CLASS 10 MATHEMATICS

Chapter 2: Polynomials

Note: These questions are designed based on NCERT concepts. Select the correct option for each question.

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1. If \( p(x) \) is a polynomial in \( x \), the highest power of \( x \) in \( p(x) \) is called the:
   a) Root
   b) Zero
   c) Degree
   d) Coefficient


2. Which of the following is NOT a polynomial?
   a) \( 4x + 2 \)
   b) \( 2y^2 - 3y + 4 \)
   c) \( \frac{1}{x-1} \)
   d) \( 7u^6 - 4 \)


3. A polynomial of degree 1 is called a:
   a) Constant polynomial
   b) Linear polynomial
   c) Quadratic polynomial
   d) Cubic polynomial


4. The word ‘quadratic’ is derived from the word ‘quadrate’, which means:
   a) Four
   b) Square
   c) Root
   d) Product


5. The general form of a quadratic polynomial in \( x \) is \( ax^2 + bx + c \), where \( a, b, c \) are real numbers and:
   a) \( a = 0 \)
   b) \( a \neq 0 \)
   c) \( b \neq 0 \)
   d) \( c \neq 0 \)


6. A cubic polynomial has a degree of:
   a) 1
   b) 2
   c) 3
   d) 4


7. If \( k \) is a real number such that \( p(k) = 0 \), then \( k \) is called a:
   a) Constant of the polynomial
   b) Degree of the polynomial
   c) Zero of the polynomial
   d) Factor of the coefficient


8. The zero of the linear polynomial \( ax + b \) is related to its coefficients as:
   a) \( b/a \)
   b) \( -a/b \)
   c) \( -b/a \)
   d) \( ab \)


9. Geometrically, the zeroes of a polynomial \( p(x) \) are the x-coordinates of the points where the graph of \( y = p(x) \) intersects the:
   a) Y-axis
   b) X-axis
   c) Origin
   d) Line \( y = x \)


10. The graph of a linear polynomial \( y = ax + b \) is a:
    a) Parabola
    b) Hyperbola
    c) Straight line
    d) Circle


11. The shape of the graph of a quadratic polynomial \( y = ax^2 + bx + c \) is called a:
    a) Linear curve
    b) Parabola
    c) Cubic curve
    d) Ellipse


12. A parabola \( y = ax^2 + bx + c \) opens upwards if:
    a) \( a < 0 \)
    b) \( a > 0 \)
    c) \( a = 0 \)
    d) \( b > 0 \)


13. The maximum number of zeroes a quadratic polynomial can have is:
    a) 1
    b) 2
    c) 3
    d) 0


14. If the graph of a quadratic polynomial touches the x-axis at exactly one point, the polynomial has:
    a) Two distinct zeroes
    b) No real zero
    c) Two equal zeroes (one zero)
    d) Infinite zeroes


15. If the graph of a quadratic polynomial is completely above the x-axis, the number of zeroes is:
    a) 1
    b) 2
    c) 0
    d) 3


16. A polynomial of degree \( n \) can have at most:
    a) \( n+1 \) zeroes
    b) \( n-1 \) zeroes
    c) \( n \) zeroes
    d) 1 zero


17. The zeroes of the quadratic polynomial \( x^2 - 3x - 4 \) are:
    a) 1 and 4
    b) -1 and 4
    c) -1 and -4
    d) 1 and -4


18. For a quadratic polynomial \( ax^2 + bx + c \), the sum of the zeroes \( \alpha + \beta \) is:
    a) \( c/a \)
    b) \( b/a \)
    c) \( -b/a \)
    d) \( -c/a \)


19. For a quadratic polynomial \( ax^2 + bx + c \), the product of the zeroes \( \alpha\beta \) is:
    a) \( -b/a \)
    b) \( c/a \)
    c) \( b/a \)
    d) \( -c/a \)


