Nature of Solutions Graphically

Q.

The slope of a line parallel to X-axis is ___ .

Q.

A polynomial $y=f\left(x\right)$ when represented on graph cuts $x-\text{axis}$ at 2 points and $y-\text{axis}$ at 3 points. What is the number of zeroes of $f\left(x\right)$ ?

1. $0$
2. $3$
3. $5$
4. $2$

Q.

Polynomials in Real Life

Polynomials are everywhere. It is found in a roller coaster of an amusement park, the slope of a hill, the curve of a bridge or the continuity of a mountain range. They play a key role in the study of algebra, in analysis and on the whole many mathematical problems involving them.

Based on the given information, answer the following question:

If the path traced by the Roller Coaster is represented by the graph $y=p\left(x\right)$, find the number of zeroes?

1. 0
2. 1
3. 2
4. 3

Q. Question 5
Classify the following as linear, quadriatic and cubic polynomials:
(i) $x^2+x$
(ii) $x-x^3$
(iii) $y+y^2+4$
(iv) 1 + x
(v) 3t
(vi) $r^2$
(viii)$7x^3$

Q. Graph of quadratic function is

1. circle
2. parabola
3. triangle
4. rectangle

Q. The graph of the equation $3x^2+4x+5$ = 0 intersects the x-axis at point(s).

1. \N
2. 1
3. 2
4. 3

Q. Which of the following graph represents $y=-x^2$?

1. 
2. 
3. 
4. 

Q.

Draw the graph of $y=x^2-4x+6$ and show that it does not have real roots.

Q.

If ${\log }_{a}x>y$ and $0<a<1$ . Then

1. $x>y$

2. $x>a^y$

3. $x=a^y$

4. $x<a^y$

Q. Choose the option having correct match regarding the shape of the parabola for the polynomial $a{x}^2+bx+c$.
I. If a > 0, then it is opening upward.

ii. If a > 0, then it is opening downward.

iii. If a < 0, then it is opening upward.

iv. If a < 0, then it is opening downward.

1. I and III
2. I and IV
3. II and IV
4. II and III

Q. Draw a graph for the polynomial $p\left(x\right)=x^2+3x-4$ and find its zeroes from the graph.

Q.

The roots of a quadratic equation $x^2-4x-{\log }_{3}a=0$ are real. Then what is the least value of $a$?

1. 64

2. $\frac{1}{81}$

3. $\frac{1}{64}$

4. 81

Q. The quadratic expression $(2x+1{)}^2-px+q\ne 0$ for any real $x$ if

1. $p^2-16p-8q<0$
2. $p^2-8p+16q<0$
3. $p^2-8p-16q<0$
4. $p^2-16p+8q<0$

Q. lf $x\in R$ and the roots of $a{x}^2+bx+c=0$ are non-real complex, then the sign of $a^2{x}^2+abx+ac$ is

1. always non-negative
2. always zero
3. always positive
4. always negative

Q. The graph of a quadratic polynomial $y={px}^2-qx+r$ is as shown in the adjacent figure, then

1. $p+q>0$
2. $r^2-4q<0$
3. $r^2-4p<0$
4. $r-p-q>0$

Q. If $a>0$ and $b^2-4ac>0$, then the graph of $y=a{x}^2+bx+c$

1. lies entirely above the x-axis.
2. is concave upwards and cuts the x-axis.
3. touches the x-axis and lies below it.
4. is concave downwards.

Q. The graph of $y=x^2+3x+4$

1. lies entirely below the x-axis.
2. lies entirely above the x-axis.
3. cuts the x-axis.
4. touches the x-axis and lies below it.

Q. If $b^2-4ac=0$ then the graph of $y=a{x}^2+bx+c$

1. touches the x-axis.
2. lies entirely above the x-axis.
3. cuts x-axis in two real points.
4. can not be determined.

Q.

Polynomials in Real Life

Polynomials are everywhere. It is found in a roller coaster of an amusement park, the slope of a hill, the curve of a bridge or the continuity of a mountain range. They play a key role in the study of algebra, in analysis and on the whole many mathematical problems involving them.

Based on the given information, answer the following question:

If the path traced by the Roller Coaster is represented by the graph $y=p\left(x\right)$, find the number of zeroes?

1. 0
2. 1
3. 2
4. 3

Q. Draw the graph of $y=12x^3-4x^2-3x+1$. Hence find the number of positive zeroes.

Q. Let $f\left(x\right)$ be a quadratic expression which is positive for all real $x$. If $g\left(x\right)=f\left(x\right)+f^{{}^{\prime }}\left(x\right)+f^{{}^{\prime \prime }}\left(x\right)$, then for any real $x$

1. $g\left(x\right)<0$
2. $g\left(x\right)>0$
3. $g\left(x\right)=0$
4. $g\left(x\right)\ge 0$

Q. If $x^2-x+a-3<0$ for atleast one negative value of $x$, then complete set of values of $a$ is

Q. If the LCM of $pq$ and $p^{3}q$ is expressible in the form of $\left(pq\right)k$, then find the value of $k$.

1. $p^3$
2. $p$
3. $p^2$

Q. For a triangle ABC it is given that $cosA+cosB+cosC=\frac{3}{2}$ Prove that the triangle is equilateral

1. True
2. False

Q. If $b>a$, then the equation $(x-a)(x-b)-1=0$ has

1. both roots in $[a,b]$
2. both roots in $(-\infty ,a)$
3. both roots in $(b,\infty )$
4. one root in $(-\infty ,a)$ and other in $(b,\infty )$

Q. The given figure shows the graph of $f\left(x\right)=a{x}^2+bx+c$, then

1. $ac<0$
2. $abc<0$
3. $bc>0$
4. $ab>0$

Q. If the LCM of $pq$ and $p^{3}q$ is expressible in the form of $\left(pq\right)k$, then find the value of $k$.

1. $p^3$
2. $p$
3. $p^2$

Q. The adjoining figure shows the graph of $y=a{x}^2+bx+c$. Then

1. $a>0$
2. $c>0$
3. $b>0$
4. $b^2<4ac$

Q. Assertion :STATEMENT-$I$ : The curve $y=\frac{-x^2}{2}+x+1$ is symmetric with respect to the line $x=1$. Reason: STATEMENT -2 : A parabola is symmetric about its axis.

1. Statement -1 is True, Statement -2 is true; Statement-2 is a correct explanation for Statement-1
2. Statement -1 is True, Statement -2 is true; Statement-2 is NOT a correct explanation for Statement-1
3. Statement -1 is True, Statement -2 is False
4. Statement -1 is False, Statement -2 is True

Q. For all $x\in R$, $x^2+2ax+(10-3a)>0$. then the interval in which $a$ lies is

1. $a<-5$
2. $-5<a<2$
3. $a>5$
4. $2<a<5$