Director Circle: Hyperbola

Q. Odd degree polynomial function have both it's domain and range as set of real numbers.

1. False
2. True

Q. Which among the following option(s) is/are correct for hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, where $n$ is the number of points on the plane through which perpendicular tangents are drawn to it and $e$ is the eccentricity.

1. If $n=1$, then $e=\sqrt{2}$
2. If $n>1$, then $0<e<\sqrt{2}$
3. If $n=0$, then $e>\sqrt{2}$
4. If $n=0$, then $e>2$

Q. Domain of even polynomial functions is always the set of all real numbers and range is always a proper subset of real numbers.

1. False
2. True

Q. The absolute value of an integer is greater than the integer.

1. True
2. False

Q. Which of the following is a polynomial function?

1. $f\left(x\right)=x^{-2}+x-1$
2. $f\left(x\right)=\frac{1}{x^2}+4x-8$
3. $f\left(x\right)=a{x}^2+bx+7$, where $a,b\in R$
4. $f\left(x\right)=\frac{1}{x}-1$

Q. Which of the following is (are) polynomial function(s)?

1. $f\left(x\right)=x^3-3$
2. $f\left(x\right)=a{x}^2+4x-3\forall a\in R$
3. $f\left(x\right)=\frac{1}{x}-3$
4. $f\left(x\right)=6$

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

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

Q. The perpendicular tangents drawn to the hyperbola
$\frac{x^2}{25}-\frac{y^2}{16}=1$
intersect on the curve $x^2+y^2=9$

1. False
2. True

Q.

For a standard hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Match the following.

$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ 1.a^2>b^2& \text{P.Director circle is real}\\ 2.a^2=b^2& \text{Q.Director circle is imaginary}\\ 3.a^2<b^2& \text{R.Centre is the only point from which}\\ & \text{two perpendicular tangents can be drawn on the}\\ & \text{hyperbola}\end{array}$

1. $1-P,2-Q,3-R$

2. $1-R,2-Q,3-P$

3. $1-P,2-R,3-Q$

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

Q.

For a standard hyperbola
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

Match the following.

$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ 1.a^2>b^2& \text{P.Director circle is real}\\ 2.a^2=b^2& \text{Q.Director circle is imaginary}\\ 3.a^2<b^2& \text{R.Centre is the only point from which}\\ & \text{two perpendicular tangents can be drawn on the}\\ & \text{hyperbola}\end{array}$

1. $1-P,2-Q,3-R$

2. $1-R,2-Q,3-P$

3. $1-P,2-R,3-Q$

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

Q.

Find the equation to the director circle of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

1. x² - y² = a² - b²

2. x² + y² = a² + b²

3. x² - y² = a² + b²

4. x² + y² = a² - b²

Q. The perpendicular tangents drawn to the hyperbola
$\frac{x^2}{25}-\frac{y^2}{16}=1$
intersect on the curve $x^2+y^2=9$

1. False
2. True

Q. $f\left(x\right)=\sin (\frac{\pi }{8})+e^2-\frac{1}{e^2}$ is a constant polynomial.

1. True
2. False

Q. Choose the correct pair of the polynomial function with it's degrees.

1. $4$
2. $5$
3. $2$
4. $x^2-4x+5$
5. ${(x-2)}^2{(9-2x)}^2$
6. $x^{5}e+3x^4-4$
7. $(x-2)(x^2+5)$
8. $3$

Q. $f\left(x\right)=(3x+6)(\frac{12}{x^3}+5x+2)$ is a polynomial function.

1. True
2. False

Q. A polynomial function $f\left(x\right)=32-2x^2$ is non-negative, then what is the domain of the function ?

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

Q. For any positive real number $a>1,$ the value of greatest integer of $a$ will belong to the interval $(0,1).$

1. True
2. False

Q. If $2f\left(x\right)-3f\left(\frac{1}{x}\right)=x^2(x\ne 0)$, then $f\left(2\right)$ is equal to :

Q. The graph given below is that of a quadratic function.

1. False
2. True

Q. $f\left(x\right)=\frac{x^3-6x^2+11x-6}{x-3}\forall x\in R$ is a polynomial.

1. True
2. False

Q. Match the polynomial with its type.

1. Quintic
2. $3x^2-1$
3. $x^4-x-1$
4. Quartic
5. Cubic
6. Quadratic
7. $e+\frac{\ln 5}{e+1}$
8. linear
9. $(x^2-1)(x+1)$
10. $5x+5$
11. Constant
12. $(x+1)(x+2)(x+3)(x+4)(x+5)$

Q.

If $f\left(x\right)+2f(1-x)=x^2+1\forall x\in R$ then $f\left(x\right)$ is

1. $\frac{2}{3}(x^2-4x+3)$

2. $\frac{2}{3}(x^2+4x-3)$

3. $\frac{1}{3}(x^2+4x-3)$

4. $\frac{1}{3}(x^2-4x+3)$

Q. $f\left(x\right)=\frac{x^3+9x^2+23x+15}{x+1}\forall x\in R$ is not a polynomial.

1. False
2. True

Q. Which of the following is/are polynomial functions?

1. $g\left(x\right)=(1+2x)(x+\frac{4}{x})$
2. $p\left(x\right)=5^{x^2+x+1}$
3. $f\left(x\right)=(x+1)(x+2)(x-1)(x-2)$
4. $h\left(x\right)=x^3+(\frac{e}{e+1})x^2-\left(\ln 6\right)x$