Question Set A

cbse 10th Standard Maths Polynomials Important Questions And Answers Study Material - QB365

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Questions:

Generic 2

5 x 2 = 10

Identify the type of the polynomials given below:
\(p(v)=\sqrt { 3v^{ 4 } } -\frac { 2 }{ 3 } v+7\)

Identify the type of the polynomial given below (on the basis of degree).
3+2z+4z⁴

Graph of a quadratic polynomial meet X-axis at .......... points.

Is x=-4, a solution of the equation 2x²+5x-12=0?

Identify the type of the polynomials given below:
\(f(p)=3-p^{ 2 }+\sqrt { 7 } p\)

Generic 5

4 x 5 = 20

If the zeroes of the polynomial ax² + bx + b = 0 are in the ratio m : n, then find the value of \(\frac{\sqrt{m}}{\sqrt{n}}+\frac{\sqrt{n}}{\sqrt{m}}\)

If the polynomial f(x)=3x⁴-9x³+x²+15x+k is completely divisible by 3x²-5, then find the value of k and hence the other two zeroes of the polynomial.

If one zero of a polynomial 2x²+x²-7x-6 is 2, then find all the zeroes.

Find the zeroes of quadratic polynomial y²+92y+1920.

Generic 3

6 x 3 = 18

10.

Find the zeroes of the quadratic polynomial x² - 15 and verify the relationship between the zeroes and the coefficients of the polynomial.

11.

Find the zeroes of the following polynomial by factorisation method and verify the relations between the zeroes and their coefficients \(\frac{-2}{\sqrt{3}}, \frac{3}{4 \sqrt{3} / 2}\)

12.

Which of the following is not the graph quadratic polynomal?

13.

If (x + a) is a factor of two polynomials x² + px + q and x² + mx + n, then prove that
a = \({\frac{n -q }{m -p } }\)

14.

The central and state government of India allotted relief fund to help the families whose suffered from Corona virus. Both governments define the fund in the form of an expression p(x) = 6x³ + 11x² - 3x - 2, where x is a contribution amount by the state government. Some part of this fund spend on their medicine Rs (3x + 1) and treatment Rs(x + 2).
(i) Find the rest of the amount, which is used for their children education.
(ii) If state government contribute the fund of 1 million, how much amount contribute by the central government?
(iii) Suppose state government released the tund of 50 lakh and central government also contribute some amounts, how much amount spend in medicine, treatment and children education?

15.

Find the zeroes of the following quadratic polynomial and verify the relationship between the zeroes and their coefficients. \(q(x)=\sqrt { 3x^{ 2 } } +10x+7\sqrt { 3 } \)

Multiple Choice Question

20 x 1 = 20

16.

The quadratic polynomial, the sum of whose zeroes is -5 and their product is 6, is

(a)

x² + 5x + 6

(b)

x² - 5x + 6

(c)

x² - 5x - 6

(d)

-x² + 5x + 6

17.

If the zeroes of the quadratic polynomial x² + (a + 1)x + b are 2 and -3, then

(a)

a = -7 and b = -1

(b)

a = 5 and b = -1

(c)

a = 2 and b = -6

(d)

a = 0 and b = -6

18.

In the given figure, graph of a polynomial f(x) is shown. The number of zeroes of polynomial f(x) is

(a)

(b)

(c)

(d)

19.

Find the zeroes of the quadratic polynomial y² - 3y + 2 with the help of the graph.

(a)

1,-2

(b)

\(\frac{-1}{4},\frac{3}{2}\)

(c)

6,-1

(d)

1,2

20.

If one of the zeroes of the cubic polynomial x³ + ax² + bx + c is -1, then the product of the other two zeroes is

(a)

b - a + 1

(b)

b - a -1

(c)

a - b + 1

(d)

a - b - 1

21.

If one of the zeroes of the quadratic polynomial (k - 1)x² + kx + 1 is -3, then the value of k is

(a)

\(\frac{4}{3}\)

(b)

\(\frac{-4}{3}\)

(c)

\(\frac{2}{3}\)

(d)

\(\frac{-2}{3}\)

22.

If α, β, γ be the zeros of the polynomial p(x) such that α+ β+ γ = 3 , αβ+ βγ+ γα = -10 and αβγ = -24 then p(x) is

(a)

x³ – 3x² – 10x – 24

(b)

x³ + 3x² – 10x + 24

(c)

x³ + 3x² + 10x – 24

(d)

x³ – 3x² – 10x + 24

23.

