MCQ test, Set 7

Answer: (B) $(12 + 5p)(12 - 5p)$

$144 - 25p^2 = 12^2 - (5p)^2$

$= (12 + 5p)(12 - 5p)$

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

Answer: (B) $(6t + 5)^2$

$36t^2 + 60t + 25 = (6t)^2 + 5^2 + 2(6t)(5)$

$= (6t + 5)^2$

$= (6t + 5)(6t + 5)$

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

Answer: (C) $(3m - 7n)^2$

$9m^2 - 42mn + 49n^2 = (3m)^2 + (7n)^2 - 2(3m)(7n)$

$= (3m - 7n)^2$

$= (3m - 7n)(3m - 7n)$

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

Answer: (D) $(2x - 2 + 5y)(2x - 2 - 5y)$

$4x^2 - 8x + 4 - 25y^2 = (2x - 2)^2 - (5y)^2$

$= (2x - 2 + 5y)(2x - 2 - 5y)$

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

Answer: (B) $(2n + 5)(n + 7)$

$2 \times 35 = 70$: $5 \times 14 = 70$ and $5 + 14 = 19$

$= 2n^2 + 5n + 14n + 35$

$= n(2n + 5) + 7(2n + 5)$

$= (2n + 5)(n + 7)$

Answer: (C) $(3z - 2)(z - 9)$

$3 \times 18 = 54$: $(-2) \times (-27) = 54$ and $-2 - 27 = -29$

$= 3z^2 - 2z - 27z + 18$

$= z(3z - 2) - 9(3z - 2)$

$= (3z - 2)(z - 9)$

Answer: (D) $-5$ and $20$

$4 \times (-25) = -100$

$(-5) \times 20 = -100$ and $-5 + 20 = 15$

So the numbers are $-5$ and $20$.

Answer: (C) $(11p - 42)(p + 2)$

$11 \times (-84) = -924$

$924 = 2 \times 2 \times 3 \times 7 \times 11 = 22 \times 42$, and $22 - 42 = -20$

$= 11p^2 + 22p - 42p - 84$

$= 11p(p + 2) - 42(p + 2)$

$= (p + 2)(11p - 42)$

Answer: (B) $(17p + 45)(p - 5)$

$17 \times (-225) = -3825$

$3825 = 3 \times 3 \times 5 \times 5 \times 17 = 45 \times 85$, and $45 - 85 = -40$

$= 17p^2 + 45p - 85p - 225$

$= p(17p + 45) - 5(17p + 45)$

$= (17p + 45)(p - 5)$

Answer: (A) $(3p - 2)(6 - p)$

Writing the terms in order: $-3p^2 + 20p - 12$

$(-3) \times (-12) = 36$: $2 \times 18 = 36$ and $2 + 18 = 20$

$= -3p^2 + 2p + 18p - 12$

$= -p(3p - 2) + 6(3p - 2)$

$= (3p - 2)(6 - p)$

Answer: (B) $(2m + 2n - 5)(m + n + 4)$

Let $m + n = p$

$2(m + n)^2 + 3(m + n) - 20 = 2p^2 + 3p - 20$

$2 \times (-20) = -40$: $(-5) \times 8 = -40$ and $-5 + 8 = 3$

$= 2p^2 - 5p + 8p - 20$

$= p(2p - 5) + 4(2p - 5)$

$= (2p - 5)(p + 4)$

$= [2(m + n) - 5][(m + n) + 4]$

$= (2m + 2n - 5)(m + n + 4)$

Answer: (A) $(3x + 4y + 1)(3x + 4y + 4)$

Let $3x + 4y = p$

$(3x + 4y)^2 + 5(3x + 4y) + 4 = p^2 + 5p + 4$

$1 \times 4 = 4$: $1 \times 4 = 4$ and $1 + 4 = 5$

$= p^2 + p + 4p + 4$

$= p(p + 1) + 4(p + 1)$

$= (p + 1)(p + 4)$

$= [(3x + 4y) + 1][(3x + 4y) + 4]$

$= (3x + 4y + 1)(3x + 4y + 4)$

Answer: (C) $-5$ and $-6$

$6 \times 5 = 30$

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

So the numbers are $-5$ and $-6$.

