1.Definitions and Basic Concepts8

Q 1C-03-Q001medium

Meaningful organization of operational signs and numerical letter symbols is called—

aequation
balgebraic expression
cfunction
didentity

Q 2C-03-Q002medium

In algebraic expressions, numbers whose values are fixed are called—

avariables
bconstants
ccoefficients
dexponents

Q 3C-03-Q003medium

Letter symbols used in algebraic expressions are—

aconstants
bvariables
coperators
dintegers

Q 4C-03-Q004medium

Algebra is the generalization of—

ageometry
btrigonometry
carithmetic
dcalculus

Q 5C-03-Q005medium

In algebra, the numbers used include—

aonly positive
bonly negative
cpositive and negative including zero
donly integers

Q 6C-03-Q006medium

Any general rule expressed by algebraic symbols is called—

acorollary
bequation
calgebraic formula
dtheorem

Q 7C-03-Q007medium

Which of the following is an algebraic expression?

a $5$
b $2a + 3b - 4c$
c $7 = 3 + 4$
d $\pi$

Q 8C-03-Q008medium

Which is true of variables?

afixed value
bvalue can change
calways positive
dalways integer

2.Formulae of Square and Corollaries12

Q 1C-03-Q009medium

$(a+b)^2 = ?$

a $a^2 - 2ab + b^2$
b $a^2 + 2ab + b^2$
c $a^2 + b^2$
d $a^2 - b^2$

Q 2C-03-Q010medium

$(a-b)^2 = ?$

a $a^2 + 2ab + b^2$
b $a^2 - 2ab + b^2$
c $a^2 - b^2$
d $a^2 + b^2$

Q 3C-03-Q011medium

Substituting $-b$ for $b$ in Formula 1 yields—

aFormula 2
bFormula 3
cFormula 4
dFormula 5

Q 4C-03-Q012medium

Adding $2ab$ to $a^2 + b^2$ gives—

a $(a-b)^2$
b $(a+b)^2$
c $a^2 - b^2$
d $2ab$

Q 5C-03-Q013medium

$a^2 + b^2 = ?$ (Corollary 1)

a $(a+b)^2 - 2ab$
b $(a+b)^2 + 2ab$
c $(a+b)^2$
d $(a-b)^2$

Q 6C-03-Q014medium

$a^2 + b^2 = ?$ (Corollary 2)

a $(a-b)^2 + 2ab$
b $(a-b)^2 - 2ab$
c $(a-b)^2$
d $(a+b)^2 + 2ab$

Q 7C-03-Q015medium

$(a+b)^2 = ?$ (Corollary 3)

a $(a-b)^2 + 4ab$
b $(a-b)^2 - 4ab$
c $(a-b)^2$
d $(a-b)^2 + 2ab$

Q 8C-03-Q016medium

$(a-b)^2 = ?$ (Corollary 4)

a $(a+b)^2 + 4ab$
b $(a+b)^2 - 4ab$
c $(a+b)^2 - 2ab$
d $(a+b)^2$

Q 9C-03-Q017medium

$a^2 + b^2 = \dfrac{(a+b)^2 + (a-b)^2}{?}$

a $1$
b $2$
c $3$
d $4$

Q 10C-03-Q018medium

$ab = \dfrac{(a+b)^2 - (a-b)^2}{?}$

a $2$
b $3$
c $4$
d $8$

Q 11C-03-Q019medium

$ab = \left(\dfrac{a+b}{2}\right)^2 - ?$

a $\left(\dfrac{a-b}{2}\right)^2$
b $\left(\dfrac{a+b}{2}\right)^2$
c $\dfrac{a-b}{2}$
d $\dfrac{ab}{4}$

Q 12C-03-Q020medium

By Corollary 6, the product of two quantities can be expressed as—

asum of two squares
bdifference of two squares
csum of two cubes
dquotient of squares

3.Difference of Squares and Product Expansion3

Q 1C-03-Q021medium

$a^2 - b^2 = ?$ (Formula 3)

a $(a-b)(a-b)$
b $(a+b)(a-b)$
c $(a+b)(a+b)$
d $a-b$

Q 2C-03-Q022medium

$(x+a)(x+b) = ?$ (Formula 4)

a $x^2 + ab$
b $x^2 + (a+b)x + ab$
c $x^2 + (a-b)x + ab$
d $x^2 - (a+b)x - ab$

Q 3C-03-Q023medium

The difference of the squares of two expressions equals—

asum $\times$ difference of the expressions
bsquare of their sum
csquare of their difference
dsum of their squares

4.Square of Three Terms6

Q 1C-03-Q024medium

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

a $a^2 + b^2 + c^2$
b $a^2 + b^2 + c^2 + 2ab + 2bc + 2ac$
c $a^2 + b^2 + c^2 + ab + bc + ac$
d $a^2 + b^2 + c^2 + 3abc$

Q 2C-03-Q025medium

$a^2 + b^2 + c^2 = ?$ (Corollary 7)

a $(a+b+c)^2$
b $(a+b+c)^2 - 2(ab+bc+ac)$
c $(a+b+c)^2 + 2(ab+bc+ac)$
d $(a+b+c)^2 - (ab+bc+ac)$

Q 3C-03-Q026medium

$2(ab+bc+ac) = ?$ (Corollary 8)

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

Q 4C-03-Q027medium

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

a $a^2+b^2+c^2 + 2ab - 2bc - 2ac$
b $a^2+b^2+c^2 + 2ab + 2bc + 2ac$
c $a^2+b^2-c^2 + 2ab$
d $a^2+b^2+c^2 - 2ab + 2bc - 2ac$

Q 5C-03-Q028medium

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

a $a^2+b^2+c^2 - 2ab - 2bc + 2ac$
b $a^2+b^2+c^2 + 2ab + 2bc + 2ac$
c $a^2+b^2+c^2 - 2ab + 2bc - 2ac$
d $a^2+b^2+c^2 + 2ab - 2bc - 2ac$

