Q1 1 mark MCQ p. 5
p(x) is a polynomial given by:p(x) = −2x + 8x² − 1At which of the following points will the graph of p(x) intersect the positive x-axis?(i) ½(ii) ¼
Reveal official answer
Correct option: 1 — only (i)
Question Bank · Mathematics
Polynomials
Source: CBSE · Competency Focused Practice Questions, Mathematics (Volume 2), Grade 10, pp. 5–17. Questions and answers are reproduced verbatim from the official CBSE document.
Try the question yourself before revealing the answer — that's how marks stick.
Q2 1 mark MCQ p. 5
Which of these are the zeros of the polynomial x(x − 7)?
Reveal official answer
Correct option: 3 — both 0 and 7
Q3 1 mark MCQ p. 5
Which of these are the quotient and the remainder when (2x³ − 9x + 3x² + 12) is divided by (x − 1)?
Reveal official answer
Correct option: 3 — quotient = (2x² + 5x − 4) and remainder = 8.
Q4 1 mark MCQ p. 5
Which of these is the coefficient of x² in the quotient when (x⁴ + x³ + x + 1) is divided by (x − 4)?
Reveal official answer
Correct option: 3 — 5
Q5 1 mark MCQ p. 5
(3a³ − 2a² − 9a + 17) is divided by (a − 2). What is the coefficient of a in the quotient?
Reveal official answer
Correct option: 4 — 4
Q6 1 mark MCQ p. 5
P(t) is a polynomial in t such that,P(t) = (t² + 5t − 14)(t² − 7t + 10)(t² + 2t − 35)Which of these is the square root of P(t)?
Reveal official answer
Correct option: 2 — (t − 2)(t − 5)(t + 7)
Q7 1 mark MCQ p. 6
Which of the following polynomials has the highest degree?
Reveal official answer
Correct option: 1 — −x⁷ + 1
Q8 1 mark MCQ p. 6
Which of these are the zeroes of x² + 7x + 12?
Reveal official answer
Correct option: 2 — (−3) and (−4)
Q9 1 mark Written p. 6
(−⅖) is one of the zeroes of the polynomial 5x² + 2x − 7. (T/F)Justify your answer.
Reveal official answer
• Writes False (F). [0.5]
• Justifies the answer. For example, substituting x = −⅖ in the given polynomial does not yield 0, so −⅖ is not a zero. [0.5]
Q10 1 mark Written p. 6
Given f(x) = x³ + 7x² + 3x − 12Find the value of f(2). Show your work.
Reveal official answer
• Finds the value of f(2) as: (2)³ + 7(2)² + 3(2) − 12 = 30 [1]
Q11 1 mark Written p. 6
Given f(x) = x⁴ + x² + 4 and g(x) = x² − 1.Find the quotient and remainder when f(x) is divided by g(x). Show your work.
Reveal official answer
• Divides x⁴ + x² + 4 by x² − 1 using the long division method to get quotient as x² + 2 and the remainder as 6. [1]
Q12 1 mark Written p. 6
f(x) = x² + 10x + 21Find the zeroes of the above polynomial. Show your work.
Reveal official answer
• Factorizes the given polynomial and finds the roots as (−3) and (−7). The working may look as follows: f(x) = x² + 10x + 21 = x² + 3x + 7x + 21 = 0 => x(x + 3) + 7(x + 3) = 0 => (x + 7)(x + 3) = 0 => (x + 7) = 0 or (x + 3) = 0 => x = (−7) or x = (−3). (Note: Award full marks if the correct roots are obtained by any alternative approach.) [1]
Q13 2 marks Written p. 6
The graph of a polynomial passes through (6, 0), (0, −2) and (−1, 0).Write two factors of the polynomial. Justify your answer.
