Sign of Quadratic Expression

Q. The range of the function $f\left(x\right)=\sqrt{9-x^2}$ is

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

Q.

Consider the two sets: $A=\{m\in R:$ both the roots of$x^2-(m+1)x+m+4=0$ are real$\}$ and$B=[-3,5)$ Which of the following is not true?

1. $A-B=(-\infty ,-3)\cup (5,\infty )$

2. $A\cap B=\{-3\}$

3. $B-A=(-3,5)$

4. $A\cup B=R$

Q.

Consider the function $f\left(x\right)=x^2+ax+b$ where $a,b\in R$. If $f\left(x\right)=0$ has all its roots imaginary, then the roots of $f\left(x\right)+f\left(x\right)+f\left(x\right)=0$ are

1. real and distinct

2. imaginary

3. equal

4. rational and equal

Q. The maximum value of the function $f\left(x\right)=2\sqrt{x-2}+\sqrt{4-x}$ is $\sqrt{K},$ then $K$ is

Q. By considering the graph of quadratic polynomial $y=a{x}^2+bx+c$ as shown below.

Which among the following conclusions is/are correct?

1. $\frac{a-b+c}{abc}=0$
2. $abc(9a+3b+c)<0$
3. $\frac{a+3b+9c}{abc}<0$
4. $abc(a-3b+9c)>0$

Q. Let $f\left(x\right)=x^5+a{x}^4+b{x}^3+c{x}^2+dx-420$ where $a,b,c,d$ are real parameters, be a polynomial. If all zeros of the polynomial $f\left(x\right)$ are integers larger than $1,$ and $f\left(4\right)$ is equal to $k,$ then $k$ is divisible by

1. $2$
2. $3$
3. $5$
4. $6$

Q. If $a,b,c$ and $d$ are real numbers such that $a^2+b^2+c^2+d^2=1$ and if $A=\left[\begin{array}{cc}a+ib& c+id\\ -c+id& a-ib\end{array}\right],$where $i^2=-1,$ then $A^{-1}=$

1. $\left[\begin{array}{cc}a+ib& -c-id\\ c-id& a-ib\end{array}\right]$
2. $\left[\begin{array}{cc}a-ib& c+id\\ -c+id& a+ib\end{array}\right]$
3. $\left[\begin{array}{cc}a-ib& -c-id\\ c-id& a+ib\end{array}\right]$
4. $\left[\begin{array}{cc}a+ib& c+id\\ c-id& a-ib\end{array}\right]$

Q. If $(\lambda -2)x^2+(\lambda -1)x+3<0,\forall x\in R,$ then the range of $\lambda$ is

1. $(-\infty ,2)$
2. $\varphi$
3. $(\frac{14-\sqrt{96}}{2},\frac{14+\sqrt{96}}{2})$
4. $(\frac{14-\sqrt{96}}{2},2)$

Q.

The equation $12x^4-56x^3+89x^2-56x+12=0$

1. has all roots integral

2. has all roots irrational

3. two roots real, two roots imaginary

4. has all roots rational

Q. If both the roots of $a{x}^2+bx+c=0,a\ne 0$ is positive, then

1. $a$ and $c$ are of the same sign.
2. $b$ and $c$ are of the same sign.
3. $a$ and $b$ are of the same sign.
4. $a,b$ and $c$ are of the same sign.

Q. The range of the function $f:R\to R$ defined as $f\left(x\right)=\frac{x^2}{1+x^2}$ is

1. $(0,1)$
2. $[0,1)$
3. $R$
4. $[0,1]$

Q. If $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ has roots which are numerically equal but of opposite sings, the value of m must be:

1. c
2. $\frac{a-b}{a+b}$
3. $\frac{1}{c}$
4. $\frac{a+b}{a-b}$

Q. Find the value of $x$ at which $f\left(x\right)=x^x$ attains the minimum value.

1. $0$
2. $\frac{1}{1+e}$
3. $e$
4. $1/e$

Q. If $a{x}^2+bx+c<0,\forall x\in R$, then $a^2{x}^2+abx+ac$ is

1. always positive
2. always negative
3. equal to zero
4. can be positive or negative

Q. The number of imaginary roots of the equation $4x(x^2+x+3)+5x(x^2-5x+3)=-\frac{3}{2}$ is

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

Q. Let $f\left(x\right)=a{x}^2+bx+c$ and $g\left(x\right)=A{x}^2+bx+\lambda ,$ where $a\ne 0,A\ne 0,a,b,c,A,\lambda \in R$. Roots of $f\left(x\right)=0$ and $g\left(x\right)=0$ are imaginary, then which of the following may be correct?

1. $f\left(x\right)+g\left(x\right)=0$ for some $x$
2. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0\forall x\in R$
3. $\frac{f\left(x\right)}{a}+\frac{g\left(x\right)}{A}>0$ for some $x$
4. Roots of equation $af\left(x\right)+Ag\left(x\right)=0$ are real

Q. The set of real values of $x$, for which $h\left(x\right)=1+2x^2+4x^4+6x^6+\cdots +100x^{100}$ is concave downward is

1. $x\in \varphi$
2. $x\in R^{-}$
3. $x\in W$
4. $x\in R^{+}$

Q. ${lim}_{x\to 0^{+}}F\left(x\right)$ equals

1. $p_{1}ln{a}_1+p_{2}lin{a}_2+...+p_{n}ln{a}_n$
2. $a_1^{p_1}+a_2^{p_2}+.....+a_n^{p_n}$
3. ${\sum }_{r=1}^n{a}_r{p}_r$
4. $a_1^{p_1}\cdot a_2^{p_2}....a_n^{p_n}$

Q. Let $f:R\to R$ be a function defined by $f\left(x\right)=\frac{x^2-3x+4}{x^2+3x+4}$ then $f$ is

1. Onto but not one - one
2. One - one but not onto
3. Onto as well as one - one
4. Neither onto nor one - one

Q.

