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Class 10 Maths Extra Questions for Polynomials





Given below are the Class 10 Maths Extra questions for Polynomials with amswers. These are important questions for the examination
Questions are of below format
a. Match the column
b. Multiple choice questions
c. True and False
d. Factorize
e. Division
f. Short answer type
g. Long Answer type
The questions are based on below concepts a. Finding Zero's Questions
b. Short Answers Questions
c. Word Problems
d. Graph Questions

Match the column

A.

Answer



B.

Answer



C.
P(x)=5x3 -3x2+7x+2

Answer



D.
Graph of polynomial
Number of Zeros
Class 10 Maths Important Questions  for Polynomials
  0

  1

2

  3

  4

  5

  6
Class 10 Maths Important Questions  for Polynomials
  7

Answer

a. it cuts the x-axis at two points ,so 2 zeroes
b. it cuts the x-axis at four points ,so 4 zeroes
c. Since it does not cut the axis, so  0 zeroes
d. it cuts the x-axis at 1 points ,so 1 zero’s
e. it cuts the x-axis at 1 points ,so 1 zero’s
f. Since it does not cut the axis, so  0 zeroes
g. Since it does not cut the axis, so  0 zeroes
h.   it cuts the x-axis at two points ,so 2 zeroes


E.
Graph of polynomial
Type of polynomial

  Linear polynomial

  Quadratic polynomial

 
Cubic polynomial

  Constant polynomial

 
 
 

Answer

a. Quadratic as parabola
b. Three zeroes,So cubic polynomial
c. Constant value polynomial
d. Linear polynomial
e. One zeroes but not straight line. So no appropriate match found
f. Quadratic as parabola
g. Quadratic as parabola
e. Cubic as has three zeroes ,two of them same



Multiple choice Questions


Question 1
Find the remainder  when x4+x3-2x2+x+1 is divided by x-1
a.1
b.5
c.2
d. 3

Answer

Answer is ( c)
Let P(x) =x4+x3-2x2+x+1
Remainder when divide by x-1
P(1) = 1+ 1-2+1+1=2


Question 2
Which of these identities is not true?
Class 10 Maths Important Questions  for Polynomials

Answer

Answer is (d)


Question 3
If a and b are the zeroes of the polynomial  x2-11x +30, Find the value of a3 + b3
a.134
b.412
c.256
d.341

Answer

a3 + b3= (a+b) (a2+b2-ab)=(a+b) {(a+b)2 -3ab}
Now a+b=-(-11)/1=11
ab=30
So a3+b3=11( 121 -90)=341


Question 4
S(x) = px2+(p-2)x +2. If  2 is the zero of this polynomial,what is the value of p
a.-1
b.1/2
c. -1/2
d. +1

Answer

S(2)=4p+0+2=0  => p=-1/2


Question 5
if the zeroes of the quadratic equation are 11 and 2 ,what is expression for quadratic
a. x2-13x+22
b. x2-11x+22
c. x2-13x-22
d. x2+13x-22

Answer

P(x) =(x-11)(x-2)


Question 6
p(x) = x4 -6x3 +16x2 -25x +10
q(x) = x2-2x+k
It is given
p(x) = r(x) q(x) + (x+a)
Find the value of k and a
a.2,-2
b. 5 ,-5
c. 7,3
d. 3,-1

Answer

Dividing p(x) by q(x) ,we get the remainder
(2k-9)x -(8-k)k +10
Comparing this with (x+a)
We get
K=5 and a=-5


Question 7
A cubic polynomial is given below
S(x) =x3 -3x2+x+1
The zeroes of the polynomial are given as (p-q) ,p  and (p+q). What is the value p and q
a. p=1,q=√2 or -√2
b. p=1,q=2 or -2
c. p=1,q=1 or -1
d. None of these

Answer

(a)


Question 8
If the zeroes of the quadratic polynomial ax2 + bx + c, c ≠ 0 are equal, then
a. c and a have opposite signs
b. c and b have opposite signs
c. c and a have the same sign
d. c and b have the same sign

Answer

Given that, the zeroes of the quadratic polynomial ax2 + bx + c , c ≠ 0 are equal
Value of discriminant (D) has to be zero
b2– 4ac = 0
b2 = 4ac
Since L.H.S. (b2) can never be negative
R.H.S. also can never be negative.
a and c must have same sign


Question 9
What are the zeroes of the Polynomial $p(x)=6x^2 - 7x -3$
a. (3/2) , (-1/3)
b. (3/2) , (1/3)
c. (-3/2) , (1/3)
d. (-3/2) , (-1/3)

