### Common Errors in Secondary Mathematics

Common Errors Committed  by the  Students  in Secondary Mathematics   Errors  that students often make in doing secondary mathematics  during their practice and during the examinations  and their remedial measures are well explained here stp by step.  Some Common Errors in Mathematics

## Maths Assignment Class 9th Chapter 2  Polynomials

Mathematics Assignment for Class IX Ch -2, Polynomials, strictly according to the CBSE syllabus. Math worksheet based on the topic Polynomial.

## Class IX Chapter 2

1. Find the value of p(x) = 3x- 2x + 8 at

a)  x = 0        Ans : 8
b)  x = -2      Ans : 24
c)  x = 3        Ans : 29
d)  x = -5      Ans : 73

2. Find the remainder when x- 3x+ 3x - 1 is  divided by

a)     x - 1         Ans 0
b)   x + 1         Ans -8

$c)\: \: x-\frac{1}{2}\: \: \:\: \: Ans\: \frac{-1}{2}$

$d)\: \: 2x+1\: \: \:\: \: Ans\: \frac{-27}{8}$

$e)\: \: 5-2x\: \: \:\: \: Ans\: \frac{27}{8}$

f)    3-x           Ans 8

Q 3. a) Show that 2x - 3 is a factor of  P(x) = 2x- 9x+ x + 12.
Ans: P(3/2) = 0  ⇒ 2x - 3 is the factor of P(x)

b) Show that x + 2 is a factor of  P(x) = x- x- 12.

Ans: P(-2) = 0  ⇒ x + 2 is the factor of P(x)

Hint: If remainder = 0 then given function is the factor of the given polynomial otherwise not.

4. Find the value of k if x + 1 is a factor of
a) 3x – kx + √3         Ans   -3 - √3
b) √2x – kx + k + 2  Ans  $-\frac{1}{2}(2+\sqrt{2})$
c) k – kx + Kx         Ans 0
d) x - 3x + kx          Ans - 4
Q5) Find “a” if x+1 is a factor of P(x) = ax3 - 9x2 + x + 6a        Ans:- 2

Solution Hint: Let x + 1 = 0 ⇒ x = - 1
Now find  P(-1) by putting x = -1 in p(x)
If P(-1) = 0 then x + 1 is the factor of p(x) otherwise not
Q 6) If (x -2)and(x + 3) are factors of p(x) = ax+ 3x- bx - 12.  Find a and b.
Ans: a = 1, b = 4

Solution Hint :
x - 2 is the factor of P(x) ⇒ P(2) = 0
a(2)3  + 3(2)2  - b(2) - 12 = 0
8a -2b = 0
4a -b = 0 ....... (i)
x + 3 is the factor of P(x) ⇒ P(-3) = 0
a(-3)3  + 3(-3)2  - b(-3) - 12 = 0
-27a + 27 + 3b - 12 = 0
-9a +b = -5 ....... (ii)
Eqn.(i) -Eqn(ii) we get

⇒ a = 1
Putting  a = 1 in eqn. (i) we get
4(1) -b = 0 ⇒ b = 4

Q 7. If x - 5 is a factor of x+ ax+ bx - 20  and leaves a remainder -2 when divided by x - 3. Find a and b.
Ans (-9, 24)

Solution Hint
x - 5 is the factor of P(x) ⇒ P(5) = 0
(5)+ a(5)+ b(5) - 20 = 0
5a + b = -21 ...... (i)
Also  P(3) = -2
(3)+ a(3)+ b(3) - 20 = 0
3a + b = -3 ........ (ii)
Eqn.(i) -Eqn.(ii) we get

⇒ a = -9
Putting  a = -9 in eqn. (ii) we get
3(-9) + b = -3 ⇒ b = 24

Q 8. Factorize:

a) x+ 11x + 30      : Ans  (x + 5)(x + 6)

b) a- 16a + 63       : Ans  (a - 7)(a - 9)

c)15 - 2x - x2           : Ans    (3 - x)(5 + x)

d) 3 + 5x - 2x2       : Ans   (1+ 2x)(3 - x)

e) 3m- 20m - 7   : Ans  (3m+1)(m - 7)

f) 24m+ m - 23      : Ans (m + 1)(24m - 23)

g) 2x+ 11xy - 21y    : Ans    (x + 7y)(2x - 3y)

h) 7(x + y)2 + 48(x + y) - 7       : Ans  (x + y + 7)(7x + 7y - 1)

