Relation between Roots and Coefficients for Quadratic

Q.

Factorize $6x^2+17x+5$ by using the factor theorem.

Q. If one of the roots of equation px^2-14x+8=0 is six times the other then p equal to

Q.

Prove that difference and quotient of $\left(3+2\sqrt{3}\right)\mathrm{and}\left(3-2\sqrt{3}\right)$ are irrational.

Q. Let $\alpha$ and $\beta$ be the roots of the equation $x^2-x-1=0$. If $p_k=(\alpha {)}^k+(\beta {)}^k,k\ge 1$, then which one of the following statements is not true ?

1. $(p_1+p_2+p_3+p_4+p_5)=26$
2. $p_5=11$
3. $p_5=p_2\cdot p_3$
4. $p_3=p_5-p_4$

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-2x+3=0$, then the equation whose roots are $P={\alpha }^3-3{\alpha }^2+5\alpha -2$ and $Q={\beta }^3-{\beta }^2+\beta +5$ is

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

Q. If $\lambda$ be the ratio of the roots of the quadratic equation in $x,3m^2{x}^2+m(m-4)x+2=0,$ then the least value of $m$ for which $\lambda +\frac{1}{\lambda }=1$, is :

1. $4-2\sqrt{3}$
2. $2-\sqrt{3}$
3. $4-3\sqrt{2}$
4. $-2+\sqrt{2}$

Q.

If we add, subtract, multiply or divide two irrational numbers, the result may be a _____ number or an irrational number.

Q. If $\alpha ,\beta$ are roots of the equation $a{x}^2+bx+c=0$ then the quadratic equation whose roots are $\frac{1}{(a\alpha +b{)}^2},\frac{1}{(a\beta +b{)}^2}$, is

1. $\left(a^2\right)x^2+(2ac-b^2)x+1=0$
2. $\left(c^2\right)x^2+(2ac-b^2)x+1=0$
3. $\left(a^2{c}^2\right)x^2+(2ac-b^2)x+1=0$
4. $\left(a^2{c}^2\right)x^2+(2ac+b^2)x+1=0$

Q. If one real root of the quadratic equation $81x^2+kx+256=0$ is cube of the other root, then a value of $k$ is :

1. $-300$
2. $100$
3. $144$
4. $-81$

Q.

If $(1-p)$ is a root of quadratic equation $x^2+px+(1-p)=0$, then its roots are

1. $0,1$

2. $–1,1$

3. $0,–1$

4. $–1,2$

Q.

If roots $\alpha ,\beta$ of the equations $x^2-px+16=0$ satisfy the relation ${\alpha }^2+{\beta }^2=9$, then write the value of P.

Q.

If in the equation $a{x}^2+bx+c=0$, the sum of the roots is equal to the sum of the squares of their reciprocals, then $\frac{c}{a},\frac{a}{b},\frac{b}{c}$ are in

1. AP

2. GP

3. HP

4. None of these

Q.

If the ratio of the roots of the equation $a{x}^2+bx+c=0$ be $p:q$, then

1. $pq{b}^2+(p+q{)}^{2}ac=0$

2. $pq{b}^2-(p+q{)}^{2}ac=0$

3. $pq{a}^2-(p+q{)}^{2}bc=0$

4. $pq{b}^2-(p-q{)}^{2}ac=0$

Q. If $\alpha$ and $\beta$ are the roots of the equation, $x^2+x\sin \theta -2\sin \theta =0,\theta \in (0,\frac{\pi }{2})$ , then $\frac{{\alpha }^{12}+{\beta }^{12}}{({\alpha }^{-12}+{\beta }^{-12})(\alpha -\beta {)}^{24}}$ is equal to

1. $\frac{2^{12}}{(\sin \theta +8{)}^{12}}$
2. $\frac{2^6}{(\sin \theta +8{)}^{12}}$
3. $\frac{2^{12}}{(\sin \theta -4{)}^{12}}$
4. $\frac{2^{12}}{(\sin \theta -8{)}^6}$

Q. If $\alpha$, $\beta$ are the roots of $x^2+px+q=0$, then the value of ${\alpha }^3+{\beta }^3$ is

1. $3pq+p^3$
2. $3pq-p^3$
3. $3pq$
4. $p^3-3pq$

Q. Let $S$ be the set of all non-zero real number $\alpha$ such that the quadratic equation $\alpha x^2-x+\alpha =0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1-x_2|<1.$ Which of the following interval is(are) a subset(s) of $S?$

1. $(-\frac{1}{2},-\frac{1}{\sqrt{5}})$
2. $(-\frac{1}{\sqrt{5}},0)$
3. $(0,\frac{1}{\sqrt{5}})$
4. $(\frac{1}{\sqrt{5}},\frac{1}{2})$

Q.

