Q) Basketball and soccer are played with a spherical ball. Even though an athlete dribbles the ball in both sports, a basketball player uses his hands and a soccer player uses his feet. Usually, soccer is played outdoors on a large field and basketball is played indoor on a court made out of wood. The projectile (path traced) of soccer ball and basketball are in the form of parabola representing quadratic polynomial.

1. The shape of the path traced shown is:
a) Spiral              b) Ellipse              c) Linear                   d) Parabola

2. The graph of parabola opens upwards, if _______
a) a = 0                b) a < 0                 c) a > 0                      d) a >= 0

3. Observe the following graph and answer

In the above graph, how many zeroes are there for the polynomial?
a) 0                      b) 1                       c) 2                             d) 3

4. The three zeroes in the above shown graph are:
a) 2, 3,-1              b) -2, 3, 1             c) -3, -1, 2                   d) -2, -3, -1

5. What will be the expression of the polynomial?
a) x³ + 2 x² – 5x – 6                         b) x³ + 2 x² – 5x + 6                      c) x³ + 2 x² + 5x – 6                       d) x³ + 2 x² + 5 x + 6

Ans:

STEP BY STEP SOLUTION

1. Shapes of path:

Let’s understand the types of images for each of given shapes by this diagram:

Here we can see that all the asanas postures shown above are parabolic and in upward direction.

Therefore, option d) is correct

2. Value of a:  

Polynomial equation is a x² + b x + c = 0

If the parabola graph opens upwards, parabolic polynomial equation has to have x² and therefore, the value of a can never be zero.

[what will happen if a = 0? Once a is zero, b x + c = 0 will make a linear equation]

∴ a > 0

Therefore, option c) is correct

3. Number of zeroes:

In the given diagram (shown in right side), it is a graph for upward parabola.

We know that in a standard quadratic polynomial expression, number of zeroes is equal to the number of times the graph intersects x-axis.

In the given diagram, the graph is intersecting X – axis 3 times, hence its number of roots are 3.

Therefore, option d) is correct

4. Values of Zeroes:

Since, Zeroes are values of x when the graph intersects X – axis.

Here, in the given diagram, the graph intersects the X-axis at x = – 3, x = – 1 and x = 2.

Hence, its roots are – 3, – 1 and 2.

Therefore, option c) is correct

5. Zeroes of the Quadratic Polynomial: 

We know that if roots are a, b, c, then the polynomial equation is p(x) = (x – a) (x – b) (x – c)

Since, here the roots are: – 3, – 1, 2; therefore the polynomial equation will be:

p(x) = (x – (- 3)) (x – (- 1)) (x – 2)

∴ p(x) = (x + 3) (x + 1) (x – 2)

∴ p(x) = (x² + 3 x + x + 3) ( x – 2)

∴ p(x) = (x² + 4 x + 3) ( x – 2)

∴ p(x) = (x³  + 4 x² + 3 x – 2 x ² – 8 x – 6)

∴ p(x) = (x³  + 2 x²  – 5 x – 6)

Therefore, option a) is correct

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