Chapter 10

Solve \(x+2 y+3 z=6\) \(2 x+4 y+z=7\) \(3 x+2 y+9 z=14\)

Solve \(x+2 y+3 z-6=0\) \(2 x+4 y+z=17\) \(3 x+9 z+2 y=2 .\)

Solve \(x+y+z=11\) \(2 x-6 y-z=0\) \(3 x+4 y+2 z=0 .\)

Solve \(x+y+z=6\) \(x-y+z=2\) \(2 x+y-z=1\)

Solve \(5 x-6 y+4 z=15\) \(7 x+4 y-3 z=19\) \(2 x+y+6 z=46\).

Solve \(x+y+z=1\) \(3 x+5 y+6 z=4\) \(9 x+2 y-36 z=17 .\)

Find \(a, b, c\) when \(f(x)=a x^{2}+b x+c\) and \(f(0)=6, f(2)=11, f(-3)=6\). Determine \(f(x)\) and find the value of \(f(1)\).

Solve \((b+c)(y+z)-a x=b-c\) \((c+a)(z+x)-b y=c-a\) \((a+b)(x+y)-c z=a-b\) where \(a+b+c \neq 0 .\left\\{\right.\) Ans. \(\left.\left(\frac{c-b}{a+b+c}, \frac{a-c}{a+b+c}, \frac{b-a}{a+b+c}\right)\right\\}\)

For what value of \(k\), the system of linear equations \(x+y+z=2,2 x+y-z=3,3 x+2 y+k z=4\) has a unique solution?

For what value of \(k\), the system of simultaneous equations \(k x+2 y-z=1,(k-1) y-2 z=2\) and \((k+2) z=3\) have a unique solution?

Solve \(x+4 y-2 z=3\) \(3 x+y+5 z=7\) \(2 x+3 y+z=5 .\)

Solve \(2 x+3 y-2 z=3\) \(x+2 y+z=4\) \(5 x+9 y+z=15\)

Find \(k\) for which the set of equations \(x+y-2 z=0\) \(2 x-3 y+z=0\) \(x-5 y+4 z=k\) are consistent and find the solution for all such values of \(k\).

Show that the system of equations \(3 x-y+4 z=3\) \(x+2 y-3 z=-2\) \(6 x+5 y+\lambda z=-3\) has at least one solution for any real number \(\lambda\). Find the set of solutions if \(\lambda=-5\).

Solve $$ \begin{aligned} &3 x-y+z=0 \\ &-15 x+6 y-5 z=0 \\ &5 x-2 y+2 z=0 \end{aligned} $$

Solve \(x+3 y-2 z=0\) \(2 x-y+4 z=0\) \(x-11 y+14 z=0\)

Find all values of \(k\) for which the following system possesses a non-trivial solution \(x+k y+3 z=0\) \(k x+2 y+2 z=0\) \(2 x+3 y+4 z=0\)

For what value of \(k\) do the following system of equations possess a non- trivial solution over the set of rationals \(x+k y+3 z=0\) \(3 x+k y-2 z=0\) \(2 x+3 y-4 z=0\) For that value of \(k\), find all the solutions of the system.

Given \(x=c y+b z\) \(y=a z+c x\) \(z=b x+a y\) where \(x, y, z\) are not all zero prove that \(a^{2}+b^{2}+c^{2}+2 a b c=1\)

If \(a=\frac{x}{y-z}, b=\frac{y}{z-x}\) and \(c=\frac{z}{x-y}\), where \(x, y, z\) are not all zero, prove that \(1+a b+b c+c a=0\).

Let \(\alpha_{1}, \alpha_{2}\), and \(\beta_{1}, \beta_{2}\) be the roots of \(a x^{2}+b x+c=0\) and \(p x^{2}+q x+r=0\) respectively. If the system of equations \(\alpha_{1} y+\alpha_{2} z=0\) \(\beta_{1} y+\beta_{2} z=0\) has non-trivial solution, then prove that \(\frac{b^{2}}{q^{2}}=\frac{a c}{p r}\).

If \(a, b, c\) are in G.P. with common ratio \(r_{1}\) and \(\alpha, \beta, \gamma\) also form in G.P.with common ratio \(r_{2}\), then find the conditions that \(r_{1}\) and \(r_{2}\) must satisfy so that the equations \(a x+\alpha y+z=0\) \(b x+\beta y+z=0\)

Solve \(2 x+3 y=2\) \(x-y=1\) \(x+2 y=2\)

Solve \(x+2 y=3\) \(2 x-y=1\) \(x-2 y=-1\)

Find the value of \(\lambda\) if the following equations are consistent \(x+y-3=0\) \((1+\lambda) x+(2+\lambda) y-8=0\) \(x-(1+\lambda) y+(2+\lambda)=0\)

If the equations \(a x+h y+g=0\) \(h x+b y+f=0\) \(g x+f y+c=\lambda\) are consistent, show that \(\lambda=\frac{a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}}{a b-h^{2}}\).

If the equations \((b+c) x+(c+a) y+(a+b)=0\) \(c x+a y+b=0\) \(a x+b y+c=0\) are consistent then show that either \(a+b+c=0\) or \(a=b=c\).

