MCQ test, Set 4

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

$25p^2 - 4q^2 = (5p)^2 - (2q)^2$

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

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

Answer: (C) $(b + 7)^2$

$b^2 + 14b + 49 = b^2 + 7^2 + 2(b)(7)$

$= (b + 7)^2$

$= (b + 7)(b + 7)$

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

Answer: (C) $(3b - 1)^2$

$9b^2 - 6b + 1 = (3b)^2 + 1^2 - 2(3b)(1)$

$= (3b - 1)^2$

$= (3b - 1)(3b - 1)$

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

Answer: (A) $(x + 4y + 6)(x + 4y - 6)$

$x^2 + 8xy + 16y^2 - 36 = (x + 4y)^2 - 6^2$

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

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

Answer: (D) $4$ and $15$

$5 \times 12 = 60$

$4 \times 15 = 60$ and $4 + 15 = 19$

So the numbers are $4$ and $15$.

Answer: (D) $(4y + 1)(y + 4)$

$4 \times 4 = 16$: $1 \times 16 = 16$ and $1 + 16 = 17$

$= 4y^2 + y + 16y + 4$

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

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

Answer: (A) $-2$ and $-3$

$6 \times 1 = 6$

$(-2) \times (-3) = 6$ and $-2 - 3 = -5$

So the numbers are $-2$ and $-3$.

Answer: (B) $(3m + 7)(m - 4)$

$3 \times (-28) = -84$: $7 \times (-12) = -84$ and $7 - 12 = -5$

$= 3m^2 + 7m - 12m - 28$

$= m(3m + 7) - 4(3m + 7)$

$= (3m + 7)(m - 4)$

Answer: (C) $-24$ and $25$

$15 \times (-40) = -600$

$(-24) \times 25 = -600$ and $-24 + 25 = 1$

So the numbers are $-24$ and $25$.

Answer: (B) $(17x - 27)(x + 7)$

$17 \times (-189) = -3213$

$3213 = 3 \times 3 \times 3 \times 7 \times 17 = 27 \times 119$, and $-27 + 119 = 92$

$= 17x^2 - 27x + 119x - 189$

$= x(17x - 27) + 7(17x - 27)$

$= (17x - 27)(x + 7)$

Answer: (D) $(3 - t)(3t - 5)$

Writing the terms in order: $-3t^2 + 14t - 15$

$(-3) \times (-15) = 45$: $5 \times 9 = 45$ and $5 + 9 = 14$

$= -3t^2 + 5t + 9t - 15$

$= -t(3t - 5) + 3(3t - 5)$

$= (3t - 5)(3 - t)$

Answer: (A) $(3x - 4z + 6)(3x - 4z - 3)$

Let $3x - 4z = p$

$(3x - 4z)^2 + 3(3x - 4z) - 18 = p^2 + 3p - 18$

$1 \times (-18) = -18$: $(-3) \times 6 = -18$ and $-3 + 6 = 3$

$= p^2 - 3p + 6p - 18$

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

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

$= [(3x - 4z) - 3][(3x - 4z) + 6]$

$= (3x - 4z - 3)(3x - 4z + 6)$

Answer: (D) $(2p + 4q - 1)(p + 2q + 7)$

Let $p + 2q = t$

$2(p + 2q)^2 + 13(p + 2q) - 7 = 2t^2 + 13t - 7$

$2 \times (-7) = -14$: $(-1) \times 14 = -14$ and $-1 + 14 = 13$

$= 2t^2 - t + 14t - 7$

$= t(2t - 1) + 7(2t - 1)$

$= (2t - 1)(t + 7)$

$= [2(p + 2q) - 1][(p + 2q) + 7]$

$= (2p + 4q - 1)(p + 2q + 7)$

Answer: (C) $(2p + 7q)(p - 4q)$

$2 \times (-28) = -56$: $7 \times (-8) = -56$ and $7 - 8 = -1$

$= 2p^2 + 7pq - 8pq - 28q^2$

$= p(2p + 7q) - 4q(2p + 7q)$

$= (2p + 7q)(p - 4q)$

Answer: (B) $(y + 4\sqrt{6})(y - 5\sqrt{6})$

Product $= -120$, sum $= -\sqrt{6}$: $4\sqrt{6} \times (-5\sqrt{6}) = -120$ and $4\sqrt{6} + (-5\sqrt{6}) = -\sqrt{6}$

