Polynomials Class 10 Mathematics — Competency Based Questions

20 Questions | 70 Marks

A group of students is creating a mathematical garden. They plant saplings in rows such that the number of saplings in each row forms a quadratic polynomial

p (x) = x^{2} - 6 x + 8

. They observe that the number of saplings in a row is zero for certain values of x.(a) Find the values of x for which the number of saplings in a row is zero. [2 marks](b) What does this observation about the zeroes of the polynomial signify in the context of the garden? [1 mark](c) If they decide to plant a triangular flower bed where the number of flowers in each row follows a cubic polynomial

q (x) = x^{3} - 6 x^{2} + 11 x - 6

, and they find that the number of flowers is zero for x=1, x=2, and x=3, interpret what these zeroes represent. 2marks

Assertion: A quadratic polynomial can have at most two zeroes.Reason: The graph of a quadratic polynomial

y = a x^{2} + b x + c

(where

a \neq 0

) is a parabola, which can intersect the x-axis at a maximum of two points.(A) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(B) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(C) Assertion is true, but reason is false.(D) Assertion is false, but reason is true. 1mark

A farmer is designing a rectangular enclosure for his sheep. The dimensions of the enclosure are given by the factors of a quadratic polynomial

p (x) = x^{2} - 15 x + 50

. The farmer observes that for certain values of 'x', the area of the enclosure becomes zero.(a) Find the values of 'x' for which the area of the enclosure is zero. [2 marks](b) What do these values of 'x' represent in terms of the dimensions of the enclosure? [1 mark](c) If the length of the enclosure is represented by

(x - 5)

metres, what is the width? 1mark

Consider the polynomial

p (x) = 2 x^{3} - x^{2} - 13 x - 5

. It is known that

(x + 1)

is a factor of

p (x)

.(a) Find the zeroes of the polynomial

p (x)

. [3 marks](b) What is the significance of these zeroes in relation to the polynomial

p (x)

? 1mark

A civil engineer is analyzing the trajectory of a projectile. The height of the projectile at time 't' is given by a quadratic polynomial

h (t) = - 5 t^{2} + 20 t

.(a) Find the times 't' at which the height of the projectile is zero. [2 marks](b) Interpret the meaning of these 't' values in the context of the projectile's motion. [1 mark](c) If another projectile's height is given by

g (t) = - 5 t^{2} + 15 t

, find the sum and product of its zeroes. 2marks

Assertion: A cubic polynomial must have at least one real zero.Reason: The graph of a cubic polynomial

y = a x^{3} + b x^{2} + c x + d

(where

a \neq 0

) is a continuous curve that extends from

- \infty

+ \infty

(or vice versa). Therefore, it must cross the x-axis at least once.(A) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(B) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(C) Assertion is true, but reason is false.(D) Assertion is false, but reason is true. 1mark

A scientist is studying the growth of a bacteria colony. The rate of growth is modeled by the polynomial

p (t) = t^{3} - 6 t^{2} + 11 t - 6

, where 't' is time in hours. The colony's growth rate becomes zero at certain specific times.(a) Find the times 't' when the growth rate of the bacteria colony is zero. [3 marks](b) What does it mean when the growth rate is zero for this polynomial? 1mark

A quadratic polynomial

p (x)

has zeroes

α

and

β

. If

α + β = \frac{1}{4}

and

α β = - \frac{1}{4}

.(a) Find the quadratic polynomial

p (x)

. [2 marks](b) If a new polynomial

q (x)

is formed such that its zeroes are

2 α

and

2 β

, find the polynomial

q (x)

. 3marks

A rectangular plot of land has an area given by the polynomial

A (x) = x^{2} + 7 x + 12

. The dimensions of the plot are expressed in terms of 'x'.(a) Find the possible dimensions of the rectangular plot by finding the zeroes of the polynomial

A (x)

. [2 marks](b) Interpret what the zeroes of the area polynomial represent in this context. 1mark

Assertion: If

α

and

β

are the zeroes of the quadratic polynomial

a x^{2} + b x + c

, then

α + β = - b / a

.Reason: For a quadratic polynomial

a x^{2} + b x + c

, the sum of the zeroes is directly proportional to the coefficient of x and inversely proportional to the coefficient of

x^{2}

.(A) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(B) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(C) Assertion is true, but reason is false.(D) Assertion is false, but reason is true. 1mark