I) 2m4n - 6m2n + 4n                 : Ans  2n(m+1)(m-1)(m2-2)

j) 36(a+3b)- 36(2a-3b)2   : Ans  (16a+3b)(33b-4a)

k) x+ 3x- 4x - 12                   : Ans (x - 2)(x + 2)(x + 3)

l) 2x- 9x +x + 12                   : Ans  (x + 1)(x - 4)(2x - 3)

m) x-3x-10x+24              : Ans   (x + 3)(x - 4)(3x - 2)

n) 2x-7x- 3x + 18               : Ans  (x - 2)(x - 3)(2x + 3)

o) 3x+ x - 38x + 24               : Ans   (x - 3)(x + 4)(3x - 2)

p)(x3 - (y - z)3)              : Ans  (x – y + z)(x2 + y2 + z2 - 2yz – xy - xz)

$q)\: \: 27p^{3}-\frac{1}{216}-\frac{9}{2}p^{2}+\frac{1}{4}p\: \: \: Ans:\; \left ( 3p-\frac{1}{6} \right )^{3}$

9.  Evaluate by using identity:

a)   92×108        Ans:- 9936

b)  6.7×5.3        Ans:- 35.51

c)104×107        Ans:- 11128

10.Factorize:

a)  8x- 125y- z- 30xyz

b)  125x- 8 + 27y+ 90xy

c)  3√3a3 - 5√5b3 - 8c3 - 6√15abc

d)  (5a - 4b)3 + (4b - 6c)+ (6c - 5a)3

e)  a+ (a - 1)+ (1 - 2a)3

f)  (3x + 3y - 5z)+ (2y + 3z - 5x)+ (2z + 2x - 5y)3

g)  (2x + y)3 - (x + y)3

h)  64m6-n6  : Ans:  (2mn + n)(4m - 2mn + n)(2m - n)(4m + 2mn + n)

11.Evaluate by using identity:

a)  (-15)3 + (-13)3 + (28)3     Ans: 16380
b)  (75)3 + (-50)3 + (-25)3    Ans: 281250

12.Simplify:
$\left ( 1-\frac{1}{x+1} \right )\left ( 1-\frac{1}{x+2} \right )\left ( 1-\frac{1}{x+3} \right )\: ......\: \left ( 1-\frac{1}{x+10} \right )$       $Ans\: \left ( \frac{x}{x+10} \right )$

Q13.If   a = 1 - √2, find  the value of $\left ( a-\frac{1}{a} \right )^{3}$   Ans: 8

Q 14.)
If a+ b+ c= 3abc  and a + b + c = 0 then show that

$\frac{(b+c)^{2}}{3bc}+\frac{(a+c)^{2}}{3ac}+\frac{(b+a)^{2}}{3ba}=1$

Q15.)
If    $x^{2}+\frac{1}{x^{2}}=14 ,\: \: find\: \: x^{3}+\frac{1}{x^{3}}$          $Ans\: \: \pm 52$
16.  Evaluate:
$\left ( \frac{x^{a}}{x^{b}} \right )^{a+b}\times \left ( \frac{x^{b}}{x^{c}} \right )^{b+c}\times \left ( \frac{x^{c}}{x^{a}} \right )^{c+a}\: \: \: Ans:\: 1$
17.  If   $x^{2}+\frac{1}{x^{2}}$  =  83, then find   $x^{3}-\frac{1}{x^{3}}$       Ans:  756
18.  If   $x=2+\sqrt{3}$, then find the value of   $x^{3}+\frac{1}{x^{3}}$     Ans: 52
19).  If    $x-\frac{1}{x}=3$,  then find the value of  $x^{3}-\frac{1}{x^{3}}$     Ans: 36
20.  If  $x=3+2\sqrt{2}$, then find the value of  $x^{3}-\frac{1}{x^{3}}$    Ans : $140\sqrt{2}$
21.  Find the value of    $\sqrt[4]{\sqrt[3]{\sqrt{2^{2}}}}$            Ans :  $2^{1/12}$
22.  If  $x=3+2\sqrt{2}$, then find the value of   $\sqrt{x}+\frac{1}{\sqrt{x}}$      Ans:  $2\sqrt{2}$
23.  If  $a=\frac{3+\sqrt{7}}{2}$ , then find   $a^{2}+\frac{1}{a^{2}}$       $\left [ Ans:\: 20-\frac{9}{2}\sqrt{7} \right ]$
24)  If    $a=\frac{4}{3-\sqrt{5}}$, then find   $a+\frac{4}{a}$         [Ans 6 ]
25.  Find the remainder when x4 + 3x2 + 3x + 1 is divided by x + 1   [Ans : 2]

26.  If x3 + ax2+ bx + 6 has (x - 2) as a factor and leaves remainder 3, when divided by (x – 3), find a and b      Ans[a = -3, b = -1]

27.  For what value of a is 2x3 + ax2 + 11x + a + 3 divided by 2x - 1     Ans[a = -7 ]

1. Very very nice assigament, thank ku so much.