$= y^2 + 4\sqrt{6}y - 5\sqrt{6}y - 120$

$= y(y + 4\sqrt{6}) - 5\sqrt{6}(y + 4\sqrt{6})$

$= (y + 4\sqrt{6})(y - 5\sqrt{6})$

Answer: (D) $\displaystyle \left(3y + \frac{2}{3}\right)\left(3y - \frac{2}{5}\right)$

$\displaystyle 9y^2 + \frac{4}{5}y - \frac{4}{15} = \frac{135y^2 + 12y - 4}{15}$

$\displaystyle = \frac{1}{15}(135y^2 + 12y - 4)$

$135 \times (-4) = -540$: $(-18) \times 30 = -540$ and $-18 + 30 = 12$

$\displaystyle = \frac{1}{15}[135y^2 - 18y + 30y - 4]$

$\displaystyle = \frac{1}{15}[9y(15y - 2) + 2(15y - 2)]$

$\displaystyle = \frac{1}{15}(15y - 2)(9y + 2)$

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

Answer: (B) $(4m - 5n - 35)(m + 6n + 13)$

Let $m - 5 = x$ and $n + 3 = y$

$4(m - 5)^2 + 19(m - 5)(n + 3) - 30(n + 3)^2 = 4x^2 + 19xy - 30y^2$

$4 \times (-30) = -120$: $(-5) \times 24 = -120$ and $-5 + 24 = 19$

$= 4x^2 - 5xy + 24xy - 30y^2$

$= x(4x - 5y) + 6y(4x - 5y)$

$= (4x - 5y)(x + 6y)$

$= [4(m - 5) - 5(n + 3)][(m - 5) + 6(n + 3)]$

$= (4m - 20 - 5n - 15)(m - 5 + 6n + 18)$

$= (4m - 5n - 35)(m + 6n + 13)$

Answer: (A) $(2a - c + 8)(4a - 3c + 18)$

Let $a + 3 = x$ and $c - 2 = y$

$8(a + 3)^2 - 10(a + 3)(c - 2) + 3(c - 2)^2 = 8x^2 - 10xy + 3y^2$

$8 \times 3 = 24$: $(-4) \times (-6) = 24$ and $-4 - 6 = -10$

$= 8x^2 - 4xy - 6xy + 3y^2$

$= 4x(2x - y) - 3y(2x - y)$

$= (2x - y)(4x - 3y)$

$= [2(a + 3) - (c - 2)][4(a + 3) - 3(c - 2)]$

$= (2a + 6 - c + 2)(4a + 12 - 3c + 6)$

$= (2a - c + 8)(4a - 3c + 18)$

Answer: (C) $(a - b + 2c)^2$

$a^2 + b^2 + 4c^2 - 2ab - 4bc + 4ca = a^2 + b^2 + (2c)^2 - 2(a)(b) - 2(b)(2c) + 2(2c)(a)$

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

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

Answer: (D) $(y - 5)^3$

$y^3 - 15y^2 + 75y - 125 = y^3 - 5^3 - 3(y)^2(5) + 3(y)(5)^2$

$= (y - 5)^3$

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

Answer: (C) $(16m^2p^2 + 12mp + 9)(4mp - 3)$

$64m^3p^3 - 27 = (4mp)^3 - 3^3$

$= (4mp - 3)[(4mp)^2 + (4mp)(3) + 3^2]$

$= (4mp - 3)(16m^2p^2 + 12mp + 9)$

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

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

$27m^3 + 125n^3 = (3m)^3 + (5n)^3$

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

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

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

Answer: (A) $3(2a - 5b)(5b - c)(c - 2a)$

Here $(2a - 5b) + (5b - c) + (c - 2a) = 0$

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

$\therefore (2a - 5b)^3 + (5b - c)^3 + (c - 2a)^3 = 3(2a - 5b)(5b - c)(c - 2a)$

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

$16m^5 - mn^4 = m(16m^4 - n^4)$

$= m[(4m^2)^2 - (n^2)^2]$

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

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

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

Answer: (A) $(2a^2 + 3)(a + 5)$

$10a^2 + (2a^2 + 3)a + 15 = 10a^2 + 2a^3 + 3a + 15$

$= 2a^3 + 10a^2 + 3a + 15$

$= 2a^2(a + 5) + 3(a + 5)$

$= (a + 5)(2a^2 + 3)$