$\begin{array}{cccc}& Column-I& & Column-II\\ \left(P\right)& Theshortestdistancebetween& \left(1\right)& 6\\ & originandthecurveis{x}^2+y^2+xy=60is& & \\ \left(Q\right)& Thevalueof{\int }_{-2011}^{2011}\frac{dx}{1+x^9+\sqrt{1+x^{18}}}-2011is& \left(2\right)& 0\\ \left(R\right)& On[0,2]themaximumvalueof& \left(3\right)& 3\\ & f\left(x\right)=max\{x^{,}x-1,3x\}is& & \\ \left(S\right)& Letf:R\to Rbegivenby& \left(4\right)& \sqrt{40}\\ & f\left(x\right)=\{\begin{array}{cccc}|x-\left[x\right]|& when\left[x\right]isodd& & \\ |x-\left[x\right]-1& when\left[x\right]iseven& & \end{array}& & \\ & Thenthevalueof{\int }_{-2}^4& & \\ & f\left(x\right)dx-3is& & \end{array}$

1. P-2, Q-1, R-3, S-4

2. P-2, Q-4, R-1, S-4

3. P-4, Q-2, R-1, S-1

4. P-4, Q-3, R-1, S-2

Q. The set of real values of $x$, for which $h\left(x\right)=1+2x^2+4x^4+6x^6+\cdots +100x^{100}$ is concave downward is

1. $x\in \varphi$
2. $x\in W$
3. $x\in R^{-}$
4. $x\in R^{+}$

Q. If $P=\left[\begin{array}{ccc}1& \alpha & 3\\ 1& 3& 3\\ 2& 4& 4\end{array}\right]$ is the ad joint of $3\times 3$ matrix $A$ and $|A|=4$ then $\alpha$ is

1. 4
2. 5
3. 0
4. 11

Q.

If the roots of the equation $a{x}^2+bx+a_1^2+b_1^2+c_1^2-a_1{b}_1-a_1{c}_1-b_1{c}_1=0$ are non real then

1. $2(a-b)+\sum (a_1-b_1{)}^2>0$

2. $2(b-a)+\sum (a_1+b_1{)}^2=0$

3. $2(b-a)+\sum (a_1+b_1{)}^2<0$

4. $2(a-b)+\sum (a_1+b_1{)}^2=0$

Q. If $a{x}^2+(1-\lambda )x+(a-1-\lambda )=0$ where $a\ne 0,$ has real roots for all $\lambda \in R,$ then

1. $a=1$
2. $a>1$
3. $0<a<1$
4. $0<a\le 1$

Q. If $f\left(x\right)=\frac{e^x}{1+e^x},I_1={\int }_{f(-a)}^{f\left(a\right)}xg\left\{x(1-x)\right\}dx$ and $I_2={\int }_{f(-a)}^{f\left(a\right)}g\left\{x(1-x)\right\}dx$, then the value of $\frac{I_2}{I_1}$ is

1. $-3$
2. $1$
3. $-1$
4. $2$

Q. Sum of real roots of $x^2+\left|x\right|-6=0$ is

1. $4$
2. $-6$
3. $-1$
4. $0$

Q. Find the sign of the quadratic polynomial.
$f\left(x\right)=x^2+5|x|+6$

1. $y>0\forall xϵ(-\infty ,2)\cup (3,\infty )y<0\forall xϵ(2,3)y=0\forall xϵ2,3$
2. $y>0\forall xϵ(-\infty ,-3)\cup (-2,2)\cup (3,\infty )y<0\forall xϵ(-3,-2)\cup (2,3)y=0\forall xϵ\{-3,-2,2,3\}$
3. $y>0\forall xϵR$
4. $y>0\forall xϵ(-\infty ,-3)\cup (-2,\infty )y<0\forall xϵ(-3,-2)y=0\forall xϵ\{-3,-2\}$

Q. If graph of $y=a{x}^2-bx+c$ is following, then sign of $a,b,c$ are

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

Q. Let $x_n=(2^n+3^n{)}^{\frac{1}{2n}}$ for all natural numbers $n.$ Then

1. $\underset{n\to \infty }{lim}{x}_n=\infty$
2. $\underset{n\to \infty }{lim}{x}_n=\sqrt{3}$
3. $\underset{n\to \infty }{lim}{x}_n=\sqrt{3}+\sqrt{2}$
4. $\underset{n\to \infty }{lim}{x}_n=\sqrt{5}$

Q. Given the graphs of $f\left(x\right)=x^2\&g\left(x\right)=(kx{)}^2.$ The possible value of $k$ is

1. $k=1$
2. $k>1$
3. $0<k<1$