 - 2ab + b^2 = (a - b)^2$ and $a^2 - b^2 = (a + b)(a - b)$

Answer: (D) $3$ and $7$

$3 \times 7 = 21$

$3 \times 7 = 21$ and $3 + 7 = 10$

So the numbers are $3$ and $7$.

Answer: (A) $(5a - 1)(a - 2)$

$5 \times 2 = 10$: $(-1) \times (-10) = 10$ and $-1 - 10 = -11$

$= 5a^2 - a - 10a + 2$

$= a(5a - 1) - 2(5a - 1)$

$= (5a - 1)(a - 2)$

Answer: (D) $-6$ and $-24$

$9 \times 16 = 144$

$(-6) \times (-24) = 144$ and $-6 - 24 = -30$

So the numbers are $-6$ and $-24$.

Answer: (C) $-2$ and $35$

$10 \times (-7) = -70$

$(-2) \times 35 = -70$ and $-2 + 35 = 33$

So the numbers are $-2$ and $35$.

Answer: (A) $(17a - 13)(a - 4)$

$17 \times 52 = 884$

$884 = 2 \times 2 \times 13 \times 17 = 13 \times 68$, and $-13 - 68 = -81$

$= 17a^2 - 13a - 68a + 52$

$= a(17a - 13) - 4(17a - 13)$

$= (17a - 13)(a - 4)$

Answer: (C) $(7 - n)(n - 5)$

Writing the terms in order: $-n^2 + 12n - 35$

$(-1) \times (-35) = 35$: $5 \times 7 = 35$ and $5 + 7 = 12$

$= -n^2 + 5n + 7n - 35$

$= -n(n - 5) + 7(n - 5)$

$= (n - 5)(7 - n)$

Answer: (B) $(a - 2b - 5)(a - 2b - 6)$

Let $a - 2b = p$

$(a - 2b)^2 - 11(a - 2b) + 30 = p^2 - 11p + 30$

$1 \times 30 = 30$: $(-5) \times (-6) = 30$ and $-5 - 6 = -11$

$= p^2 - 5p - 6p + 30$

$= p(p - 5) - 6(p - 5)$

$= (p - 5)(p - 6)$

$= [(a - 2b) - 5][(a - 2b) - 6]$

$= (a - 2b - 5)(a - 2b - 6)$

Answer: (C) $(4x - y)(x - 6y)$

$4 \times 6 = 24$: $(-1) \times (-24) = 24$ and $-1 - 24 = -25$

$= 4x^2 - xy - 24xy + 6y^2$

$= x(4x - y) - 6y(4x - y)$

$= (4x - y)(x - 6y)$

Answer: (D) $(7t + 2\sqrt{3})(\sqrt{3}t + 6)$

$7\sqrt{3} \times 12\sqrt{3} = 7 \times 12 \times 3 = 252$: $42 \times 6 = 252$ and $42 + 6 = 48$

