CBSE Class 10 Polynomials Sure Shot Questions Set 09

• “Polynomial” comes from the word ‘Poly’ (Meaning Many) and ‘nomial’ (in this case meaning Term)- so it means many terms.
• A polynomial is made up of terms that are only added, subtracted or multiplied.
• A quadratic polynomial in x with real coefficients is of the form \( ax^2 + bx + c \), where a, b, c are real numbers with \( a \neq 0 \).
• Degree – The highest exponent of the variable in the polynomial is called the degree of polynomial. Example: \( 3x^3 + 4 \), here degree = 3.
• Polynomials of degrees 1, 2 and 3 are called linear, quadratic and cubic polynomial respectively.
• A polynomial can have terms which have Constants like 3, -20, etc., Variables like x and y and Exponents like 2 in \( y^2 \).
• These can be combined using addition, subtraction and multiplication but NOT DIVISION.
• The zeroes of a polynomial \( p(x) \) are precisely the x-coordinates of the points, where the graph of \( y = p(x) \) intersects the x-axis.
• If \( \alpha \) and \( \beta \) are the zeroes of the quadratic polynomial \( ax^2 + bx + c \), then
  sum of zeros, \( \alpha+\beta = -b/a = -\text{coefficient of } x / \text{coefficient of } x^2 \)
  product of zeros, \( \alpha\beta = c/a = \text{constant term} / \text{coefficient of } x^2 \)
• If \( \alpha, \beta, \gamma \) are the zeroes of the cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), then
  \( \alpha+\beta+\gamma = -b/a = -\text{coefficient of } x^2 / \text{coefficient of } x^3 \)
  \( \alpha\beta+\beta\gamma+\gamma\alpha = c/a = \text{coefficient of } x / \text{coefficient of } x^3 \)
  \( \alpha\beta\gamma = -d/a = -\text{constant term} / \text{coefficient of } x^3 \)

RELATIONSHIP BETWEEN ZEROES & COEFFICIENTS OF POLYNOMIALS

Linear Polynomial
General Form: \( ax + b, a \neq 0 \)
No. of Zeros: 1
Relationship: \( K = -b/a \), ie, k = Constant term / Coefficient of x

Quadratic Polynomial
General Form: \( ax^2 + bx + c, a \neq 0 \)
No. of Zeros: 2
Relationship: Sum of zeroes \( (\alpha + \beta) = \text{Coefficient of } x / \text{Coefficient of } x^2 = -b/a \)
Product of zeroes \( (\alpha \times \beta) = \text{constant term} / \text{Coefficient of } x^2 = c/a \)

Cubic Polynomial
General Form: \( ax^3 + bx^2 + cx + d, a \neq 0 \)
No. of Zeros: 3
Relationship: Sum of zeroes \( (\alpha + \beta + \gamma) = -b/a \)
Product of sum of zeroes taken two at a time \( (\alpha\beta + \beta\gamma + \gamma\alpha) = \text{Coefficient of } x / \text{Coefficient of } x^3 = c/a \)
Product of zeroes \( (\alpha \times \beta \times \gamma) = -\text{constant term} / \text{Coefficient of } x^3 = -d/a \)

A quadratic polynomial whose zeroes are \( \alpha \) and \( \beta \) is given by \( p(x) = x^2 - (\alpha + \beta)x + \alpha \beta \)
i.e. \( x^2 - (\text{Sum of zeroes})x + (\text{Product of zeroes}) \)

A cubic polynomial whose zeroes are \( \alpha \), \( \beta \) and \( \gamma \) is given by \( p(x) = x^3 - (\alpha + \beta + \gamma)x^2 + (\alpha\beta + \beta\gamma + \gamma\alpha)x - \alpha \beta \gamma \)

Question. The zeroes of the quadratic polynomial \( x^2 + 7x + 10 \) are
(a) -4, -3
(b) 2, 5
(c) -2, -5
(d) -2, 5
Answer: (c) -2, -5

Question. The zeroes of the quadratic polynomial \( x^2 - 27 \) are
(a) \( +3\sqrt{3}, -3\sqrt{3} \)
(b) 3, 3
(c) 9, 9
(d) \( +\sqrt{3}, -\sqrt{3} \)
Answer: (a) \( +3\sqrt{3}, -3\sqrt{3} \)

Question. A quadratic polynomial can have at most ______ zeroes.
(a) 0
(b) 1
(c) 2
(d) infinite
Answer: (c) 2

Question. A quadratic polynomial, whose zeroes are -2 and 4, is
(a) \( x^2 - 2x + 8 \)
(b) \( x^2 + 2x + 8 \)
(c) \( x^2 - 2x - 8 \)
(d) \( 2x^2 + 2x - 24 \)
Answer: (c) \( x^2 - 2x - 8 \)

