Introduction to Linear Polynomials Class 9 MCQ Maths Chapter 2

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MCQ on Introduction to Linear Polynomials Class 9

Class 9 Maths Introduction to Linear Polynomials MCQ

Choose the correct option from the given options:

Question 1.
Which of the following is a polynomial?
(a) z^{\(\frac{2}{3}\)} + 1
(b) √x + x²
(c) x² – xy
(d) x – \(\frac{3}{x^2}\)
Solution:
(c) x² – xy

Explanation:
An algebraic expression is called a polynomial if all degrees on variables occurring in that polynomial are whole numbers, i.e., 0. 1, 2, 3, 4, … .
Here, z^{\(\frac{2}{3}\)} + 1 , √x + x² and x – \(\frac{3}{x^2}\), have not all powers as whole numbers, while x² – xy has all powers as whole numbers. So, amongst the given options only x² – xy is a polynomial.

Question 2.
Which of the following is not a polynomial?
(a) x + x²
(b) x² + x³
(c) x³ – y³
(d) x² – \(\frac{1}{x^2}\)
Solution:
(d) x² – \(\frac{1}{x^2}\)

Explanation:
x² – \(\frac{1}{x^2}\) is not a polynomial as x² – \(\frac{1}{x^2}\) = x² – x^-2.
An algebraic expression is called a polynomial if all degrees on variables occurring in that polynomial are whole numbers, i.e., 0, 1, 2, 3, 4
Here, x² – \(\frac{1}{x^2}\) has a power -2. So, it is not a polynomial.

Question 3.
Which of the following polynomials is univariate polynomial?
(a) xz +1
(b) xy + yx + 2zx
(c) x² – xy
(d) x⁴ – 3x
Solution:
(d) x⁴ – 3x

Explanation:
The polynomial x⁴ – 3x has only one variable x, so it is a univariate polynomial. A polynomial having only one kind of letter-number (variable) is called a univariate polynomial.

Question 4.
Which of the following polynomials is a monomial?
(a) x
(b) x + y + 2z
(c) x² – 1
(d) x⁴ – 3x
Solution:
(a) x

Explanation:
A polynomial having only one term is called a monomial.

Question 5.
Which of the following polynomials is a monomial?
(a) x² + 3x + 10
(b) 7x + 3y + 2z
(c) x² – 3x²
(d) x² – 3x³ – 1
Solution:
(c) x² – 3x²

Explanation:
A polynomial having only one term is called a monomial. Here, the terms must be counted after writing the polynomial in simplest form in which there are no two like terms. In this case x² – 3x²= -2xz, which is a single term, so it is a monomial.

Introduction to Linear Polynomials MCQ Class 9

Question 6.
Which of the following polynomials is a binomial?
(a) 3x²
(b) x + 3y – 2x
(c) x² – x + y
(d) x⁴ – 3x² + 5
Solution:
(b) x + 3y – 2x

Explanation:
A polynomial having two terms is called a binomial. Here, after simplifying like terms we get x + 3y – 2x = 3y – x, which is a binomial.

Question 7.
Which of the following polynomials is a trinomial?
(a) 3x² + 3y -2xz
(b) x + 3y – 2x
(c) x² – x + y
(d) x⁴ – 3x⁴ + 5
Solution:
(c) x² – x + y

Explanation:
A polynomial having three terms is called a trinomial. Here, after simplifying like terms in all alternatives provided, we get x² – x + y, is a trinomial.

Question 8.
Which of the following polynomials has degree zero?
(a) x + 2
(b) 7
(c) 3t
(d) 2 + x + x²
Solution:
(b) 7

Explanation:
Any constant polynomial has degree zero as any constant polynomial can be written in the form of (constant) × x° or (constant) × (variable)0 or (constant) × (letter – number)0. Here, we can write 7 as 7x°, which means its degree is zero.

Question 9.
Which of the following polynomials has degree one?
(a) x⁴ + 2
(b) 7
(c) 3t
(d) 2 + x + x²
Solution:
(c) 3t

Explanation:
Here, 3t = 3t¹, so it has degree one.

Question 10.
Which of the following polynomials is a linear polynomial?
(a) x² – 2x + 1
(b) 2x + 1
(c) x³ – 12x + 7
(d) x⁴
Solution:
(b) 2x + 1

Explanation:
Here, 2x + 1 has degree 1, so it is a linear polynomial. A polynomial having degree equal to one is called a linear polynomial as its graph is a straight line.

