Polynomial Functions

Q.

What is a degree of a polynomial?

Q. The minimum positive numerical value of $x$ for which the polynomial function $(x^2-1)(x^3-1)(x^4-1)$ is non-negative, is

Q. If the roots of $10x^3-c{x}^2-54x-27=0$ are in harmonic progression, then the value of c is

1. 7
2. 8
3. 9
4. 0

Q.

The degree of $4x^3-12x^2+3x+9$ is

1. $0$

2. $1$

3. $2$

4. $3$

Q.

$\sqrt{2}$ is a polynomial of which degree?

Q. The value of $\sqrt{9998\times 10002+4}$ is

Q. Let $f:R\to R$ and $g:R\to R$ be two bijective functions such that they are the mirror images of each other about the line $y=a.$ If $h:R\to R$ given by $h\left(x\right)=f\left(x\right)+g\left(x\right),$ then $h\left(x\right)$ is

1. a one-one and an onto function
2. a one-one but not an onto function
3. an onto but not a one-one function
4. Neither a one-one nor an onto function

Q.

If the function $f\left(x\right)=a{x}^3+b{x}^2+11x-6$ satisfies the condition of Rolles theorem in $\left[1,3\right]$ and $f\left(2+\frac{1}{\sqrt{3}}\right)=0$, then the values of $a$ and $b$ respectively are:

1. $-1,6$

2. $-2,1$

3. $1,-6$

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

Q. Consider a function $f\left(x\right)=\frac{1}{\left\{x\right\}+\frac{1}{\left\{x\right\}}},x\notin Z.$ Then which of the following options is/are true?

1. $f\left(x\right)=\frac{1}{3}$ has exactly one real solution in $(2,3)$
2. $f\left(x\right)=\frac{1}{3}$ has four real solutions in $(-2,2)$
3. $f\left(x\right)=\frac{1}{3}$ has at least two real solutions in $(2,3)$
4. $f\left(x\right)=\frac{1}{3}$ has no real solution in $(0,1)$

Q. The degree of the polynomial function $f\left(x\right)=(1-x^2)(x-1)$ is

Q.

The maximum value of the function $x^3+x^2+x-4$ is

1. 127
2. 4
3. Does not have a maximum value
4. None of these

Q.

The function $f\left(x\right)=\log \left(x+1\right)-\frac{2x}{2+x}$ is increasing on

1. $\left(1,\infty \right)$

2. $\left(-\infty ,0\right)$

3. $\left(-\infty ,\infty \right)$

4. None of these

Q. If the polynomial function $f\left(x\right)=32-2x^2$ is non-negative, then $x\in$

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

Q. The graph of $y=x+\cos x$ is

1. 
2. 
3. 
4. None of these

Q. If a function satisfies $(x-y)f(x+y)-(x+y)f(x-y)=2(x^{2}y-y^3),\forall x,y\in R$ and $f\left(1\right)=2,$ then

1. $f\left(x\right)$ must be polynomial function
2. $f\left(3\right)=12$
3. $f\left(0\right)=0$
4. $f\left(x\right)$ may not be differentiable

Q. Let $P\left(x\right)$ be a polynomial of degree $2010$. Suppose $P\left(n\right)=\frac{n}{1+n}$ for all $n=0,1,2,\dots ,2010.$ Find $P\left(2012\right).$
(correct answer + 5, wrong answer 0)

Q. For real constants $a,b,c,d,$ suppose $f\left(x\right)$ is a function of the form $f\left(x\right)=\frac{a{x}^8+b{x}^6+c{x}^4+d{x}^2+15x+1}{x}$ for $x\ne 0.$ If $f\left(5\right)=2,$ then the value of $f(–5)$ is

Q.

Write down the degree of following polynomials in x .

Q.

Write the degree of polynomial$\pi$.

Q. $1+2+3+4+.....+n=?$
$1+2+3+4+.....+(n-1)=?$
$1+1+1+1+.....+n=?$
$1+1+1+1+......+(n-1)=?$
$1^2+2^2+3^2+4^2+....+n^2=?$
$1^2+2^2+3^2+4^2+....+{(n-1)}^2=?$

Q. Simplify:

Q. $Iff\left(x\right)=a{x}^2+bx+ca,b,cϵ$ R and the equation $f\left(x\right)-x=0$ has imaginary roots $\alpha$ and $\beta$ and $\gamma$ and$\delta$ be the roots of $f\left(f\right(x\left)\right)-x=0$, then$\left|\begin{array}{ccc}2& \alpha & \delta \\ \beta & 0& \alpha \\ \gamma & \beta & 1\end{array}\right|is$

1. purely real
2. purely imaginary
3. none of these
4. 0

Q. Let $S=\{1,2,3,.....,9\}$. For $k=1,2,.....,5$, let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

1. $125$
2. $210$
3. $252$
4. $126$

Q. Consider a polynomial of degree $4$ with leading coefficient $2$ such that $P\left(1\right)=1,P\left(2\right)=16,P\left(3\right)=81,P\left(4\right)=256$ then $P\left(5\right)$ is :

1. $673$
2. $637$
3. $727$
4. $Noneofthese$

Q. Let $P\left(x\right)$ be a polynomial, which when divided by $x-3$ and $x-5$ leaves remainders $10$ and $6$ respectively. If the polynomial is divided by $(x-3)(x-5)$ then the remainder is

1. $-2x+16$
2. $16$
3. $2x-16$
4. $60$

Q. If the polynomial function $f\left(x\right)=32-2x^2$ is non-negative, then $x\in$

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

Q. Explain nature of the roots: $h^2$ > ab ?

Q. If f (x) is a function satisfying $f(x+y)=f\left(x\right).f\left(y\right)$ for all $x,yϵ$ N such that f(1) = 3 and ${\sum }_{x=1}^{n}f\left(x\right)=120$. Then, the value of n is

1. 4
2. 5
3. 6
4. None of these

Q. Let $a,b,cϵR$. If $f\left(x\right)=a{x}^2+bx+c$ be such that $a+b+c=3$ and $f(x+y)=f\left(x\right)+f\left(y\right)+xy,\forall x,yϵR,\text{then}{\sum }_{n=1}^{10}f\left(n\right)$ is equal to

1. 330
2. 165
3. 190
4. 255

Q. Number of solutions of the equation $\tan 2x=2\cos (x+\pi )$ in $(-\pi ,\pi )$ is/are