If one zero of the polynomial x²+ kx+18 is double the other zero then k =?

(a)

±3

(b)

(c)

(d)

±9

24.

When the polynomial f(x) = 4x³ + 8x² + 8x + 7 is divided by the polynomial g(x) = 2x² – x + 1, the quotient and the remainder are

(a)

Quotient = 2x – 5, Remainder = 11x

(b)

Quotient = 2x + 5, Remainder = 11x + 2

(c)

Quotient = x² – 5, Remainder = 15

(d)

Quotient = x -5, Remainder = 13

25.

value of ‘a’ so that (x + 6) is a factor of the polynomial x³ + 5x² – 4x + a

(a)

(b)

(c)

(d)

26.

If one root of polynomial equation ax²+bx+c=0 be reciprocal of other, then

(a)

a = c

(b)

a = 0

(c)

b = 0

(d)

b = c

27.

The number of zeroes for the polynomial y = p (x) from the given graph is :

(a)

(b)

(c)

(d)

28.

Sum and the product of zeroes of the polynomial x² +7x +10 is

(a)

7 and -10

(b)

-7 and 10

(c)

10/7 and -10/7

(d)

7/10 and -7/10

29.

If “1” is a zero of the polynomial P(a) = x²a² – 2xa + 3x – 2 , then x =

(a)

-2

(b)

-2, 0

(c)

+2, 2

(d)

30.

If sum of the squares of zeros of the quadratic polynomial f(x) = x² – 8x + k is 40, find the value of k.

(a)

(b)

(c)

-14

(d)

-12

31.

If one zero of 2x² – 3x + k is reciprocal to the other, then the value of k is :

(a)

-3

(b)

(c)

-3/2

(d)

-2/3

32.

The graph y= p(x) is shown below. How many zeroes does the polynomial p(x) have?

(a)

(b)

(c)

(d)

33.

The graph of y = p(x) is given below. The number of zeroes of p(x) are

(a)

(b)

(c)

(d)

34.

The graph of the polynomial f(x) = 2x – 5 crosses the X-axis at the point

(a)

(1, -3)

(b)

(5/2, 0)

(c)

(0, 0)

(d)

(4, 3)

35.

If the degree of the dividend is 5 and the degree of the divisor is 3, then the degree of the quotient will be

(a)

(b)

(c)

(d)

-2

Case Study Questions

3 x 4 = 12

36.

One day, due to heavy storm an electric wire got bent as shown in the figure. It followed some mathematical shape of curve. Answer the following questions below.

(i) How many zeroes are there for the polynomial (shape of the wire)

(a) 2

(b) 3

(c) 4

(d) 5

(b) Find the zeroes of the polynomial.

(a) 2, 0, -2

(b) 2, -2, -5

(c) -2, 2, -5.5

(d) None of these

37.

Shruti is very good in painting. So she thought of exhibiting her paintings in which she want to display her latest painting which is in the form of a graph of a polynomial as shown below:

Based on the above information, answer the following questions.
(i) The number of zeroes of the polynomial represented by the graph is

(a) 1

(b) 2

(c) 3

(d) can't be determined

(ii) The sum of zeroes of the polynomial represented by the graph is

(a) -4

(b) -3

(c) 2

(d) -5

(iii) Find the value of the polynomial represented by the graph when x = 0.

(a) -6

(b) -8

(c) 6

(d) 8

(iv) The polynomial representing the graph drawn in the painting by Shruti is a

(a) quadratic polynomial	(b) cubic polynomial
(c) bi-quadratic polynomial	(d) linear polynomial

(v) The sum of product of zeroes, taken two at a time, of the polynomial represented by the graph is

(a) 2

(b) 3

(c) -2

(d) -3

38.

Priya visited a temple in Gwalior. On the way she sees the Agra Fort. The entrance gate of the fort has a shape of quadratic.polynomial (parabolic). The mathematical representation of the gate is shown in the figure .

Based on the above information, answer the following questions.
(i) Find the zeroes of the polynomial represented by the graph.

(a) -1,3

(b) 1,3

(c) 1,-3

(d) 0,1

(ii) What will be the expression for the polynomial represented by the graph?

\((a) x^{2}+4 x-5\)

\((b) x^{2}-4 x+5\)

\((c) -x^{2}+4 x-3\)

\((d) x^{2}+5 x-4\)

(iii) What will be the value of polynomial, represented by the graph, when x = 4?

