
Case Study  1 : Q. 1 to 5 
The below pictures show few natural examples of parabolic shape which can be represented by a quadratic polynomial. A parabolic arch is an arch in the shape of a parabola. In structures, their curve represents an efficient method of load, and so can be found in bridges and in architecture in a variety of forms. Based on the above information, answer the following questions: 

In the standard form of quadratic polynomial, \[a{{x}^{2}}+bx+c,\] a, b and c:
A)
all are real numbers done
clear
B)
all are rational numbers done
clear
C)
'a' is a nonzero real number and b and c are any real numbers done
clear
D)
all are integers done
clear
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If the roots of the quadratic polynomial are equal, and the discriminant \[D={{b}^{2}}4ac,\] then:
A)
\[D>0\] done
clear
B)
\[D<0\] done
clear
C)
\[D\ge 0\] done
clear
D)
\[D=0\] done
clear
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If \[\alpha \] and \[\frac{1}{\alpha }\]are the zeroes of the quadratic polynomial \[2{{x}^{2}}x+8k,\] then k is:
A)
\[4\] done
clear
B)
\[\frac{1}{4}\] done
clear
C)
\[\frac{1}{4}\] done
clear
D)
\[2\] done
clear
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The graph of \[{{\text{x}}^{\text{2}}}+\text{4}=0:\]
A)
intersects Xaxis at two distinct points done
clear
B)
touches Xaxis at a point done
clear
C)
neither touches nor intersects Xaxis done
clear
D)
either touches or intersects Xaxis. done
clear
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If the sum of the roots is \[p\] and product of the roots \[\frac{1}{p}\] is, then the quadratic polynomial is:
A)
\[k\left( p{{x}^{2}}+\frac{x}{p}+1 \right)\] done
clear
B)
\[k\left( p{{x}^{2}}\frac{x}{p}1 \right)\] done
clear
C)
\[k\left( {{x}^{2}}+px\frac{1}{p} \right)\] done
clear
D)
\[k\left( {{x}^{2}}px+\frac{1}{p} \right)\] done
clear
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Case Study : Q. 6 to 10 
Manya is a body posture, originally and still a general term for a sitting meditation pose, and later extended in hatha yoga and modem yoga as exercise, to any type of pose or position, adding reclining, standing, inverted, twisting, and balancing poses. In the figure, one can observe that poses can be related to representation of quadratic polynomial. 

Based on the above information, answer the following questions: 
The shape of the poses shown is:
A)
spiral done
clear
B)
ellipse done
clear
C)
linear done
clear
D)
parabola done
clear
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The graph of parabola opens downwards, if..........
A)
\[a\ge 0\] done
clear
B)
\[a=0\] done
clear
C)
\[a<0\] done
clear
D)
\[a>0\] done
clear
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In the graph, how many zeroes are there for the polynomial? 

A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
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The quadratic polynomial of the two zeroes in the above shown graph are:
A)
\[k({{x}^{2}}2x8)\] done
clear
B)
\[k({{x}^{2}}+2x8)\] done
clear
C)
\[k({{x}^{2}}+2x+8)\] done
clear
D)
\[k({{x}^{2}}2x+8)\] done
clear
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The zeroes of the quadratic polynomial \[4\sqrt{3}\,{{x}^{2}}+5x2\sqrt{3}\] are:
A)
\[\frac{2}{\sqrt{3}}.\frac{\sqrt{3}}{4}\] done
clear
B)
\[\frac{2}{\sqrt{3}}.\frac{\sqrt{3}}{4}\] done
clear
C)
\[\frac{2}{\sqrt{3}}.\frac{\sqrt{3}}{4}\] done
clear
D)
\[\frac{2}{\sqrt{3}}.\frac{\sqrt{3}}{4}\] done
clear
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Case Study : Q. 11 to 15 
Applications of parabolas  highway overpasses/ underpasses. A highway underpass is parabolic in shape. 

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. 

Parabolic chamber \[y=2{{x}^{2}}/nw\] 
The axis of symmetry runs through the maximum or minimum point of the parabola which is called the vertex. 