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

Product $= 4$, sum $= 3\sqrt{2}$: $\sqrt{2} \times 2\sqrt{2} = 4$ and $\sqrt{2} + 2\sqrt{2} = 3\sqrt{2}$

$= t^2 + \sqrt{2}t + 2\sqrt{2}t + 4$

$= t(t + \sqrt{2}) + 2\sqrt{2}(t + \sqrt{2})$

$= (t + \sqrt{2})(t + 2\sqrt{2})$

Answer: (A) $\displaystyle \left(2x - \frac{1}{5}\right)\left(x + \frac{1}{3}\right)$

$\displaystyle 2x^2 + \frac{7}{15}x - \frac{1}{15} = \frac{30x^2 + 7x - 1}{15}$

$\displaystyle = \frac{1}{15}(30x^2 + 7x - 1)$

$30 \times (-1) = -30$: $(-3) \times 10 = -30$ and $-3 + 10 = 7$

$\displaystyle = \frac{1}{15}[30x^2 - 3x + 10x - 1]$

$\displaystyle = \frac{1}{15}[3x(10x - 1) + 1(10x - 1)]$

$\displaystyle = \frac{1}{15}(10x - 1)(3x + 1)$

$\displaystyle = \left(x + \frac{1}{3}\right)\left(2x - \frac{1}{5}\right)$

Answer: (B) $(3p + 4q + 3)(p - 5q - 18)$

Let $p - 3 = x$ and $q + 3 = y$

$3(p - 3)^2 - 11(p - 3)(q + 3) - 20(q + 3)^2 = 3x^2 - 11xy - 20y^2$

$3 \times (-20) = -60$: $4 \times (-15) = -60$ and $4 - 15 = -11$

$= 3x^2 + 4xy - 15xy - 20y^2$

$= x(3x + 4y) - 5y(3x + 4y)$

$= (3x + 4y)(x - 5y)$

$= [3(p - 3) + 4(q + 3)][(p - 3) - 5(q + 3)]$

$= (3p - 9 + 4q + 12)(p - 3 - 5q - 15)$

$= (3p + 4q + 3)(p - 5q - 18)$

Answer: (B) $(2p - q - 7)(p - 5q + 1)$

Let $p - 4 = x$ and $q - 1 = y$

$2(p - 4)^2 - 11(p - 4)(q - 1) + 5(q - 1)^2 = 2x^2 - 11xy + 5y^2$

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

$= 2x^2 - xy - 10xy + 5y^2$

$= x(2x - y) - 5y(2x - y)$

$= (2x - y)(x - 5y)$

$= [2(p - 4) - (q - 1)][(p - 4) - 5(q - 1)]$

$= (2p - 8 - q + 1)(p - 4 - 5q + 5)$

$= (2p - q - 7)(p - 5q + 1)$

Answer: (A) $(3p - 3q + 4r)^2$

$9p^2 + 9q^2 + 16r^2 - 18pq - 24qr + 24rp = (3p)^2 + (3q)^2 + (4r)^2 - 2(3p)(3q) - 2(3q)(4r) + 2(4r)(3p)$

$= (3p - 3q + 4r)^2$

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

Answer: (D) $(m + 4p)^3$

$m^3 + 12m^2p + 48mp^2 + 64p^3 = m^3 + (4p)^3 + 3(m)^2(4p) + 3(m)(4p)^2$

$= (m + 4p)^3$

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

Answer: (D) $(25m^4 + 15m^2n^2 + 9n^4)(5m^2 - 3n^2)$

$125m^6 - 27n^6 = (5m^2)^3 - (3n^2)^3$

$= (5m^2 - 3n^2)[(5m^2)^2 + (5m^2)(3n^2) + (3n^2)^2]$

$= (5m^2 - 3n^2)(25m^4 + 15m^2n^2 + 9n^4)$

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

Answer: (A) $(2xy + 3)(4x^2y^2 - 6xy + 9)$

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

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

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

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

Answer: (D) $(9m^2 + 12m + 8)(9m^2 - 12m + 8)$

$81m^4 + 64 = (9m^2)^2 + 8^2 + 2(9m^2)(8) - 144m^2$

$= (9m^2 + 8)^2 - (12m)^2$

$= (9m^2 + 8 + 12m)(9m^2 + 8 - 12m)$

$= (9m^2 + 12m + 8)(9m^2 - 12m + 8)$

Answer: (C) $3(5p - 3q)(3q - 2)(2 - 5p)$

Here $(5p - 3q) + (3q - 2) + (2 - 5p) = 0$

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

$\therefore (5p - 3q)^3 + (3q - 2)^3 + (2 - 5p)^3 = 3(5p - 3q)(3q - 2)(2 - 5p)$

Answer: (A) $2t(t + 2)(t - 2)(t^2 + 4)$

$2t^5 - 32t = 2t(t^4 - 16)$

$= 2t[(t^2)^2 - 4^2]$

$= 2t(t^2 + 4)(t^2 - 4)$

$= 2t(t^2 + 4)[t^2 - 2^2]$

$= 2t(t^2 + 4)(t + 2)(t - 2)$

Answer: (C) $(3x^2 + 1)(x + 1)$

$3x^2 + (3x^2 + 1)x + 1 = 3x^2 + 3x^3 + x + 1$

$= 3x^3 + 3x^2 + x + 1$

$= 3x^2(x + 1) + 1(x + 1)$

$= (x + 1)(3x^2 + 1)$