Answer

$p(x)=6x^2 - 7x -3$
$p(x) = 6x^2 -9x + 2x -3 = 3x(2x -3) + 1(2x-3)=(3x+1)(2x-3)
So roots are (3/2) , (-1/3)
Hence option(a) is correct


Question 10
What are the zeroes of the Polynomial $p(x)=x^2 + 7x +10$
a. (2) , (-5)
b. (-2) , (5)
c. (2) , (5)
d. (-2) , (-5)

Answer

"> $p(x)=x^2 + 7x +10$
$p(x) = x^2 + 5x +2x +10 = x(x +5) + 2(x+5)=(x+2)(x+5)
So roots are (-2) , (-5)
Hence option(d) is correct


True or False statement

  1. P(x) =x-1 and g(x) =x2-2x +1 .  p(x) is a factor of g(x)
  2. The factor of 3x2 -x-4 are  (x+1)(3x-4)
  3. Every linear polynomial has only one zero
  4. Every real number is the zero’s of zero polynomial
  5. A binomial may have degree 6
  6. 1,2 are the zeroes of x2-3x+2
  7. The degree of zero polynomial is not defined
  8.  Graph of polynomial (x2-1) meets the  x-axis at one point
  9. Graph of constant polynomial never meets x axis

Answer

  1. True, as g(1)=0
  2. True, we can get this by split method
  3. True
  4. True
  5. True , example x6 +1
  6. True
  7. True
  8. False as it meets at two points
  9.  True


Factorize following

  1. x2 +9x+18
  2. 3x3 -x2-3x+1
  3. x3-23x2+142x-120
  4. 1+8x3

Answer

a.(x+6)(x+3)
b.(3x-1)(x-1)(x+1)
c.(x-1)(x-10)(x-12)
d. (2x+1)(4x2-2x+1)


Division of polynomial

s(x) =r(x) s(x) + w(x)
Find the value of r(x) and w(x) in each case
a. p(x) =x4+x3+2x2+3x+4
s(x) =x+2
b. p(x) =x4+4
  s(x)=x2+x+1

Answer

a. r(x)=x3-x2 +4x-5   w(x)=14
b. r(x)=x2-x  w(x) =x+4


Short Answer type

Question 1
Find a quadratic polynomial whose zeroes are $5 + \sqrt {2}$ and $5 - \sqrt {2}$

Answer

If $\alpha$ and $\beta$ are the zeroes of the polynomials
$x^2 - (\alpha + \beta) + \alpha \beta$
Here $\alpha + \beta =5 + \sqrt {2} + 5 - \sqrt {2} =10$
$\alpha \beta =(5 + \sqrt {2})(5 - \sqrt {2}) = 25 -4 =21$
Therefore quadratic Polynomial is
$x^2 - 10x + 21$


Question 2
If a and b are zeroes of quadratic polynomial $kx^2 + 4x + 4$, find the value of k such that $(a+b )^2- 2ab= 24$.

Answer

Here $a + b= -\frac {4}{k}$
$ab=\frac {4}{k}$
Now $(a+b )^2- 2ab= 24$
$(-\frac {4}{k})^2 -2 \frac {4}{k}=24$
$16-8k=24k^2$ or $3K^2 +k-2=0$
$k=-1$ or $k=\frac {2}{3}$


Question 3
If one zero of $3x^2 - 4x + p$ is reciprocal to the other, then find the value of p

Answer

Let a be one of zero, then 1/a be the another zero
Now $a \times \frac {1}{a} =\frac {p}{3}$
or p=3


Question 4
If p and q are the zeroes of $p(x) = kx^2 - 3x + 2k$ and $p+q=pq$ then find the value of k.

Answer

Here $ p + q=\frac {3}{k}$
$pq=\frac {2k}{k}$
Now
$p+q=pq$
$\frac {3}{k} = 2$
$k= \frac {3}{2}$


Question 5
If a and b are zeroes of $x^2 - 6x + k$. What is the value of k, if $3a+2b= 20$?

Answer

Here $a + b= - \frac {-6}{1} = 6$ --(1)
$ab= k$ -(3)
Given $3a+2b= 20$ --(2)
Solving (1) and (2)
a=14 and b=-8
Now substituting these values in (3)
k=-112


Question 6
If one zero of the quadratic polynomial $x^2 + 3x + k$ is 2, then find the value of k.

Answer

Since 2 is zero of the quadratic polynomial
$2^2 + 3 \times 2 + k=0$
k=-10


Question 7
Find the zeroes of the quadratic polynomial $x^2 + 8x + 16$ and verify the relationship between the zeroes and the coefficients.

