CBSE MATHEMATICS WORKSHEET

Chapter 2: Polynomials

Class: X | Worksheet No: 02 | Difficulty: Medium | Time: 3 Hours | Total Marks: 80

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Chapter Introduction

In this chapter, we explore polynomials and their degrees. A polynomial \( p(x) \) is an algebraic expression where the highest power of the variable is called its degree [1]. We study linear (degree 1), quadratic (degree 2), and cubic (degree 3) polynomials [1-3]. A key concept is the "zero" of a polynomial—a real number \( k \) such that \( p(k) = 0 \) [4]. Geometrically, these zeroes represent the x-coordinates of points where the graph of the polynomial intersects the x-axis [5-7]. We also examine the fundamental relationship between the zeroes of a polynomial and its coefficients [8, 9].

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Section A: Multiple Choice Questions (10 × 1 Mark)

1. What is the degree of the polynomial \( 7u^6 - \frac{3}{2}u^4 + 4u^2 + u - 8 \)? [1]
   (a) 4
   (b) 2
   (c) 6
   (d) 1
   
   
2. A polynomial of degree 2 is specifically called a: [2]
   (a) Linear polynomial
   (b) Quadratic polynomial
   (c) Cubic polynomial
   (d) Constant polynomial
   
   
3. The name 'quadratic' is derived from 'quadrate', which means: [2]
   (a) Four
   (b) Root
   (c) Square
   (d) Variable
   
   
4. The shape of the graph of a quadratic polynomial \( ax^2 + bx + c \) is a: [6]
   (a) Straight line
   (b) Circle
   (c) Parabola
   (d) Hyperbola
   
   
5. If a polynomial \( p(x) \) has degree \( n \), what is the maximum number of zeroes it can have? [7]
   (a) \( n-1 \)
   (b) \( n \)
   (c) \( n+1 \)
   (d) 1
   
   
6. For a linear polynomial \( ax + b \), the zero is related to its coefficients as: [10]
   (a) \( b/a \)
   (b) \( -a/b \)
   (c) \( -b/a \)
   (d) \( ab \)
   
   
7. In the quadratic polynomial \( x^2 - 3x - 4 \), the zeroes are: [4]
   (a) 1 and 4
   (b) -1 and 4
   (c) -1 and -4
   (d) 0 and 4
   
   
8. The sum of the zeroes of the polynomial \( 2x^2 - 8x + 6 \) is: [11]
   (a) 3
   (b) -4
   (c) 4
   (d) -3
   
   
9. A cubic polynomial can have at most how many zeroes? [12]
   (a) 1
   (b) 2
   (c) 3
   (d) 4
   
   
10. The graph of \( y = x^3 - 4x \) intersects the x-axis at how many points? [13]
    (a) 1
    (b) 2
    (c) 3
    (d) 0
    
    

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Section B: Very Short Answer Questions (10 × 1 Mark)

1. Define the degree of a polynomial \( p(x) \). [1]
2. Is \( \frac{1}{x-1} \) a polynomial? Explain why or why not. [1]
3. Write the general form of a cubic polynomial. [3]
4. If \( k \) is a zero of \( p(x) \), what is the value of \( p(k) \)? [4]
5. Find the zero of the linear polynomial \( 2x + 3 \). [4]
6. What does the graph of a linear polynomial \( ax + b \) represent geometrically? [14]
7. If a quadratic graph is completely above the x-axis, how many zeroes does it have? [15]
8. What is the relationship between the product of zeroes and coefficients for \( ax^2 + bx + c \)? [16]
9. Find the value of \( p(x) = x^2 - 3x - 4 \) at \( x = 0 \). [17]
10. State the Fundamental Theorem mentioned regarding the number of zeroes of a degree \( n \) polynomial. [7]

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Section C: Short Answer Questions (10 × 2 Marks)

1. Find the zeroes of \( x^2 + 7x + 10 \) and verify the sum relationship. [18]
2. Find a quadratic polynomial if the sum and product of zeroes are \( -3 \) and \( 2 \) respectively. [19]
3. Verify that the product of zeroes of \( 3x^2 + 5x - 2 \) is \( c/a \). [20]
4. Identify the zeroes of \( x^2 - 3 \) and verify their product. [21]
5. Explain why the graph of \( y = ax^2 + bx + c \) opens upwards when \( a > 0 \). [6]
6. Factorise \( 2x^2 - 8x + 6 \) by splitting the middle term. [11]
7. If \( \alpha \) and \( \beta \) are zeroes of \( ax^2 + bx + c \), prove \( \alpha + \beta = -b/a \). [16]
8. Using Source [22], find the zeroes of the cubic polynomial \( x^3 - x^2 \).
9. Check if \( -1 \) is a zero of \( p(x) = x^3 - 3x^2 + x + 1 \). [22]
10. Find the \( HCF \) of the zeroes of \( 4u^2 + 8u \). [23]

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Section D: Long/Application-Based Questions (10 × 3 Marks)

