Relations Between Roots and Coefficients

Q.

If $\alpha ,\beta$ are the roots of $a{x}^2+2bx+c=0$ and $a+\delta ,\beta +\delta$ are the roots of $A{x}^2+2Bx+C=0$ then $\frac{b^2–ac}{B^2-AC}$ is

1. $\frac{A}{a}$

2. $\frac{a}{A}$

3. ${\left(\frac{A}{a}\right)}^2$

4. ${\left(\frac{a}{A}\right)}^2$

Q. The value of $p$ in the equation $x^2-4x+p=0,$ if it is known that the sum of the squares of its roots is equal to $16$ is

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

Q.

If $\sin \alpha ,\cos \alpha$ are the roots of the equation $a{x}^2+bx+c=0$, then

1. $a^2+b^2-2ac=0$

2. $(a-c{)}^2=b^2+c^2$

3. $a^2+b^2+2ac=0$

4. $a^2-b^2+2ac=0$

Q.

Let $p$ and $q$ be real numbers such that$p\ne 0,p^3\ne q\text{,}{p}^3\ne -q.$ If $\alpha$ and$\beta$ are non - zero complex numbers satisfying $\alpha +\beta =-p\text{,}{\alpha }^3+{\beta }^3=q,$ then a quadratic equation having $\frac{\alpha }{\beta }$and $\frac{\beta }{\alpha }$ as its roots is.

1. $(p^3+q)x^2–(p^3+2q)x+(p^3+q)=0$

2. $(p^3+q)x^2–(p^3–2q)x+(p^3+q)=0$

3. $(p^3-q)x^2–(5p^3–2q)x+(p^3–q)=0$

4. $(p^3-q)x^2–(5p^3+2q)x+(p^3–q)=0$

Q. If $\alpha ,\beta$ be the roots of the equation $x^2+px+q=0,$ then the equation whose roots are $\frac{1}{\alpha +\beta }$ and $\frac{1}{\alpha }+\frac{1}{\beta }$ will be:

1. None of the above
2. $pq{x}^2+(p^2+q^2)x+p=0$
3. $pq{x}^2+(p^2+q)x+p=0$
4. $pq{x}^2+(q^2+p)x+p=0$

Q. If one root of $6x^2+13x+b+1=0$ is the reciprocal of the other root, then the value of $b$ is

Q. If $\alpha ,\beta$ are the roots of $x^2+3x+4=0,$ then the quadratic equation whose roots are ${\alpha }^3+4{\alpha }^2+7\alpha +7$ and ${\beta }^3-{\beta }^2-8\beta -14$ is

1. $x^2-5x+6=0$
2. $x^2+5x+6=0$
3. $x^2-3x+4=0$
4. $x^2+3x+4=0$

Q.

If the roots of the equation $x^2+ax+b=0$ are $c$ and $d$, then one of the roots of equation $x^2+(2c+a)x+c^2+ac+b=0$ is

1. $2c$

2. $2d$

3. $d-c$

4. $c$

Q.

The value of $c$ for which $|{\alpha }^2-{\beta }^2|=\frac{7}{4}$ , where $\alpha$ and $\beta$ are the roots of $2x^2+7x+c=0$ , is

1. 0

2. 4

3. 6

4. 2

Q.

If two persons $A$ and $B$ solve the equation, $x^2+ax+b=0$, while solving, $A$ commits a mistake the coefficient of $x$ was taken as in $15$place of $-9$and finds the roots as $-7$and $-2$ Then, the equation is.

1. $x^2+9x+14=0$

2. $x^2–9x+14=0$

3. $x^2+9x–14=0$

4. $x^2–9x–14=0$

5. None of these

Q. The value of '$3a$' for which one root of the quadratic equation
$(a^2-5a+3)x^2-3(a-1)x+2=0$ is twice as large as other is

Q.

If $(\alpha +\surd \beta )$ and $(\alpha -\surd \beta )$ are the roots of the equation $x^2+px+q=0$, where $\alpha ,\beta ,p$ and $q$ are real, then the roots of the equation $(p^2-4q)(p^2{x}^2+4px)-16q=0$ are

1. $\frac{1}{ɑ}+\frac{1}{\surd \beta }$ and $\frac{1}{ɑ}-\frac{1}{\surd \beta }$

2. $\frac{1}{\surd ɑ}+\frac{1}{\beta }$ and $\frac{1}{\surd ɑ}-\frac{1}{\beta }$

3. $\frac{1}{\surd ɑ}+\frac{1}{\surd \beta }$ and $\frac{1}{\surd ɑ}-\frac{1}{\surd \beta }$

4. $\surd ɑ+\surd \beta$ and $\surd ɑ-\surd \beta$

Q.