Question 11.
The degree of the polynomial x³ – 3x² + 4 is :
(a) 0
(b) 1
(c) 2
(d) 3
Solution:
(d) 3

Explanation:
The polynomial x³ – 3x² + 4 has highest power of variables equals to 3, so it is a polynomial of degree 3. Degree 3 polynomials are called cubic polynomials.

Question 12.
The coefficient of x in the polynomial 3x³ – 2x² + x – 5 is :
(a) 3
(b) -2
(c) 1
(d) -5
Solution:
(c) 1

Explanation:
The polynomial 3x³ – 2x² + x – 5 has term containing letter – number x as + x. So, its coefficient is 1.

Question 13.
The coefficient of x² in the polynomial 3x³ – 2x² + x – 5 is :
(a) 3
(b) -2
(c) 1
(d) -5
Solution:
(b) -2

Explanation:
The polynomial 3x³ – 2x² + x – 5 has term containing letter – number x² as – 2x². So, its coefficient is -2.

Question 14.
The constant term in the polynomial 5x² – 2x – 4 is :
(a) 5
(b) -2
(c) 1
(d) -4
Solution:
(d) -4

Explanation:
The polynomial 5x² – 2x – 4 has term containing no letter – number as – 4. So, its constant term is -4.

Question 15.
Alinear polynomial has degree equals to:
(a) 1
(b) -2
(c) -1
(d) 0
Solution:
(a) 1

Explanation:
The polynomials having degrees equal to 1, are called linear polynomials.

Question 16.
What kind of graph we get for a linear polynomial?
(a) circle
(b) a point
(c) a straight line
(d) a triangle
Solution:
(c) a straight line

Explanation:
The polynomial whose degree is one, is called a linear polynomial. Its graph is a straight line.

Question 17.
A quadratic polynomial has the degree :
(a) 1
(b) -2
(c) -1
(d) 2
Solution:
(d) 2

Explanation:
The polynomials having degrees equal to 2, are called quadratic polynomials.

Question 18.
The value of the polynomial 7x – 4 at x = 2 is :
(a) 10
(b) -11
(c) -10
(d) -16
Solution:
(a) 10

Explanation:
The value of the polynomial is given by substituting the value of the variable in it. Here, given polynomial is 7x – 4 and given value of the variable is x = 2.
Substituting x = 2 in the polynomial 7x – 4 we obtain 7 × (2) – 4 = 14 — 4= 10, which gives the required value for the given polynomial at the given value.

Question 19.
The value of the polynomial 5 – 4x at x = – 1 is :
(a) 9
(b) -9
(c) -1
(d) 1
Solution:
(a) 9

Explanation:
The value of the polynomial is given by substituting the value of the variable in it. Here, given polynomial is 5 – 4x and given value of the variable is x = -1.
Substituting x = -1 in the polynomial 5 – 4x we obtain 5 – 4 × (-1) = 5 + 4 = 9, which gives the required value for the given polynomial at the given value.

Question 20.
The value of the polynomial 2x² – 4x + 5 at x = – 3 is :
(a) 11
(b) 35
(c) -35
(d) -11
Solution:
(b) 35

Explanation:
The value of the polynomial is given by substituting the value of the variable in it. Here, given polynomial is 2x² – 4x + 5 and given value of the variable is x = – 3. Substituting x = – 3 in the polynomial 2x² – 4x + 5, we obtain 2 × ( -3)² — 4( -3) + 5 = 2 × 9 + 12 + 5= 18 + 12 + 5 = 35, which gives the required value of the given polynomial at the given value.

Question 21.
What will come next in the following sequence?
12, 15, 18, 21, …
(a) 23
(b) 24
(c) -24
(d) 28
Solution:
(b) 24

Explanation:
Here, there is a linear growth. Each term increases by 3 to give its next term. So, after 21, the next term will be 21 + 3 = 24.

Question 22.
What will come next in the following sequence?
34, 28, 22, 16, …
(a) 12
(b) 8
(c) 10
(d) 14
Solution:
(c) 10

Explanation:
Here, there is a linear decay. Each term decreases by 6 to give its next term. So, after 16, the next term will be 16 – 6 = 10.