Q 6C-03-Q029medium

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

a $a^2+b^2+c^2 - 2ab - 2bc - 2ac$
b $a^2+b^2+c^2 - 2ab + 2bc - 2ac$
c $a^2+b^2+c^2 + 2ab + 2bc + 2ac$
d $a^2+b^2+c^2 - 2ab - 2bc + 2ac$

5.Worked Examples on Squares16

Q 1C-03-Q030medium

Square of $(4x+5y) = ?$

a $16x^2 + 40xy + 25y^2$
b $16x^2 + 25y^2$
c $16x^2 - 40xy + 25y^2$
d $20x^2 + 40xy + 25y^2$

Q 2C-03-Q031medium

Square of $(3a-7b) = ?$

a $9a^2 - 42ab + 49b^2$
b $9a^2 + 42ab + 49b^2$
c $9a^2 - 42ab - 49b^2$
d $9a^2 - 21ab + 49b^2$

Q 3C-03-Q032medium

Using $(1000-4)^2$ , the value of $996^2 = ?$

a $992016$
b $992004$
c $1000016$
d $984016$

Q 4C-03-Q033medium

$(a+b+c+d)^2 = ?$

a $a^2+b^2+c^2+d^2$
b $a^2+b^2+c^2+d^2 + 2(ab+ac+ad+bc+bd+cd)$
c $a^2+b^2+c^2+d^2 + 2(ab+bc+cd)$
d $a^2+b^2+c^2+d^2 + ab+bc+cd$

Q 5C-03-Q034medium

Simplify: $(5x+7y+3z)^2 + 2(7x-7y-3z)(5x+7y+3z) + (7x-7y-3z)^2$

a $(12x)^2$
b $(12y)^2$
c $(7x)^2$
d $0$

Q 6C-03-Q035medium

If $x-y=2$ and $xy=24$ , $(x+y)^2 = ?$

a $96$
b $100$
c $48$
d $50$

Q 7C-03-Q036medium

With same data, $x+y = ?$

a $\pm 8$
b $\pm 10$
c $\pm 12$
d $\pm 6$

Q 8C-03-Q037medium

If $a^4 + a^2 b^2 + b^4 = 3$ and $a^2 + ab + b^2 = 3$ , $a^2 - ab + b^2 = ?$

a $0$
b $1$
c $2$
d $3$

Q 9C-03-Q038medium

With same data, $a^2 + b^2 = ?$

a $1$
b $2$
c $3$
d $4$

Q 10C-03-Q039medium

With same data, $ab = ?$

a $0$
b $1$
c $2$
d $3$

Q 11C-03-Q040medium

$(a+b)^4 - (a-b)^4 = ?$

a $8ab(a^2+b^2)$
b $4ab(a^2+b^2)$
c $2ab(a+b)$
d $16ab$

Q 12C-03-Q041medium

If $a+b+c=15$ and $a^2+b^2+c^2=83$ , $ab+bc+ca = ?$

a $71$
b $142$
c $35$
d $50$

Q 13C-03-Q042medium

If $a+b+c=2$ and $ab+bc+ca=1$ , $(a+b)^2+(b+c)^2+(c+a)^2 = ?$

a $4$
b $6$
c $8$
d $10$

Q 14C-03-Q043medium

Express $(2x+3y)(4x-5y)$ as the difference of two squares:

a $(3x-y)^2 - (4y-x)^2$
b $(3x+y)^2 - (4y-x)^2$
c $(3x-y)^2 + (4y-x)^2$
d $(2x+3y)^2 - (4x-5y)^2$

Q 15C-03-Q044medium

Simplify: $(4x+3y)^2 + 2(4x+3y)(4x-3y) + (4x-3y)^2$

a $64x^2$
b $36y^2$
c $16x^2$
d $25xy$

Q 16C-03-Q045medium

If $x+y+z=12$ and $x^2+y^2+z^2=50$ , $(x-y)^2+(y-z)^2+(z-x)^2 = ?$

a $6$
b $50$
c $94$
d $144$

6.Exercise 3.1 Problems22

Q 1C-03-Q046medium

Square of $(2a+3b) = ?$

a $4a^2 + 12ab + 9b^2$
b $4a^2 - 12ab + 9b^2$
c $4a^2 + 9b^2$
d $2a^2 + 12ab + 3b^2$

Q 2C-03-Q047medium

Square of $(4y-5x) = ?$

a $16y^2 + 40xy + 25x^2$
b $16y^2 - 40xy + 25x^2$
c $16y^2 - 25x^2$
d $4y^2 - 40xy + 5x^2$

Q 3C-03-Q048medium

Square of $(5a^2 - y) = ?$

a $25a^4 - 10a^2 y + y^2$
b $25a^4 + 10a^2 y + y^2$
c $25a^2 - 10ay + y^2$
d $25a^4 - 5ay + y^2$

Q 4C-03-Q049medium

Square of $(3b - 5c - 2a) = ?$

a $9b^2 + 25c^2 + 4a^2 - 30bc + 10ac - 12ab$
b $9b^2 + 25c^2 + 4a^2 + 30bc - 10ac + 12ab$
c $9b^2 - 25c^2 - 4a^2 - 30bc + 10ac - 12ab$
d $9b^2 + 25c^2 + 4a^2 - 30bc - 10ac - 12ab$

Q 5C-03-Q050medium

Square of $(ax - by - cz) = ?$

a $a^2 x^2 + b^2 y^2 + c^2 z^2 - 2abxy + 2bcyz - 2acxz$
b $a^2 x^2 + b^2 y^2 + c^2 z^2 + 2abxy + 2bcyz + 2acxz$
c $a^2 x^2 + b^2 y^2 + c^2 z^2 - 2abxy - 2bcyz + 2acxz$
d $a^2 x^2 - b^2 y^2 - c^2 z^2$

