Matrices and Determinants jee mains questions | JEE Main Maths Matrices and Determinants Previous Year Question

Q 11 :

Let S denote the set of all real values of $λ$ such that the system of equations $λ x + y + z = 1, x + λ y + z = 1, x + y + λ z = 1$ is inconsistent. Then $\sum_{λ \in S} (| λ |^{2} + | λ |)$ is equal to [2023]

12
4
2
6

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(4)

Since, the given system of equations is inconsistent

$\Rightarrow$ $| \begin{matrix} λ & 1 & 1 \\ 1 & λ & 1 \\ 1 & 1 & λ \end{matrix} |$ $= 0$ $\Rightarrow λ (λ^{2} - 1) - 1 (λ - 1) + 1 (1 - λ) = 0$

$\Rightarrow λ^{3} - λ - λ + 1 + 1 - λ = 0 \Rightarrow (λ - 1) (λ^{2} + λ - 2) = 0$

$\Rightarrow λ = 1, λ = \frac{- 1 \pm \sqrt{1 + 8}}{2} \Rightarrow λ = 1, λ = \frac{- 1 \pm 3}{2}$

$\Rightarrow λ = 1, 1, - 2$

Now, for $λ = 1$ , we have infinite solutions as we have only one equation $x + y + z = 1$

For $λ = - 2$ , $D_{1} =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix} |$ $= 9 \neq 0$
$\Rightarrow for λ = - 2, we have no solution.$

$\sum_{λ \in S} (| λ^{2} | + | λ |) = (4 + 2) = 6$

Q 12 :

For the system of linear equations $α x + y + z = 1, x + α y + z = 1, x + y + α z = β,$ which one of the following statements is NOT correct? [2023]

It has infinitely many solutions if $α = 2$ and $β = - 1$
$x + y + z = \frac{3}{4}$ if $α = 2$ and $β = 1$
It has infinitely many solutions if $α = 1$ and $β = 1$
It has no solution if $α = - 2$ and $β = 1$

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(1)

$| \begin{matrix} α & 1 & 1 \\ 1 & α & 1 \\ 1 & 1 & α \end{matrix} |$ $= 0$

$\Rightarrow α (α^{2} - 1) - 1 (α - 1) + (1 - α) = 0$

$\Rightarrow α^{3} - α - α + 1 + 1 - α = 0 \Rightarrow α^{2} (α - 1) + α (α - 1) - 2 (α - 1) = 0$

$(α - 1) (α^{2} + α - 2) = 0 \Rightarrow α = 1, α = - 2, 1$

For $α = 2, β = 1;$ $Δ = 4$

$Δ_{1} =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{matrix} |$ $= 3 - 1 - 1 = 1 \Rightarrow x = \frac{1}{4}$

$Δ_{2} =$ $| \begin{matrix} 2 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 2 \end{matrix} |$ $= 2 - 1 = 1 \Rightarrow y = \frac{1}{4}$

$Δ_{3} =$ $| \begin{matrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{matrix} |$ $= 2 - 1 = 1 \Rightarrow z = \frac{1}{4}$

For $α = 2 \Rightarrow$ unique solution and $x + y + z = \frac{3}{4}$

For $α = - 2, β = 1$

$Δ =$ $| \begin{matrix} - 2 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix} |$ $= - 2 (4 - 1) - 1 (- 2 - 1) + 1 (1 + 2)$

$= - 6 + 3 + 3 = 0$

$Δ_{1} =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & - 2 \end{matrix} |$ $= 3 + 3 + 3 = 9 \Rightarrow x = \frac{9}{0}$

$Δ_{2} =$ $| \begin{matrix} - 2 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & - 2 \end{matrix} |$ $= 6 + 3 + 0 = 9 \Rightarrow y = \frac{9}{0}$

$Δ_{3} =$ $| \begin{matrix} - 2 & 1 & 1 \\ 1 & - 2 & 1 \\ 1 & 1 & 1 \end{matrix} |$ $= 6 + 0 + 3 = 9 \Rightarrow z = \frac{9}{0}$

For $α = - 2 \Rightarrow$ no solution

For $α = 1$ and $β = 1$

$Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0$

$∴ It has infinitely many solutions if α = 1 and β = 1$

For $α = 2$ and $β = - 1$

$Δ = 0$

$Δ_{1} =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ - 1 & 1 & 2 \end{matrix} |$ $= 1 (3) - 1 (3) + 1 (3) = 3$

$Δ_{2} =$ $| \begin{matrix} 2 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & - 1 & 2 \end{matrix} |$ $= 2 (3) - 1 (1) + 1 (- 2) = 3$

$Δ_{3} =$ $| \begin{matrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & - 1 \end{matrix} |$ $= 2 (- 3) - 1 (- 2) + 1 (- 1) = - 5$

Hence, it does not have infinitely many solutions if $α = 2$ and $β = - 1$ .

