Homogeneous System of Equations

Q.

Let $\alpha ,\beta ,\gamma$be the roots of the equations, $x^3+a{x}^2+bx+c=0,$ ($a,b,c\in R$and $a,b\&a,b\ne 0$). The system of the equations (in $u,v,w$) given by $\alpha u+\beta v+\gamma w=0;\beta u+\gamma v+\alpha w=0;\gamma u+\alpha v+\beta w=0$ has non-trivial solutions, then the value of $\frac{a^2}{b}$ is

1. $5$

2. $1$

3. $0$

4. $3$

Q. Let $\lambda$ be a real number for which the system of linear equations
$x+y+z=6$
$4x+\lambda y-\lambda z=\lambda -2$
$3x+2y-4z=-5$
has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation :

1. ${\lambda }^2+3\lambda -4=0$
2. ${\lambda }^2-3\lambda -4=0$
3. ${\lambda }^2+\lambda -6=0$
4. ${\lambda }^2-\lambda -6=0$

Q. Let ${\alpha }_1,{\alpha }_2$ and ${\beta }_1,{\beta }_2$ be the roots of $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0$ and ${\beta }_{1}y+{\beta }_{2}z=0$ has a non-trivial solution, then which of the following options is CORRECT ?

1. $abc=pqr$
2. $a^{2}pr=q^{2}bc$
3. $b^{2}pr=q^{2}ac$
4. $a^{2}qr=p^{2}bc$

Q. Let $f\left(x\right)=x^3-3x^2+2x.$ If the equation $f\left(x\right)=k$ has exactly one positive and one negative solution, then the value of $k$ is equal to

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

Q. The set of all values of $\lambda$ for which the system of linear equations
$x-2y-2z=\lambda x$
$x+2y+z=\lambda y$
$-x-y=\lambda z$
has a non-trivial solution :

1. is a singleton
2. is an empty set
3. contains exactly two elements
4. contains more than two elements

Q. The least positive value of $t$ so that the lines $x=t+\alpha ,y+16=0$ and $y=\alpha x$ are concurrent is

1. $2$
2. $4$
3. $16$
4. $8$

Q. The set of equations
$\lambda x-y+\left(\cos \theta \right)z=0$
$3x+y+2z=0$
$\left(\cos \theta \right)x+y+2z=0,$
where $0\le \theta <2\pi ,$ has non-trivial solutions for

1. all values of $\lambda$ and $\theta$
2. no value of $\lambda$ and $\theta$
3. all values of $\lambda$ and only two values of $\theta$
4. only one value of $\lambda$ and all values of $\theta$

Q.

The sum of distinct values of $\lambda$ for which the systems of equations
$(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$
$(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$
$2x+(3\lambda +1)y+3(\lambda -1)z=0$
has non-zero solutions, is

Q. The system of linear equations
$x+\lambda y-z=0$
$\lambda x-y-z=0$
$x+y-\lambda z=0$ has a non-trivial solution for:

1. infinitely many values of $\lambda$
2. exactly one value of $\lambda$
3. exactly two values of $\lambda$
4. exactly three values of $\lambda$

Q. The system of equations
$\lambda x+y+3z=0$
$2x+\mu y-z=0$
$5x+7y+z=0$
has infinitely many solutions in $R$. Then,

1. $\lambda =2,\mu =3$
2. $\lambda =1,\mu =2$
3. $\lambda =1,\mu =3$
4. $\lambda =3,\mu =1$

Q. Determine the value of $\lambda$ for which the following system of equations fail to have a unique solution.
$\lambda x+3y-z=1$
$x+2y+z=2$
$-\lambda x+y+2z=-1$
Does it have any solution for $\lambda =1$ ?

Q. Consider the planes
$P_1:cy+bz=x$
$P_2:az+cx=y$
$P_3:bx+ay=z.$
$P_1,P_2$ and $P_3$ pass through one line, if

1. $a^2+b^2+c^2=ab+bc+ca$
2. $a^2+b^2+c^2+2abc=1$
3. $a^2+b^2+c^2=1$
4. $a^2+b^2+c^2+2ab+2bc+2ca+2abc=1$

Q. The system of equations $ax+4y+z=0,bx+3y+z=0,cx+2y+z=0$ has non-trivial solutions if $a,b,c$ are in

1. $AP$
2. $GP$
3. $HP$
4. none of these

Q. Consider the planes
$P_1:cy+bz=x$
$P_2:az+cx=y$
$P_3:bx+ay=z.$
$P_1,P_2$ and $P_3$ pass through one line, if

1. $a^2+b^2+c^2=ab+bc+ca$
2. $a^2+b^2+c^2+2abc=1$
3. $a^2+b^2+c^2=1$
4. $a^2+b^2+c^2+2ab+2bc+2ca+2abc=1$

Q. Let ${\alpha }_1,{\alpha }_2$ and ${\beta }_1,{\beta }_2$ be the roots of $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0$ and ${\beta }_{1}y+{\beta }_{2}z=0$ has a non-trivial solution, then

1. $b^{2}pr=q^{2}ac$
2. $bp{r}^2=qa{c}^2$
3. $b{p}^{2}r=q{a}^{2}c$
4. None of these

Q. The set of all values of $\lambda$ for which the system of linear equations
$x-2y-2z=\lambda x$
$x+2y+z=\lambda y$
$-x-y=\lambda z$
has a non-trivial solution :

1. is a singleton
2. is an empty set
3. contains exactly two elements
4. contains more than two elements

Q. Let ${\alpha }_1,{\alpha }_2$ and ${\beta }_1,{\beta }_2$ be the roots of $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0$ and ${\beta }_{1}y+{\beta }_{2}z=0$ has a non-trivial solution, then which of the following option is CORRECT ?

1. $abc=pqr$
2. $a^{2}pr=q^{2}bc$
3. $b^{2}pr=q^{2}ac$
4. $a^{2}qr=p^{2}bc$

Q. If $\alpha$ and $\beta ,(\alpha <\beta )$ are two different roots of the equation $p{x}^2+qx+r=0,$ then

1. $\alpha >-\frac{q}{2p}$
2. $\alpha <-\frac{q}{2p}<\beta$
3. $-\frac{q}{2p}>\beta$
4. None of the above

Q. The set of equations
$\lambda x-y+\left(\cos \theta \right)z=0$
$3x+y+2z=0$
$\left(\cos \theta \right)x+y+2z=0,$
where $0\le \theta <2\pi ,$ has non-trivial solutions for

1. all values of $\lambda$ and $\theta$
2. no value of $\lambda$ and $\theta$
3. all values of $\lambda$ and only two values of $\theta$
4. only one value of $\lambda$ and all values of $\theta$

Q. The values of $\lambda$ for which the system of equations
$(\lambda +5)x+(\lambda -4)y+z=0$
$(\lambda -2)x+(\lambda +3)y+z=0$
$\lambda x+\lambda y+z=0$
has a non-trivial solution is (are)

1. $-1,2$
2. $0$
3. $0,-1$
4. none of these