1. Verify that \( 3, -1, \) and \( -1/3 \) are zeroes of \( p(x) = 3x^3 - 5x^2 - 11x - 3 \). [24]
2. Discuss the three possible cases for the intersection of a quadratic parabola with the x-axis. [15, 25]
3. For the polynomial \( 2x^3 - 5x^2 - 14x + 8 \), verify all three relationships between zeroes and coefficients. [9, 26]
4. Explain the geometrical meaning of the zeroes of a cubic polynomial using the example of \( x^3 - 4x \). [13, 27]
5. Exercise 2.2: Find zeroes of \( 6x^2 - 3 - 7x \) and verify relationships. [23]
6. If zeroes of a quadratic are \( \sqrt{2} \) and \( 1/3 \), find the polynomial. [23]
7. Draw a rough sketch (based on description in [14]) showing the zero of \( 2x+3 \).
8. Show that a quadratic polynomial can have exactly one zero and provide an example from the source. [25]
9. Verify that \( 0 \) is the only zero of the polynomial \( x^3 \). [22]
10. Explain how the zero of a linear polynomial is related to its constant term and coefficient of \( x \). [10]

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Section E: HOTS (Higher Order Thinking Skills) (5 × 4 Marks)

1. If \( \alpha, \beta, \gamma \) are zeroes of \( ax^3 + bx^2 + cx + d \), derive the formula for the sum of products taken two at a time. [26]
2. Prove that for any real number \( k \), \( k(x^2 + 3x + 2) \) has the same zeroes. [19]
3. Analyze the graph of \( y = x^2 - 3x - 4 \) to explain why zeroes are at points where \( y=0 \). [6]
4. If the sum of zeroes of \( kx^2 + 2x + 3k \) is equal to their product, find \( k \). [Concept from 20]
5. Can a cubic polynomial have no real zeroes? Justify using the geometrical concepts in Source [7, 13, 22, 27].

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Section F: Case Study Questions (3 × 4 Marks)

Case Study 1: The Projectile Path
The path of a ball thrown in the air follows the curve \( y = -x^2 + 3x + 4 \).
i) What is the name of this curve? [6]
ii) Does the curve open upwards or downwards? Why? [6]
iii) Find the points where the ball hits the ground (zeroes). [4, 6]
iv) How many zeroes does this path represent? [25]

Case Study 2: Designing a Cubic Slide
An architect uses the polynomial \( p(x) = x^3 - 4x \) to design a slide section.
i) Identify the zeroes of this design. [13]
ii) At how many points will the slide touch the ground level? [27]
iii) What is the maximum number of times a degree 3 curve can change direction? [7]
iv) Verify the sum of zeroes for this slide. [12, 26]

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Section G: Assertion–Reason Questions (5 × 1 Mark)

(a) Both A and R are true, R explains A. (b) Both true, R doesn't explain A. (c) A is true, R false. (d) A is false, R true.

1. A: \( 4x + 2 \) is a linear polynomial. R: The highest power of \( x \) is 1. [1]
2. A: A quadratic polynomial has exactly 2 zeroes. R: A degree 2 polynomial has at most 2 zeroes. [15]
3. A: Zero of \( 2x+3 \) is \( -3/2 \). R: Zero of \( ax+b \) is \( -b/a \). [4, 10]
4. A: \( \sqrt{x} + 2 \) is a polynomial. R: Exponents must be non-negative integers. [1]
5. A: Sum of zeroes of \( x^2+3x+2 \) is \( -3 \). R: Sum \( = -b/a \). [19]

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Section H: Match the Following (5 × 1 Mark)

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Section I: Activity-Based Questions (5 × 2 Marks)

1. Draw a factor tree for a quadratic polynomial of your choice. [28]
2. Plot points for \( y = 2x+3 \) and find where it crosses the x-axis. [14]
3. Using Table 2.1, plot the parabola for \( x^2 - 3x - 4 \). [28]
4. Verify the zeroes of \( x^3 - 4x \) by plugging in \( -2, 0, 2 \). [13]
5. Collect 3 different algebraic expressions and categorize them as linear, quadratic, or non-polynomials. [1, 2]

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Section J: Challenge/Olympiad-style Questions (5 × 4 Marks)

1. If \( \alpha, \beta \) are zeroes of \( ax^2+bx+c \), find \( \alpha^2 + \beta^2 \) in terms of coefficients. [16]
2. Find a cubic polynomial where sum of zeroes is \( 2 \), sum of product taken two at a time is \( -7 \), and product is \( -14 \). [12]
3. If one zero of \( 3x^2 + 5x - 2 \) is \( -2 \), find the other using the product formula. [20]
4. Prove that a linear polynomial cannot have more than one zero. [5]
5. Show that the graph of \( y = x^3 - x^2 \) touches the x-axis at the origin. [22]

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Important Formulas:
- Quadratic: \( \alpha + \beta = -b/a \), \( \alpha\beta = c/a \) [16]
- Cubic: \( \alpha + \beta + \gamma = -b/a \), \( \alpha\beta + \beta\gamma + \gamma\alpha = c/a \), \( \alpha\beta\gamma = -d/a \) [12]

Common Mistakes:
- Forgetting to change the sign of 'b' in the sum formula.
- Not checking if the expression is a polynomial (powers must be integers) [1].

Exam Tips:
- Zeroes are ALWAYS the x-coordinates on the graph where \( y=0 \) [6].
- Always verify relationships if asked; it ensures your zeroes are correct.