Q 6C-03-Q051medium

$1007^2 = ?$

a $1014049$
b $1007049$
c $1014007$
d $1004900$

Q 7C-03-Q052medium

$6.35 \times 6.35 + 2 \times 6.35 \times 3.65 + 3.65 \times 3.65 = ?$

a $100$
b $10$
c $25.5$
d $40$

Q 8C-03-Q053medium

$\dfrac{2345^2 - 759^2}{2345 - 759} = ?$

a $3104$
b $1586$
c $2345$
d $759$

Q 9C-03-Q054medium

If $a-b=4$ and $ab=60$ , $(a+b)^2 = ?$

a $256$
b $244$
c $272$
d $64$

Q 10C-03-Q055medium

If $a+b=9m$ and $ab=18m^2$ , $(a-b)^2 = ?$

a $9m^2$
b $81m^2$
c $0$
d $27m^2$

Q 11C-03-Q056medium

If $a+b+c=9$ and $ab+bc+ca=31$ , $a^2+b^2+c^2 = ?$

a $19$
b $50$
c $40$
d $112$

Q 12C-03-Q057medium

If $a^2+b^2+c^2=9$ and $ab+bc+ca=8$ , $(a+b+c)^2 = ?$

a $25$
b $17$
c $1$
d $5$

Q 13C-03-Q058medium

If $a+b+c=6$ and $a^2+b^2+c^2=14$ , $(a-b)^2+(b-c)^2+(c-a)^2 = ?$

a $6$
b $14$
c $36$
d $22$

Q 14C-03-Q059medium

If $x=3$ , $y=4$ , $z=5$ , $9x^2 + 16y^2 + 4z^2 - 24xy - 16yz + 12zx = ?$

a $9$
b $0$
c $1$
d $25$

Q 15C-03-Q060medium

Express $(a+2b)(3a+2c)$ as the difference of two squares:

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

Q 16C-03-Q061medium

Express $x^2 + 10x + 24$ as the difference of two squares:

a $(x+5)^2 - 1^2$
b $(x-5)^2 - 1$
c $(x+5)^2 + 1$
d $(x+12)(x-2)$

Q 17C-03-Q062medium

If $a^4 + a^2 b^2 + b^4 = 8$ and $a^2 + ab + b^2 = 4$ , $a^2 + b^2 = ?$

a $3$
b $4$
c $5$
d $6$

Q 18C-03-Q063medium

With same data, $ab = ?$

a $1$
b $2$
c $3$
d $0$

Q 19C-03-Q064medium

If $a+b = \sqrt{7}$ and $a-b = \sqrt{5}$ , $8ab(a^2+b^2) = ?$

a $24$
b $12$
c $48$
d $35$

Q 20C-03-Q065medium

If $x + \dfrac{1}{x} = 2$ , $x^2 + \dfrac{1}{x^2} = ?$

a $2$
b $4$
c $6$
d $0$

Q 21C-03-Q066medium

If $x - \dfrac{1}{x} = 4$ , $x^4 + \dfrac{1}{x^4} = ?$

a $322$
b $324$
c $256$
d $144$

Q 22C-03-Q067medium

If $2x + \dfrac{1}{x} = 3$ , $x^2 + \dfrac{1}{4x^2} = ?$

a $\dfrac{5}{4}$
b $\dfrac{9}{4}$
c $9$
d $5$

7.Formulae of Cubes8

Q 1C-03-Q068medium

$(a+b)^3 = ?$ (Formula 6)

a $a^3 + b^3$
b $a^3 + 3a^2 b + 3ab^2 + b^3$
c $a^3 + 3a^2 b - 3ab^2 + b^3$
d $a^3 + b^3 + 3ab$

Q 2C-03-Q069medium

$(a+b)^3 = a^3 + b^3 + ?$

a $3ab(a-b)$
b $3ab(a+b)$
c $3a^2 b^2$
d $ab$

Q 3C-03-Q070medium

$a^3 + b^3 = ?$ (Corollary 9)

a $(a+b)^3 - 3ab(a+b)$
b $(a+b)^3 + 3ab(a+b)$
c $(a+b)^3$
d $(a-b)^3$

Q 4C-03-Q071medium

$(a-b)^3 = ?$ (Formula 7)

a $a^3 - 3a^2 b + 3ab^2 - b^3$
b $a^3 + 3a^2 b - 3ab^2 - b^3$
c $a^3 - 3a^2 b + 3ab^2 + b^3$
d $a^3 - b^3$

Q 5C-03-Q072medium

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

a $3ab(a+b)$
b $3ab(a-b)$
c $3a^2 b^2$
d $-3ab$

Q 6C-03-Q073medium

$a^3 - b^3 = ?$ (Corollary 10)

a $(a-b)^3 + 3ab(a-b)$
b $(a-b)^3 - 3ab(a-b)$
c $(a+b)^3 - 3ab$
d $(a-b)^3$

Q 7C-03-Q074medium

$a^3 + b^3 = ?$ (Formula 8)

a $(a+b)(a^2 - ab + b^2)$
b $(a-b)(a^2 + ab + b^2)$
c $(a+b)(a^2 + ab + b^2)$
d $(a-b)(a^2 - ab + b^2)$

Q 8C-03-Q075medium

$a^3 - b^3 = ?$ (Formula 9)

a $(a-b)(a^2 - ab + b^2)$
b $(a-b)(a^2 + ab + b^2)$
c $(a+b)(a^2 - ab + b^2)$
d $(a+b)(a^2 + ab + b^2)$

8.Worked Examples on Cubes12

Q 1C-03-Q076medium

Cube of $(2x+3y) = ?$

a $8x^3 + 36x^2 y + 54xy^2 + 27y^3$
b $8x^3 + 18x^2 y + 27xy^2 + 27y^3$
c $6x^3 + 36x^2 y + 54xy^2 + 27y^3$
d $8x^3 + 27y^3$