(a) -2

(b) 3

(c) -3

(d) 2

(iv) If one zero of a polynomial p(x) is 7 and product of its zeroes is -35, then p(x) =

\((a) -x^{2}+{2 x}+35 \)

\((b) x^{2}+2 x+35\)

\((c) x^{2}+12 x-35\)

\((d) x^{2}-12 x-35\)

(v) If the gate is represented by the polynomial \(-x^{2}+5 x-6\) then its zeroes are

(a) 2,-3

(b) 2,3

(c) -2,3

(d) -2,-3

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Answers Key:

Generic 2

5 x 2 = 10

Biquadratic

Here, the highest power of z in the given polynomial is 4, so it is a biquadratic polynomial.

two

Yes

Quadratic

Generic 5

4 x 5 = 20

Let the zeroes of the given polynomial ax² + bx +b be m \(\alpha\) and n \(\alpha\)
\(\therefore\) Sum of zeroes
m\(\alpha\)+n\(\alpha\) = \( -\frac{Coefficient of x}{Coefficient of x^{2}}\) = \( -\frac{b }{a }\)
and product of zeroes,
m \(\alpha\)x n\(\alpha\) = \( \frac{Coefficient of x}{Coefficient of x^{2}}\) = \( -\frac{b }{a}\)

\(-\sqrt{\frac{b}{a } }\)

On dividing f(x) by 3x²-5, we get
Quotient=x²-3x+2 and remainder =k+10.
Since, f(x) is exactly divisible by 3x²-5, so remainder =0.
⇒ k+10=0⇒ k-10
Now, for other zeroes, put x²-3x+2=0
⇒ (x2)(x1)=0⇒x=1,2

\(-1,\quad -\frac { 3 }{ 2 } \)

Let p(y)=y²+92y+1920=(y+32)(y+60)
Now, for zeroes of p(y), put p(y)=0
Zeroes y=-32, -60

Generic 3

6 x 3 = 18

10.

Let f(x) = x² - 15
\(\Rightarrow f(x)=(x-\sqrt{15})(x+\sqrt{15}) \quad\left[\because a^2-b^2=(a-b)(a+b)\right]\)
On putting f(x) = 0, we get
\(\begin{aligned} & (x-\sqrt{15})(x+\sqrt{15})=0 \\ \end{aligned}\)
\(\begin{aligned} & \Rightarrow x-\sqrt{15}=0 \text { or } x+\sqrt{15}=0 \\ \end{aligned}\)
\(\begin{aligned} & \Rightarrow \quad x=\sqrt{15} \text { or } x=-\sqrt{15} \end{aligned}\)
Thus, the zeros of given polynomial are \(\alpha=\sqrt{15}\) and \(\beta=-\sqrt{15}\)
Verification
Here, the sum of zeroes, \(\alpha+\beta=\sqrt{15}-\sqrt{15}=0\)
\(=-\frac{0}{1}=-\frac{\text { Coefficient of } x}{\text { Coefficient of } x^2}\)
and product of zeroes,
\(\begin{aligned} \alpha \beta=\sqrt{15} \times(-\sqrt{15}) & =-15=\frac{-15}{1} \end{aligned}\)
\(\begin{aligned} & =\frac{\text { Constant term }}{\text { Coefficient of } x^2} \end{aligned}\)
So, the relationship between the zeroes and the coefficients is verified.

11.

= \(\frac{-2}{\sqrt{3}}, \frac{3}{4 \sqrt{3} / 2}\)

12.

(d) We know that, for any quadratic polynomial ax² +bx +c, \(a \neq 0\), the graph of the corresponding equation y = {[x² +bx + c has one of the two shapes ei ther open upwards like or open downwards like depending on whether a> 0 or a < 0. So, option (d) cannot be possible. Also, the curve of a quadratic polynomial crosses the X-axis atrnost two points, but in option (d) the curve crosses the X-axis at three points, so it does not represent the quadratic polynomial.

13.

If x +a is a factor of x² +px +q and x²+mx +n
then (-a)² +p(-a)+ q = (-a)² +m(-a)+n [\( \because\)x = -a]

14.

(i) Rs (2x -1)
(ii) 12 million
(iii) Medicine = 151 lakh
Treatment = 52 lakh
Children Education = 99 lakh

15.

