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CBSE 9th Standard Mathematics Polynomials Important Questions And Answers Study Material - QB365 Set D

Published on: 29/10/2025

Question Set D

CBSE 9th Standard Mathematics Polynomials Important Questions And Answers Study Material - QB365

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Std: 9th Standard CBSE - Mathematics

chapter: Polynomials - Important Questions And Answers

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Questions:

Generic 2

35 x 2 = 70

Evaluate: 249\(\times\) 251 by using an identity.

For what value of k, is the polynomial p(x)=2x³-kx²+3x+10 exactly divisible by (x+2)?

Show that (x-1) is a factor of the polynomial f(x) = 2x³-3x²+7x-6

Find the value of 'a' for which (x-1) is a factor of the polynomial a²x³-4ax+4a-1.

Find the value of k, if x-2 is a factor of p(x) = x²+kx+2k.

If the point (2k-3, k+2) lies on the graph of the equation 2x+3y+15=0, find value of k.

A part of monthly expense of a family on milk is fixed which is Rs 700 and the remaining varies with the quantity of milk taken extra at the rate of Rs 25 per litre. taking the quantity of milk required extra as x litre and total expenditure on milk is rs y. Write a linear equation representing the above information.

Write the coordinates of the point,
(i) whose ordinate is - 5 and which lies on Y-axis
(ii) which lies on X and Y-axes both.
(iii) whose abscissa is - 3 and which lies on X-axis.

Express the linear equation 5 = 2y in the form ax + by + c = a and indicate the values of a, b and c. Also, give the geometrical representation of above equation in two variables.

10.

Check whether x = 5 and y = 2 is a solution of following equations or not.
(i) 5x- + 2y = 7 (ii) 5x + y = 7

11.

Factorise
(i) 2x²-7x-15
(ii) 84-2r-2r²

12.

Find the zeroes of the polynomial
p(x)=(x-2)²-(x+2)².

13.

Find p(0),p(1) and p(-2) for the following polynomials.
(i) p(x) = 10x-4x²-3
(ii) p(y) = (y+2)(y-2)

14.

What are the factors of the polynomial p(x) = X²+ X- 6?

15.

Factorise the following by using factor theorem.
x²+5x-66

16.

Factorise the following by using splitting the middle terms.
(ii) 3x²+7x-6

17.

Factorise the following by using splitting the middle terms.
x²+2x=3

18.

If f(t) = 4t² - 3t + 6, find
(i) f(4).
(ii) f(-5).

19.

If p (x) = 4x⁵- 3x⁴-5x³ + X² - 8, then find p (-1).

20.

If f(x) = X - 9, find the value of f(x) - f(-x).

21.

The monthly hostel charges for Apeksha comprises Rs 1000 per month as fixed boarding charges and remaining charges at the rate of Rs 50 per day (for the number, for which the food has been availed by Apeksha).
(i) Form a linear equation in two variables to represent the above situation.
(ii) Find two solutions possible for equation formed.
(iii) What are the monthly charges to be paid by Apeksha who availed meals for 21 days in a given month?

22.

The cost of a toy telephone is the same as cost of 4 balls. Express the statement as a linear equation in two variables. Also, find the cost of getting 2 toy telephones packed, if cost of 1 ball is Rs 5 and there is a fixed cost of 2 for packing a toy telephone.

23.

Two points with coordinates (4, 3) and (4, - 2) lie on a line, parallel to which axis?

24.

Find the point which two coordinate axes meet

25.

If p = 14. find the degree of the polynomial g(x) = (p - X)³ + 14.

26.

Find the coefficient of x in the expansion of (x + 3)³

27.

(i) Plot the points A(-5,3), B(3,3), C(3,0) and D(-5,0).
(ii) Name the figure ABCD.
(iii) Find the ratio of areas of two parts of ABCD in the Ist quadrant and IInd quadrant.

28.

(i) Plot the points A(-5,-2), B(1,-2), C(6,4) and D(o,4).
(ii) Join the points to get AB, BC, CD and DA.Name the figures so obtained.

29.

If \(x+\frac{1}{x}=3\) then find \(x^3+\frac{1}{x^3}\)

30.

if x-y=2 and xy=15, find \(x^2+y^2\) and \(x^2-y^2\)

31.

Find the product of \((3x+2y)(3x-2y)(9x^2+4y^2)\).

32.

If a+b=10 and \(a^2+b^2=58,\) find the value of \(a^3+b^3\)

33.

Factorise: \(216x^3+\frac{1}{125}\)

34.

Factorise:\(x^4y^4-256z^4\)

35.

Factorise \(x^3-22x^2+142x-120\)

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Answers Key:

Generic 2

35 x 2 = 70

249\(\times\)251 =(250-1)(250+1)
= (250)²-(1)²
= 62500-1
= 62499.