Q 2C-03-Q077medium

Cube of $(2x-y) = ?$

a $8x^3 - 12x^2 y + 6xy^2 - y^3$
b $8x^3 + 12x^2 y + 6xy^2 + y^3$
c $8x^3 - y^3$
d $8x^3 - 6x^2 y + 12xy^2 - y^3$

Q 3C-03-Q078medium

If $x=37$ , $8x^3 + 72x^2 + 216x + 216 = ?$

a $512000$
b $216000$
c $343000$
d $64000$

Q 4C-03-Q079medium

If $x-y=8$ and $xy=5$ , $x^3 - y^3 + 8(x+y)^2 = ?$

a $1304$
b $632$
c $672$
d $1024$

Q 5C-03-Q080medium

If $a = \sqrt{3} + \sqrt{2}$ , $a + \dfrac{1}{a} = ?$

a $2\sqrt{3}$
b $\sqrt{5}$
c $2\sqrt{2}$
d $\sqrt{3}$

Q 6C-03-Q081medium

With same data, $a^3 + \dfrac{1}{a^3} = ?$

a $18\sqrt{3}$
b $24\sqrt{3}$
c $6\sqrt{3}$
d $8\sqrt{3}$

Q 7C-03-Q082medium

If $x+y=5$ and $xy=6$ , $2(x^2+y^2) = ?$

a $26$
b $13$
c $50$
d $36$

Q 8C-03-Q083medium

With same data and $x>y$ , $x-y = ?$

a $1$
b $2$
c $-1$
d $\sqrt{5}$

Q 9C-03-Q084medium

With same data and $x>y$ , $x^3 + y^3 = ?$

a $35$
b $275$
c $90$
d $125$

Q 10C-03-Q085medium

With same data and $x>y$ , $x^5 + y^5 = ?$

a $275$
b $35$
c $243$
d $32$

Q 11C-03-Q086medium

If $x = -2$ , $27x^3 - 54x^2 + 36x - 8 = ?$

a $-512$
b $512$
c $-216$
d $216$

Q 12C-03-Q087medium

If $a+b=5$ and $ab=6$ , $a^3 + b^3 + 4(a-b)^2 = ?$

a $39$
b $35$
c $40$
d $4$

9.Exercise 3.2 Problems15

Q 1C-03-Q088medium

Cube of $(2x^2 + 3y^2) = ?$

a $8x^6 + 36x^4 y^2 + 54x^2 y^4 + 27y^6$
b $8x^6 + 27y^6$
c $8x^6 + 18x^4 y^2 + 27x^2 y^4 + 27y^6$
d $6x^6 + 36x^4 y^2 + 54x^2 y^4 + 27y^6$

Q 2C-03-Q089medium

Cube of $(7m^2 - 2n) = ?$

a $343m^6 - 294m^4 n + 84m^2 n^2 - 8n^3$
b $343m^6 + 294m^4 n + 84m^2 n^2 + 8n^3$
c $343m^6 - 8n^3$
d $7m^6 - 294m^4 n + 84m^2 n^2 - 2n^3$

Q 3C-03-Q090medium

$(x+y)(x^2 - xy + y^2) + (y+z)(y^2 - yz + z^2) + (z+x)(z^2 - zx + x^2) = ?$

a $2(x^3 + y^3 + z^3)$
b $x^3 + y^3 + z^3$
c $3xyz$
d $0$

Q 4C-03-Q091medium

If $a-b=5$ and $ab=36$ , $a^3 - b^3 = ?$

a $665$
b $125$
c $540$
d $216$

Q 5C-03-Q092medium

If $a^3 - b^3 = 513$ and $a-b = 3$ , $ab = ?$

a $54$
b $27$
c $36$
d $18$

Q 6C-03-Q093medium

If $x=19$ , $y=-12$ , $8x^3 + 36x^2 y + 54xy^2 + 27y^3 = ?$

a $8$
b $2$
c $216$
d $0$

Q 7C-03-Q094medium

If $a=15$ , $8a^3 + 60a^2 + 150a + 130 = ?$

a $42880$
b $42875$
c $35000$
d $42870$

Q 8C-03-Q095medium

If $a+b=83$ and $ab=2$ , $a^2 - ab + b^2 = ?$

a $6883$
b $6885$
c $6889$
d $6887$

Q 9C-03-Q096medium

With same data, $a^3 + b^3 = ?$

a $571289$
b $571189$
c $691083$
d $570289$

Q 10C-03-Q097medium

If $a-b=5$ and $ab=36$ , $a^2 + ab + b^2 = ?$

a $133$
b $97$
c $25$
d $36$

Q 11C-03-Q098medium

If $m + \dfrac{1}{m} = a$ , $m^3 + \dfrac{1}{m^3} = ?$

a $a^3 - 3a$
b $a^3 + 3a$
c $a^3 - 3$
d $a^3$

Q 12C-03-Q099medium

If $x - \dfrac{1}{x} = p$ , $x^3 - \dfrac{1}{x^3} = ?$

a $p^3 + 3p$
b $p^3 - 3p$
c $p^3$
d $p$

Q 13C-03-Q100medium

If $a - \dfrac{1}{a} = 1$ , $a^3 - \dfrac{1}{a^3} = ?$

a $4$
b $1$
c $-1$
d $3$

Q 14C-03-Q101medium

If $a+b+c = 0$ , $a^3 + b^3 + c^3 = ?$

a $3abc$
b $0$
c $abc$
d $-3abc$

Q 15C-03-Q102medium

If $p - q = r$ , $p^3 - q^3 - r^3 = ?$

a $3pqr$
b $0$
c $-3pqr$
d $pqr$

10.Common Factor and Perfect Square Factorization5

Q 1C-03-Q103medium

Factorize $3a^2 b + 6ab^2 + 12a^2 b^2$ :

a $3ab(a + 2b + 4ab)$
b $3a^2 b(1 + 2b + 4ab)$
c $3ab(a + 2b)$
d $ab(3a + 6b + 12ab)$

