Class 10 Maths Chapter 2 Polynomials Sample Paper

SAMPLE PAPER 02 - CHAPTER 02 POLYNOMIALS (2025-26) SUBJECT: MATHEMATICS MAX. MARKS: 40 CLASS: X DURATION: 1½ hrs

General Instructions:

1. All questions are compulsory.

2. This question paper contains 20 questions divided into five Sections A, B, C, D and E.

3. Section A comprises of 10 MCQs of 1 mark each. Section B comprises of 4 questions of 2 marks each. Section C comprises of 3 questions of 3 marks each. Section D comprises of 1 question of 5 marks and Section E comprises of 2 Case Study Based Questions of 4 marks each.

4. There is no overall choice.

5. Use of Calculators is not permitted. SECTION – A Questions 1 to 10 carry 1 mark each. 2

1. If α and β are zeroes of polynomial x - 7x + 10, then 1/α + 1/β equals:
(a) 7/10
(b) 10/7
(c) -7/10
(d) -10/7

2. The number of zeroes of a polynomial y = f(x) from the given graph is: [Graph shows parabola touching x-axis at one point]
(a) 0
(b) 1
(c) 2
(d) 3 2

3. If the sum of zeroes of polynomial 3x - kx + 6 is 3, then k equals:
(a) 3
(b) 6
(c) 9
(d) 12

4. A quadratic polynomial whose product of zeroes is -3 and sum is 0 is: 2 2 2 2
(a) x - 3
(b) x + 3
(c) x - 3x
(d) x + 3x 2

5. If α, β are zeroes of x + 7x + 12, then the value of (1/α + 1/β) is:
(a) -7/12
(b) 7/12
(c) 12/7
(d) -12/7 2

6. If one zero of 2x + 3x + k is reciprocal of the other, then k equals:
(a) 1
(b) 2
(c) 3
(d) 4

7. The degree of zero polynomial is:
(a) 0
(b) 1
(c) any natural number
(d) not defined

8. If sum of zeroes of kx 2 + 2x + 3k is equal to their product, then k equals:
(a) 1/3
(b) -1/3
(c) 2/3
(d) -2/3

9. Assertion
(a) : If the graph of polynomial touches x-axis at only one point, it has only one zero. Reason (R): A polynomial of degree n can have at most n zeroes.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true. 2

10. Assertion
(a) : x - 1 is a quadratic polynomial. 2 Reason (R): Any polynomial of the form ax + bx + c, where a ≠ 0 is a quadratic polynomial.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true. SECTION – B Questions 11 to 14 carry 2 marks each.

11. Find a quadratic polynomial whose zeroes are 2 + √3 and 2 - √3. 2

12. If α and β are zeroes of x - x - 2, find the value of 1/α + 1/β - 2αβ. 2

13. If one zero of polynomial p(x) = (k + 4)x + 13x + 4k is reciprocal of the other, find k. 2

14. Find the zeroes of polynomial 4u + 8u. SECTION – C Questions 15 to 17 carry 3 marks each. 2

15. Find the zeroes of quadratic polynomial 6x - 3 - 7x and verify the relationship between zeroes and coefficients. 2

16. If α and β are zeroes of polynomial x - 6x + k. Find k if 3α + 2β = 20. 2

17. If α and β are zeroes of polynomial x + 7x + 10, find the polynomial whose zeroes are α/β and β/α. SECTION – D Question 18 carries 5 marks. 2 2 2

18. If α and β are zeroes of polynomial p(x) = 2x + 5x + k satisfying relation α + β + αβ = 21/4, then find:
(a) Value of k (3 marks)
(b) Value of α + β + 3αβ (2 marks) SECTION – E (Case Study Based Questions) Questions 19 to 20 carry 4 marks each.

19. An athletics coach is preparing a rectangular training ground. The length of the ground exceeds 2 its breadth by 10 m. The area of the ground is represented by polynomial x + 10x, where x is the breadth in meters. 2 (i) Find the zeroes of polynomial x + 10x. (1 mark) (ii) What is the sum of zeroes of this polynomial? (1 mark) (iii) If the breadth is 20 m, find the area of the training ground. (2 marks)

20. A bakery owner finds that the profit P(x) from selling x cakes per day is given by polynomial P(x) = 2 -x + 40x - 300 (in rupees). (i) What is the degree of profit polynomial? (1 mark) (ii) Find the zeroes of profit polynomial. (2 marks) (iii) What do these zeroes represent in real context? (1 mark) ✓ DETAILED SOLUTIONS - SAMPLE PAPER 02 SECTION – A (SOLUTIONS)

Solution 1:

2 For x - 7x + 10: α + β = 7, αβ = 10 1/α + 1/β = (α + β)/αβ = 7/10 Answer:
(a) 7/10

Solution 2:

When parabola touches x-axis at exactly one point, it has 2 equal zeroes (repeated root). So technically 2 zeroes but since they're equal, graph shows 1 point. Answer:
(c) 2

Solution 3:

2 For 3x - kx + 6: Sum = k/3 Given: k/3 = 3 k = 9 Answer:
(c) 9

Solution 4:

Sum = 0, Product = -3 Polynomial = x 2 - (sum)x + product = x 2 - 0·x + (-3) = x 2 - 3 2 Answer:
(a) x - 3

Solution 5:

2 For x + 7x + 12: α + β = -7, αβ = 12 1/α + 1/β = (α + β)/αβ = -7/12 Answer:
(a) -7/12

Solution 6:

Let zeroes be α and 1/α Product = α × 1/α = 1 = k/2 k = 2 Answer:
(b) 2

Solution 7:

Zero polynomial is p(x) = 0 for all x Its degree is not defined Answer:
(d) not defined

Solution 8:

2 For kx + 2x + 3k: Sum = -2/k, Product = 3k/k = 3 Given: Sum = Product -2/k = 3 k = -2/3 Answer:
(d) -2/3

Solution 9:

Assertion is FALSE - when graph touches at one point, polynomial has 2 equal zeroes (counted as 2) Reason is TRUE - polynomial of degree n can have at most n zeroes Answer:
(d) A is false but R is true

Solution 10:

2 2 x - 1 is of form ax + bx + c where a = 1, b = 0, c = -1 Both A and R are true, and R correctly explains A Answer:
(a) Both A and R are true and R is the correct explanation of A SECTION – B (SOLUTIONS)

Solution 11:

Zeroes: α = 2 + √3, β = 2 - √3 Sum = (2 + √3) + (2 - √3) = 4 Product = (2 + √3)(2 - √3) = 4 - 3 = 1 2 Polynomial = x - 4x + 1

Solution 12:

2 For x - x - 2: α + β = 1, αβ = -2 1/α + 1/β = (α + β)/αβ = 1/(-2) = -1/2 1/α + 1/β - 2αβ = -1/2 - 2(-2) = -1/2 + 4 = 7/2 Value = 7/2

Solution 13:

Let zeroes be α and 1/α Product = 4k/(k + 4) α × 1/α = 1 = 4k/(k + 4) k + 4 = 4k 3k = 4 k = 4/3 k = 4/3

Solution 14:

2 4u + 8u = 0 4u(u + 2) = 0 u = 0 or u = -2 Zeroes: 0 and -2 SECTION – C (SOLUTIONS)

Solution 15:

2 6x - 7x - 3 = 0 2 6x - 9x + 2x - 3 = 0 3x(2x - 3) + 1(2x - 3) = 0 (3x + 1)(2x - 3) = 0 x = -1/3 or x = 3/2 Verification: Sum = -1/3 + 3/2 = 7/6 = -(-7)/6 ✓ Product = (-1/3)(3/2) = -1/2 = -3/6 ✓ Zeroes: -1/3 and 3/2

Solution 16:

2 For x - 6x + k: α + β = 6, αβ = k Given: 3α + 2β = 20 ...(1) Also: α + β = 6 ...(2) From (2): β = 6 - α Substituting in (1): 3α + 2(6 - α) = 20 3α + 12 - 2α = 20 α = 8, β = -2 k = αβ = 8 × (-2) = -16 k = -16

Solution 17:

2 For x + 7x + 10: α + β = -7, αβ = 10 New zeroes: α/β and β/α 2 2 Sum = α/β + β/α = (α + β )/αβ 2 2 2 α + β = (α + β) - 2αβ = 49 - 20 = 29 Sum = 29/10 Product = (α/β)(β/α) = 1 Polynomial = x 2 - (29/10)x + 1 or 10x 2 - 29x + 10 SECTION – D (SOLUTIONS)

Solution 18(a):

2 For 2x + 5x + k: α + β = -5/2, αβ = k/2 2 2 Given: α + β + αβ = 21/4 2 (α + β) - 2αβ + αβ = 21/4 2 (α + β) - αβ = 21/4 25/4 - k/2 = 21/4 k/2 = 1 k = 2 k = 2

Solution 18(b):

α + β = -5/2 αβ = k/2 = 2/2 = 1 α + β + 3αβ = -5/2 + 3(1) = -5/2 + 3 = 1/2 Value = 1/2 SECTION – E (SOLUTIONS)

Solution 19(i):

2 x + 10x = 0 x(x + 10) = 0 x = 0 or x = -10 Zeroes: 0 and -10

Solution 19(ii):

Sum of zeroes = 0 + (-10) = -10 Sum = -10

Solution 19(iii):

Breadth = x = 20 m Length = x + 10 = 30 m 2 2 2 Area = x + 10x = 20 + 10(20) = 400 + 200 = 600 m 2 Area = 600 m

Solution 20(i):

2 P(x) = -x + 40x - 300 Highest power of x is 2 Degree = 2

Solution 20(ii):

2 -x + 40x - 300 = 0 2 x - 40x + 300 = 0 2 x - 30x - 10x + 300 = 0 x(x - 30) - 10(x - 30) = 0 (x - 10)(x - 30) = 0 x = 10 or x = 30 Zeroes: 10 and 30

Solution 20(iii):

The zeroes 10 and 30 represent break-even points where profit is zero. Selling 10 or 30 cakes results in no profit (profit = ₹0) Break-even points: 10 and 30 cakes