Higher Order Equations

Q.

Given that - 4 is a root of the equation $2x^3+6x^2+7x+60=0$. Find the other roots.

1. 

2. 

3. 

4. 

Q. The number of positive integral values of $m$ less than $17$ for which the equation $(x^2+x+1{)}^2-(m-3\left)\right(x^2+x+1)+m=0,m\in R$ has $4$ distinct real roots is

Q. Let $f\left(x\right)=x^2+6x+c,c\in R.$ If $f\left(f\right(x\left)\right)=0$ has exactly three distinct real roots, then the value of $c$ can be

1. $9$
2. $-3$
3. $\frac{11-\sqrt{13}}{2}$
4. $\frac{11+\sqrt{13}}{2}$

Q. The value of $\sqrt{2+\sqrt{2+\sqrt{2+\dots \dots \infty }}}$ is

Q. If $\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$is the A.M. between $a$ and $b$, then find the value of $n$.

Q. The linear factor(s) of the equation $x^2+4xy+4y^2+3x+6y-4=0$ is/are

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

Q. If $f(x)=(ax^2+b)^3 , b\in \mathbb R, a\in \mathbb R -\{0\}$ and $g(x)$ is a function such that $f(g(x))=g(f(x))=x, $ then $g(x)=$
(Given that $f$ and $g$ are bijective functions)

Q.

If the equation $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ has roots equal in magnitude but opposite in sign, then m =

1. 

2. 

3. 

4. 

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

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

Q.

If the equation $x^4-4x^3+a{x}^2+bx+1=0$ has four roots. All of them are positive real roots then value of a and b are given.

1. a = - 4, b = 8

2. a = 8, b = 6

3. a = 6, b = - 4

4. a = - 8, b = - 6

Q. If $p\left(x\right)$ is a polynomial of degree greater than $2$ such that $p\left(x\right)$ leaves remainder $a$ and $-a$ when divided by $x+a$ and $x-a$ respectively. If $p\left(x\right)$ is divided by $x^2-a^2$ then remainder is

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

Q. Let $p,q,r$ be roots of cubic equation $x^3+2x^2+3x+3=0$, then

1. $\frac{p}{p+1}+\frac{q}{q+1}+\frac{r}{r+1}=5$
2. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=44$
3. $\frac{p}{p+1}+\frac{q}{q+1}+\frac{r}{r+1}=6$
4. ${\left(\frac{p}{p+1}\right)}^3+{\left(\frac{q}{q+1}\right)}^3+{\left(\frac{r}{r+1}\right)}^3=38$

Q. If $\sqrt{2}$ and $3i$ are two roots of a biquadratic equation with rational coefficients, then its equation is, (where $i^2=-1$)

1. $x^4-7x^2-18=0$
2. $x^4-7x^2+18=0$
3. $x^4+7x^2-18=0$
4. $x^4+7x^2+18=0$

Q. For the equation $\frac{4x^2+x+4}{x^2+1}+\frac{x^2+1}{x^2+x+1}=\frac{31}{6}$
Which of the following statement(s) is/are correct?

1. The equation has $4$ real and distinct roots.
2. The equation has $3$ real and distinct roots.
3. The sum of all real and distinct roots is $-2.$
4. The sum of all real and distinct roots is $-1.$

Q. For the equation $|x-4{|}^{\left(\frac{x^2-10x+24}{x-3}\right)}=1$, which among the following statement(s) is/are true?

1. Sum of the real roots is $18.$
2. Sum of the real roots is $11.$
3. Product of the real roots is $90.$
4. Product of the real roots is $30.$

Q. If $x^2-3x+2$ is a factor of $x^4-a{x}^2+b$ then the equation whose roots are $a,b$ is

1. $x^2-9x-20=0$
2. $x^2-9x+20=0$
3. $x^2+9x+20=0$
4. $x^2+9x-20=0$

Q. If two roots of the equation $x^5-x^4+8x^2-9x-15=0$ are $-\sqrt{3},1-2i$ then number of positive real roots are

Q. The linear factor(s) of the equation $9x^2-24xy+16y^2-12x+16y-12=0$ is/are

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

Q. If the equation $(a-2)(x-[x]{)}^2+2(x-\left[x\right])+a^2=0,a\in R$ has no integral solution and has exactly one solution in $[2,3)$, then $a$ lies in the interval
(where $\left[x\right]$ denotes the greatest integer function)

1. $(-1,2)$
2. $(0,1)$
3. $(-1,0)$
4. $(2,3)$

Q.

If the equation $x^4-3x^3+m{x}^2-7x+3=0$ has four positive real roots. Then the value of m should always be greater than zero.

1. True

2. False

Q. If $x^2-3x+2$ is a factor of $x^4-a{x}^2+b$ then the equation whose roots are $a,b$ is

1. $x^2-9x+20=0$
2. $x^2+9x-20=0$
3. $x^2-9x-20=0$
4. $x^2+9x+20=0$

Q. If $\alpha ,\beta ,\gamma$ are non zero roots of $x^3+p{x}^2+qx+r=0$, then the equation whose roots are $\alpha (\beta +\gamma ),\beta (\gamma +\alpha ),\gamma (\alpha +\beta )$

1. $x^3-2q{x}^2+(pr+q^2)x+(r^2-pqr)=0$
2. $x^3-2q{x}^2+(pr+q^2)x+(r^2+pqr)=0$
3. $x^3+2q{x}^2+(pr+q^2)x+(r^2-pqr)=0$
4. $x^3+2q{x}^2+(pr+q^2)x+(r^2+pqr)=0$

Q. If $2x^2+7xy+3y^2+8x+14y+\lambda =0$ can be resolved into $2$ linear factors, then the value of $\lambda$ is

Q.

If ∝ and β are the roots of the equation a$x^2$ + bx + c = 0, (a, b, c ​ R) , then (1+α+${\alpha }^2$) (1+β+${\beta }^2$) is :

1. < 0

2. >0

3. =0

4. None of these

Q.

If p, q and r are in GP and the equations $p{x}^2+2qx+r=0$ and $d{x}^2+2ex+f=0$ have a common root, show that $\frac{d}{p},\frac{e}{q}$ and $\frac{f}{r}$ are in AP.

Q. The number of real solutions of $(x-1)(x+1)(2x+1)(2x-3)=15$ is

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

Q. If $\alpha ,\beta$ are the roots of the equation $x^2-px+r=0$ and $\frac{\alpha }{2},2\beta$ are the roots of the equation $x^2-qx+r=0$, then the value of $r$ in terms of $p$ and $q$ is

1. $\frac{2}{9}(p-q)(2q-p)$
2. $\frac{2}{9}(q-p)(2p-q)$
3. $\frac{2}{9}(q-2p)(2q-p)$
4. $\frac{2}{9}(2p-q)(2q-p)$

Q. If two roots of $4x^3-12x^2+9x-2=0$ are equal, then the roots are

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

Q. If ${\log }_x(7x-10)=2$, then find the value(s) of $x$.

1. $2$
2. $3$
3. $5$
4. $4$

Q. If $f\left(x\right)=(x-1)(x-3)(x-4)(x-6)+10$, then which of the following statements(s) is/are correct?

1. $f\left(x\right)=0$ has $4$ distinct real roots.
2. $f\left(x\right)=0$ has no real roots.
3. $f\left(x\right)$ is always positive for all $x\in R.$
4. $f\left(x\right)$ has negative values for some real values of $x.$