Q 2C-03-Q104medium

Factorize $2ab(x-y) + 2bc(x-y) + 3ca(x-y)$ :

a $(x-y)(2ab + 2bc + 3ca)$
b $(x+y)(2ab + 2bc + 3ca)$
c $(x-y)(ab + bc + ca)$
d $(x-y)(2ab + bc + 3ca)$

Q 3C-03-Q105medium

Factorize $4x^2 + 12x + 9$ :

a $(2x+3)^2$
b $(2x-3)^2$
c $(2x+3)(2x-3)$
d $(4x+9)(x+1)$

Q 4C-03-Q106medium

Factorize $9x^2 - 30xy + 25y^2$ :

a $(3x - 5y)^2$
b $(3x + 5y)^2$
c $(3x - 5y)(3x + 5y)$
d $9x^2 + 25y^2$

Q 5C-03-Q107medium

Factorize $9x^2 + 24x + 16$ :

a $(3x+4)^2$
b $(3x-4)^2$
c $(3x+4)(3x-4)$
d $(9x+16)(x+1)$

11.Difference of Two Squares Factorization5

Q 1C-03-Q108medium

Factorize $a^2 - 1 + 2b - b^2$ :

a $(a+b-1)(a-b+1)$
b $(a-b-1)(a+b+1)$
c $(a+1)(a-1)(b-1)$
d $(a+b)(a-b)$

Q 2C-03-Q109medium

Factorize $a^4 + 64 b^4$ :

a $(a^2 + 4ab + 8b^2)(a^2 - 4ab + 8b^2)$
b $(a^2 + 8b^2)^2$
c $(a^2 - 8b^2)^2$
d $(a^2 + 4ab)(a^2 - 4ab)$

Q 3C-03-Q110medium

Factorize $4a^2 - 12ab + 9b^2 - 4c^2$ :

a $(2a-3b+2c)(2a-3b-2c)$
b $(2a+3b+2c)(2a+3b-2c)$
c $(2a-3b-2c)^2$
d $(2a-3b)^2$

Q 4C-03-Q111medium

Factorize $x^2 + 6x + 8 - y^2 + 2y$ :

a $(x+y+2)(x-y+4)$
b $(x-y+2)(x+y+4)$
c $(x+y-2)(x-y-4)$
d $(x+y+4)(x-y+2)$

Q 5C-03-Q112medium

Factorize $16x^2 - 25y^2 - 8xz + 10yz$ :

a $(4x - 5y)(4x + 5y - 2z)$
b $(4x + 5y)(4x - 5y + 2z)$
c $(4x - 5y)(4x - 5y + 2z)$
d $(4x + 5y)(4x + 5y - 2z)$

12.Middle Term Factorization11

Q 1C-03-Q113medium

Factorize $x^2 + 12x + 35$ :

a $(x+5)(x+7)$
b $(x-5)(x-7)$
c $(x+1)(x+35)$
d $(x-7)(x+5)$

Q 2C-03-Q114medium

Factorize $x^2 + x - 20$ :

a $(x+5)(x-4)$
b $(x+4)(x-5)$
c $(x-5)(x-4)$
d $(x+10)(x-2)$

Q 3C-03-Q115medium

Factorize $3x^2 - x - 14$ :

a $(3x-7)(x+2)$
b $(3x+7)(x-2)$
c $(3x-2)(x+7)$
d $(3x-14)(x+1)$

Q 4C-03-Q116medium

Factorize $x^2 + 13x + 36$ :

a $(x+4)(x+9)$
b $(x-4)(x-9)$
c $(x+6)^2$
d $(x+12)(x+3)$

Q 5C-03-Q117medium

Factorize $a^2 - 30a + 216$ :

a $(a-12)(a-18)$
b $(a-6)(a-36)$
c $(a-9)(a-24)$
d $(a+12)(a+18)$

Q 6C-03-Q118medium

Factorize $x^2 - 37x - 650$ :

a $(x-50)(x+13)$
b $(x+50)(x-13)$
c $(x-26)(x-25)$
d $(x-65)(x-10)$

Q 7C-03-Q119medium

Factorize $x^4 + x^2 - 20$ :

a $(x^2+5)(x-2)(x+2)$
b $(x^2-5)(x^2+4)$
c $(x+5)(x-4)$
d $(x^2+5)(x+4)$

Q 8C-03-Q120medium

Factorize $4x^4 - 27x^2 - 81$ :

a $(4x^2 + 9)(x-3)(x+3)$
b $(4x^2 - 9)(x^2 + 9)$
c $(2x-3)^2 (2x+3)^2$
d $(4x^2 + 9)(x^2 + 9)$

Q 9C-03-Q121medium

Factorize $ax^2 + (a^2+1)x + a$ :

a $(x+a)(ax+1)$
b $(x-a)(ax-1)$
c $(ax+a)(x+1)$
d $(x+a)(x+a)$

Q 10C-03-Q122medium

Factorize $3(a^2 + 2a)^2 - 22(a^2 + 2a) + 40$ :

a $(3a^2 + 6a - 10)(a^2 + 2a - 4)$
b $(3a^2 - 6a + 10)(a^2 - 2a + 4)$
c $(3a-10)(a-4)$
d $(a^2 + 2a - 10)(3a^2 + 6a - 4)$

Q 11C-03-Q123medium

Factorize $x^4 - 6x^2 y^2 + y^4$ :

a $(x^2 - 2xy - y^2)(x^2 + 2xy - y^2)$
b $(x^2 - y^2)^2$
c $(x-y)^4$
d $(x^2 - 3y^2)(x^2 - 2y^2)$