If one root of $5x^2+13x+k=0$ is reciprocal of the other, then k=

1. $0$

2. $5$

3. $\frac{1}{6}$

4. $6$

Q. If the difference of the roots of the equation $(k-2)x^2-(k-4)x-2=0,k\ne 2$ is $3$, then the sum of all the values of $k$ is

1. $0$
2. $\frac{9}{2}$
3. $\frac{7}{2}$
4. $-\frac{3}{2}$

Q. If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation,
$x^2+(2-\lambda )x+(10-\lambda )=0$ is minimum, then the magnitude of the difference of the roots of this equation is :

1. $4\sqrt{2}$
2. $2\sqrt{5}$
3. $2\sqrt{7}$
4. $20$

Q. If $\alpha ,\beta$ are the roots of $2x^2+6x+b=0$ and $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }<2$ then b lies in the interval

1. $(0,\infty )$
2. $(-1,\infty )$
3. $(-\infty ,0)$
4. $(-\infty ,0]$

Q.

If ${\alpha }_1,{\alpha }_2,{\alpha }_{3}and{\alpha }_4$are the roots of the equation$x^4+(2-\sqrt{3})x^2+(2+\sqrt{3})=0$, then the value of $(1-{\alpha }_1)(1-{\alpha }_2)(1-{\alpha }_3)(1-{\alpha }_4)$ is

1. $1$

2. $4$

3. $2+\sqrt{3}$

4. $5$

5. $0$

Q. If $\alpha ,\beta$ are the roots of $2x^2+3x+1=0$, then the equation whose roots are $\frac{1}{\alpha },\frac{1}{\beta }$ is

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

Q. Prove by the Principle of Mathematical Induction: $4^n-1$ is divisible by 3, for each natural number n.

Q.

HM between the roots of the equation $x^2-10x+11=0$ is

1. $\frac{1}{5}$

2. $\frac{5}{21}$

3. $\frac{21}{20}$

4. $\frac{11}{5}$

Q. The value of x obtained from the equation $\left|\begin{array}{ccc}x+\alpha & \beta & \gamma \\ \gamma & x+\beta & \alpha \\ \alpha & \beta & x+\gamma \end{array}\right|=0$ will be

1. 0 and $-(\alpha +\beta +\gamma )$
2. 0 and $(\alpha +\beta +\gamma )$
3. 1 and $(\alpha -\beta -\gamma )$
4. 0 and $({\alpha }^2+{\beta }^2+{\gamma }^2)$

Q. The equation formed by decreasing the roots of the quadractic equation $a{x}^2+bx+c=0$ by $1$ is $2x^2+8x+2=0$, then which of the following statement(s) is/are true?

1. $a:b:c=1:2:-2$
2. $a:b:c=2:1:-2$
3. $b=-c$
4. $a=-c$

Q. If $n$ is a integer which lies in $[5,100]$, then the number of integral roots of the equation $x^2+2x-n=0$ is

Q.

If one root of the equation $x^2+px+q=0$ is $2+\sqrt{3}$, then the values of $p\text{and}q$ are respectively

1. $-4,1$

2. $4,-1$

3. $2,\sqrt{3}$

4. $-2,-\sqrt{3}$

Q. If $f(x^3-6x^2+11x-3)=x-1,$ then $f\left(3\right)=$

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

Q. If p and q are the roots of the quadratic equation
ax2+bx+c=0, $\mathrm{(c}\ne 0)$
then $\frac{1}{ap+b}+\frac{1}{aq+b}$=
.

1. $\frac{c}{ab}$
2. $\frac{b}{ac}$
3. $\frac{a}{bc}$
4. 1