$= 7\sqrt{3}t^2 + 42t + 6t + 12\sqrt{3}$

$= 7t(\sqrt{3}t + 6) + 2\sqrt{3}(\sqrt{3}t + 6)$

$= (\sqrt{3}t + 6)(7t + 2\sqrt{3})$

Answer: (B) $\displaystyle (10y + 3)\left(y + \frac{3}{10}\right)$

$\displaystyle 10y^2 + 6y + \frac{9}{10} = \frac{100y^2 + 60y + 9}{10}$

$\displaystyle = \frac{1}{10}(100y^2 + 60y + 9)$

$\displaystyle = \frac{1}{10}[(10y)^2 + 3^2 + 2(10y)(3)]$

$\displaystyle = \frac{1}{10}(10y + 3)^2$

$\displaystyle = \left(y + \frac{3}{10}\right)(10y + 3)$

Answer: (A) $(5a + 3c + 27)(a - 6c - 21)$

Let $a + 3 = x$ and $c + 4 = y$

$5(a + 3)^2 - 27(a + 3)(c + 4) - 18(c + 4)^2 = 5x^2 - 27xy - 18y^2$

$5 \times (-18) = -90$: $3 \times (-30) = -90$ and $3 - 30 = -27$

$= 5x^2 + 3xy - 30xy - 18y^2$

$= x(5x + 3y) - 6y(5x + 3y)$

$= (5x + 3y)(x - 6y)$

$= [5(a + 3) + 3(c + 4)][(a + 3) - 6(c + 4)]$

$= (5a + 15 + 3c + 12)(a + 3 - 6c - 24)$

$= (5a + 3c + 27)(a - 6c - 21)$

Answer: (B) $(3p - q + 3r)^2$

$9p^2 + q^2 + 9r^2 - 6pq - 6qr + 18rp = (3p)^2 + q^2 + (3r)^2 - 2(3p)(q) - 2(q)(3r) + 2(3r)(3p)$

$= (3p - q + 3r)^2$

using $a^2 + b^2 + c^2 - 2ab - 2bc + 2ca = (a - b + c)^2$

Answer: (B) $(3y + 4)^3$

$27y^3 + 108y^2 + 144y + 64 = (3y)^3 + 4^3 + 3(3y)^2(4) + 3(3y)(4)^2$

$= (3y + 4)^3$

using $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$

Answer: (D) $(x - 5)(x^2 + 5x + 25)$

$x^3 - 125 = x^3 - 5^3$

$= (x - 5)[x^2 + (x)(5) + 5^2]$

$= (x - 5)(x^2 + 5x + 25)$

using $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Answer: (C) $(2y + 3)(4y^2 - 6y + 9)$

$8y^3 + 27 = (2y)^3 + 3^3$

$= (2y + 3)[(2y)^2 - (2y)(3) + 3^2]$

$= (2y + 3)(4y^2 - 6y + 9)$

using $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$

Answer: (B) $(x + 1)(x + 4)(x - 5)$

$x^3 - 21x - 20 = x^3 - x - 20x - 20$

$= x(x^2 - 1) - 20(x + 1)$

$= x(x + 1)(x - 1) - 20(x + 1)$

$= (x + 1)[x(x - 1) - 20]$

$= (x + 1)(x^2 - x - 20)$

Now $x^2 - x - 20$: $4 \times (-5) = -20$ and $4 - 5 = -1$

$x^2 - x - 20 = x^2 + 4x - 5x - 20$

$= x(x + 4) - 5(x + 4)$

$= (x + 4)(x - 5)$

So $x^3 - 21x - 20 = (x + 1)(x + 4)(x - 5)$

Answer: (C) $(x^2 + 5xz + 25z^2)(x^2 - 5xz + 25z^2)$

$x^4 + 25x^2z^2 + 625z^4 = (x^2)^2 + 2(x^2)(25z^2) + (25z^2)^2 - 25x^2z^2$

$= (x^2 + 25z^2)^2 - (5xz)^2$

$= (x^2 + 25z^2 + 5xz)(x^2 + 25z^2 - 5xz)$

$= (x^2 + 5xz + 25z^2)(x^2 - 5xz + 25z^2)$

Answer: (B) $3(4p - 5q)(5q - 3)(3 - 4p)$

Here $(4p - 5q) + (5q - 3) + (3 - 4p) = 0$

If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$

$\therefore (4p - 5q)^3 + (5q - 3)^3 + (3 - 4p)^3 = 3(4p - 5q)(5q - 3)(3 - 4p)$

Answer: (D) $2ac(a + 3c)(a - 3c)(a^2 + 9c^2)$

$2a^5c - 162ac^5 = 2ac(a^4 - 81c^4)$

$= 2ac[(a^2)^2 - (9c^2)^2]$

$= 2ac(a^2 + 9c^2)(a^2 - 9c^2)$

$= 2ac(a^2 + 9c^2)[a^2 - (3c)^2]$

$= 2ac(a^2 + 9c^2)(a + 3c)(a - 3c)$

Answer: (D) $(3m^2 + 8)(m + 6)$

$18m^2 + (3m^2 + 8)m + 48 = 18m^2 + 3m^3 + 8m + 48$

$= 3m^3 + 18m^2 + 8m + 48$

$= 3m^2(m + 6) + 8(m + 6)$

$= (m + 6)(3m^2 + 8)$