(x+2) is a factor of p(x)=2x³-kx²+3x+10
\(\Rightarrow\) p(-2)=0
\(\Rightarrow\) 2(-2)³-k(-2)²+3(-2)+10=0
\(\Rightarrow\) -16-4k-6+10=0
k = -3

f(x)=2x³-3x²+7x-6
Put, x-a = 0 or x=1 in f(x)
Thus, f(1)=2\(\times\)1³-3\(\times\)1²+7\(\times\)1-6
= 2-3+7-6 = 0
Hence, (x-1) is a factor of f(x).

(x-1) is a factor of f(x) = a²x³-4ax+4a-1
f(1) = 0
\(\Rightarrow\) a²(1)³-4a(1)+4a-1 = 0
\(\Rightarrow\) a²-4a+4a-1 = 0
\(\Rightarrow\) a²-1 = 0
\(\Rightarrow\) a = \(\pm \) 1

(x-2) is a factor of f(x) = x²+kx+2k
f(2) = 0
\(\Rightarrow\) (2)²+k(2)+2k = 0
\(\Rightarrow\) 4 + 2k + 2 = 0
\(\Rightarrow\) 4+4k = 0
\(\Rightarrow\) k = -1

Putting x=2k-3, y=k+2 in 2x+3y=15=0, we get
2(2k-3)+3(k+2)+15=0
\(\Rightarrow 4k-6+3k+6+15=0\)
\(\Rightarrow k=\frac{-15}{7}\)

According to equation,
700+25x=y
25x-y+700=0

(i) (0, -5) (ii) (0,0) (iii) (-3,0)

0.x+2.y-5=0;a=0,b=2,c=-5

10.

(i) No
(ii) No

11.

(i) 2x²-7x-15=2x²-10x+3x-15 [by splitting the middle term]
=2x(x-5)+3*x-5)=(2x+3)(x-5)
(ii) 84-2r-2r²=-2(r²+r-42)
=-2(r²+7r-6r-42)
=-2[r(r+7)-6(r+7)]
-2(r-6)(r+7)=2(6-r)(r+7)

12.

Given, p(x)=(x-2)²-(x+2)²
=[(x-2)+(x+2)][(x-2)-(x+2)] \(\left[ \because { a }^{ 2 }-{ b }^{ 2 }=(a+b)(a-b) \right] \)
=(x-2+x+2)(x-2-x-2)
=(2x)(-4)=-8x
For finding the zeroes, put p(x)=0
\(\therefore\) -8x=0 \(\Rightarrow\) x=0
Hence, the required zero of p(x) is zero.

13.

Given polynomial is p(x)=10x-4x²-3 ...(i)
On putting x=0,1 and -2 respectively in Eq. (i), we get
p(0)=10(0)-4(0)²-3=0-0-3=-3
p(1)=10(1)-4(1)²-3
=10-4-3=10-7=3
and p(-2)=10(-2)-4(-2)²-3
=-20-4 x 4-3
=-20-16-3=-39
Hence, the values of p90),p(1) and p(2) are respectively -3,3 and -39.
(ii) Given polynomial p(y)=(y+2)(y-2) ...(i)
On putting x=0,1 and -2 respectively in Eq. (i), we get
p(0)=(0+2)(0-2)=-4
p(1)=(1+2)(1-2)
=3 x (-1)=-3
p(-2)=(-2+2)(-2-2)
=0(-4)=0
Hence the value of p(0), p(1) and p9-2) are respectively -4,-3 and 0.

14.

(x+3),(x-2)

15.

(x+11)(x-6)

16.

(x+3)(3x-2)

17.

(x-2)(x-2)

18.

(i) 58
(ii) 121

19.

-9

20.

f(x) - f(-x) = (x - 9)- (-x - 9) = 2x

21.

(i) Let the number of days for which the food has been availed by Apeksha be x and total charges be y.
Then, 1000 + 50x = y \(\Rightarrow\)y - 50x = 1000
(ii) (1200,4)(1300,6)
(iii) For 21 days, put x = 21, then y = 1000+ 50x 21 = 2050

22.

Let the cost of a toy telephone and a ball be Rs x and Rs y, respectively. Then, x = 4y As y = 5, therefore 2x = 4 x 5 = 20.
\(\therefore\) Cost of packing of two telephone toys = 2x + 4 = 20 x 2 + 4 = 44

23.

On joining the points (4, 3) and (4, -2) by a straight line, we get the line which is parallel to Y·axis.

24.

Origin

25.

g(x) = (3- x)³ + 14 = 27 - 2 - 27x + 9x² + 14 =-x³ + 9x² - 27x + 41
= 3

26.

(x + 3)³ = x³ + 27 + 9x² + 27x
\(\Rightarrow \) Coefficient of x is 27

27.

(i)

(ii) Rectangle
(iii) 3 : 5