Vidaara.orgClass 10 · Mathematics
CodeVID-M10-14-CT
Factorization — Full Chapter Test
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- This is a full-length test covering the whole chapter — every topic is included.
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
$(x-a)$ is a factor of $p(x)$ iff:
- A.$p(a)=1$
- B.$p(a)=0$
- C.$p(0)=0$
- D.$p(a)>0$
2.
The remainder of $p(x)\div(x-a)$ is:
- A.$p(0)$
- B.$p(a)$
- C.$0$
- D.$a$
3.
To factorise a cubic, first use the:
- A.quadratic formula
- B.factor theorem
- C.binomial theorem
- D.graph
4.
If $p(2)=0$, a factor is:
- A.$(x+2)$
- B.$(x-2)$
- C.$2x$
- D.$(x-1)$
5.
The remainder of $p(x)\div(x+a)$ is:
- A.$p(a)$
- B.$p(-a)$
- C.$0$
- D.$a$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Is $(x-1)$ a factor of $x^2-3x+2$?
7.
Find the remainder when $x^2+1$ is divided by $(x-2)$.
8.
Factorise $x^3-1$.
9.
Is $(x-2)$ a factor of $x^2-4$?
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Show that $(x-2)$ is a factor of $x^3-3x^2+4x-4$.
11.
Find the remainder when $x^3-3x^2+2x+5$ is divided by $(x-2)$.
12.
Factorise $x^3-3x^2-x+3$.
13.
Find $k$ if $(x-1)$ is a factor of $x^2+kx+2$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Using the factor theorem, show that $(x-1),(x+2)$ and $(x-3)$ are factors of $x^3-2x^2-5x+6$.
15.
If $x^3+ax^2+bx-12$ is exactly divisible by $(x-2)$ and $(x+3)$, find $a$ and $b$.
Answer Key
Section A — Multiple Choice Questions
- (B) $p(a)=0$
- (B) $p(a)$
- (B) factor theorem
- (B) $(x-2)$
- (B) $p(-a)$
Section B — Short Answer (2 marks)
- Yes ($p(1)=0$).
- $5$.
- $(x-1)(x^2+x+1)$.
- Yes.
Section C — Short Answer (3 marks)
- Yes ($p(2)=0$).
- $5$.
- $(x-1)(x+1)(x-3)$.
- $k=-3$.
Section D — Long Answer (5 marks)
- All three are factors (each gives $p=0$).
- $a=3,\ b=-4$.
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