Question. The number of polynomials having zeroes as -2 and 5 is
(a) 1
(b) 2
(c) 3
(d) more than 3
Answer: (d) more than 3

Question. The sum and the product of the zeroes of polynomial \( 6x^2 - 5 \) respectively are
(a) 0, (-6)/5
(b) 0, 6/5
(c) 0, 5/6
(d) 0, (-5)/6
Answer: (d) 0, (-5)/6

Question. The zeroes of the quadratic polynomial \( x^2 + kx + k \) where \( k \neq 0 \),
(a) cannot both be positive
(b) cannot both be negative
(c) are always unequal
(d) are always equal
Answer: (a) cannot both be positive

Question. If the zeroes of the quadratic polynomial \( ax^2 + bx + c \), where \( c \neq 0 \), are equal, then
(a) c and a have opposite signs
(b) c and b have opposite signs
(c) c and a have same signs
(d) c and b have the same signs
Answer: (c) c and a have same signs

Question. If one of the zeroes of a quadratic polynomial of the form \( x^2 + ax + b \) is the negative of the other, then it
(a) has no linear term and the constant term is negative
(b) has no linear term and the constant term is positive
(c) can have a linear term but the constant term is negative
(d) can have a linear term but the constant term is positive
Answer: (a) has no linear term and the constant term is negative

Question. If one zero of the quadratic polynomial \( x^2 + 3x + k \) is 2, then the value of k is
(a) 10
(b) -10
(c) 5
(d) -5
Answer: (b) -10

Question. If one zero of the quadratic polynomial \( x^2 - 4x + 1 \) is \( 2 + \sqrt{3} \), then the other zero is
(a) -2 + \( \sqrt{3} \)
(b) -\( \sqrt{3} \) -2
(c) 2 - \( \sqrt{3} \)
(d) \( \sqrt{3} \) + 1
Answer: (c) 2 - \( \sqrt{3} \)

Question. If 2 is a zero of the polynomial \( p(x) = k x^2 + 3x + k \), then the value of k is
(a) 5/6
(b) (-5)/6
(c) 6/5
(d) (-6)/5
Answer: (d) (-6)/5

Question. If one of the zeroes of the quadratic polynomial \( (k – 1)x^2 + kx + 1 \) is – 3, then the value of k is
(a) 4/3
(b) (-4)/3
(c) 2/3
(d) (-2)/3
Answer: (a) 4/3

Question. If the sum of the zeroes of the quadratic polynomial \( kx^2 + 2x + 3k \) is equal to their product, then k is equal to
(a) 1/3
(b) (-2)/3
(c) (-1)/3
(d) 2/3
Answer: (b) (-2)/3

Question. If zeroes of \( p(x) = 2x^2 - 7x + k \) reciprocal of each other
(a) 1
(b) 2
(c) 3
(d) -7
Answer: (b) 2

ASSERTION REASONING QUESTIONS

DIRECTION: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) if both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) if both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) if Assertion (A) is true but reason (R) is false.
(d) if Assertion (A) is false but reason (R) is true.

Question. Assertion: \( x^2 + 4x + 5 \) has two real zeroes.
Reason: A quadratic polynomial can have at the most two zeroes.
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion (A) is false but reason (R) is true.

Question. Assertion: \( y^3 + 3y \) has only one real zero.
Reason: A polynomial of nth degree must have n real zeroes.
(a) A
(b) B
(c) C
(d) D
Answer: (c) Assertion (A) is true but reason (R) is false.

Question. Assertion: Degree of a zero polynomial is not defined.
Reason: Degree of a non-zero constant polynomial is ‘0’.
(a) A
(b) B
(c) C
(d) D
Answer: (b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

Question. Assertion: \( x^2 + 11x + 30 \) has no real zeroes.
Reason: A quadratic polynomial can have at the most two zeroes.
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion (A) is false but reason (R) is true.

Question. Assertion: If the sum of the zeroes of the quadratic polynomial \( x^2 – 2kx + 8 \) is 2, then value of k is 1.
Reason: Sum of zeroes of a quadratic polynomial \( ax^2 + bx + c \) is \( (-b)/a \).
(a) A
(b) B
(c) C
(d) D
Answer: (a) both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion: A quadratic polynomial, sum of whose zeroes is 6 and their product is 8 is \( x^2 – 14x + 48 \).
Reason: If \( \alpha \) and \( \beta \) be the zeroes of the polynomial \( f(x) \), then polynomial is given by \( f(x) = x^2 - (\alpha + \beta) x + \alpha \beta \).
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion (A) is false but reason (R) is true.

Question. Assertion: \( P(x) = 3x^3 – 2x^2 + 4x^4 + x – 2 \) is a polynomial of degree 3.
Reason: The highest power of x in the polynomial \( P(x) \) is the degree of the polynomial.
(a) A
(b) B
(c) C
(d) D
Answer: (d) Assertion (A) is false but reason (R) is true.