20. If the sum and product of zeroes of a quadratic polynomial are \( -3 \) and \( 2 \) respectively, the polynomial is:
    a) \( x^2 - 3x + 2 \)
    b) \( x^2 + 3x + 2 \)
    c) \( x^2 + 3x - 2 \)
    d) \( x^2 - 3x - 2 \)


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


22. The zeroes of the polynomial \( x^2 - 3 \) are:
    a) 3, -3
    b) \( \sqrt{3}, -\sqrt{3} \)
    c) 3, 0
    d) \( \sqrt{3}, 0 \)


23. The number of zeroes for the cubic polynomial \( y = x^3 - 4x \) is:
    a) 1
    b) 2
    c) 3
    d) 4


24. For the cubic polynomial \( ax^3 + bx^2 + cx + d \), the sum of the zeroes \( \alpha + \beta + \gamma \) is:
    a) \( c/a \)
    b) \( -b/a \)
    c) \( -d/a \)
    d) \( b/a \)


25. For a cubic polynomial, the product of the zeroes \( \alpha\beta\gamma \) is:
    a) \( c/a \)
    b) \( -b/a \)
    c) \( -d/a \)
    d) \( d/a \)


26. In a cubic polynomial, the sum of the products of zeroes taken two at a time (\( \alpha\beta + \beta\gamma + \gamma\alpha \)) is:
    a) \( -b/a \)
    b) \( c/a \)
    c) \( -d/a \)
    d) \( -c/a \)


27. If a cubic polynomial is \( x^3 - x^2 \), its zeroes are:
    a) 1 only
    b) 0 only
    c) 0 and 1
    d) 0, 1, and -1


28. The number of zeroes of a polynomial whose graph is parallel to the x-axis and does not touch it is:
    a) 1
    b) 0
    c) Infinite
    d) 2


29. Competency-Based: A student observes that the graph of a polynomial intersects the x-axis at 4 distinct points. This polynomial must have a degree of at least:
    a) 3
    b) 4
    c) 5
    d) 2


30. The constant term of a linear polynomial \( ax + b \) is \( b \). Its zero is:
    a) \( \frac{\text{Constant term}}{\text{Coefficient of } x} \)
    b) \( -\frac{\text{Constant term}}{\text{Coefficient of } x} \)
    c) \( \frac{\text{Coefficient of } x}{\text{Constant term}} \)
    d) \( -\frac{\text{Coefficient of } x}{\text{Constant term}} \)


31. If \( \alpha, \beta \) are the zeroes of \( 2x^2 - 8x + 6 \), then \( \alpha + \beta \) is:
    a) 4
    b) -4
    c) 3
    d) -3


32. The product of zeroes of the polynomial \( 3x^2 + 5x - 2 \) is:
    a) \( 5/3 \)
    b) \( -5/3 \)
    c) \( -2/3 \)
    d) \( 2/3 \)


33. Which of the following polynomials has \( -2 \) and \( 4 \) as its zeroes?
    a) \( x^2 - 2x - 8 \)
    b) \( x^2 + 2x - 8 \)
    c) \( x^2 - 2x + 8 \)
    d) \( x^2 + 2x + 8 \)


34. A quadratic polynomial whose sum of zeroes is \( 0 \) and one zero is \( 3 \) is:
    a) \( x^2 - 9 \)
    b) \( x^2 + 9 \)
    c) \( x^2 - 3 \)
    d) \( x^2 + 3 \)


35. In the cubic polynomial \( p(x) = 3x^3 - 5x^2 - 11x - 3 \), the value of \( \alpha\beta\gamma \) is:
    a) 1
    b) -1
    c) 3
    d) -3


36. If a polynomial \( p(x) \) has zeroes \( 4, -2, \) and \( 1/2 \), it is most likely a:
    a) Linear polynomial
    b) Quadratic polynomial
    c) Cubic polynomial
    d) Bi-quadratic polynomial


37. Competency-Based: If the graph of \( y = p(x) \) passes through the origin, then one of the zeroes of \( p(x) \) is:
    a) 1
    b) -1
    c) 0
    d) Cannot be determined