Based on the above information, answer the following questions. 
If the highway overpass is represented by \[{{x}^{2}}2x8\].  
Then its zeroes are:  
A)
\[(2,4)\] done
clear
B)
\[(4,2)\] done
clear
C)
\[(2,2)\] done
clear
D)
\[(4,4)\] done
clear
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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 Xaxis done
clear
B)
Intersects Vaxis done
clear
C)
Intersects /axis or Xaxis done
clear
D)
None of these done
clear
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Graph of a quadratic polynomial is a:
A)
straight line done
clear
B)
circle done
clear
C)
parabola done
clear
D)
ellipse done
clear
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The representation of Highway Underpass whose one zero is 6 and sum of the zeroes is 0, is:
A)
\[{{x}^{2}}6x+2\] done
clear
B)
\[{{x}^{2}}36\] done
clear
C)
\[{{x}^{2}}6\] done
clear
D)
\[{{x}^{2}}3\] done
clear
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The number of zeroes that polynomial \[f(x)={{(x2)}^{2}}+4\] can have is:
A)
1 done
clear
B)
2 done
clear
C)
0 done
clear
D)
3 done
clear
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Case Study : Q. 16 to 20 
A student was given the task to prepare a graph of quadratic polynomial \[y=82x+{{x}^{2}}.\]. To draw this graph he take seven values of y corresponding to different values of x. After plotting the points on the graph paper with suitable scale, he obtain the graph as shown below. 

Based on the above graph, answer the following questions: 
What is the graph of a quadratic polynomial called?
A)
Parabola done
clear
B)
Hyperbola done
clear
C)
Ellipse done
clear
D)
None of these done
clear
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The zeroes of given quadratic polynomial are:
A)
\[2,4\] done
clear
B)
\[2,4\] done
clear
C)
\[3,4\] done
clear
D)
\[3,4\] done
clear
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Read from the graph the value of y corresponding to \[x=1\] is:
A)
\[8\] done
clear
B)
\[6\] done
clear
C)
\[5\] done
clear
D)
\[2\] done
clear
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The graph of the given quadratic polynomial cut at which points on the Xaxis?
A)
\[(2,0),\,\,(4,0)\] done
clear
B)
\[(0,2),\,\,(0,4)\] done
clear
C)
\[(0,2),\,\,(0,8)\] done
clear
D)
\[None\,\, of\,\, the\,\,above\] done
clear
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The graph of the given quadratic polynomial cut at which point on Xaxis?
A)
\[(8,0)\] done
clear
B)
\[(0,8)\] done
clear
C)
\[(10,0)\] done
clear
D)
\[(10,0)\] done
clear
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Case Study : Q. 21 to 25 
In a classroom, four students Anil, Jay, Richa and Suresh were asked to draw the graph of \[p(x)=a{{x}^{2}}+bx+c\]. 
Following graphs are drawn by the students: 

Based on the above graphs, answer the following questions: 
How many students have drawn the graph correctly?
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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Which type of polynomial is represented by Jay's graph?
A)
Linear done
clear
B)
Parabola done
clear
C)
Zigzag done
clear
D)
None of these done
clear
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How many zeroes are there for the Richa's graph?
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
4 done
clear
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If \[p(x)=a{{x}^{2}}+bx+c\] and \[a+b+c=0,\] then one zero is:
A)
\[\frac{b}{a}\] done
clear
B)
\[\frac{c}{a}\] done
clear
C)
\[\frac{b}{c}\] done
clear
D)
\[\frac{c}{a}\] done
clear
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If \[p(x)=a{{x}^{2}}+bx+c\] and \[a+c=b,\] then one of the zeroes is:
A)
\[\frac{b}{a}\] done
clear
B)
\[\frac{c}{a}\] done
clear
C)
\[\frac{c}{a}\] done
clear
D)
\[\frac{b}{a}\] done
clear
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Case Study : Q. 26 to 30 
Ramesh was asked by one of his friends to find the polynomial whose zeroes are \[\frac{2}{\sqrt{3}}\] and \[\frac{\sqrt{3}}{4}\]. He obtained the polynomial as shown below: 
Let \[\alpha =\frac{2}{\sqrt{3}}\] and \[\alpha =\frac{\sqrt{3}}{4}\] 
\[\Rightarrow \,\,\,\,\,\,\,\alpha +\beta =\frac{2}{\sqrt{3}}+\frac{\sqrt{3}}{4}=\frac{8+1}{4\sqrt{3}}=\frac{7}{4\sqrt{3}}\] 
\[\Rightarrow \,\,\,\,\,\,\,\alpha \beta =\frac{2}{\sqrt{3}}\times \frac{\sqrt{3}}{4}=\frac{1}{2}\] 
Required polynomial \[={{x}^{2}}(\alpha +\beta )x+\alpha \beta \] 
\[={{x}^{2}}\left( \frac{7}{4\sqrt{3}} \right)x+\left( \frac{1}{2} \right)\] 
\[={{x}^{2}}+\frac{7x}{4\sqrt{3}}\frac{1}{2}\] 
\[=4\sqrt{3}{{x}^{2}}+7x2\sqrt{3}\] 
His another friend Kavita pointed out that the polynomial obtained is not correct. 
Based on the above situation, answer the following questions: 
Is the claim of Kavita correct?
A)
Yes done
clear
B)
No done
clear
C)
Can't say done
clear
D)
None of these done
clear
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If yes, then the correct quadratic polynomial is:
A)
\[4\sqrt{3}{{x}^{2}}5x+2\sqrt{3}\] done
clear
B)
\[4\sqrt{3}{{x}^{2}}+5x2\sqrt{3}\] done
clear
C)
\[4\sqrt{3}{{x}^{2}}+5x+2\sqrt{3}\] done
clear
D)
\[4\sqrt{3}{{x}^{2}}5x2\sqrt{3}\] done
clear
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The value of \[{{\alpha }^{2}}+{{\beta }^{2}}\] is:
A)
\[\frac{53}{48}\] done
clear
B)
\[\frac{59}{48}\] done
clear
C)
\[\frac{73}{48}\] done
clear
D)
\[\frac{71}{48}\] done
clear
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What is the value of the correct polynomial if \[x=1\]?
A)
\[5+2\sqrt{3}\] done
clear
B)
\[52\sqrt{3}\] done
clear
C)
\[56\sqrt{3}\] done
clear
D)
\[5+6\sqrt{3}\] done
clear
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The value of \[{{\alpha }^{3}}{{\beta }^{3}}\]is:
A)
\[\frac{539}{192\sqrt{3}}\] done
clear
B)
\[\frac{539}{192\sqrt{3}}\] done
clear
C)
\[\frac{539\sqrt{3}}{192}\] done
clear
D)
\[\frac{539\sqrt{3}}{192}\] done
clear
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Case Study : Q. 31 to 35 
A group of school friends went on an expedition to see caves. One person remarked that the entrance of the caves resembles a parabola, and can be represented by a quadratic polynomial \[f(x)=a{{x}^{2}}+bx+c,\] \[a\ne 0,\] where a, b and c are real numbers. 

