Polynomials — Chapter Test | Class 10 IMO

Q1

The product of the zeroes of 2x² − 8x + 6 is:

Product = c/a = 6/2 = 3.

Q2

A polynomial whose graph cuts the x-axis at exactly three points has:

Each x-intercept is a real zero, so there are 3 real zeros.

Q3

The product of the zeroes of 3x² − x − 4 is:

Product = c/a = −4/3.

Q4

A quadratic polynomial whose zeroes have sum 3 and product −10 is:

x² − (sum)x + product = x² − 3x − 10.

Q5

A rectangular garden has area x² + 7x + 10 square metres. Its dimensions are:

x² + 7x + 10 = (x + 2)(x + 5).

Q6

A hall's length is 3 m more than its breadth and its area is 40 m². The breadth is:

b(b + 3) = 40 gives b² + 3b − 40 = (b + 8)(b − 5) = 0, so b = 5 m.

Q7

The value of x² − 3x + 2 at x = 0 is:

At x = 0 the value is the constant term, 2.

Q8

If a quadratic has discriminant equal to 0, its parabola meets the x-axis at exactly:

Discriminant 0 means two equal zeroes, so the parabola just touches the axis at one point.

Q9

If one zero of x² − 4 is 2, the other zero is:

x² − 4 = (x − 2)(x + 2), so the other zero is −2.

Q10

If α and β are the zeroes of x² − 7x + 10, then α² + β² equals:

α + β = 7, αβ = 10, so α² + β² = 49 − 20 = 29.

Q11

The graph of a quadratic polynomial that opens upward has a:

An upward parabola (a > 0) has a lowest point, i.e. a minimum.

Q12

If α and β are the zeroes of x² − 6x + 8, then 1/α + 1/β equals:

1/α + 1/β = (α + β)/(αβ) = 6/8 = 3/4.

Q13

A firm's profit (in ₹ thousand) is P = −x² + 10x − 16. It breaks even (P = 0) when x equals:

−x² + 10x − 16 = 0 gives x² − 10x + 16 = (x − 2)(x − 8) = 0, so x = 2 or 8.

Q14

If α, β, γ are the zeroes of ax³ + bx² + cx + d, then α + β + γ equals:

For a cubic, the sum of the zeroes is −b/a.

Q15

The maximum number of zeroes a cubic polynomial can have is:

A polynomial of degree n has at most n zeroes, so a cubic has at most 3.