Quadratic Equation

Q.

If $\sec \left(\theta \right)=m$ and $\tan \left(\theta \right)=n$, then $\frac{1}{m}\left[\left(m+n\right)+\frac{1}{\left(m+n\right)}\right]$ is equals to

1. $2$

2. $2m$

3. $2n$

4. $mn$

Q. The set of values of $a$ for which $(a-1)x^2-(a+1)x+a-1\ge 0$ is true for all $x\ge 2$ is

1. $(1,\frac{7}{3})$
2. $(-\infty ,1)$
3. None of these
4. $[\frac{7}{3},\infty )$

Q. The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :

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

Q. The value of $a$ for which the equation $(a^2-a-2)x^2+(a^2-4)x+(a^2-3a+2)=0$ have more than two roots

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

Q. The sum of all real values of $x$ satisfying the equation $(x^2-5x+5{)}^{(x^2+4x-60)}=1$ is

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

Q.

If $k_1,k_2,k_3$ are real numbers and the equation $k_1{x}^2+k_{2}x+k_3=0$ have three roots, then value of $k_1+k_2+k_3$ is

1. 0

2. 1

3. 2

4. 3

Q.

Check whether the following are quadratic equations $(\mathrm{x}-3)(2\mathrm{x}+1)=\mathrm{x}(\mathrm{x}+5)$.

Q. Consider the equation $\sqrt{3x^2-8x+1}+\sqrt{9x^2-24x-8}=3.$ It is known that the largest root of the equation is $-k$ times the smallest root. The value of $k$ is
(correct answer + 3, wrong answer 0)

Q. If $x,y\in R$ and $2x^2+6xy+5y^2=1$, then

1. $|x|\le \sqrt{5}$
2. $|x|\ge \sqrt{5}$
3. $y^2\le 2$
4. $y^2\ge 2$

Q.

If $k_1,k_2,k_3$ are real numbers and the equation $k_1{x}^2+k_{2}x+k_3=0$ have three roots, then value of $k_1+k_2+k_3$ is

1. 1

2. 2

3. 3

4. 0

Q. Identify which of the following statement(s) is(are) correct for the equation ${\log }_{(x^2+2x-3)}(4-{\log }_2(|x-1|+|x-3|))=0$

1. The equation has exactly two solution
2. The equation has more than two solutions
3. The sum of all the solutions of the equation is $6$
4. The sum of all the solutions of the equation is $4$

Q.

Check whether the following are quadratic equations$(\mathrm{x}-2)(\mathrm{x}+1)=(\mathrm{x}-1)(\mathrm{x}+3)$.

Q. The value of $3+{\log }_{9/4}\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\cdots \cdots \infty }}}$ is

Q. For non-zero distinct real numbers $a_1$ and $a_2$, let $f\left(x\right)=a_1{x}^2+b_{1}x+c_1,$$g\left(x\right)=a_2{x}^2+b_{2}x+c_2$ and $p\left(x\right)=f\left(x\right)-g\left(x\right)$. If $p\left(x\right)=0$ only at $x=-1$ and $p(-2)=2$, then the value of $p\left(2\right)$ is

1. $9$
2. $6$
3. $18$
4. $3$

Q. The sum of all real values of $x$ satisfying the equation $(x^2-5x+5{)}^{(x^2+4x-60)}=1$ is

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

Q. Solve the following equations:
$x^4+\frac{8}{9}{x}^2+1=3x^3+3x$.

Q.

If $tan{\theta }_1,tan{\theta }_2,tan{\theta }_3$ are the real roots of $x^3-(a+1)x^2+(b-a)x-b=0where{\theta }_1,{\theta }_2,{\theta }_3$ are acute then ${\theta }_1+{\theta }_2+{\theta }_3=$

1. $\frac{\pi }{2}$

2. $\frac{5\pi }{6}$

3. $\frac{6\pi }{5}$

4. $\frac{\pi }{4}$

Q.

How do you find the $x-\mathrm{intercepts}$ of a quadratic function?

Q. Which of the following option(s) is/are true for $f\left(x\right)=7$?

1. It is a constant polynomial
2. It is a polynomial of degree $0$
3. It is a monomial
4. The number of zeroes of the polynomial is $0$

Q. Identify which of the following statement(s) is(are) correct for the equation ${\log }_{(x^2+2x-3)}(4-{\log }_2(|x-1|+|x-3|))=0$

1. The equation has exactly two solution
2. The equation has more than two solutions
3. The sum of all the solutions of the equation is $6$
4. The sum of all the solutions of the equation is $4$

Q. If $x$ is a solution of the equation $\sqrt{2x+1}-\sqrt{2x–1}=1,(x\ge \frac{1}{2}),$ then $\sqrt{4x^2–1}$ is eaual to :

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

Q. If $x$ is a solution of the equation $\sqrt{2x+1}-\sqrt{2x–1}=1,(x\ge \frac{1}{2}),$ then $\sqrt{4x^2–1}$ is eaual to :

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

Q. Which of the following option(s) is/are true for $f\left(x\right)=7$?

1. It is a constant polynomial
2. It is a polynomial of degree $0$
3. It is a monomial
4. The number of zeroes of the polynomial is $0$

Q. The set of values of $a$ for which $(a-1)x^2-(a+1)x+a-1\ge 0$ is true for all $x\ge 2$ is

1. $(-\infty ,1)$
2. $(1,\frac{7}{3})$
3. $[\frac{7}{3},\infty )$
4. None of these

Q. If $x,y\in R$ and $2x^2+6xy+5y^2=1$, then

1. $|x|\le \sqrt{5}$
2. $|x|\ge \sqrt{5}$
3. $y^2\le 2$
4. $y^2\ge 2$

Q. The value of $3+{\log }_{9/4}\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\sqrt{4-\frac{1}{3\sqrt{2}}\cdots \cdots \infty }}}$ is