Q 13 :

Let $N$ denote the number that turns up when a fair die is rolled. If the probability that the system of equations

$x + y + z = 1$

$2 x + N y + 2 z = 2$

$3 x + 3 y + N z = 3$

has a unique solution is $\frac{k}{6}$ , then the sum of the value of $k$ and all possible values of $N$ is [2023]

18
19
20
21

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(3)

For unique solution, determinant $\neq 0$

$| \begin{matrix} 1 & 1 & 1 \\ 2 & N & 2 \\ 3 & 3 & N \end{matrix} |$ $\neq 0$

$\Rightarrow N^{2} - 5 N + 6 \neq 0$

$\Rightarrow (N - 2) (N - 3) \neq 0 \Rightarrow N \neq 2, 3$

$∴ Possible values of N are 1, 4, 5, 6 and \frac{k}{6} = \frac{4}{6} \Rightarrow k = 4$

$Required sum = 1 + 4 + 5 + 6 + 4 = 20$

Q 14 :

If the system of equations

$x + 2 y + 3 z = 3$

$4 x + 3 y - 4 z = 4$

$8 x + 4 y - λ z = 9 + μ$

has infinitely many solutions, then the ordered pair $(λ, μ)$ is equal to [2023]

$(\frac{72}{5}, - \frac{21}{5})$
$(\frac{72}{5}, \frac{21}{5})$
$(- \frac{72}{5}, - \frac{21}{5})$
$(- \frac{72}{5}, \frac{21}{5})$

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(1)

$x + 2 y + 3 z = 3; 4 x + 3 y - 4 z = 4; 8 x + 4 y - λ z = 9 + μ$

$For infinite many solutions, Δ = 0 and Δ_{x} = 0$

$Δ =$ $| \begin{matrix} 1 & 2 & 3 \\ 4 & 3 & - 4 \\ 8 & 4 & - λ \end{matrix} |$ $= 0$

$\Rightarrow 1 (- 3 λ + 16) - 2 (- 4 λ + 32) + 3 (16 - 24) = 0$

$\Rightarrow 16 - 3 λ + 8 λ - 64 - 24 = 0 \Rightarrow 5 λ = 72$

$∴$ $λ = \frac{72}{5}$

$Δ_{x} =$ $| \begin{matrix} 3 & 2 & 3 \\ 4 & 3 & - 4 \\ 9 + μ & 4 & - λ \end{matrix} |$ $= 0$

$\Rightarrow 3 (- 3 λ + 16) - 2 (- 4 λ + 36 + 4 μ) + 3 (16 - 27 - 3 μ) = 0$

$\Rightarrow - 9 λ + 48 + 8 λ - 72 - 8 μ - 33 - 9 μ = 0$

$\Rightarrow - λ - 17 μ = 57 \Rightarrow - 17 μ = 57 + λ$

$∴$ $- μ = \frac{57 + \frac{72}{5}}{17} \Rightarrow μ = \frac{- 357}{85} = \frac{- 21}{5}$

$Thus, (λ, μ) \equiv (\frac{72}{5}, \frac{- 21}{5})$

Q 15 :

Let $S_{1}$ and $S_{2}$ be respectively the sets of all $a \in R - {0}$ for which the system of linear equations

$a x + 2 a y - 3 a z = 1$

$(2 a + 1) x + (2 a + 3) y + (a + 1) z = 2$

$(3 a + 5) x + (a + 5) y + (a + 2) z = 3$

has a unique solution and infinitely many solutions. Then [2023]

$S_{1} = Φ$ and $S_{2} = R - {0}$
$S_{1}$ is an infinite set and $n (S_{2}) = 2$
$S_{1} = R - {0}$ and $S_{2} = Φ$
$n (S_{1}) = 2$ and $S_{2}$ is an infinite set.