Based on the above information give the answer of the following questions. 
If one of the zeroes of the quadratic polynomial \[(p1){{x}^{2}}+px+1\]is 4, then the value of p is:
A)
\[\frac{3}{5}\] done
clear
B)
\[\frac{3}{4}\] done
clear
C)
\[\frac{2}{5}\] done
clear
D)
\[\frac{1}{5}\] done
clear
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The zeroes of the quadratic polynomial \[{{x}^{2}}+20x+96\]are:
A)
both positive done
clear
B)
both negative done
clear
C)
one positive and other negative done
clear
D)
both equal done
clear
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The quadratic polynomial whose zeroes are 5 and \[12\] is given by:
A)
\[{{x}^{2}}+7x60\] done
clear
B)
\[15{{x}^{2}}x6\] done
clear
C)
\[{{x}^{2}}7x+60\] done
clear
D)
\[15{{x}^{2}}+x+6\] done
clear
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If one zero of the polynomial \[f(x)=5{{x}^{2}}+13x+m\] is reciprocal of the other, then the value of m is:
A)
6 done
clear
B)
0 done
clear
C)
5 done
clear
D)
1 done
clear
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Which of the following cannot be the graph, of a quadratic polynomial?
A)
B)
C)
D)
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Case Study : Q. 36 to 40 
Naveeka everyday goes to swimming. One day, Naveeka noticed the water coming out of the pipes to fill the pool. 
She then told her brother that the shape of the path of the water falling is like that of a parabola and also that a parabola can be represented by a quadratic polynomial which has atmost two zeroes. 

Based on the given information, answer the following questions. 
The number of zeroes that polynomial \[f(x)={{(x+2)}^{2}}+6\]can have is:
A)
0 done
clear
B)
1 done
clear
C)
2 done
clear
D)
3 done
clear
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If the product of the zeroes of the quadratic polynomial \[f(x)=a{{x}^{2}}6x6\] is 4, then the value of a is:
A)
\[\frac{3}{2}\] done
clear
B)
\[\frac{3}{2}\] done
clear
C)
\[\frac{2}{3}\] done
clear
D)
\[\frac{2}{3}\] done
clear
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The flow of the water in the pool is represented by \[{{x}^{2}}2x8,\] then its zeroes are:
A)
\[2,4\] done
clear
B)
\[4,2\] done
clear
C)
\[2,2\] done
clear
D)
\[4,4\] done
clear
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If a and p be the zeroes of the polynomial \[{{x}^{2}}1,\]then the value of \[\frac{1}{\alpha }+\frac{1}{\beta }\]is:
A)
\[0\] done
clear
B)
\[\frac{1}{2}\] done
clear
C)
\[1\] done
clear
D)
\[1\] done
clear
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A quadratic polynomial whose one zero is \[3\] and product of zeroes is 0, is:
A)
\[3{{x}^{2}}+3\] done
clear
B)
\[{{x}^{2}}3x\] done
clear
C)
\[{{x}^{2}}+3x\] done
clear
D)
\[3{{x}^{2}}3\] done
clear
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