Answer

Now $x^2 + 8x + 16$
$= x^2 + 4x + 4x + 16 = x(x+4) + 4(x+ 4) = (x+4)^2$
So zeroes are -4 ,-4

Now if $k_1$ ,$k_2$ are the roots of quadratic polynomials $ax^2 + bx + c$, then relationship between them is given by
$k_1 + k_2 = -\frac {b}{a}$ and $k_1 k_2= \frac {c}{a}$

Now lets verify the relationship in the above polynomial
$-4 + (-4) = -\frac {8}{1}$
$-8=-8$

Also
$(-4) \times (-4) = \frac {16}{1}$
16=16


Question 8
Find the value of a and b, if they are the zeroes of polynomial x2 + ax + b.

Answer

Here
$a + b= -a$ or $2a + b=0$ -(1)
Also $ab= b$ or $a=1$
Substituting the value of a in (1)
b=-2
Hence a=1 and b=-2


Question 9
If m and n are the zeroes of the polynomial $3x^2 + 11x - 4$, find the value of $ \frac {m}{n} + \frac {n}{m}$

Answer

Here $m + n= = \frac {11}{3}$
$mn= \frac {-4}{3}$
Now
$ \frac {m}{n} + \frac {n}{m} = \frac {m^2 + n^2}{mn} = \frac {(m+n)^2 -2mn}{mn} = \frac {-145}{12}$


Question 10
Show that 2, -1 and 1/2 are the zeroes of the cubic polynomial
$p(x) = 2x^3 - 3x^2 - 3x + 2$


and then verify that the sum of the zeroes =-(Coeff of x2/Coeff of x3)


Question 11
Find the zeroes of the polynomial $f(x) = 4 \sqrt {3} x^2 + 5x - 2 \sqrt {3}$ , and verify the relationship between the zeroes and its coefficients.

Answer

$4 \sqrt {3} x^2 + 5x - 2 \sqrt {3}$
$= 4 \sqrt {3} x^2 + 8x -3x - 2 \sqrt {3}$
$= 4x (\sqrt {3}x + 2) - \sqrt {3}(\sqrt {3}x + 2)$
$= (4x - \sqrt {3})(\sqrt {3}x + 2)$


Question 12
If p and q are the zeroes of the polynomial $f(x) = x^2 - 5x + k$ such that $p-q= 1$, find the value of k.

Answer

Here $ p + q = 5$ -(1)
$pq= k$ -(2)
Given $p-q= 1$ - (3)
From (1) and (3)
p=3, q=2
Substituting these values in (2)
k=6



Long Answer type

Question 14
If a and b are the zeroes of the quadratic polynomial $f(x) = x^2 - 4x + 3$, find the value of $a^4 b^3 + a^3 b^4$.

Answer

Here $ a + b=-\frac {-4}{1} =4$
$ab=3$
$a^4 b^3 + a^3 b^4 = a^3b^3(a +b) = 27 \times 4 = 108$


Question 15 If a and b are the zeroes of the quadratic polynomial $f(x) = x^2 - px + q$, prove that
Class 10 Maths Extra Questions  for Polynomials

Answer

Here
$a + b =p$ and $ab=q$
Now LHS
$= \frac {a^2}{b^2} + \frac {b^2}{a^2}$
$= (\frac {a}{b} + \frac {b}{a})^2 -2$
$= ( \frac {a^2 + b^2}{ab})^2 -2 $
$=(\frac {(a+b)^2 - 2ab}{ab})^2 -2 = (\frac {p^2 -2q}{q})^2 -2$
$=\frac {p^4 +4q^2 -4p^2q}{q^2} -2 = \frac {p^4}{q^2} +4 - 4\frac {p^2}{q} -2$
$= \frac {p^4}{q^2} - 4\frac {p^2}{q} +2$
=RHS


Question 16
If two zeroes of the polynomial $f(x) =x^3 - 4x^2 - 3x + 12$ are √3 and -√3 then find its third zero.

Answer

If √3 and -√3 are zeroes of the polynomial, then $(x- \sqrt {3})(x+ \sqrt {3}) = x^2 -3$ should be factor of it.
$x^3 - 4x^2 - 3x + 12= x^2(x-4) -3(x -4) = (x^2 -3)(x-4)$
Clearly the other zero is 4


Question 17
If l and m are zeroes of the polynomial $p(x) = 2x^2 - 5x + 7$, find a polynomial whose zeroes are $2l+ 3$ and $2m+ 3$.

Answer

Here $l+m= \frac {5}{2}$
$lm= \frac {7}{2}$

Now the polynomial whose zeroes are $2l+ 3$ and $2m+ 3$ is given by
$x^2 - (2l+3 + 2m + 3)x +(2l+3)(2m+3)=x^2 -2(l+m)x -6x + 4lm + 6l + 6m +9$
$=x^2 -10x -6x + 14 + 6(l+m) +9$
$=x^2 -16x +14+15+9=x^2 -16x + 38$


Question 18
If p ,q and r are zeroes of polynomial $6x^3+ 3x^2 - 5x + 1$, then find the value of $\frac {1}{p} + \frac {1}{q} + \frac {1}{r}$.

