Solution:

Correct option: (d).

(ii) The graph of parabola opens upwards, if

(a) $a=0$

(b) $a<0$

(c) $a>0$

(d) $a\ge 0$

Sol. Correct option: (c).

(iii) 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

Sol. Correct option: (d).

Explanation: The number of zeroes of polynomial is the number of times the curve intersects the $x$-axis, i.e. attains the value 0 .

Here, the polynomial meets the $x$-axis at 3 points. So, number of zeroes $=3$.

(iv) The three zeroes in the above shown graph are

(a) $2,3,-1$

(b) $-2,3,1$

(c) $-3,-1,2$

(d) $-2,-3,-1$

Sol. Correct option: (c).

(v) What will be the expression of the polynomial?

(a) $x^3+2x^2-5x-6$

(b) $x^3+2x^2-5x-6$

(c) $x^3+2x^2+5x-6$

(d) $x^3+2x^2+5x+6$

Sol. Correct option: (a).

Explanation: Since, the three zeroes $=-3,-1,2$

Hence, the expression is $(x+3)(x+1)(x-2)$

$$ \begin{aligned} & =\leftx^{2}+x+3 x+3\right \ & =x^{3}+4 x^{2}+3 x-2 x^{2}-8 x-6 \ & =x^{3}+2 x^{2}-5 x-6 \end{aligned} $$

[II Q. 4. Applications of Parabolas: Highway Overpasses/ Underpasses

A highway underpass is parabolic in shape.

Shape of Cross Slope :

a. Paraboilc camber

$$y=2x^2/\mathrm{nw}$$

Parabola

A parabola is the graph that results from

$$p(x)=a{x}^2+bx+c.$$

Parabolas are symmetric about a vertical line known as the Axis of Symmetry.

The Axis of Symmetry runs through the maximum or minimum point of the parabola which is called the vertex.

C + AE [CBSE SQP, 2020-21]

(i) If the highway overpass is represented by $x^2-2x-8$. Then its zeroes are

(a) $(2,-4)$

(b) $(4,-2)$

(c) $(-2,-2)$

(d) $(-4,-4)$

Sol. Correct option: (b) $(4,-2)$

[CBSE Marking Scheme, 2020]

Detailed Solution:

$$\begin{array}{}\text{(1)}& \begin{array}{rlrllllllllll}{x}^2-2x-8& =0& & \text{ or, }& x^2-4x+2x-8& =0& \text{ or, }& x(x-4)+2(x-4)& =0\text{ or, }& (x-4)(x+2)& =0\text{ or, }& x& =4,x=-2\end{array}\end{array}$$

(ii) The highway overpass is represented graphically. Zeroes of a polynomial can be expressed graphically. Number of zeroes of polynomial is equal to number of points where the graph of polynomial:

(a) Intersects $X$-axis

(b) Intersects Y-axis

(c) Intersects $\mathrm{Y}$-axis or $\mathrm{X}$-axis

(d) None of the above

Sol. Correct option: (a)

Explanation: We know that the number of zeroes of polynomial is equal to number of points where the graph of polynomial intersects X-axis.

(iii) Graph of a quadratic polynomial is a:

(a) straight line

(b) circle

(c) parabola

(d) ellipse

Sol. Correct option: (c)

Explanation: Here, the given graph of a quadratic polynomial is a parabola.

1

(iv) The representation of Highway Underpass whose one zero is 6 and sum of the zeroes is 0 , is:

(a) $x^2-6x+2$

(b) $x^2-36$

(c) $x^2-6$

(d) $x^2-3$

Sol. Correct option: (b)

$$x^2-36$$

[CBSE Marking Scheme, 2020]

Detailed Solution:

$$\begin{array}{rlr}& x^2-36=0\Rightarrow x^2=36& \Rightarrow x=6,-6.\Rightarrow x=\pm \sqrt{36}\end{array}$$

(v) The number of zeroes that polynomial $f(x)=(x-2{)}^2$ +4 can have is:

(a) 1

(b) 2

(c) 0

(d) 3

Sol. Correct option: (c)

Explanation: We have,

$$\begin{array}{rlrl}f(x)& =(x-2{)}^2+4& =x^2+4-4x+4& =x^2-4x+8.\end{array}$$

i.e., It has no factorisation.

Hence no real value of $x$ is possible, i.e., no zero. 1