Graph of Quadratic Expression

Q.

The angle between the two vectors $\overrightarrow{A}=5\hat{i}+5\hat{j},\overrightarrow{B}=5\hat{i}-5\hat{j}$ will be

1. Zero

2. ${45}^{°}$

3. ${90}^{°}$

4. ${180}^{°}$

Q. An equation $a_0+a_{1}x+a_2{x}^2+...+a_{99}{x}^{99}+x^{100}=0$ has root ${}^{99}{C}_0,{}^{99}{C}_1,{}^{99}{C}_2,...{}^{99}{C}_{99}$ then

1. $({}^{99}{C}_0{)}^2+({}^{99}{C}_1{)}^2+({}^{99}{C}_2{)}^2+...({}^{99}{C}_{99}{)}^2=(a_{99}{)}^2-2a_{98}$
2. $({}^{99}{C}_0{)}^2+({}^{99}{C}_1{)}^2+({}^{99}{C}_2{)}^2+...({}^{99}{C}_{99}{)}^2=(a_{99}{)}^2+2a_{98}$
3. $a_{98}=2^{197}+\frac{1}{2}{}^{198}{C}_{99}$
4. $a_{98}=2^{197}-\frac{1}{2}{}^{198}{C}_{99}$

Q. The point on the curve $3y=6x-5x^3,$ at which the normal passes through origin, is

1. $(1,\frac{1}{3})$
2. $(\frac{1}{3},1)$
3. $(2,-\frac{28}{3})$
4. None of these.

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

1. a > 0
2. b < 0
3. b² - 4ac = 0
4. c < 0

Q.

The number of solutions of $x^2+|x-1|=1is$

1. 0

2. 1

3. 2

4. 3

Q. The graph of a quadratic polynomial $f\left(x\right)=a{x}^2+bx+c$ is shown below. Then which of the following option(s) is/are correct ?

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

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. The graph of $f\left(x\right)=2x^2-3x+2$ is

1. 
2. 
3. 
4. 

Q. If $a(p+q{)}^2+2bpq+c=0$ and $a(p+r{)}^2+2bpr+c=0$, then $p^2+\frac{c}{a}$ is

1. $pr$
2. $rq$
3. $r^2$
4. $q^2$

Q. If $x^4+3\cos (a{x}^2+bx+c)=2(x^2-2)$ has two solutions with $a,b,c\in (2,5),$ then which of the following can be TRUE ?

1. $a+b+c=\pi$
2. $a-b+c=\pi$
3. $a+b+c=3\pi$
4. $a-b+c=3\pi$

Q. Define the function $f:R\to R$ by $y=f\left(x\right)=x^2,x\in R$. Complete the table given below by using this defintion. What is the domain and range of this function ? Draw the graph of $f$.

Q. The most general solutions of $2^{1+|\cos x|+{\cos }^{2}x+|\cos x{|}^3+...\infty }=4$ are given by

1. 
2. 
3. 
4. None of these

Q. If the quadratic expression
$a{x}^2+(a-2+3^{\sqrt{{\log }_{3}5}}-5^{\sqrt{{\log }_{5}3}})x+(5^{{\log }_{5}3}-3^{{\log }_{3}5})$ is negative for exactly two integral values of $x$, then the possible value(s) of $a$ is/are

1. $-\frac{2}{3}$
2. $1$
3. $2$
4. $-\frac{1}{2}$

Q.

The coordinate of the point on y^2 =8x which is closest from x^2 +(y+6)^2 =1 is

Q. The least positive integral value of $a$ for which the equation $x^2-2(a-1)x+2a+1=0$ has both roots positive is

Q. If $a<0$ and $D<0$, then the graph of $y=a{x}^2+bx+c$
$(\text{where}D=b^2-4ac)$

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

Q. The graph of the quadratic polynomial $y=a{x}^2+bx+c$ has its vertex at $(4,-5)$ and two $x$ intercepts, one positive and one negative. Which of the following hold(s) good?

1. $a>0$
2. $bc>0$
3. $2b+c+5=0$
4. $b+8a=0$

Q. If $y=f\left(x\right)=a{x}^2+bx+c$ is a quadratic polynomial, then

1. Its graph is concave upward if $a>0$
2. Its graph is concave downward if $a<0$
3. $\frac{d^{2}y}{d{x}^2}=2a$
4. $\frac{d^{2}y}{d{x}^2}=c$

Q. A, B, C are three points on the curve xy - x - y - 3 = 0 which are not collinear. D, E, F are foot of perpendiculars from vertices A, B, C to the sides BC, CA and AB of $△ABC$ respectively. If $(\alpha ,\alpha$\) is incentre of $△DEF$ then '$\alpha$' can be

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

Q. The quadratic polynomial $p\left(x\right)$ has the following properties: $p\left(x\right)\ge 0$ for all real numbers, $p\left(1\right)=0$ and $p\left(2\right)=2$. Then the value of $p\left(3\right)$ is

Q. The number of positive integral values of $x$ satisfying $\frac{x^2-7x+12}{x^2-13x+22}\le 1$, is

Q. The number of solutions of the equation $x^3+2x^2+5x+2cosx=0$ in $[0,2\pi ]$ is

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

Q.

Verify Rolle's theorem for the function $f\left(x\right)=x^2+2x-8,xϵ[-4,2]$

Q.

If $\left\{\begin{array}{l}\frac{1}{\left|x\right|},x\ge 1\\ a{x}^2+b,x<1\end{array}\right.$is differentiable at every point of the domain, then the values of $a$ and $b$ are respectively:

1. $\frac{5}{2},-\frac{3}{2}$

2. $-\frac{1}{2},\frac{3}{2}$

3. $\frac{1}{2},\frac{1}{2}$

4. $\frac{1}{2},-\frac{3}{2}$

Q.

The parametric representation of a parabola is x=3+t² , y=2t-1. Its focus is at

Q. If $(1,3)$ is the point of inflection of the curve $y=a{x}^3+b{x}^2,$ then the value of $4(a^2+b^2)$ is

Q. If parabola y=(x−2)2 is shifted by 3 units toward right, then will be the equation of new parabola obtained.

1. $y=(x-3{)}^2$
2. $y=(x+1{)}^2$
3. $y=(x-5{)}^2$
4. $y=(x+3{)}^2$

Q. If $2a+3b+6c=0$, then atleast one root of the equation $a{x}^2+bx+c=0$ lies in the interval

1. $(0,1)$
2. $(1,2)$
3. $(2,3)$
4. $(1,3)$

Q. The vertex of the parabola represented by the quadratic equation $y=x^2+2x+6$ is

1. (1, -5)
2. (-1, 5)
3. (1, -6)
4. (-1, 6)

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. $7$
3. $6$
4. $3$