If $\alpha$ is a root of $4x^2+2x-1=0$ then root is:

1. $3{\alpha }^3-4\alpha$

2. $4{\alpha }^3-3\alpha$

3. None of these

4. $4{\alpha }^3+3\alpha$

Q. If the roots of a quadratic equation are $-6$ and $7$, then the quadratic equation is

1. $x^2-x-42=0$
2. $x^2-x+21=0$
3. $x^2-x+42=0$
4. $x^2+x+42=0$

Q.

The equation $x^2+px-q=0$ has one root as a square root of the other.
If $p^3+q^2=q(1+kp)$, find the value of k.

1. 4

2. -1

3. -2

4. -3

Q. If \(\alpha, \beta\) are the roots of \(x^2+px-q=0\) and \(\gamma, \delta\) are the roots of \(x^2+px+r=0\), then the value of \(\dfrac{(\alpha-\gamma)(\alpha-\delta)}{(\beta-\gamma)(\beta-\delta)}\) is

Q. If the product of the roots of the quadratic equation $m{x}^2-2x+(2m-1)=0$ is $3$, then the value of $m$ is

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

Q. Suppose $a,b$ denotes the distinct real roots of the quadratic polynomial $x^2+20x-2020$ and suppose $c,d$ denotes the distinct complex roots of the quadratic polynomial $x^2-20x+2020.$
Then the value of $ac(a-c)+ad(a-d)+bc(b-c)+bd(b-d)$ is

1. $8080$
2. $0$
3. $8000$
4. $16000$

Q. If $a$ and $b$ are the non-zero distinct roots of $x^2+ax+b=0$, then the least value of quadratic polynomial $x^2+ax+b$ is

1. $1$
2. $\frac{2}{3}$
3. $\frac{9}{4}$
4. $-\frac{9}{4}$

Q. If $\alpha$ and $\beta$ be two roots of the equation $x^2-64x+256=0.$ Then the value of ${\left(\frac{{\alpha }^3}{{\beta }^5}\right)}^{\frac{1}{8}}+{\left(\frac{{\beta }^3}{{\alpha }^5}\right)}^{\frac{1}{8}}$ is:

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

Q. If $\alpha$ and $\beta$ are the roots of the equation $8x^2-3x+27=0,$ then the value of ${\left(\frac{{\alpha }^2}{\beta }\right)}^{1/3}+{\left(\frac{{\beta }^2}{\alpha }\right)}^{1/3}$ is

1. $\frac{1}{6}$
2. $\frac{1}{3}$
3. $\frac{1}{4}$
4. $\frac{1}{5}$

Q.

In a triangle ABC the value of $\angle A$ is given by $5cosA+3=0$, then the equation whose roots are sin A and tan A will be

1. $15x^2-8\sqrt{2}x+16=0$

2. $15x^2-8x-16=0$

3. $15x^2+8x-16=0$

4. $15x^2-8x+16=0$

Q. The value of $m$ for which one root of the quadratic equation $x^2-3x+2m=0$ is twice the other is

Q. The set of values of $a$ for which the sum and product of roots of the equation $x^2-(a^2-5a+5)x+(2a^2-3a-4)=0$ is less than $1$, is

1. $(-1,\frac{5}{2})$
2. $(1,\frac{5}{2})$
3. $(1,4)$
4. $(-1,4)$

Q. If one root of the quadratic equation $x^2+qx+2=0$ is double the other, then the value of $q=$

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

Q. The value of $p$ in the equation $x^2-4x+p=0,$ if it is known that the sum of the squares of its roots is equal to $16$ is

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

Q.

If $\alpha ,\beta$ are the roots of the equation $4x^2+3x+7=0$, then find the value of ${\alpha }^2\beta +{\beta }^2\alpha$

1. $\frac{(-16)}{21}$

2. $\frac{21}{16}$

3. $\frac{16}{21}$

4. $\frac{(-21)}{16}$

Q. If $\alpha \text{and}\beta$ are the the roots of $x^2-ax+b^2=0$, then ${\alpha }^2+{\beta }^2$ is equal to

1. $2a^2-b^2$
2. $a^2-b^2$
3. $a^2+b^2$
4. $a^2-2b^2$

Q. If $7\alpha ={\alpha }^2+3$ and ${\beta }^2=7\beta -3,$ then the value of $\alpha \beta$ is

Q.

If ∝ and β are roots of $x^2$ + ax - b = 0 and γ, δ​ arethe roots of $x^2$ + ax + b = 0, then (α-γ) (β-δ) (α-δ) (β-γ)

1. 4$b^2$

2. b

3. 2b

4. 2$b^2$