13.Perfect Cube and Sum/Difference of Cubes7

Q 1C-03-Q124medium

Factorize $8x^3 + 36x^2 y + 54xy^2 + 27y^3$ :

a $(2x + 3y)^3$
b $(2x - 3y)^3$
c $(2x + 3y)(4x^2 - 6xy + 9y^2)$
d $(2x)^3 + (3y)^3$

Q 2C-03-Q125medium

Factorize $8a^3 + 27b^3$ :

a $(2a + 3b)(4a^2 - 6ab + 9b^2)$
b $(2a - 3b)(4a^2 + 6ab + 9b^2)$
c $(2a + 3b)^3$
d $(2a + 3b)(4a^2 + 6ab + 9b^2)$

Q 3C-03-Q126medium

Factorize $a^6 - 64$ :

a $(a+2)(a-2)(a^2 + 2a + 4)(a^2 - 2a + 4)$
b $(a-2)^6$
c $(a^2 - 4)(a^4 + 4a^2 + 16)$
d $(a^3 - 8)^2$

Q 4C-03-Q127medium

Factorize $x^3 + 6x^2 y + 11xy^2 + 6y^3$ :

a $(x+y)(x+2y)(x+3y)$
b $(x+y)^3$
c $(x+2y)^3$
d $(x+y)(x+y)(x+6y)$

Q 5C-03-Q128medium

Factorize $8a^3 + \dfrac{1}{27}$ :

a $\left(2a + \dfrac{1}{3}\right)\left(4a^2 - \dfrac{2a}{3} + \dfrac{1}{9}\right)$
b $\left(2a - \dfrac{1}{3}\right)\left(4a^2 + \dfrac{2a}{3} + \dfrac{1}{9}\right)$
c $(2a)^3 - \left(\dfrac{1}{3}\right)^3$
d $\left(2a + \dfrac{1}{3}\right)^3$

Q 6C-03-Q129medium

Factorize $x^3 + 3x^2 + 3x + 2$ :

a $(x+2)(x^2 + x + 1)$
b $(x+1)(x^2 + 2x + 2)$
c $(x+2)^3$
d $(x-2)(x^2 - x + 1)$

Q 7C-03-Q130medium

Factorize $a^3 - 6a^2 + 12a - 9$ :

a $(a-3)(a^2 - 3a + 3)$
b $(a-3)^3$
c $(a-2)^3 - 1$
d $(a-3)(a^2 + 3a + 3)$

14.Remainder and Factor Theorem6

Q 1C-03-Q131medium

If $f(x)$ is divided by $(x-a)$ , the remainder is—

a $f(0)$
b $f(a)$
c $f(-a)$
d $0$

Q 2C-03-Q132medium

By factor theorem, $(x-a)$ is a factor of $f(x)$ iff—

a $f(0) = 0$
b $f(a) = 0$
c $f(-a) = 0$
d $f(1) = 0$

Q 3C-03-Q133medium

If $f(x) = 6x^2 - 7x + 5$ is divided by $(x-1)$ , remainder = ?

a $4$
b $0$
c $5$
d $-4$

Q 4C-03-Q134medium

If $f(x)$ is divided by $(ax + b)$ where $a \ne 0$ , remainder = ?

a $f\left(\dfrac{b}{a}\right)$
b $f\left(-\dfrac{b}{a}\right)$
c $f\left(\dfrac{a}{b}\right)$
d $f\left(-\dfrac{a}{b}\right)$

Q 5C-03-Q135medium

$(ax+b)$ is a factor of $f(x)$ iff—

a $f\left(\dfrac{b}{a}\right) = 0$
b $f\left(-\dfrac{b}{a}\right) = 0$
c $f(0) = 0$
d $f(a) = 0$

Q 6C-03-Q136medium

The vanishing method of factoring uses—

aquadratic formula
bremainder/factor theorem
ccompleting square
dsynthetic division

15.Vanishing Method Examples9

Q 1C-03-Q137medium

Factorize $x^3 - x - 6$ :

a $(x-2)(x^2 + 2x + 3)$
b $(x+2)(x^2 - 2x + 3)$
c $(x-2)(x+1)(x+3)$
d $(x-1)(x^2 + x + 6)$

Q 2C-03-Q138medium

Factorize $x^3 - 3xy^2 + 2y^3$ :

a $(x-y)^2 (x + 2y)$
b $(x+y)^2 (x - 2y)$
c $(x-y)(x+2y)$
d $(x-y)(x-2y)^2$

Q 3C-03-Q139medium

Factorize $x^2 + xy - 2y^2$ :

a $(x-y)(x+2y)$
b $(x+y)(x-2y)$
c $(x-y)(x-2y)$
d $(x+y)(x+2y)$

Q 4C-03-Q140medium

Factorize $54x^4 + 27x^3 a - 16x - 8a$ :

a $(2x+a)(3x-2)(9x^2 + 6x + 4)$
b $(2x-a)(3x+2)(9x^2 - 6x + 4)$
c $(2x+a)(27x^3 - 8)$
d $(2x+a)(3x+2)(9x^2 - 6x + 4)$

Q 5C-03-Q141medium

If $g(a) = a^3 + a^2 + 10a - 8$ is divided by $(a-2)$ , remainder = ?

a $24$
b $14$
c $-8$
d $0$

Q 6C-03-Q142medium

Factorize $f(a) = a^3 - 9 + (a+1)^3$ :

a $(a-1)(2a^2 + 5a + 8)$
b $(a+1)(2a^2 - 5a + 8)$
c $(a-1)(a^2 + 5a - 8)$
d $(a-1)^3$