Reveal official answer
• Writes that P(x) = 0 at x = 6 or P(6) = 0 and hence (x − 6) is a factor of the polynomial. (Award 0.5 marks if only the factor is written.) [1]
• Writes that P(x) = 0 at x = −1 or P(−1) = 0 and hence (x + 1) is a factor of the polynomial. (Award 0.5 marks if only the factor is written.) [1]
Q14 2 marks Written p. 6
p(x) = (x + 5)² − 7(x − k); where k is a constant.If p(x) is divisible by x, find the value of k. Show your steps.
Reveal official answer
• Simplifies the given polynomial as: p(x) = x² + 3x + 7k + 25 [0.5]
• Writes that, if p(x) is divisible by x, p(0) = 0. OR Writes that the remainder of p(x)/x, which is 7k + 25, should be 0. [1]
• Finds the value of k as −25/7. [0.5]
Q15 2 marks Written p. 6
p and q are zeroes of the polynomial 3x² + 4x − 4.Without finding the actual values of p and q, evaluate (1 − p)(1 − q). Show your steps.
Reveal official answer
• Expands (1 − p)(1 − q) to get 1 − (p + q) + pq. [0.5]
• Finds the sum of the zeroes i.e. p + q = (−4/3). [0.5]
• Finds the product of the zeroes i.e. pq = (−4/3). [0.5]
• Uses the above steps to find the value of (1 − p)(1 − q) as 1 − (−4/3) + (−4/3) = 1. [0.5]
Q16 2 marks Written p. 7
Shown below is an expression:(x² − 2√3x − 9) / (x + √3); x ≠ −√3At how many points does the graph of the above expression intersect the x-axis? Show your work.
Reveal official answer
• Factorises the numerator to rewrite the given expression as: (x − 3√3)(x + √3) / (x + √3) [1]
• Writes that the graph of the above expression, (x − 3√3), intersects the x-axis at exactly one point i.e. (3√3, 0). [1]
Q17 2 marks Written p. 7
When a polynomial is divided by (2x − 1), the quotient is (3x − 2) and the remainder is (x − 3).Find the polynomial. Show your work.
Reveal official answer
• Applies the remainder theorem to write the polynomial as: (2x − 1)(3x − 2) + (x − 3). [1]
• Simplifies the above expression to find the polynomial as 6x² − 6x − 1. [1]
Q18 3 marks Written p. 7
p(x) is a polynomial given by ax² − 4x + 3, where a is a non-zero real number. One of the zeroes of p(x) is 3 times the other zero.i) Find the value of a. Show your work.ii) Based on the value of a, what would be the shape of the graph of p(x)? Give a reason for your answer.
Reveal official answer
• i) Assumes the roots of p(x) to be m and n to write the relation as m = 3n. [0.5]
• Finds the relation between β and a using the sum of the roots as: m + n = 3n + n = 4n = 4/a => n = 1/a [0.5]
• Finds the value of a using the product of the roots as: m.n = 3n² = 3/a => a = 1. [1]
• ii) Writes that, since a is positive, the graph of p(x) is an open upward parabola or open upwards like U. (Note: Award half mark if the student just writes parabola instead of upward parabola.) [1]
Q19 3 marks Written p. 7
A polynomial is given by p(x) = x³ + 3x² − 4x + c, where c is a constant.The sum of two zeroes of p(x) is zero.Using the relationship between the zeroes and coefficients of a polynomial, find the:i) zeroes of p(x).ii) value of c.Show your steps.
Reveal official answer
• i) Assumes the values of zeroes of p(x) as (−α), α and β. [0.5]
• Writes the sum of zeroes as: −α + α + β = −3. Finds β as −3. [0.5]
• Writes the equation for the sum of the products of zeroes taken two at a time as: −α² − αβ + βα = −4. Finds α² as 4. [1]
• Finds the zeroes of p(x) as (−2), 2 and (−3). [0.5]
• ii) Writes the equation for the product of zeroes as (−α²β) = (−c) and finds the value of c as (−12). [0.5]
Q20 3 marks Written p. 7
Anand multiplied a variable with 6, subtracted 27 and added the square of the original variable. He expressed the final expression as a product of 2 factors.His friend, Amit, said that the factors will always have a difference of 6.Is Amit right? Show your work.