Question 23.
The height of water in a cylindrical tank is 275 cm on a particular day. If its height decreases by 25 cm every day due to consumption and no refilling. The height h cm at the end of d days is given by the linear function :
(a) h(d) = 275 – d
(b) h(d) = 275 – 25d
(c) h(d) = 275 – 10d
(d) h(d) = 25 – 275d
Solution:
(b) h(d) = 275 – 25d

Explanation:
The height h m after days will be given by the sequence of numbers as 275 – 25, 275 – 2 × 25, 275-3 × 25, …
As a linear function this sequence can be written as h(d) = 275 – 25d.

Question 24.
An auto – rickshaw fare starts at ?85 and remains the same for the initial 2 km. Then it increases by ₹ 20 per km. What will be the linear function representing the fare f(d)?
(a) f(d) = 35 – 20d
(b) f(d) = 35 + 20d
(c) f(d) = 35 + 20(d – 1)
(d) f(d) = 35 + 20(d – 2)
Solution:
(d) f(d) = 35 + 20(d – 2)

Explanation:
The fare f after d km of journey is given by f(d) = 35 + 20(d – 2) as ₹ 35 is for the first 2 km, which is fixed and for the journey of 3rd km and onwards it will be charged ₹ 20 for each km.

Question 25.
The initial population of a village is 1200. Every year, 80 people move from a nearby city to the village. The population of the village after 5 years is :
(a) 1200
(b) 1600
(c) 1080
(d) 1440
Solution:
(b) 1600

Explanation:
The population of the village after t years is given by p(t) = 1200 + 801. For t = 5, p(5) = 1200 + 80 × 5 = 1200 + 400 = 1600.

Question 26.
Following is the graph of y = 2x – 1.

Which of the following points does not lie on the line?
(a) (0, -1)
(b) (1, 1)
(c) (2, 3)
(d) (3, 4)
Solution:
(d) (3, 4)

Explanation:
From the graph it is quite clear that (0, – 1),(1, 1), (2, 3) lie on the given line. But we can see that (3, 4) does not lie on the given line.

Question 27.
What is the slope of the line whose equation is given by y = 4x + 3?
(a) 1
(b) 3
(c) 4
(d) -4
Solution:
(c) 4

Explanation:
The slope of a line whose equation is given as y = mx + c, is given by ‘m’.
So, slope of the line whose equation is given by y = 4x + 3, is 4.

Question 28.
What is the slope of the line whose equation is given by y = \(\frac{1}{3}\) x – 2?
(a) 1
(b) 3
(c) -3
(d) y
Solution:
(a) 1

Explanation:
The slope of a line whose equation is given as y = mx + c, is given by ‘m’
So, slope of the line whose equation is given
by y = \(\frac{1}{3}\) x – 2, is \(\frac{1}{3}\).

Question 29.
What is the slope of the line whose equation is given by y = -5?
(a) \(\frac{1}{2}\)
(b) 2
(c) -1
(d) 0
Solution:
(d) 0

Explanation:
The slope of a line whose equation is given as y = mx + c, is given by ‘m’.
On comparing the given equation with y = mx + c, we have m = 0.
So, the slope of the line whose equation is given by y = -5, is 0.

Question 30.
What is the y-intercept of the line whose equation is given by y = 4x + 3?
(a) -3
(b) 3
(c) 4
(d) -4
Solution:
(b) 3

Explanation:
The y-intercept of a line whose equation is given as y = mx + c, is given by ‘c’
So, y-intercept of the line whose equation is given by y = 4x + 3, is 3.

Question 31.
What is the equation of a straight line whose slope is – 3 and y-intercept is 1?
(a) y = -3x – 1
(b) y = 3x – 1
(c) y = -3x + 1
(d) y = 3x + 1
Solution:
(c) y = -3x + 1

Explanation:
The slope – intercept form of a line is given by the equation y = mx + c, where m is its slope and c is its y-intercept. Here, m = – 3 and c = 1, so the required equation of the line is given by y = – 3x + 1.