Q 7C-03-Q143medium

Factorize $x^3 - 21x - 20$ :

a $(x+1)(x-5)(x+4)$
b $(x-1)(x+5)(x+4)$
c $(x+1)(x+5)(x-4)$
d $(x-1)(x-5)(x+4)$

Q 8C-03-Q144medium

Factorize $2x^3 - 3x^2 + 3x - 1$ :

a $(2x-1)(x^2 - x + 1)$
b $(2x+1)(x^2 + x + 1)$
c $(x-1)^3$
d $(2x-1)(x^2 + x + 1)$

Q 9C-03-Q145medium

Factorize $x^3 + 6x^2 + 11x + 6$ :

a $(x+1)(x+2)(x+3)$
b $(x-1)(x-2)(x-3)$
c $(x+6)(x^2 + 1)$
d $(x+1)(x+5)(x+6)$

16.Exercise 3.4 Problems13

Q 1C-03-Q146medium

Factorize $3x^3 + 2x + 5$ :

a $(x+1)(3x^2 - 3x + 5)$
b $(x-1)(3x^2 + 3x + 5)$
c $(x+1)(3x^2 + 3x + 5)$
d $(x+1)^3$

Q 2C-03-Q147medium

Factorize $x^3 - 7xy^2 - 6y^3$ :

a $(x+y)(x-3y)(x+2y)$
b $(x-y)(x+3y)(x-2y)$
c $(x+y)(x+3y)(x-2y)$
d $(x-y)(x-3y)(x-2y)$

Q 3C-03-Q148medium

Factorize $x^3 + 2x^2 - 5x - 6$ :

a $(x+1)(x+3)(x-2)$
b $(x-1)(x+3)(x-2)$
c $(x+1)(x-3)(x+2)$
d $(x+1)^3$

Q 4C-03-Q149medium

Factorize $x^3 + 4x^2 + x - 6$ :

a $(x-1)(x+2)(x+3)$
b $(x+1)(x-2)(x-3)$
c $(x+1)(x+2)(x+3)$
d $(x-1)(x-2)(x-3)$

Q 5C-03-Q150medium

Factorize $a^3 + 3a + 36$ :

a $(a+3)(a^2 - 3a + 12)$
b $(a-3)(a^2 + 3a + 12)$
c $(a+3)^3$
d $(a+3)(a^2 + 3a + 12)$

Q 6C-03-Q151medium

Factorize $a^4 - 4a + 3$ :

a $(a-1)^2 (a^2 + 2a + 3)$
b $(a+1)^2 (a^2 - 2a + 3)$
c $(a-1)(a^3 + a^2 + a - 3)$
d $(a-1)^4$

Q 7C-03-Q152medium

Factorize $a^3 - a^2 - 10a - 8$ :

a $(a+1)(a-4)(a+2)$
b $(a-1)(a+4)(a-2)$
c $(a+1)(a+4)(a-2)$
d $(a-1)(a-4)(a+2)$

Q 8C-03-Q153medium

Factorize $x^3 - 3x^2 + 4x - 4$ :

a $(x-2)(x^2 - x + 2)$
b $(x+2)(x^2 + x + 2)$
c $(x-2)^3$
d $(x-2)(x^2 + x + 2)$

Q 9C-03-Q154medium

Factorize $a^3 - 7a^2 b + 7ab^2 - b^3$ :

a $(a-b)(a^2 - 6ab + b^2)$
b $(a-b)^3$
c $(a-b)(a^2 + 6ab + b^2)$
d $(a+b)(a^2 - 6ab + b^2)$

Q 10C-03-Q155medium

Factorize $2x^4 - 3x^3 - 3x - 2$ :

a $(x-2)(2x+1)(x^2 + 1)$
b $(x+2)(2x-1)(x^2 + 1)$
c $(x-2)(2x-1)(x^2 + 1)$
d $(x+2)(2x+1)(x^2 + 1)$

Q 11C-03-Q156medium

Factorize $4x^4 + 12x^3 + 7x^2 - 3x - 2$ :

a $(x+1)(x+2)(2x-1)(2x+1)$
b $(x-1)(x-2)(2x+1)(2x-1)$
c $(x+1)(x+2)(4x^2 - 1)$
d $(x+1)^2 (2x-1)(2x+1)$

Q 12C-03-Q157medium

Factorize $4a^3 - 5a^2 + 5a - 1$ :

a $(4a-1)(a^2 - a + 1)$
b $(4a+1)(a^2 + a + 1)$
c $(4a-1)(a^2 + a + 1)$
d $(a-1)(4a^2 - a + 1)$

Q 13C-03-Q158medium

Factorize $18x^3 + 15x^2 - x - 2$ :

a $(3x-1)(3x+2)(2x+1)$
b $(3x+1)(3x-2)(2x-1)$
c $(3x-1)(2x+1)(3x-2)$
d $(3x+1)(3x+2)(2x+1)$

17.Real-Life Algebraic Formulae9

Q 1C-03-Q159medium

Money payable to $n$ persons at rate $q$ each is—

a $A = q + n$
b $A = qn$
c $A = q/n$
d $A = n/q$

Q 2C-03-Q160medium

If $q$ = portion of work per person per unit time, $n$ persons in time $x$ do—

a $W = q + n + x$
b $W = qnx$
c $W = q/(nx)$
d $W = qn/x$

Q 3C-03-Q161medium

Distance covered at speed $v$ in time $t$ is—

a $d = v + t$
b $d = vt$
c $d = v/t$
d $d = t/v$

Q 4C-03-Q162medium

Pipe-tank: with initial $Q_0$ and inflow rate $q$ , $Q(t) = ?$

a $Q_0 - qt$
b $Q_0 + qt$
c $Q_0 \cdot qt$
d $Q_0 / q + t$

Q 5C-03-Q163medium

Rate-of-percentage $r$ for $s\%$ :