Reveal official answer
• Assumes the original variable as x and frames the expression as 6x − 27 + x². [1]
• Factorises the above expression as (x − 3)(x + 9). [1]
• Concludes that Amit was wrong as the above factors have a difference of 12. [1]
Q22 5 marks Written p. 8
p(x) = x³ + (k − 3)x² − (k + 4)x − 6, where k is a non-zero real number and (x + 2) is a factor of p(x).Find the zeroes of p(x). Show your work.
Reveal official answer
• Writes that, since p(x) is divisible by (x + 2), p(−2) = 0 and finds the value of k as 3. [1]
• Uses the above step and writes p(x) as x³ − 7x − 6. [1]
• Divides p(x) by (x + 2) and finds the quotient as x² − 2x − 3. [1]
• Factorizes the quotient as (x + 1)(x − 3). [1]
• Finds the zeroes of p(x) as (−2), (−1) and 3. [1]
Q23 5 marks Written p. 8
f(x) = ax² + bx + 325 is a polynomial where a and b are real numbers. The zeroes of f(x) are distinct prime numbers. Find the:i) zeroes of f(x).ii) values of a and b.Show your work and give valid reasons.
Reveal official answer
• i) Writes the equation for the product of zeroes as: product of zeroes = 325/a. [1]
• Writes the prime factorisation of 325 as 5² × 13. [0.5]
• Writes that since the zeroes are distinct prime numbers, finds the zeroes of f(x) as 5 and 13. [1]
• Finds the value of a as 325/65 = 5. [0.5]
• ii) Writes the equation for the sum of zeroes as: 5 + 13 = −b/5. [1]
• Solves the above equation to find the value of b as (−90). [1]
Q25 1 mark Written p. 9
Answer the questions based on the given information.The revenue (in Rs) of a firm is represented by the polynomial R(x) = 5x³ + 4x² + 7, and the expenditure (in Rs) by the firm is represented by the polynomial E(x) = 3x³ + 2x − 1 where x is the number of items produced by the firm in a year.Find the profit polynomial P(x). Show your work.
Reveal official answer
• Subtracts E(x) from R(x) to find P(x) as 2x³ + 4x² − 2x + 8. [1]
Q26 2 marks Written p. 9
Answer the questions based on the given information.The revenue (in Rs) of a firm is represented by the polynomial R(x) = 5x³ + 4x² + 7, and the expenditure (in Rs) by the firm is represented by the polynomial E(x) = 3x³ + 2x − 1 where x is the number of items produced by the firm in a year.If the firm produces 100 products in a year, find the revenue and profit (in Rs) for the firm using the polynomials. Show your work.
Reveal official answer
• Finds the revenue made by the company from 100 products as: R(100) = 5(100)³ + 4(100)² + 7 => R(100) = 5000000 + 40000 + 7 = Rs 50,40,007. [1]
• Finds the profit made by the company from 100 products as: P(100) = 2(100)³ + 4(100)² − 2(100) + 8 => P(100) = 2000000 + 40000 − 200 + 8 = Rs 20,39,808. [1]
Q27 2 marks Written p. 9
Answer the questions based on the given information.The revenue (in Rs) of a firm is represented by the polynomial R(x) = 5x³ + 4x² + 7, and the expenditure (in Rs) by the firm is represented by the polynomial E(x) = 3x³ + 2x − 1 where x is the number of items produced by the firm in a year.Tax is calculated on the profit using the polynomial T(y) = 0.3y + 100, where y represents the profit earned.Determine the tax amount (in Rs) to be paid on the profit generated from 10 items. Show your work.
Reveal official answer
• Finds profit for 10 items as: P(10) = 2(10)³ + 4(10)² − 2(10) + 8 => P(10) = 2000 + 400 − 20 + 8 = Rs 2388. [1]
• Finds tax as: T(2388) = 0.3(2388) + 100 = Rs 816.4. [1]