Graph of Quadratic Expression

Q. The number of integral values of $m$ for which the quadratic expression,
$(1+2m)x^2-2(1+3m)x+4(1+m),x\in R,$ is always positive, is

1. $8$
2. $3$
3. $7$
4. $6$

Q.

Let $\left[t\right]$ denote the greatest integer $\le t$ Then the equation in $x$, ${\left[x\right]}^2+2[x+2]-7=0$ has:

1. exactly four integral solutions.

2. infinitely many solutions.

3. no integral solution.

4. exactly two solutions

Q. If the roots of the equation $(m-2)x^2-(8-2m)x-(8-3m)=0$ are real and opposite in sign, then the number of integral value(s) of $m$ is

Q. Let the graph of $f\left(x\right)=a{x}^2+bx+c$ passes through origin and makes an intercept of $10$ units on $x-$axis. If the maximum value of $f\left(x\right)$ is $25$, then the least value of $|a+b+c|$ is

Q. Let $y=a{x}^2+bx+c(a\ne 0)$ and $a,b,c\in R$. If $abc>0$, then which of the following graph(s) satisfy the given condition?

1. 
2. 
3. 
4. 

Q. The range of $k$ for which both the roots of the quadratic equation $(k+1)x^2-3kx+4k=0$ are greater than $1$, is

1. $[-\frac{16}{7},-1)$
2. $(-\infty ,-1)$
3. $(-\frac{16}{7},-1)$
4. $[-\frac{16}{7},-1]$

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

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

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

Q. If the curve $y=a{x}^2+bx+c=0$ has $y$-intercept $6$ and vertex as $(\frac{5}{2},\frac{49}{4}),$ then the value of $a+b+c$ is

Q. Which of the following is NOT the graph of a quadratic polynomial ?

1. 
2. 
3. 
4. 

Q. If $f\left(x\right)$ is a quadratic polynomial such that graph of $y=f\left(x\right)$ touches at $(4,0)$ and intersects the positive $y-$axis at $4$, then which of the following is/are correct?

1. $f\left(2\right)=1$
2. $f\left(3\right)=\frac{1}{4}$
3. $f\left(x\right)=\frac{1}{4}{x}^2-2x+4$
4. $f\left(x\right)=\frac{1}{2}{x}^2-x+\frac{5}{2}$

Q. If $c>0$ and the equation $3a{x}^2+4bx+c=0$ has no real root, then

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

Q. Set of values of $a$ for which both the roots of the quadratic polynomial $f\left(x\right)=a{x}^2+(a-3)x+1$ lie on one side of the $y-$axis is

1. $[0,1]\cup [9,\infty )$
2. $[0,1)\cup [3,\infty )$
3. $(0,1)\cup (3,\infty )$
4. $(0,1]\cup [9,\infty )$

Q. If the length of $x-$ intercept made by the graph of $f\left(x\right)=4x^2-10x+4$ is $k$, then the value of $\left[k\right]$ is
(where [.] denotes the greatest integer function)

Q. If both the roots of $a{x}^2+bx+c=0$ are negative and $b<0$, then which of the following statements is always true?

1. $a<0,c>0$
2. $a<0,c<0$
3. $a>0,c>0$
4. $a>0,c<0$

Q. If both the roots of $a{x}^2+bx+c=0$ are real, positive and distinct, then
(where $\Delta =b^2-4ax$)

1. $\Delta >0,ab>0,bc>0$
2. $\Delta >0,ab<0,ac<0$
3. $\Delta >0,ab<0,ac>0$
4. $\Delta >0,ab<0,bc>0$

Q. The equation of the axis of symmetry of the quadratic polynomial $y=3x^2+6x-1$ is

1. $y=4$
2. $y=-4$
3. $x=-1$
4. $x=1$

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. contains exactly two elements.
4. is a singleton.

Q. The set of values of $m$ for which $f\left(x\right)=x^2-(m-3)x+m$ intersects the positive direction of $x-$axis atleast once, is

1. $(-\infty ,0)\cup (3,\infty )$
2. $(-\infty ,1]\cup (9,\infty )$
3. $(-\infty ,0]\cup (3,\infty )$
4. $(-\infty ,0)\cup [9,\infty )$

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

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

Q. The number of integral value(s) of $a$ for which ${\log }_e(x^2+5x)={\log }_e(x+a+3)$ has exactly one solution is

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

Q. If a $ϵ$ R and the equation $-3(x-[x]{)}^2+2(x-\left[x\right])+a^2=0$
(where [x] denotes the greatest integer $\le x)$ has no integral solution, then all posssible values of $a$ lie in the interval:

1. $(-2,-1)$
2. $(-\infty ,-2)\cup (2,\infty )$
3. $(-1,0)\cup (0,1)$
4. (1, 2)

Q.

If a, b, c $\in$ R and a $\ne$ 0, c < 0, and if the quadratic equation $a{x}^2+bx+c$ = 0 has imaginary roots, then a + b + c is

1. Negative

2. Zero

3. Can't say

Q.

Which among the following is the correct graphical representation of $y=-x^2+4x+1$ ?

1. 

2. 

3. 

4. 

Q.

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

1. 

2. 

3. 

4. 

Q. Which of the following represents the graph of $f\left(x\right)=a{x}^2+bx+c$ where $a<b<0<c$ ?

1. 
2. 
3. 
4. 

Q. The number of integral values of $x$ for which $f\left(x\right)=2x^2-20x+42$ is negative is

Q. If a $ϵ$ R and the equation $-3(x-[x]{)}^2+2(x-\left[x\right])+a^2=0$
(where [x] denotes the greatest integer $\le x)$ has no integral solution, then all posssible values of $a$ lie in the interval:

1. $(-2,-1)$
2. $(-\infty ,-2)\cup (2,\infty )$
3. $(-1,0)\cup (0,1)$
4. (1, 2)

Q.

If a, b, c $ϵ$ R and a $\ne$ 0, c > 0, the graph of $f\left(x\right)$ = $a{x}^2+bx+c$ for which $f\left(x\right)=0$ has only imaginary roots, will look like

1. 

2. 

3. 

4.