On dividing $p\left(4 p^2-16\right)$ by $4 p(p-2)$, we get
- A$2 p+4$
- B$2 p-4$
- ✓$p+2$
- D$p-2$
Answer: C.
View full solution →Question types
Factorisation question types
69 questions across 9 question groups — pick any mix to generate a MATHS paper with step-by-step answer keys.
69
Questions
9
Question groups
5
Question types
01
M.C.Q. [1 Marks Each]
15 Q→02TRUE-FLASE [1 Marks Each]
5 Q→03Assertion (A) & Reason (B) MCQ
4 Q→042 Marks Questions
5 Q→053 Marks Question
24 Q→065 Marks Questions
7 Q→07MATCH THE FOLLOWING.
2 Q→081 Marks Question
2 Q→09BLANKS [1 Marks Each]
5 Q→Sample Questions
Factorisation questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
Which of the following is true for all the factors of 75?
- ✓They are odd numbers.
- BThey are divisible by 3 .
- CThey are divisible by 5 .
- DThey are prime numbers.
Answer: A.
View full solution →Which of the following divides $25\left(x^2 y+y^2 x\right)$ completely?
- A$x^2 y$
- ✓$5 x y$
- C$5 x^2 y$
- D$5 y^2 x$
Answer: B.
View full solution →Rajat factorises the given algebraic expression. $a^4-r^4$, which of the following would be his next step?
- A$a^4-r^4-2 a^2 r^2$
- B$(a+r)(a-r)$
- ✓$\left(a^2\right)^2-\left(r^2\right)^2$
- D$a^2-r^2$
Answer: C.
View full solution →Which of the following is not a factor of $18 p^2 q^2$ ?
- A3
- B$p^2$
- C$p q$
- ✓$q^3$
Answer: D.
View full solution →The irreducible factorisation of $3 a^3+6 a$ is $3\left(a^2+2\right)$.
View full solution →On dividing $\frac{p}{3}$ by $\frac{3}{p}$, the quotient is 9 .
View full solution →An identity is true for all values of its variables.
The value of $(a+1)(a-1)\left(a^2+1\right)$ is $\left(a^4+1\right)$.
View full solution →An equation is true for all the values of its variables.
Assertion (A) The value of $593 \times 607$ is 359951 .
Reason (R) $(a+b)(a-b)=a^2-b^2$.
Reason (R) $(a+b)(a-b)=a^2-b^2$.
- ✓Both A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A .
- CA is true but R is false.
- DA is false but R is true.
Answer: A.
View full solution →Assertion (A) To factorise $a x^2+b x+c$, write $b$ as sum of two numbers whose product is ac.
Reason (R) $3 x^2+x-1=(x+1)(3 x-2)+1$
Reason (R) $3 x^2+x-1=(x+1)(3 x-2)+1$
- ✓Both A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A .
- CA is true but R is false.
- DA is false but R is true.
Answer: A.
View full solution →Assertion (A) A factor which occurs in each term is called the common factor.
Reason (R) Expression $x^2-7 x+12$ has $(x-3)$ as a common factor.
Reason (R) Expression $x^2-7 x+12$ has $(x-3)$ as a common factor.
- ABoth A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A .
- CA is true but R is false.
- ✓A is false but R is true.
Answer: D.
View full solution →Assertion (A) The common factor of $a^2 m^4$ and $a^4 m^2$ is $a^2 m^4$.
Reason (R) A common factor is a number that can be divided into two different numbers, without leaving a remainder.
Reason (R) A common factor is a number that can be divided into two different numbers, without leaving a remainder.
- ABoth A and R are true and R is the correct explanation of A.
- BBoth A and R are true but R is not the correct explanation of A .
- CA is true but R is false.
- ✓A is false but R is true.
Answer: D.
View full solution →Factorise $15 x^2 y+6 x^4 y^2-9 x y$ and find its irreducible form.
View full solution →Build a square garden. Divide the square garden into four rectangular flower beds in such a way that each flower bed is as long as one side of the square. The perimeter of each flower bed is 40 m .
(i) Draw a diagram to represent the above information.
(ii) Mention the expression for perimeter of the entire garden.
(i) Draw a diagram to represent the above information.
(ii) Mention the expression for perimeter of the entire garden.
Algebraic Tiles
(i) Cut the following tiles from a graph sheet. Now, colour the tiles as per the colour code. Arrange these algebraic tiles to form a square.
Find the length of the side of the square so formed. Also find the area of the square. Using the above result factorise $x^2+4 x+4$.
(ii)
Find the length of the side of the rectangle so formed. Also, find the area of the rectangle. Using the above result factorise $x^2+5 x+4$.
Now, choose and cut more algebraic tiles from the graph sheet. Create your own colour code and colour the tiles. Arrange them to form square/rectangle. Find the area of the figure so formed using it to factorise
(a) $x^2+4 x+3$ $\qquad$ (b) $x^2+9 x+18$
(iii) Build a square garden. Divide the square garden into four rectangular flower beds in such a way that each flower bed is as long as one side of the square. The perimeter of each flower bed is 40 m .