a $r = s$
b $r = \dfrac{s}{100}$
c $r = \dfrac{100}{s}$
d $r = \dfrac{s}{10}$

Q 6C-03-Q164medium

Selling price at profit rate $r$ is—

a $S = C(1+r)$
b $S = C(1-r)$
c $S = Cr$
d $S = C + r$

Q 7C-03-Q165medium

Selling price at loss rate $r$ is—

a $S = C(1+r)$
b $S = C(1-r)$
c $S = Cr$
d $S = C - r$

Q 8C-03-Q166medium

Simple profit: $I = ?$

a $Pnr$
b $P + nr$
c $P/(nr)$
d $P(1+r)^n$

Q 9C-03-Q167medium

Compound principal after $n$ periods is—

a $P(1+nr)$
b $P(1+r)^n$
c $P + nr$
d $Pnr$

18.Time-Work, Distance, Pipe-Tank Examples14

Q 1C-03-Q168medium

Budget Tk 45000; 5 members refused, fee per head increased by Tk 15. Number of original members?

a $125$
b $100$
c $150$
d $120$

Q 2C-03-Q169medium

Rafiq alone 10 days, Shafiq alone 15 days. Together they finish in—

a $6$ days
b $5$ days
c $25$ days
d $12.5$ days

Q 3C-03-Q170medium

Boatman covers $x$ km against current in $t_1$ hr, with current in $t_2$ hr. Speed of boat = ?

a $\dfrac{x}{2}\left(\dfrac{1}{t_1} + \dfrac{1}{t_2}\right)$
b $\dfrac{x}{2}\left(\dfrac{1}{t_2} - \dfrac{1}{t_1}\right)$
c $\dfrac{x}{2}\left(\dfrac{1}{t_1} - \dfrac{1}{t_2}\right)$
d $\dfrac{x(t_1+t_2)}{2}$

Q 4C-03-Q171medium

Same setup. Speed of current = ?

a $\dfrac{x}{2}\left(\dfrac{1}{t_2} - \dfrac{1}{t_1}\right)$
b $\dfrac{x}{2}\left(\dfrac{1}{t_1} + \dfrac{1}{t_2}\right)$
c $\dfrac{x}{2}\left(\dfrac{1}{t_1} - \dfrac{1}{t_2}\right)$
d $x(t_1 - t_2)$

Q 5C-03-Q172medium

Pipe fills empty tank in 12 min; another flows out 14 L/min. Together fills empty tank in 96 min. Tank capacity?

a $192$ L
b $168$ L
c $200$ L
d $96$ L

Q 6C-03-Q173medium

Bus Tk 2400; 10 absent → fare per head +Tk 8. Number of passengers actually went?

a $50$
b $40$
c $60$
d $80$

Q 7C-03-Q174medium

Same setup. Fare each paid?

aTk 48
bTk 50
cTk 40
dTk 52

Q 8C-03-Q175medium

A and B together $p$ days; A alone $q$ days. B alone takes—

a $\dfrac{pq}{q-p}$ days
b $\dfrac{pq}{p-q}$ days
c $p+q$ days
d $pq$ days

Q 9C-03-Q176medium

Speed against current 2 km/h, current 3 km/h. Time to row 32 km along current?

a $4$ hr
b $5$ hr
c $6$ hr
d $8$ hr

Q 10C-03-Q177medium

10 persons in 7 days at 6 hr/day. 14 persons in 6 days at how many hr/day?

a $5$
b $6$
c $4$
d $7$

Q 11C-03-Q178medium

Two pipes; first fills in $t_1$ min, second empties in $t_2$ min ( $t_2 > t_1$ ). Together fills in—

a $\dfrac{t_1 t_2}{t_2 - t_1}$ min
b $\dfrac{t_1 t_2}{t_1 + t_2}$ min
c $\dfrac{t_2 - t_1}{t_1 t_2}$ min
d $t_1 + t_2$ min

Q 12C-03-Q179medium

Pipe fills tank in 12 min; outlet 15 L/min. Together fills empty tank in 48 min. Tank capacity?

a $240$ L
b $192$ L
c $180$ L
d $300$ L

Q 13C-03-Q180medium

Boatman covers $d$ km against current in $p$ hr, along current in $q$ hr. Speed of boat = ?

a $\dfrac{d}{2}\left(\dfrac{1}{p} + \dfrac{1}{q}\right)$
b $\dfrac{d}{2}\left(\dfrac{1}{p} - \dfrac{1}{q}\right)$
c $\dfrac{d}{2}\left(\dfrac{1}{q} - \dfrac{1}{p}\right)$
d $d(p+q)$

Q 14C-03-Q181medium

Same setup. Speed of current = ?

a $\dfrac{d}{2}\left(\dfrac{1}{q} - \dfrac{1}{p}\right)$
b $\dfrac{d}{2}\left(\dfrac{1}{p} - \dfrac{1}{q}\right)$
c $\dfrac{d}{2}\left(\dfrac{1}{p} + \dfrac{1}{q}\right)$
d $d(q - p)$

19.Profit, Loss, Percentage Examples9

Q 1C-03-Q182medium

Book price Tk 24 = 80% of actual price. Actual price?

aTk 30
bTk 20
cTk 36
dTk 25

Q 2C-03-Q183medium

Same setup. Government subsidy per book?

aTk 6
bTk 4
cTk 5
dTk 8

Q 3C-03-Q184medium

Loss is $r\%$ when $n$ oranges/taka. To gain $s\%$ profit, oranges per taka =

a $\dfrac{n(100 - r)}{100 + s}$
b $\dfrac{n(100 + s)}{100 - r}$
c $\dfrac{(100-r)(100+s)}{n}$
d $\dfrac{n}{100 + s}$

Q 4C-03-Q185medium

Loss is $z\%$ when 10 ice-cream sticks/taka. To gain $z\%$ profit, sticks/taka = $\dfrac{1000 - 10z}{100 + z}$ . With this formula, choose the matching expression.