Question 32.
Which of he following is a polynomial?
(a) \(z^{\frac{1}{3}}+z^{\frac{-1}{3}}\)
(b) \(\sqrt{x}+\sqrt[3]{x}\)
(c) \(z^{\frac{3}{2}}\) – x + 2
(d) x² – x + 3
Solution:
(d) x² – x + 3

Question 33.
Which of the following is not a polynomial?
(a) \(x+\frac{1}{x}\)
(b) \(x^4+\frac{2}{t^{-2}}-1\)
(c) \(x^2-y^2-\frac{1}{t^{-2}}\)
(d) \(x^2+\frac{1}{t^{-2}}\)
Solution:
(a) \(x+\frac{1}{x}\)

Question 34.
Which of the following polynomials is univariate polynomial?
(a) xyz +1
(b) x² + y² + 2z²
(c) u² – uv + w
(d) t² – 3t + 4
Solution:
(d) t² – 3t + 4

Question 35.
Which of the following polynomials is a monomial?
(a) t + 4
(b) 3x + 2y – z
(c) x² – 7x + 8
(d) xyz
Solution:
(d) xyz

Question 36.
Which of the following polynomials is a monomial?
(a) x² + 3x²
(b) 2x – 3y + 20
(c) x² – 3x² – 2x
(d) x² – 3x – 1
Solution:
(a) x² + 3x²

Question 37.
Which of the following polynomials is a binomial?
(a) 3x² + 3y²
(b) x² + 3y² – 2xy
(c) x² – x + y
(d) 4x⁵ – 2x³ + 5x² – 11
Solution:
(a) 3x² + 3y²

Question 38.
Which of the following polynomials is a trinomial?
(a) 3x² + 3y² – 2
(b) x² + 3y – 2x²
(c) x² – 4x² + y
(d) x² – 3x² + 5x³
Solution:
(a) 3x² + 3y² – 2

Question 39.
Which of the following polynomials has degree zero?
(a) x° + x² – 1
(b) 3x³
(c) 3
(d) 2 + 3y + y²
Solution:
(c) 3

Question 40.
Which of the following polynomials has degree not equal to one?
(a) z + 2
(b) 2x – 1
(c) 3t – 3
(d) 2x² – x + 1
Solution:
(d) 2x² – x + 1

Question 41.
Which of the following polynomials is a linear polynomial?
(a) x³ – 2x² + 1
(b) 2x + 1
(c) x² – 2x + 4
(d) x⁷
Solution:
(b) 2x + 1

Introduction to Linear Polynomials Class 9 Assertion and Reason Questions

Direction: A statement of Assertion (A) is followed by a statement of Reason (R). Choose the correct option from the following options.
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
(c) Assertion (A) is true but Reason (R) is false.
(d) Assertion (A) is false but Reason (R) is true.

Question 1.
Assertion (A): The polynomial 3x – x is a binomial.
Reason (R): A polynomial is a binomial if it has two terms.
Solution:
(d) Assertion (A) is false but Reason (R) is true.

Explanation:
Here, given polynomial is 3x – x = 2x, which is a monomial. So, Assertion (A) is false. A polynomial having two terms are said to be a binomial. So, Reason (R) is true.

Question 2.
Assertion (A): The degree of the polynomial 3x³ – 2x + 74 is 4.
Reason (R): Highest degree on letter- number in a polynomial is its degree.
Solution:
(d) Assertion (A) is false but Reason (R) is true.

Explanation:
Here, given polynomial is 3x³ – 2x + 74, whose highest degree on letter- number is 3. So, Assertion (A) is false.
The degree of a polynomial is given by the power or index of variables in the polynomial which is highest. So, Reason (R) is true.

Question 3.
Assertion (A): The coefficient of x in the polynomial 7x² – \(\frac{1}{2}\) x + 12 is – \(\frac{1}{2}\).
Reason (R): To get the coefficient of any variable in a polynomial we need to take all in the term containing that variable with its power for which coefficient is needed, except that variable with its power.
Solution:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation:
Here, given polynomial is 7x² – \(\frac{1}{2}\) x + 12 is – \(\frac{1}{2}\), in which the term containing x is 2
—\(\frac{1}{2}\) x . Clearly, —\(\frac{1}{2}\) is the coefficient of x in this term. So, Assertion (A) is true.

To get the coefficient of any variable in a polynomial we exactly do the same as
explained in the Reason (R). So, Reason (R) is true and it explains the existence of the Assertion (A).

Question 4.
Assertion (A): The slope of the line whose equation is given by y = -2x – 3, is 2.
Reason (R): In the equation of a line y = mx + c, m gives the slope.
Solution:
(d) Assertion (A) is false but Reason (R) is true.