Question. Assertion: If the sum and product of zeroes of a quadratic polynomial are 3 and -2 respectively, then quadratic polynomial is \( x^2 – 3x – 2 \).
Reason: If S is the sum of the zeroes and P is the product of the zeroes of a quadratic polynomial, then the corresponding quadratic polynomial is \( x^2 – Sx + P \).
(a) A
(b) B
(c) C
(d) D
Answer: (a) both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Question. Assertion: If \( \alpha \) and \( \beta \) are the zeroes of the polynomial \( x^2 + 2x-15 \), then \( 1/\alpha + 1/\beta \) is 2/15.
Reason: If \( \alpha \) and \( \beta \) are the zeroes of a quadratic polynomial \( ax^2 + bx + c \), then \( \alpha + \beta \) is \( (-b)/a \) and \( \alpha\beta = c/a \)
(a) A
(b) B
(c) C
(d) D
Answer: (a) both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

Short Answer Type Questions

Question. If a fifth degree polynomial is divided by a quadratic polynomial, write the possible degree of the quotient
Answer: 3

Question. What is the value of \( p(x) = x^2 – 3x – 4 \) at \( x = –1 \)?
Answer: 0

Question. For what value of k, (–4) is a zero of the polynomial \( x^2 – x – (2k + 2) \)?
Answer: 9

Question. If 1 is a zero of the polynomial \( p(x) = ax^2 – 3 (a – 1) x – 1 \), then find the value of a
Answer: 1

Question. If the sum of zeroes of the quadratic polynomial \( 3x^2 – kx + 6 \) is 3, then find the value of k ?
Answer: 9

Question. Find a quadratic polynomial whose zeroes are –12 and 4 and verify the relationship between the zeroes and the coefficients
Answer: \( x^2 - 8x - 48 \)

Question. If the sum of the zeroes of the polynomial \( p(x) = (k^2 – 14) x^2 – 2x – 12 \) is 1, then find the value of k.
Answer: 4

Question. Find the value of ‘k’ such that the quadratic polynomial \( 3x^2 + 2kx + x – k – 5 \) has the sum of zeroes as half of their product.
Answer: 1

Question. Find the product of sum and product of zeroes of the quadratic polynomial \( 3x^2 + 5x – 2 \)
Answer: 10/3

Question. Write a quadratic polynomial, sum of whose zeroes is \( 2\sqrt{3} \) and product is 5.
Answer: \( x^2 - 2\sqrt{3}x + 5 \)

Short Answer - II Type Questions

Question. If the zeroes of the polynomial \( x^2 + px + q \) are double in value to the zeroes of \( 2x^2 – 5x – 3 \), find the value of p and q.
Answer: \( p = -5, q = -6 \)

Question. Find the zeroes of the quadratic polynomial \( 6x^2 – 3 – 7x \) and verify the relationship between the zeroes and the coefficients of the polynomial.
Answer: 3/2, -1/2

Question. If \( \alpha \) and \( \beta \) are zeroes of \( p(x) = kx^2 + 4x + 4 \), such that \( \alpha^2 + \beta^2 = 24 \), find k.
Answer: -1

Question. If \( \alpha \) and \( \beta \) are the zeroes of a quadratic polynomial \( x^2 + x – 2 \) then find the value of \( \left( \frac{1}{\alpha} - \frac{1}{\beta} \right) \).
Answer: -3/2

Question. If the zeroes of the polynomial \( x^2 + px + q \) are double in value to the zeroes of \( 2x^2 - 5x - 3 \), find the value of p and q.
Answer: \( p = -5, q = -6 \)

Case Study Questions

CASE STUDY 1: The below picture are few natural examples of parabolic shape which is represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms.

Question. In the standard form of quadratic polynomial, \( ax^2 + bx + c \), what are a, b and c ?
Answer: a is a non zero real number and b and c are any real numbers.

Question. If the roots of the quadratic polynomial are equal, what is the discriminant D ?
Answer: D = 0

Question. If \( \alpha \) and \( 1/\alpha \) are the zeroes of the quadratic polynomial \( 2x^2 – x + 8k \), then find the value of k ?
Answer: 1/4

CASE STUDY 2: Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball in both sports, a basketball player uses his hands and a soccer player uses his feet. Usually, soccer is played outdoors on a large field and basketball is played indoor on a court made out of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola representing quadratic polynomial.

Question. Which type the shape of the path traced shown in given figure?
Answer: parabola

Question. Why the graph of parabola opens upwards,?
Answer: a > 0

CASE STUDY 3: An asana is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modern yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to representation of quadratic polynomial.

Question. Which type the shape of the poses shown in figure?
Answer: parabola

Question. Write two zeroes in the above shown graph ?
Answer: -2, 4