We have, \(q(x)=\sqrt { 3x^{ 2 } } +10x+7\sqrt { 3 } \)
On splitting the middle term, i.e.10x into two parts, we get
\(q(x)=\sqrt { 3x^{ 2 } } +3x+7+7\sqrt { 3 } \)
\(=\sqrt { 3x } (x+\sqrt { 3 } )+7(x+\sqrt { 3 } )\)
\(=(x+\sqrt { 3 } )\sqrt { 3x } +7)\)
For the zeroes of q(x), put q(x)=0
\(\therefore \quad (x+\sqrt { 3 } )\sqrt { 3x } +7)=0\)
\(\Rightarrow \quad x+\sqrt { 3 } =0\) and \(\sqrt { 3x } +7=0\)
\(\Rightarrow \quad x=-\sqrt { 3 } \) and \(x=\frac { -7 }{ \sqrt { 3 } } \)
Hence, the zeroes of q(x) are \(\alpha =-\sqrt { 3 } \) and \(\beta =\frac { -7 }{ \sqrt { 3 } } \)
Verification
Here, \(\alpha +\beta =-\sqrt { 3 } +\left( \frac { -7 }{ \sqrt { 3 } } \right) =\frac { -3-7 }{ \sqrt { 3 } } =-\frac { 10 }{ \sqrt { 3 } } \)
\(=-\frac { Coefficient\quad of\quad x }{ Coefficient\quad of\quad x^{ 2 } } \)
and \(\alpha \beta =-\sqrt { 3 } \times \left( \frac { -7 }{ \sqrt { 3 } } \right) =\frac { 7\sqrt { 3 } }{ \sqrt { 3 } } =-\frac { Constant \ term }{ Coefficient \ of \ x^{ 2 } } \)
Hence, the relations between the zeroes and the coefficients of the polynomial is verified.

Multiple Choice Question

20 x 1 = 20

16.

(a)
x² + 5x + 6

17.

(a)
a = -7 and b = -1

18.

(d)
2

19.

(d)
1,2

20.

(a)
b - a + 1

21.

Given that, one of the zeroes of the quadratic polynomial say p(x) = (k -1)x² + kx + 1 is -3, then
p(-3)=0
\( \Rightarrow\) (k -1)(-3)2 + k(-3) + 1= 0

22.

(d)
x³ – 3x² – 10x + 24

23.

(d)
±9

24.

(b)
Quotient = 2x + 5, Remainder = 11x + 2

25.

(b)
12

26.

(a)
a = c

27.

(d)
1

28.

(b)
-7 and 10

29.

30.

(b)
12

31.

(b)
2

32.

(b)
4

33.

34.

(b)
(5/2, 0)

35.

(b)
2

Case Study Questions

3 x 4 = 12

36.

(i) (c): 3
(ii) (a): 2, 0, -2

37.

(i) (c) :Since the graph intersect the x-axis at 3 points, therefore the polynomial has 3 zeroes.
(ii) (d): Clearly the graph intersect the x-axis at x = -4, x = -2 and x = 1, therefore the zeroes are -4, -2 and 1. Now, the sum of zeroes = -4 - 2 + 1 = -5
(iii) (b): From the graph, it can be seen that When x = 0, then y = -8.
(iv) (b): Since there are 3 zeroes, therefore the graph represents a cubic polynomial.
(v) (a): The sum of product of zeroes taken two at a time = (-4)(-2) + (-2)(1) + (1)(-4) = 8 - 2 - 4 = 2

38.

(i) (b): Since, the graph of the polynomial intersect the x-axis at x = 1, 3 therefore required zeroes of the polynomial are 1 and 3.
(ii) (c)
(iii) (c): Let \(f(x)=-x^{2}+4 x-3\)
then \(f(4) =-4^{2}+4 \times 4-3 \)
\(=-16+16-3=-3\)
(iv) (a): Clearly, other zero \(=\frac{-35}{7}=-5\)
Thus, the zeroes are 7 and -5. From the options, 7 and -5 satisfies only \(-x^{2}+2 x+35\)
\(\text { So, } p(x)=-x^{2}+2 x+35\)
(v) (b): Let \(p(x)=-x^{2}+5 x-6\)
For zeroes, consider p(x) = 0
\(\begin{array}{l} \Rightarrow \quad-x^{2}+5 x-6=0 \Rightarrow x^{2}-5 x+6=0 \\ \Rightarrow \quad x^{2}-3 x-2 x+6=0 \\ \Rightarrow \quad(x-3)(x-2)=0 \Rightarrow x=3,2 \end{array}\)
Thus, the required zeroes are 3 and 2.

cbse 10th Standard Maths Polynomials Important Questions And Answers Study Material - QB365 Set A

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Std: 10th Standard CBSE - Maths

chapter: Polynomials - Important Questions And Answers

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