(a) Draw a diagram to represent the above information.
(b) Mention the expression for perimeter of the entire garden.
View full solution →(i) Cut the following tiles from a graph sheet. Now, colour the tiles as per the colour code. Arrange these algebraic tiles to form a square.
Find the length of the side of the square so formed. Also find the area of the square. Using the above result factorise $x^2+4 x+4$.
(ii)
Find the length of the side of the rectangle so formed. Also, find the area of the rectangle. Using the above result factorise $x^2+5 x+4$.
Now, choose and cut more algebraic tiles from the graph sheet. Create your own colour code and colour the tiles. Arrange them to form square/rectangle. Find the area of the figure so formed using it to factorise
(a) $x^2+4 x+3$ $\qquad$ (b) $x^2+9 x+18$
(iii) Build a square garden. Divide the square garden into four rectangular flower beds in such a way that each flower bed is as long as one side of the square. The perimeter of each flower bed is 40 m .
(a) Draw a diagram to represent the above information.
(b) Mention the expression for perimeter of the entire garden.
Divide $\left(-27 x^2 y\right)+(-3 x y)$.
View full solution →Divide $\left(-121 p^3 q^3 r^3\right)$ by $\left(-11 x y^2 z^3\right)$.
View full solution →Divide
(i) $24 x y^2 z^3$ by $6 y z^2$
(ii) $63 a^2 b^4 c^6$ by $7 a^2 b^2 c^3$
View full solution →(i) $24 x y^2 z^3$ by $6 y z^2$
(ii) $63 a^2 b^4 c^6$ by $7 a^2 b^2 c^3$
Factorise
(i) $12 x+36$
(ii) $22 y-33 z$
(iii) $14 p q+35 p q r$
View full solution →(i) $12 x+36$
(ii) $22 y-33 z$
(iii) $14 p q+35 p q r$
Factorise the expressions and divide them as directed.
(i) $\left(y^2+7 y+10\right)÷(y+5)$
(ii) $\left(m^2-14 m-32\right)÷(m+2)$
(iii) $\left(5 p^2-25 p+20\right)÷(p-1)$
(iv) $4 y z\left(z^2+6 z-16\right)÷2 y(z+8)$
(v) $5 p q\left(p^2-q^2\right)÷2 p(p+q)$
(vi) $12 x y\left(9 x^2-16 y^2\right) ÷ 4 x y(3 x+4 y)$
(vii) $39 y^3\left(50 y^2-98\right)÷26 y^2(5 y+7)$
View full solution →(i) $\left(y^2+7 y+10\right)÷(y+5)$
(ii) $\left(m^2-14 m-32\right)÷(m+2)$
(iii) $\left(5 p^2-25 p+20\right)÷(p-1)$
(iv) $4 y z\left(z^2+6 z-16\right)÷2 y(z+8)$
(v) $5 p q\left(p^2-q^2\right)÷2 p(p+q)$
(vi) $12 x y\left(9 x^2-16 y^2\right) ÷ 4 x y(3 x+4 y)$
(vii) $39 y^3\left(50 y^2-98\right)÷26 y^2(5 y+7)$
Divide as directed.
(i) $5(2 x+1)(3 x+5)÷(2 x+1)$
(ii) $26 x y(x+5)(y-4)÷13 x(y-4)$
(iii) $52 p q r(p+q)(q+r)(r+p)÷104 p q(q+r)(r+p)$
(iv) $20(y+4)\left(y^2+5 y+3\right)÷5(y+4)$
(v) $x(x+1)(x+2)(x+3)÷x(x+1)$
View full solution →(i) $5(2 x+1)(3 x+5)÷(2 x+1)$
(ii) $26 x y(x+5)(y-4)÷13 x(y-4)$
(iii) $52 p q r(p+q)(q+r)(r+p)÷104 p q(q+r)(r+p)$
(iv) $20(y+4)\left(y^2+5 y+3\right)÷5(y+4)$
(v) $x(x+1)(x+2)(x+3)÷x(x+1)$
Work out the following divisions
(i) $(10 x-25) \div 5$
(ii) $(10 x-25) \div(2 x-5)$
(iii) $10 y(6 y+21) \div 5(2 y+7)$
(iv) $9 x^2 y^2(3 z-24) \div 27 x y(z-8)$
(v) $96 a b c(3 a-12)(5 b-30) \div 144(a-4)(b-6)$
View full solution →(i) $(10 x-25) \div 5$
(ii) $(10 x-25) \div(2 x-5)$
(iii) $10 y(6 y+21) \div 5(2 y+7)$
(iv) $9 x^2 y^2(3 z-24) \div 27 x y(z-8)$
(v) $96 a b c(3 a-12)(5 b-30) \div 144(a-4)(b-6)$
Salma and Abid factorise the algebraic expression $p^4+9 p^2+18$
Salma
$
\begin{aligned}
p^4+9 p^2+18 & =p^4+6 \rho^2+3 p^2+18 \\
& =p^2\left(p^2+6\right)+3\left(p^2+6\right) \\
& =\left(p^2+3\right)\left(p^2+6\right)
\end{aligned}
$
Abid
$
\begin{aligned}
p^4+9 p^2+18 & =p^4+6 p+3 p+18 \\
& =p\left(p^3+6\right)+3(p+6) \\
& =\left(p^3+6\right)(p+6)(p+3)
\end{aligned}
$
Who is correct? Give a reason to justify your answer.