A student is given a quadratic polynomial

p (x) = 3 x^{2} - k x + 6

. The student is told that one of the zeroes of this polynomial is 2.(a) Find the value of k. [2 marks](b) Find the other zero of the polynomial. 2marks

A manufacturing company produces widgets. The profit function is given by

P (x) = - x^{2} + 8 x - 12

, where 'x' is the number of widgets produced (in thousands) and

P (x)

is the profit in lakhs of rupees.(a) Find the number of widgets (in thousands) that need to be produced for the profit to be zero. [2 marks](b) Interpret the meaning of these values of 'x' for the company. [1 mark](c) For what number of widgets produced is the profit maximum? (Hint: Vertex of the parabola) 2marks

Consider the polynomial

p (x) = x^{3} - 3 x^{2} + x - 3

.(a) Find the zeroes of this polynomial. [3 marks](b) What does the geometrical interpretation of these zeroes signify? 1mark

Assertion: A polynomial of degree n has exactly n zeroes.Reason: The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n roots (counting multiplicity) in the complex number system.(A) Both assertion and reason are true, and the reason is the correct explanation of the assertion.(B) Both assertion and reason are true, but the reason is not the correct explanation of the assertion.(C) Assertion is true, but reason is false.(D) Assertion is false, but reason is true. 1mark

A teacher is explaining the relationship between the zeroes and coefficients of a cubic polynomial. She uses the example

p (x) = 2 x^{3} + x^{2} - 6 x - 3

.(a) Find the zeroes of

p (x)

. [3 marks](b) Verify the relationship between the sum of the zeroes, the sum of the products of the zeroes taken two at a time, and the product of the zeroes with their respective coefficients. 2marks

A farmer is analyzing the yield of his crops. The yield function is given by a quadratic polynomial

Y (x) = - x^{2} + 10 x - 21

, where 'x' is the amount of fertilizer used.(a) Find the amounts of fertilizer 'x' for which the crop yield is zero. [2 marks](b) What does it mean when the yield is zero in this context? [1 mark](c) If a new fertilizer is introduced, and its yield function is

Z (x) = - x^{2} + 12 x - 35

, find the sum and product of its zeroes. 2marks

A researcher is studying the population dynamics of a species. The population model is given by

P (t) = t^{3} - 5 t^{2} + 2 t + 8

, where

t

is time in years. The population becomes zero at certain times.(a) Find the zeroes of the polynomial

P (t)

. [3 marks](b) Interpret the meaning of these zeroes in the context of population dynamics. 1mark

Assertion: If

α

and

β

are the zeroes of

x^{2} - 5 x + 6

, then

α β = 6

.Reason: For a quadratic polynomial

a x^{2} + b x + c

, the product of the zeroes is

c / a

A landscape artist is designing a rectangular lawn. The length and width of the lawn are represented by the factors of the polynomial

p (x) = x^{2} - 13 x + 30

.(a) Find the possible dimensions of the lawn by finding the zeroes of the polynomial. [2 marks](b) If 'x' represents a length in meters, what are the possible lengths of the lawn? [1 mark](c) If the artist decides to create a circular fountain whose area is modeled by

A (r) = π r^{2}

, and she wants the area to be zero, what would that imply for the radius 'r'? 1mark

A teacher gives a problem to the students: "Form a quadratic polynomial whose zeroes are

3 + \sqrt{2}

and

3 - \sqrt{2}

."(a) Calculate the sum and product of the given zeroes. [2 marks](b) Form the quadratic polynomial using the sum and product of its zeroes. 2marks

Answers

(a) The values of x are 2 and 4. (

x^{2} - 6 x + 8 = (x - 2) (x - 4) = 0 \Rightarrow x = 2, x = 4

)(b) These zeroes signify the points where the number of saplings in a row is zero, which might indicate the boundaries or specific points of interest in the garden layout.(c) The zeroes x=1, x=2, and x=3 represent the specific times or conditions when the growth rate of the flower bed is zero.

(A) Both assertion and reason are true, and the reason is the correct explanation of the assertion.

