Graphs of Quadratic Equation for different values of D when a>0

Q. If $2a{x}^2+3bx+5c=0,a\in R-\left\{0\right\},c>0$ does not have any real roots, then which of the following is/are always true?

1. $a<0$
2. $4a-3b+3c>0$
3. $4a-3b+3c<0$
4. $a>0$

Q.

The range of the expression $f\left(x\right)=3x^2-12x+5$ is

1. $[-7,\infty )$

2. $(-\infty ,-4]$

3. $[-4,\infty )$

4. None of these

Q. The graph of $y=a{x}^2+bx+c$ is shown in the figure below, where $Q$ is the vertex of the parabola. If the length of $PQ$ and $OR$ are $9\text{units}$ and $5\text{units}$ respectively and area of $△OBQ$ is $\frac{45}{4}\text{sq. units}$, then which of the following is/are correct?

1. length of $AB$ is $4\text{units}$
2. $\left|\frac{b}{a}\right|<1$
3. length of $AB$ is $3\text{units}$
4. $\left|\frac{b}{a}\right|>1$

Q. Consider the function $f\left(x\right)=x^2-4x+17.$ If $M$ and $m$ are the maximum and minimum values of $f$ in $[0,3]$ respectively, then the value of $2M-m$ is

Q. Number of roots of the equation $x^2-2x-{\log }_2|1-x|=3$ is

Q. For the given graph of the quadratic polynomial $y=a{x}^2+bx+c$ as shown below. Select the correct options.

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

Q. For the quadratic expression $y=a{x}^{2}bx+c,$ choose the correct pair.

1. $a<0,D>0$
2. $a>0,D=0$
3. $g\left(x\right)$
4. $a>0,D<0$
5. $a<0,D<0$
6. $f\left(x\right)$
7. $h\left(x\right)$
8. $p\left(x\right)$

Q.

A: The minimum value of '$y$' for the expression $y=2x^2+4x+5$ occur at $x=$______

B: The maximum value of expression $-3x^2+12x+5$ is _______

1. $-1,27$

2. $2,17$

3. $-1,17$

4. $2,27$

Q. The maximum value of $f\left(x\right)=x^2+7x+9$ is

1. $\frac{13}{4}$
2. Infinity
3. $9$
4. $-\frac{13}{4}$

Q. The number of real solution(s) does this graph (representing a quadratic polynomial) have is

1. $1$
2. $0$
3. $2$
4. cannot be determined

Q. The graph of $y=a{x}^2+bx+c$ is shown in the figure below, where $Q$ is the vertex of the parabola. If the length of $PQ$ and $OR$ are $9\text{units}$ and $5\text{units}$ respectively and area of $△OBQ$ is $\frac{45}{4}\text{sq. units}$, then which of the following is/are correct?

1. length of $AB$ is $3\text{units}$
2. $\left|\frac{b}{a}\right|<1$
3. length of $AB$ is $4\text{units}$
4. $\left|\frac{b}{a}\right|>1$

Q. Let $S$ be the set of all real roots of the equation, $3^x(3^x-1)+2=|3^x-1|+|3^x-2|$. Then $S$ :

1. is an empty set.
2. contains at least four elements.
3. is a singleton.
4. contains exactly two elements.

Q.

Graph of $y=x^2+7x+3$ intersects the $x$-axis at __ points

Q. If the graph of quadratic polynomial $a{x}^2+bx+c$ is


then which of the following is/are correct?

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

Q. If $2a{x}^2+3bx+5c=0,a\in R-\left\{0\right\},c>0$ does not have any real roots, then which of the following is/are always true?

1. $a<0$
2. $4a-3b+3c<0$
3. $a>0$
4. $4a-3b+3c>0$

Q. The graph of a quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ is shown below


Which of the following options is/are true for the graph?

1. $4a-2b+c<0$
2. $9a-3b+c<0$
3. $a-b+c>0$
4. $4a-6b+9c>0$

Q.

Plot the graph of $y=(x-3{)}^2+7$

1. 

2. 

3. 

4. 

Q.

One or more options can be correct:

The condition for the roots of the quadratic equation $f\left(x\right)=a{x}^2+bx+c$ to be less than a real value $x_0$ is/are

1. $f\left(x_0\right)>0$

2. $D\ge 0$

3. $f\left(x_0\right)<0$

4. $x_0>\frac{-b}{2a}$

Q. Which of the following cannot be true for the given quadratic equation $ax^2+bx+c = 0 ; c\ne0$ if ratio of roots of this equation is equal to its reciprocal?

Q. The maximum value of $f\left(x\right)=x^2+7x+9$ is

1. $\frac{13}{4}$
2. $9$
3. Infinity
4. $-\frac{13}{4}$

Q. Which of the following is correct for the quadratic polynomial $f\left(x\right)=2x^2+8x-12$

1. the graph is concave downward
2. $y-$ intercept $=2$
3. vertex $=(-2,-20)$
4. $f\left(x\right)=0$ has no real solutions

Q.

Pick the correct plot for the function $y=x^2-2x+6$

1. 

2. 

3. 

4. 
   
   

Q.

What can be the shape of graph of $y=2x^2+bx+c$ where $b,c\in R$?

1. 

2. Can't say

3. 

4. 

Q.

The range of the expression $f\left(x\right)=3x^2-12x+5$ is

1. $[-4,\infty )$

2. None of these

3. $[-7,\infty )$

4. $(-\infty ,-4]$

Q.

What can be the shape of graph of \(y = 2 {x}^{2} + bx + c\) where \( b, c \in~ \mathbb R\)?

Q. The values of $a$ for which the quadratic equation $3x^2+2(a^2+1)x+(a^2-3a+2)=0$ has roots of opposite sign, is

1. $a\in (2,3)$
2. $a\in (1,2)$
3. $a\in (2,\infty )$
4. $a\in (1,\infty )$

Q. Number of roots of the equation $x^2-2x-{\log }_2|1-x|=3$ is

Q. For the given quadratic expression $y=a{x}^2+bx+c,$ if $a>0\&D=0,$ then the graph will intersect the $x-$ axis at

1. $1$ point.
2. $2$ points.
3. $0$ points.

Q.

A: The minimum value of '$y$' for the expression $y=2x^2+4x+5$ occur at $x=$______

B: The maximum value of expression $-3x^2+12x+5$ is _______

1. $-1,17$

2. $-1,27$

3. $2,27$

4. $2,17$

Q. Let $S$ be the set of all real roots of the equation, \(3^x(3^x-1)+2=\left | 3^x-1 \right |+\left | 3^x-2 \right|\). Then $S$ :