If \(a, b, c\) are all different and the equations \(a x+a^{2} y+\left(a^{3}+1\right)=0\) \(b x+b^{2} y+\left(b^{3}+1\right)=0\) \(c x+c^{2} y+\left(c^{3}+1\right)=0\) are consistent then prove that \(a b c+1=0\).

Find the value of \(a\) if the three equations are consistent \((a+1)^{3} x+(a+2)^{3} y=(a+3)^{3}\) \((a+1) x+(a+2) y=(a+3)\) \(x+y=1\)

Are the equations \(x+a y=b+c\) \(x+b y=c+a\) \(x+c y=a+b\) where \(a, b\) and \(c\) are real numbers such that \(a^{2}+b^{2}+c^{2}=1\), consistent?

SYSTEM OF LINEAR EQUATIONS WITH PARAMETERS. $$ \left\\{\begin{array}{l} a x+y=2 \\ x+a y=2 a \end{array}\right\\} $$

SYSTEM OF LINEAR EQUATIONS WITH PARAMETERS. $$ \left\\{\begin{array}{l} x+a y-1=0 \\ a x-3 a y-(2 a+3)=0 \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} 3 x+a y=5 a^{2} \\ 3 x-a y=a^{2} \end{array}\right\\} $$

SYSTEM OF LINEAR EQUATIONS WITH PARAMETERS. $$ \left\\{\begin{array}{l} (a+5) x+(2 a+3) y-(3 a+2)=0 \\ (3 a+10) x+(5 a+6) y-(2 a+4)=0 \end{array}\right\\} $$

SYSTEM OF LINEAR EQUATIONS WITH PARAMETERS. $$ \left\\{\begin{array}{l} a(a-1) x+a(a+1) y=a^{3}+2 \\ \left(a^{2}-1\right) x+\left(a^{3}+1\right) y=a^{4}-1 \end{array}\right\\} $$

SYSTEM OF LINEAR EQUATIONS WITH PARAMETERS. $$ \left\\{\begin{array}{l} a x-y=b \\ b x+y=a \end{array}\right\\} $$

SYSTEM OF LINEAR EQUATIONS WITH PARAMETERS. $$ \left\\{\begin{array}{l} \left(a^{2}+b^{2}\right) x+\left(a^{2}-b^{2}\right) y=a^{2} \\ (a+b) x+(a-b) y=a \end{array}\right\\} $$

For what values of \(m\) does the system of equations \(3 x+m y=m\) \(2 x-5 y=20\) has a solution satisfying the conditions \(x>0, y>0\).

For what values of \(a\) does the system \(a^{2} x+(2-a) y=4+a^{2}\) \(a x+(2 a-1) y=a^{5}-2\)

For what values of the parameter \(a\) does the system of equations \(a x-4 y=a+1\) \(2 x+(a+6) y=a+3\) possess no solutions?

For what values of the parameter \(a\) does the system \(2 x+a y=a+2\) \((a+1) x+2 a y=2 a+4\) possess infinitely many solutions?

Find the values of the parameters \(m\) and \(p\) such that the system \((3 m-5 p+1) x+(8 m-3 p-1) y=1\) \((2 m-3 p+1) x+(4 m-p) y=2\) possesses infinite solutions.

Find the parameters \(a\) and \(b\) such that the system \(a^{2} x-a y=1-a\) \(b x+(3-2 b) y=3+a\) possesses a unique solution \(x=1, y=1 .\)

For what values of \(a\) and \(b\) does the system \(a^{2} x-b y=a^{2}-b\) \(b x-b^{2} y=2+4 b\) possesses an infinite number of solutions?

Find the values of \(\lambda\) and \(\mu\) so that the system of equation \(x+y+z=6\) \(x+2 y+3 z=10\) \(x+2 y+\lambda z=\mu\) has i. Unique solution \\{Ans. \(\lambda \neq 3, \mu \in R\\}\) ii. Infinite solutions \\{Ans. \(\lambda=3, \mu=10\\}\) iii. No solution \\{Ans. \(\lambda=3, \mu \neq 10\\}\)

For what values of \(a, b\) and \(c\), the system \(a x-b y=2 a-b\) \((c+1) x+c y=10-a+3 b\) has infinitely many solutions and \(x=1, y=3\) is one of the solutions?

Find a \(3 \times 4\) matrix \(A\) given that \(a_{i j}=\frac{(i+j)^{2}}{2}\).

Find a matrix \(A_{2 \times 3}\) given that \(a_{i j}=\left[\frac{i}{j}\right]\), where [] denotes greatest integer function.

If \(A=\left[\begin{array}{ccc}1 & -5 & 7 \\ 0 & 7 & 9 \\ 11 & 8 & 9\end{array}\right]\), then find the trace of matrix \(A\)

Find the value of \(x, y, z\) and \(a\) which satisfy the matrix equation \(\left[\begin{array}{cc}x+3 & 2 y+x \\ z-1 & 4 a-6\end{array}\right]=\left[\begin{array}{cc}0 & -7 \\ 3 & 2 a\end{array}\right]\)