38. The degree of the polynomial \( 5x^3 - 4x^2 + x - 2 \) is:
    a) 2
    b) 3
    c) 1
    d) 0


39. A quadratic polynomial can have at most how many points of intersection with the x-axis?
    a) 0
    b) 1
    c) 2
    d) 3


40. The zeroes of the polynomial \( 4u^2 + 8u \) are:
    a) 0, 2
    b) 0, -2
    c) 4, 8
    d) -4, -8


41. If \( \alpha + \beta = 4 \) and \( \alpha\beta = 1 \) for a quadratic polynomial, the polynomial is:
    a) \( x^2 - 4x + 1 \)
    b) \( x^2 + 4x + 1 \)
    c) \( x^2 - 4x - 1 \)
    d) \( x^2 + 4x - 1 \)


42. The sum of the zeroes of the cubic polynomial \( 2x^3 - 5x^2 - 14x + 8 \) is:
    a) \( -5/2 \)
    b) \( 5/2 \)
    c) \( 7 \)
    d) \( -7 \)


43. If one zero of the quadratic polynomial \( ax^2 + bx + c \) is the reciprocal of the other, then:
    a) \( a = b \)
    b) \( b = c \)
    c) \( a = c \)
    d) \( a = 0 \)


44. Competency-Based: For which value of \( a \) will the graph of \( y = ax^2 + bx + c \) be a downward parabola?
    a) \( a = 5 \)
    b) \( a = 0 \)
    c) \( a = -3 \)
    d) \( a = 1 \)


45. The degree of a constant polynomial (like \( p(x) = 5 \)) is:
    a) 1
    b) 0
    c) Undefined
    d) 2


46. If the zeroes of a quadratic polynomial are \( \alpha \) and \( \beta \), then the polynomial can be written as:
    a) \( k[x^2 + (\alpha+\beta)x + \alpha\beta] \)
    b) \( k[x^2 - (\alpha+\beta)x + \alpha\beta] \)
    c) \( k[x^2 - (\alpha+\beta)x - \alpha\beta] \)
    d) \( k[x^2 + (\alpha+\beta)x - \alpha\beta] \)


47. The zeroes of the polynomial \( t^2 - 15 \) are:
    a) 15, -15
    b) \( \sqrt{15}, -\sqrt{15} \)
    c) 0, 15
    d) \( \sqrt{15}, 0 \)


48. In the quadratic polynomial \( 4s^2 - 4s + 1 \), the sum of zeroes is:
    a) 1
    b) -1
    c) 1/4
    d) 4


49. A cubic polynomial has zeroes \( \alpha, \beta, \gamma \). If \( \alpha + \beta + \gamma = 2 \), \( \alpha\beta + \beta\gamma + \gamma\alpha = -7 \), and \( \alpha\beta\gamma = -14 \), the polynomial is:
    a) \( x^3 - 2x^2 - 7x + 14 \)
    b) \( x^3 - 2x^2 - 7x - 14 \)
    c) \( x^3 + 2x^2 - 7x + 14 \)
    d) \( x^3 - 2x^2 + 7x + 14 \)


50. If a polynomial has no real zeroes, its graph:
    a) Intersects the x-axis at two points
    b) Touches the x-axis at one point
    c) Does not intersect the x-axis at all
    d) Is a straight line through the origin

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Answer Key

1(c), 2(c), 3(b), 4(b), 5(b), 6(c), 7(c), 8(c), 9(b), 10(c),
11(b), 12(b), 13(b), 14(c), 15(c), 16(c), 17(b), 18(c), 19(b), 20(b),
21(b), 22(b), 23(c), 24(b), 25(c), 26(b), 27(c), 28(b), 29(b), 30(b),
31(a), 32(c), 33(a), 34(a), 35(a), 36(c), 37(c), 38(b), 39(c), 40(b),
41(a), 42(b), 43(c), 44(c), 45(b), 46(b), 47(b), 48(a), 49(a), 50(c)