Quadratic Identity

Q. If $(P^2-1)x^2+(P-1)x+(P^2-4P+3)=0$ is an identity in $x$, then the value of $P$ is

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

Q. The sum of value(s) of $k$ for which the equation $(({\log }_{5}k{)}^2+({\log }_{5}k)-2)x^2-(2^{2k}-34\cdot 2^k+64)x+(k^2+7k-60)=0$ possesses more than two roots, is

1. $5$
2. $7$
3. $-10$
4. $-12$

Q. The sum of value(s) of $k$ for which the equation $(({\log }_{5}k{)}^2+({\log }_{5}k)-2)x^2-(2^{2k}-34\cdot 2^k+64)x+(k^2+7k-60)=0$ possesses more than two roots, is

1. $5$
2. $7$
3. $-10$
4. $-12$

Q. Consider $\frac{a^2(x-b)(x-c)}{(a-b)(a-c)}+\frac{b^2(x-c)(x-a)}{(b-c)(b-a)}+\frac{c^2(x-a)(x-b)}{(c-a)(c-b)}=x^2,$ then which of the following is/are true

1. Given equation is an identity
2. $a,b,c$ are the roots of the given equation
3. Given equation is not an identity
4. Roots of the given equation are equal

Q. For the polynomial equation
$\frac{(x+q)(x+r)}{(q-p)(r-p)}+\frac{(x+r)(x+p)}{(r-q)(p-q)}+\frac{(x+p)(x+q)}{(p-r)(q-r)}-1=0$, where $p,q,r$ are distinct real numbers.
Which of the following statement is correct?

1. It has real and equal roots.
2. It has more than $2$ real roots.
3. It has no real roots.
4. It has $2$ distinct real roots.

Q. The sum of value(s) of $k$ for which the equation $(({\log }_{5}k{)}^2+({\log }_{5}k)-2)x^2-(2^{2k}-34\cdot 2^k+64)x+(k^2+7k-60)=0$ possesses more than two roots, is

1. $7$
2. $5$
3. $-10$
4. $-12$

Q. Let $\alpha ,\beta ,\gamma$ be distinct real numbers such that
$a{\alpha }^2+b\alpha +c=\left(\sin \theta \right){\alpha }^2+\left(\cos \theta \right)\alpha a{\beta }^2+b\beta +c=\left(\sin \theta \right){\beta }^2+\left(\cos \theta \right)\beta a{\gamma }^2+b\gamma +c=\left(\sin \theta \right){\gamma }^2+\left(\cos \theta \right)\gamma$
where $a,b,c,\theta \in R$
Then which of the following is/are correct?

1. The maximum value of $\frac{a^2+b^2}{a^2+3ab+5b^2}$ is $2$
2. The minimum value of $\frac{a^2+b^2}{a^2+3ab+5b^2}$ is $\frac{2}{11}$
3. If $\overrightarrow{V_1}=a\hat{i}+b\hat{j}+c\hat{k}$ makes an angle $\frac{\pi }{3}$ with $\overrightarrow{V_2}=\hat{i}+\hat{j}+\sqrt{2}\hat{k}$, then the number of values of $\theta \in [0,2\pi ]$ is $3$.
4. If $\overrightarrow{V_1}=a\hat{i}+b\hat{j}+c\hat{k}$ makes an angle $\frac{\pi }{3}$ with $\overrightarrow{V_2}=\hat{i}+\hat{j}+\sqrt{2}\hat{k}$, then the number of values of $\theta \in [0,2\pi ]$ is $5$.