Answer

Here $pq + qr+ pr=\frac {-5}{6}=- \frac {5}{6}$
$pqr= -\frac {1}{6}$
Now
$\frac {1}{p} + \frac {1}{q} + \frac {1}{r}$
$= \frac {pq+ qr + pr}{pqr} =5$


Question 19
If the zeroes of the polynomial $f(x) = ax^3 + 3bx^2 + 3cx + d$ are in A. P., prove that $2b^3 - 3abc + a^2d = 0$.

Answer

Given Roots of given polynomial is in AP
let p-q, p, and p+q are the roots of
$f( x) = ax^3 + 3bx^2 + 3cx + d$
Now we know that
$sum \; of \; roots = - \frac {coefficient \; of \; x^2} {coefficient \;of\; x^3}$
$p-q +p + p+q = - \frac {3b}{a}$
or $3p = - \frac {3b}{a}$
or $p = - \frac {b}{a}$ -----------(1)
Also,
$sum \; of \; products \; of \; two roots = \frac {coefficient \; of\; x}{coefficient \; of \; x^3}$
$ (p -q)\times p + p \times (p+q) + (p-q)(p+q)=\frac {3c}{a}$
$3p^2 - q^2 = \frac {3c}{a}$ ------------------(2)

Also,
$products \; of \; all \; roots = -\frac {constant }{coefficient \; of \; x^3}$
$(p-q)\times p \times (p+q) = -\frac {d}{a}$
$p^3 - pq^2 = -\frac {d}{a}$ ----------(3)

Substituting the value of p from (1) in (2), we get the value of q as
$ 3 (-\frac {b}{a})^2 - q^2 = \frac {3c}{a}$
or
$q^2 = 3 \frac {b^2}{a^2} - 3 \frac {c}{a}$ ----------(4)

Now that we have got the values of p and q, substituting these in equation (3)

$(- \frac {b}{a})^3 +\frac {b}{a}( 3 \frac {b^2}{a^2} -3\frac {c}{a}) = -\frac {d}{a}$

$- \frac {b^3}{a^3} + \frac {b}{a}(3 \frac {b^2}{a^2} -3\frac {c}{a} ) = -\frac {d}{a}$

$- \frac {b^3}{a^3} +3 \frac {b^3}{a^3} -3 \frac {bc}{a^2} = -\frac {d}{a}$

$2\frac {b^3}{a^3} -3 \frac {bc}{a^2} = -\frac {d}{a}$

$2b^3 -3abc + a^2d = 0$


Question 20
Obtain all zeroes of the polynomial $f(x) = 2x^4 - 2x^3 - 7x^2 + 3x + 6$, if its two zeros are √3/2 and -√3/2

Answer

If √(3/2) and -√(3/2) are zeroes of the polynomial, then $(x- \sqrt {3/2})(x+ \sqrt {3/2}) = 2x^2 -3$ should be factor of it.
$f(x) = 2x^4 - 2x^3 - 7x^2 + 3x + 6$
Lets use division algorithm to find the other factors
Class 10 Maths Extra Questions  for Polynomials with answers
So,
$2x^4 - 2x^3 - 7x^2 + 3x + 6= (2x^2 -3)(x^2 -x -2) = (2x^2 -3)(x^2 -2x + x-2) =(2x^2 -3)(x-2)(x+1)$
So other zeros are x=2 and x=-1


Question 21
The graphs of y = p(x) are given in below figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case. Polynomial graph questions

Answer

i. 1
ii. 2
iii. 3
iv 1
v. 4



Summary

This Class 10 Maths Extra Questions for Polynomials with answers is prepared keeping in mind the latest syllabus of CBSE . This has been designed in a way to improve the academic performance of the students. If you find mistakes , please do provide the feedback on the mail.



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Practice Question

Question 1 What is $1 - \sqrt {3}$ ?
A) Non terminating repeating
B) Non terminating non repeating
C) Terminating
D) None of the above
Question 2 The volume of the largest right circular cone that can be cut out from a cube of edge 4.2 cm is?
A) 19.4 cm3
B) 12 cm3
C) 78.6 cm3
D) 58.2 cm3
Question 3 The sum of the first three terms of an AP is 33. If the product of the first and the third term exceeds the second term by 29, the AP is ?
A) 2 ,21,11
B) 1,10,19
C) -1 ,8,17
D) 2 ,11,20



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