View full solution →Salma
$
\begin{aligned}
p^4+9 p^2+18 & =p^4+6 \rho^2+3 p^2+18 \\
& =p^2\left(p^2+6\right)+3\left(p^2+6\right) \\
& =\left(p^2+3\right)\left(p^2+6\right)
\end{aligned}
$
Abid
$
\begin{aligned}
p^4+9 p^2+18 & =p^4+6 p+3 p+18 \\
& =p\left(p^3+6\right)+3(p+6) \\
& =\left(p^3+6\right)(p+6)(p+3)
\end{aligned}
$
Who is correct? Give a reason to justify your answer.
Perform the following divisions.
(i) $\left(3 p q r-6 p^2 q^2 r^2\right)÷ 3 p q$
(ii) $\left(a x^3-b x^2+c x\right)÷ (-d x)$
(iii) $\left(x^3 y^3+x^2 y^3-x y^4 + x y\right)÷x y$
(iv) $(-q r x y+p r y z-n y z)÷ (-x y z)$
View full solution →(i) $\left(3 p q r-6 p^2 q^2 r^2\right)÷ 3 p q$
(ii) $\left(a x^3-b x^2+c x\right)÷ (-d x)$
(iii) $\left(x^3 y^3+x^2 y^3-x y^4 + x y\right)÷x y$
(iv) $(-q r x y+p r y z-n y z)÷ (-x y z)$
Carry out the following divisions.
(i) $51 x^3 y^2 z÷17 x y z$
(ii) $76 x^3 y z^3÷19 x^2 y^2$
(iii) $17 a b^2 c^3÷\left(-a b c^2\right)$
(iv) $-121 p^3 q^3 r^3÷\left(-11 x y^2 z^3\right)$
View full solution →(i) $51 x^3 y^2 z÷17 x y z$
(ii) $76 x^3 y z^3÷19 x^2 y^2$
(iii) $17 a b^2 c^3÷\left(-a b c^2\right)$
(iv) $-121 p^3 q^3 r^3÷\left(-11 x y^2 z^3\right)$
Factorise the following, using the identity $\left(a^2-b^2\right)=(a-b)(a+b)$.
(i) $4 x^2-25 y^2$
(ii) $\frac{2 p^2}{25}-32 q^2$
(iii) $\frac{x^3 y}{9}-\frac{x y^3}{16}$
(iv) $16 x^4-81$
View full solution →(i) $4 x^2-25 y^2$
(ii) $\frac{2 p^2}{25}-32 q^2$
(iii) $\frac{x^3 y}{9}-\frac{x y^3}{16}$
(iv) $16 x^4-81$
Factorise the following, using the identity $\left(a^2-2 a b+b^2\right)=(a-b)^2$.
(i) $y^2-14 y+49$
(ii) $\frac{x^2}{4}-2 x+4$
(iii) $a^2 y^3-2 a b y^2+b^2 y$
(iv) $9 y^2-4 x y+\frac{4 x^2}{9}$
View full solution →(i) $y^2-14 y+49$
(ii) $\frac{x^2}{4}-2 x+4$
(iii) $a^2 y^3-2 a b y^2+b^2 y$
(iv) $9 y^2-4 x y+\frac{4 x^2}{9}$
| Column I | Column II |
| (i) $\quad\left(x^2+5 x+6\right)$ | (a) $\quad(x+1)(x+1)$ |
| (ii) $\quad\left(x^2+4 x+4\right)$ | (b) $\quad(x+2)(x-2)$ |
| (iii) $\quad\left(x^2+2 x+1\right)$ | (c) $\quad(x+2)(x+3)$ |
| (iv) $\quad\left(x^2-6 x+9\right)$ | (d) $\quad(x+2)(x+2)$ |
| (e) $\quad(x-3)(x-3)$ |
| Column I | Column II |
| (i) $\quad(21 x+13 y)^2$ | (a) $441 x^2-169 y^2$ |
| (ii) $(21 x-13 y)^2$ | (b) $441 x^2+169 y^2+546 x y$ |
| (iii) $(21 x-13 y) \times(21 x+13 y)$ | (c) $441 x^2+169 y^2-546 x y$ |
Factorise the expression $2 a^3-3 a^2 b+5 a b^2-a b$.
View full solution →Find the common factor of $x^2 y$ and $-x y^2$.
View full solution →$(x+a)(x+b)=x^2+(a+b) x+$ __________
View full solution →$(a-b)^2+$ __________ $=a^2-b^2$
View full solution →$105^2-103^2=$ __________ $\times(105-103)=$ __________
View full solution →The factorisation of $3 x+15 z$ is. __________
View full solution →The common factor method of factorisation for a polynomial is based on __________ law.
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