Consistency and solution of simultaneous linear equations — practice questions

Question 1 · difficulty L2 · understanding

If the system of linear equations : $$\begin{aligned} & x+y+z=6 \\ & x+2 y+5 z=10 \\ & 2 x+3 y+\lambda z=\mu \end{aligned}$$ has infinitely many solutions, then the value of $\lambda+\mu$ equals:

• A. 12
• B. 16
• C. 22
• D. 28

Question 2 · difficulty L2 · understanding

If the system of equations : $$\begin{aligned} & x+y+z=5 \\ & x+2 y+3 z=9 \\ & x+3 y+\lambda z=\mu \end{aligned}$$ has infinitely many solutions, then the value of $\lambda+\mu$ is :

• A. 16
• B. 18
• C. 19
• D. 21

Question 3 · difficulty L2 · understanding

The following system of linear equations 2x + 3y + 2z = 9 3x + 2y + 2z = 9 x $$-$$ y + 4z = 8

• A. does not have any solution
• B. has a solution ($$\alpha$$, $$\beta$$, $$\gamma$$) satisfying $$\alpha$$ + $$\beta$$ 2 + $$\gamma$$ 3 = 12
• C. has a unique solution
• D. has infinitely many solutions

Question 4 · difficulty L2 · understanding

Let $$\alpha \beta \gamma=45 ; \alpha, \beta, \gamma \in \mathbb{R}$$. If $$x(\alpha, 1,2)+y(1, \beta, 2)+z(2,3, \gamma)=(0,0,0)$$ for some $$x, y, z \in \mathbb{R}, x y z \neq 0$$, then $$6 \alpha+4 \beta+\gamma$$ is equal to _________.

Question 5 · difficulty L2 · understanding

The number of $3 \times 3$ matrices A whose entries are either 0 or 1 and for which the system $\mathrm{A}\left[$$\begin{array}{l}x \\ y \\ z\end{array}$$\right]=\left[$$\begin{array}{l}1 \\ 0 \\ 0\end{array}$$\right]$ has exactly two distinct solutions, is

• A. 0
• B. $2^9-1$
• C. 168
• D. 2

Question 6 · difficulty L2 · understanding

Let a, $$\lambda$$, m $$\in$$ R. Consider the system of linear equations ax + 2y = $$\lambda$$ 3x $$-$$ 2y = $$\mu$$ Which of the following statements is(are) correct?

• A. If a = $$-$$3, then the system has infinitely many solutions for all values of $$\lambda$$ and $$\mu$$.
• B. If a $$\ne$$ $$-$$3, then the system has a unique solution for all values of $$\lambda$$ and $$\mu$$.
• C. If $$\lambda$$ + $$\mu$$ = 0, then the system has infinitely many solutions for a = $$-$$3.
• D. If $$\lambda$$ + $$\mu$$ $$\ne$$ 0, then the system has no solution for a = -3.

Question 7 · difficulty L3 · understanding

The sum of all possible values of $\theta \in[0,2 \pi]$, for which the system of equations : $$\begin{aligned} & x \cos 3 \theta-8 y-12 z=0 \\ & x \cos 2 \theta+3 y+3 z=0 \\ & x+y+3 z=0 \end{aligned}$$ has a non-trivial solution, is equal to :

• A. ${ }\pi$
• B. $2 \pi$
• C. $3 \pi$
• D. $4 \pi$

Question 8 · difficulty L3 · understanding

Let M be a $3 \times 3$ matrix such that $\mathrm{M}\left($$\begin{array}{l}1 \\ 0 \\ 0\end{array}$$\right)=\left($$\begin{array}{l}1 \\ 2 \\ 3\end{array}$$\right), \mathrm{M}\left($$\begin{array}{l}0 \\ 1 \\ 0\end{array}$$\right)=\left($$\begin{array}{l}0 \\ 1 \\ 2\end{array}$$\right)$and$\mathrm{M}\left($$\begin{array}{l}0 \\ 0 \\ 1\end{array}$$\right)=\left($$\begin{array}{c}-1 \\ 1 \\ 1\end{array}$$\right)$. If$\mathrm{M}\left($$\begin{array}{l}x \\ y \\ z\end{array}$$\right)=\left($$\begin{array}{c}1 \\ 7 \\ 11\end{array}$$\right)$, then$x+y+z$ equals :

• A. 4
• B. 5
• C. 7
• D. 11

Question 9 · difficulty L3 · understanding

Consider the system of linear equations in $x, y, z$ : $$\begin{aligned} & x+2 y+t z=0 \\ & 6 x+y+5 t z=0 \\ & 3 x+t^2 y+f(t) z=0 \end{aligned}$$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $t \in \mathbb{R}$, then $f$

• A. is a constant function
• B. is strictly increasing on $\mathbb{R}$
• C. is strictly decreasing on $R$
• D. has two critical points

Question 10 · difficulty L3 · understanding

If the system of equations $x + 5y + 6z = 4$ $2x + 3y + 4z = 7$ $x + 6y + az = b$ has infinitely many solutions, then the point $(a, b)$ lies on the line

• A. $y - x = 3$
• B. $x - y = 3$
• C. $x + y = 11$
• D. $x + y = 12$