Explanation:
Here, given equation is y = – 2x – 3, on comparing it with y = mx + c, we get m = -2, so slope of the given line is – 2. So, Assertion (A) is false.
In the equation of a line y = mx + c, m gives the slope. So, Reason (R) is true.

Question 5.
Assertion (A): The y-intercept of the line whose equation is given by y = 4x – \(\frac{1}{2}\), is –\(\frac{1}{2}\).
Reason (R): In the equation of a line y = mx + c, c gives the y-intercept.
Solution:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation:
Here, given equation is y = 4x – \(\frac{1}{2}\), on comparing it with y = mx + c, we get c = – \(\frac{1}{2}\), so y-intercept of the given line is .
So, Assertion (A) is true.
In the equation of a line y = mx + c, c gives the y-intercept. So, Reason (R) is true and it explains the existence of the Assertion (A).

Question 6.
Assertion (A): The slope of the line whose equation is given by y = x –
Reason (R): In the equation of a line y = mx + c, c gives the y-intercept.
Solution:
(b) Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

Explanation:
Here, given equation is y = –\(\frac{1}{3}\)x – \(\frac{1}{2}\) on comparing it with y = mx + c, we
get m = –\(\frac{1}{3}\), so slope of the given line is – \(\frac{1}{3}\). So, Assertion (A) is true.
In the equation of a line y = mx + c, c gives the y-intercept, which is a true statement. So, Reason (R) is true but it does not explains the existence of the Assertion (A).

Question 7.
Assertion (A): The line whose equation is given by y = 4x – 10, cuts the x-axis at the point (\(\frac{5}{2}\), 0)
Reason (R): A line cuts x-axis if its y-coordinate is zero.
Solution:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation:
Here, given equation is y = 4x – 10, on substituting y = 0 in the given equation, we obtain 4x – 10 = 0 ⇒ x = x = \(\frac{10}{4}\) ⇒ x = \(\frac{5}{2}\). The corresponding point on the x-axis will be (\(\frac{5}{2}\), 0). So, Assertion (A) is true.
A line cuts x-axis if its y – coordinate is zero, which is a true statement. So, Reason (R) is true and it explains the existence of the Assertion (A).

Question 8.
Assertion (A): The line whose equation is given by y = 7x + 3, cuts the y-axis at the point (3, 0).
Reason (R): A line cuts y-axis if its x — coordinate is zero.
Solution:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is tne correct explanation of Assertion (A).
Explanation: Here, given equation is y = 7x + 3, on substituting x = 0 in the given equation, we obtain y = 7 × 0 + 3 ⇒ y = 3. The corresponding point on the y-axis will be (0, 3).
So, Assertion (A) is true.
A line cuts y-axis if its x – coordinate is zero, which is a true statement. So, Reason (R) is true and it explains the existence of the Assertion (A).

Question 9.
Assertion (A): The polynomial 6x – x is a monomial.
Reason (R): A polynomial is a binomial if it has two terms.
Solution:
(b) Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).

Question 10.
Assertion (A): The degree of the polynomial 19x³ – 17x + 47 is 3.
Reason (R): Highest degree on letter-number in a polynomial is its degree.
Solution:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question 11.
Assertion (A): The coefficient of x2 in the polynomial 7x² – \(\frac{1}{2}\) x + 12 is 7.
Reason (R): To get the coefficient of any variable in a polynomial we need to take all in the term containing that variable with its power for which coefficient is needed, except that variable with its power.
Solution:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question 12.
Assertion (A): The slope of the line whose equation is given by y = – 3x + 2, is 2.
Reason (R): In the equation of a line y = mx + c, m gives the slope.
Solution:
(d) Assertion (A) is false but Reason (R) is true.

Question 13.
Assertion (A): The y-intercept of the line whose equation is given by y = 4x – 7, is – 7.
Reason (R): In the equation of a line y = mx + c, c gives the y-intercept.
Solution:
(a) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Question 14.
Assertion (A): The point at which the line y = 2x – 3 cuts x-axis is (- 3, 0).
Reason (R): A line y = mx + c, cuts the x-axis if y = 0.
Solution:
(d) Assertion (A) is false but Reason (R) is true.

Question 15.
Assertion (A): The sequence 20, 25, 32.5, 42.5, 55, …, is an example of a linear growth.
Reason (R): A linear growth refers to a pattern in which a quantity increases by a fixed amount over equal intervals.
Solution:
(d) Assertion (A) is false but Reason (R) is true.