Quadratic Identity

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 $5$.

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. 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 general value of the real angle $\theta ,$ which satisfies the equation, $(\cos \theta +i\sin \theta )(\cos 2\theta +i\sin 2\theta )...(\cos n\theta +i\sin n\theta )=1$ is given by, (assuming $k$ is an integer)

1. $\frac{2k\pi }{n+2}$
2. $\frac{4k\pi }{n(n+1)}$
3. $\frac{4k\pi }{n+1}$
4. $\frac{6k\pi }{n(n+1)}$

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

Q. Let $p\left(x\right)$ be any polynomial. When it is divided by $(x-19)$ and $(x-91)$, then the remainders are $91$ and $19$ respectively. The remainder, when $p\left(x\right)$ is divided by $(x-19)(x-91)$, is:

1. $x-110$
2. $110$
3. $110-x$
4. $4x+88$

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. 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. 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. If equation $(a-1)x^2+(a^2-2a+1)x+(a^2–1)=0$ has more than two roots then $a$ is

Q. The polynomials $a{x}^3-3x^2+4$ & $2x^3-5x+a$ when $(x-2)$ leaves remainders $p$ and $q$ & if $p-2q=4$ find $a$.

1. $a=2$
2. $a=4$
3. $a=6$
4. both a and b

Q.

Number of values of a for which the equation ( $a^2$ - 5a + 4) $x^2$ + ( $a^2$ - 1) x + ( $a^2$ - 8a + 7) = 0 possesses more than two roots, is__.

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. 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. If $x$ is real and $k=\frac{x^2-x+1}{x^2+x+1}$ the:

1. $k\ge 3$
2. $k\le \frac{1}{3}$
3. None of these
4. $k\in [\frac{1}{3},3]$

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 the roots of the equation, $x+1-2{\log }_2(3+2^x)+2{\log }_4(10-2^{-x})=0,$ is

1. ${\log }_{2}14$
2. ${\log }_{2}12$
3. ${\log }_{2}11$
4. ${\log }_{2}13$