Nature of Roots

Q. Prove that roots of the equation (x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0 are always real and can be equal unless a=b=c

Q.

Let $\alpha$ and $\beta$ be the roots of $x^2–6x–2=0$. If$a_n={\alpha }^n–{\beta }^n$ for $n\ge 1$, then the value of $\frac{(a_{10}–2a_8)}{(3a_9)}$ is

1. $4$

2. $1$

3. $2$

4. $3$

Q.

Find the square root of $7-24i$.

Q.

A value of $k$ for which the quadratic equation $x^2-2x\left(1+3k\right)+7\left(2k+3\right)=0$ has equal roots, is

1. $1$

2. $2$

3. $3$

4. $4$

Q. Two students while solving a quadratic equation in $x$, one copied the constant term incorrectly and got the roots as $3$ and $2$. The other copied coefficient of $x$ incorrectly and got roots as $-6$ and $1$ respectively. The correct root(s) is/are (Assume the leading coefficient of the quadratic equation as $1$)

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

Q.

If one root of the equation $x^2+px+12=0$~is~4, while the equation

$x^2+px+q=0$ has equal roots, the value of q is

1. none of these

2. $\frac{49}{4}$

3. $\frac{4}{49}$

4. 4

Q.

The polynomial $(a{x}^2+bx+c)(a{x}^2-dx-c)$, $ac\ne 0$ has

1. Four real roots

2. At least two real roots

3. At most two real roots

4. No real roots

Q.

The value of $a$ so that the sum of the squares of the roots of the equation $x^2-(a-2)x-a-1=0$ assume the least value, is

1. $2$

2. $1$

3. $3$

4. $0$

Q.

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\frac{2}{3}$and $\frac{1}{3}$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2-\alpha x+\beta$ are real and equal, is

1. $\frac{40}{81}$

2. $\frac{20}{81}$

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

4. $\frac{1}{4}$

Q.

Find $k$ for which the equation $x^2-2kx+7k-12=0$ has real and equal roots.

Q.

Which of the following equations has $2$ as a root?

(A) $x^2-4x+5=0$

(B) $x^2+3x-12=0$

(C) $2x^2-7x+6=0$

(D) $3x^2-6x-2=0$

1. (A)

2. (B)

3. (C)

4. (D)

Q.

If the equation ax^2+bx+c=0 does not have 2 distinct real roots and a+c>b,

then prove that f(x)>=0, for all x belongs to real number.

Q.

If a, b, c are odd integers and $a{x}^2+bx+c=0$ , has real roots then:

1. Roots are of opposite signs.

2. Both roots are rational

3. Both roots are irrational

4. Both roots are positive

Q. If the given expression $x^2-(5m-2)x+(4m^2+10m+25)$ can be expressed as a perfect square, then the value(s) of $m$ is/are

1. $-\frac{4}{3}$
2. $\frac{4}{3}$
3. $8$
4. $-8$

Q. If one root of the quadratic equation $a{x}^2-bx-c=0;a,b,c\in R$ is reciprocal of the other, then which one of the following is correct?

1. $a+b=0$
2. $b=c$
3. $a=c$
4. $a+c=0$

Q. The domain of the function f defined by $f\left(x\right)=\sqrt{(4–x)}+\frac{1}{\sqrt{x^2-1}}$ is equal to

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

Q. Evaluate : $\int \frac{3x-1}{\sqrt{x^2+9}}dx$

Q.

If $\sin A,\sin B$ and $\cos A$ are in GP, then the roots of $x^2+2x\cot B+1=0$ are always.

1. real

2. imaginary

3. greater than $1$

4. equal

Q. If $a{x}^2+bx+6=0$ does not have distinct real roots, then the least value of $3a+b$ is

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

Q. If both the roots of the quadratic equation $x^2-(2n+18)x-n-11=0,n\in Z$ are rational, then the value(s) of $n$ is/are

1. $-8$
2. $-10$
3. $-11$
4. $-12$

Q. The number of distinct real roots of $x^4-4x+1=0$ is:

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

Q.

Minimum value of $x^2+2xy+3y^2-6x-2y$ $x,y,ϵR$ is

1. - 9

2. - 11

3. - 12

4. - 10

Q. If K>0 and the product of the roots of the equation x^2 - 3kx + 2e^2logk - 1 = 0 is 7 then the sum of the roots is
(A) 1. (B) 4. (C) 6. (D) 8.

Q.

State whether the following quadratic equations have two distinct real roots. Justify your answer.

$x(1-x)-2=0$

Q. The domain of $f\left(x\right)={\left(\frac{x^4-2x^3+3x^2-2x+2}{2x^2-2x+1}\right)}^{1/2022}$ is

1. $R$
2. $R-\{\frac{1}{2},1\}$
3. $[0,\infty )$
4. $R-\left\{\frac{1}{2}\right\}$

Q.

If equation $a{x}^2+bx+c$ = 0 has only imaginary roots and c < 0, then a is ______.

1. positive

2. negative

3. Zero

4. Can’t say

Q. If $A.M.$ and $G.M.$ of two positive numbers $a$ and $b$ are $10$ and $8$ respectively, then find the numbers.

Q. For the quadratic equation $a{x}^2+bx+c=0$, if $b^2-4ac>0$ and $a,c$ have same sign, then which of the following is true?
$(a,b,c\in R,a\ne 0)$

1. Roots are always positive
2. Roots are of same sign
3. Roots are always negative
4. Roots are of opposite sign

Q.

If $2+i\sqrt{3}$ is a root of the equation $x^2+px+q=0$, where p and q are real, then (p, q) =

1. (-4, -7)

2. (4, 7)

3. (4, 7)

4. (-4, 7)

Q. If the roots of the equation $x^2-(p+4)x+2p+5=0$ are equal, then the value(s) of $p$ is/are

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