(a) The values of x are 5 and 10. (

x^{2} - 15 x + 50 = (x - 5) (x - 10) = 0 \Rightarrow x = 5, x = 10

)(b) These values of x represent the conditions under which the area of the enclosure would be zero, which implies one of the dimensions must be zero.(c) The width is

(x - 10)

metres.

(a)

p (x) = (x + 1) (2 x^{2} - 3 x - 5) = (x + 1) (x + 1) (2 x - 5)

. The zeroes are

x = - 1

(multiplicity 2) and

x = 5 / 2

.(b) These zeroes represent the values of x for which the value of the polynomial

p (x)

is zero.

(a) The times are

t = 0

and

t = 4

seconds. (

- 5 t^{2} + 20 t = - 5 t (t - 4) = 0 \Rightarrow t = 0, t = 4

)(b)

t = 0

represents the initial time when the projectile is launched (at ground level).

t = 4

represents the time when the projectile returns to the ground.(c) For

g (t) = - 5 t^{2} + 15 t

, sum of zeroes =

- 15 / (- 5) = 3

. Product of zeroes =

0 / (- 5) = 0

(A) Both assertion and reason are true, and the reason is the correct explanation of the assertion.

(a)

t^{3} - 6 t^{2} + 11 t - 6 = (t - 1) (t - 2) (t - 3) = 0

. The zeroes are

t = 1, t = 2, t = 3

.(b) When the growth rate is zero, it means the population is neither increasing nor decreasing at that particular instant.

(a) Sum of zeroes =

\frac{1}{4}

, Product of zeroes =

- \frac{1}{4}

. The polynomial is

k (x^{2} - (sum of zeroes) x + product of zeroes) = k (x^{2} - \frac{1}{4} x - \frac{1}{4})

. For

k = 4

p (x) = 4 x^{2} - x - 1

.(b) For

q (x)

, the sum of zeroes is

2 α + 2 β = 2 (α + β) = 2 (\frac{1}{4}) = \frac{1}{2}

. The product of zeroes is

(2 α) (2 β) = 4 α β = 4 (- \frac{1}{4}) = - 1

. The polynomial

q (x)

k (x^{2} - \frac{1}{2} x - 1)

. For

k = 2

q (x) = 2 x^{2} - x - 2

(a)

A (x) = x^{2} + 7 x + 12 = (x + 3) (x + 4)

. The zeroes are

x = - 3

and

x = - 4

. Since dimensions cannot be negative, we consider the factors

(x + 3)

and

(x + 4)

as representations of dimensions.(b) The zeroes of the area polynomial themselves do not have a direct physical meaning as dimensions since they are negative. However, the factors derived from these zeroes,

(x + 3)

and

(x + 4)

, represent the possible dimensions (length and width) of the rectangular plot, assuming

x > 0

(B) Both assertion and reason are true, but the reason is not the correct explanation of the assertion. (The reason states "directly proportional" and "inversely proportional", which is true, but the assertion is a specific formula, not a general relationship explanation).

(a) Since 2 is a zero,

p (2) = 0

3 (2)^{2} - k (2) + 6 = 0 \Rightarrow 12 - 2 k + 6 = 0 \Rightarrow 18 = 2 k \Rightarrow k = 9

.(b) The polynomial is

3 x^{2} - 9 x + 6

. Sum of zeroes =

- (- 9) / 3 = 3

. If one zero is 2, let the other be

β

. Then

2 + β = 3 \Rightarrow β = 1

(a)

- x^{2} + 8 x - 12 = - (x^{2} - 8 x + 12) = - (x - 2) (x - 6) = 0

. The profit is zero when

x = 2

x = 6

(thousand widgets).(b) These values represent the break-even points where the company neither makes a profit nor incurs a loss.(c) The maximum profit occurs at the vertex of the parabola. The x-coordinate of the vertex is

- b / (2 a) = - 8 / (2 \times - 1) = - 8 / (- 2) = 4

. So, producing 4 thousand widgets maximizes profit.

(a)

p (x) = x^{3} - 3 x^{2} + x - 3 = x^{2} (x - 3) + 1 (x - 3) = (x^{2} + 1) (x - 3)

. The zeroes are

x = 3

and

x = \pm i

(imaginary zeroes). The real zero is

x = 3

.(b) The real zero

x = 3

represents the x-intercept of the graph of

y = p (x)

(D) Assertion is false, but reason is true. (A polynomial of degree n has at most n real zeroes, but exactly n zeroes in the complex number system).

(a)

p (x) = 2 x^{3} + x^{2} - 6 x - 3

. By grouping:

x^{2} (2 x + 1) - 3 (2 x + 1) = (x^{2} - 3) (2 x + 1)

. The zeroes are

x = \sqrt{3}

x = - \sqrt{3}

, and

x = - 1 / 2

.(b) Sum of zeroes:

\sqrt{3} + (- \sqrt{3}) + (- \frac{1}{2}) = - \frac{1}{2}

. From polynomial:

- b / a = - (1) / 2 = - 1 / 2

. Verified.Sum of products of zeroes taken two at a time:

(\sqrt{3}) (- \sqrt{3}) + (- \sqrt{3}) (- \frac{1}{2}) + (- \frac{1}{2}) (\sqrt{3}) = - 3 + \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = - 3

. From polynomial:

c / a = - 6 / 2 = - 3

. Verified.Product of zeroes:

(\sqrt{3}) (- \sqrt{3}) (- \frac{1}{2}) = (- 3) (- \frac{1}{2}) = \frac{3}{2}

. From polynomial:

- d / a = - (- 3) / 2 = 3 / 2

. Verified.

(a)

Y (x) = - x^{2} + 10 x - 21 = - (x^{2} - 10 x + 21) = - (x - 3) (x - 7) = 0

. The amounts of fertilizer are

x = 3

and

x = 7

.(b) This means that if the farmer uses 3 units or 7 units of fertilizer, the crop yield will be zero. This could indicate an optimal range for fertilizer use or a point of toxicity.(c) For

Z (x) = - x^{2} + 12 x - 35

, sum of zeroes =

- 12 / (- 1) = 12

. Product of zeroes =

- 35 / (- 1) = 35

(a)

P (t) = t^{3} - 5 t^{2} + 2 t + 8

. By inspection, we find that

P (- 1) = (- 1)^{3} - 5 (- 1)^{2} + 2 (- 1) + 8 = - 1 - 5 - 2 + 8 = 0

. So,

(t + 1)

is a factor. Using polynomial division or synthetic division, we get

P (t) = (t + 1) (t^{2} - 6 t + 8)

. Factoring the quadratic:

t^{2} - 6 t + 8 = (t - 2) (t - 4)

. Thus, the zeroes are

t = - 1, t = 2, t = 4

. Since time cannot be negative in this context, the relevant zeroes are

t = 2

and

t = 4

.(b) These zeroes represent the times when the population of the species becomes zero, indicating extinction or a complete absence of the species at those specific time points.

(A) Both assertion and reason are true, and the reason is the correct explanation of the assertion.

(a)

p (x) = x^{2} - 13 x + 30 = (x - 3) (x - 10)

. The zeroes are

x = 3

and

x = 10

. These represent the possible values for 'x' which, when used in the expressions for length and width, result in zero area.(b) The possible lengths of the lawn would be 3 meters and 10 meters, assuming 'x' itself represents one of the dimensions or a parameter related to dimensions. If 'x' is a parameter, then the dimensions are

(x - 3)

and

(x - 10)

. For positive dimensions,

x > 10

. The possible dimensions are

(10 - 3) = 7

m and

(10 - 10) = 0

m (not possible for a rectangle) or

(3 - 3) = 0

m and

(3 - 10) = - 7

m (not possible). Reinterpreting: If 'x' is such that the dimensions are

L = x

and

W = y

, and

p (x)

represents area, then the factors of

p (x)

represent the dimensions. Thus, possible dimensions are 3m and 10m.(c) If the area of a circular fountain

A (r) = π r^{2}

is zero, it implies that the radius

r

must be zero. This means there is no fountain.

(a) Sum of zeroes =

(3 + \sqrt{2}) + (3 - \sqrt{2}) = 6

. Product of zeroes =

(3 + \sqrt{2}) (3 - \sqrt{2}) = 3^{2} - (\sqrt{2})^{2} = 9 - 2 = 7

.(b) The quadratic polynomial is

x^{2} - (sum of zeroes) x + (product of zeroes) = x^{2} - 6 x + 7

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