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(3)

$| A |$ $=$ $| \begin{matrix} a & 2 a & - 3 a \\ 2 a + 1 & 2 a + 3 & a + 1 \\ 3 a + 5 & a + 5 & a + 2 \end{matrix} |$

$= a \times$ $| \begin{matrix} 1 & 2 & - 3 \\ 2 a + 1 & 2 a + 3 & a + 1 \\ 3 a + 5 & a + 5 & a + 2 \end{matrix} |$

Applying $(C_{2} \to C_{2} - C_{1})$ , we get

$| A |$ $= a \times$ $| \begin{matrix} 1 & 1 & - 3 \\ 2 a + 1 & 2 & a + 1 \\ 3 a + 5 & - 2 a & a + 2 \end{matrix} |$

$= a \times$ $| \begin{matrix} 1 & 1 & - 3 \\ - a - 4 & 2 + 2 a & - 1 \\ 3 a + 5 & - 2 a & a + 2 \end{matrix} |$ Applying $(R_{2} \to R_{2} - R_{3})$

$= a \times [2 a^{2} + 6 a + 4 - 2 a - 3 a - 5 + (a + 4) (a + 2) - 3 [(a + 4) 2 a - (2 + 2 a) (3 a + 5)]]$

$= a \times (2 a^{2} + 6 a + 4 - 2 a - 3 a - 5 + a^{2} + 6 a + 8) - 3 (2 a^{2} + 8 a - 6 a^{2} - 16 a - 10)$

$| A |$ $= a (15 a^{2} + 31 a + 37)$

$| A |$ $\neq 0 \forall a \in R as 15 a^{2} + 31 a + 37 \neq 0 \forall a \in R & a \neq 0$

$S_{1} = R - {0}$

Since $| A |$ can never be 0.

There cannot be infinitely many solutions for any value of $a$ .

$∴$ $S_{2} = ϕ$

Q 16 :

Consider the following system of equations

$α x + 2 y + z = 1$

$2 α x + 3 y + z = 1$

$3 x + α y + 2 z = β$

for some $α, β \in ℝ$ . Then which of the following is NOT correct? [2023]

It has no solution if $α = - 1$ and $β \neq 2$
It has no solution for $α = - 1$ and for all $β \in ℝ$
It has a solution for all $α \neq - 1$ and $β = 2$
It has no solution for $α = 3$ and for all $β \neq 2$

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(2)

Given system of equations has no solution if

$| \begin{matrix} α & 2 & 1 \\ 2 α & 3 & 1 \\ 3 & α & 2 \end{matrix} |$ $= 0$

$\Rightarrow α (6 - α) - 2 (4 α - 3) + (2 α^{2} - 9) = 0 \Rightarrow α = 3, - 1$

For no solution, any one of $Δ_{x}, Δ_{y}, Δ_{z}$ should be non-zero.

Now, $Δ_{x} =$ $| \begin{matrix} 2 & 1 & 1 \\ 3 & 1 & 1 \\ α & 2 & β \end{matrix} |$ $\neq 0 \Rightarrow β \neq 2$

Now, $Δ_{y} =$ $| \begin{matrix} α & 1 & 1 \\ 2 α & 1 & 1 \\ 3 & 2 & β \end{matrix} |$ $\neq 0 \Rightarrow - α β + 2 α \neq 0$

For $α = - 1 \Rightarrow β \neq 2$

So, option (2) is incorrect as it has no solution for $α = - 1$ and $β \in R - {2}$ .

Q 17 :

Let the system of linear equations

$x + y + k z = 2$

$2 x + 3 y - z = 1$

$3 x + 4 y + 2 z = k$

have infinitely many solutions. Then the system

$(k + 1) x + (2 k - 1) y = 7$

$(2 k + 1) x + (k + 5) y = 10$

has [2023]

infinitely many solutions
unique solution satisfying $x - y = 1$
no solution
unique solution satisfying $x + y = 1$

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(4)

We have system of linear equations,

$x + y + k z = 2,$

$2 x + 3 y - z = 1,$

$3 x + 4 y + 2 z = k$

System has infinitely many solutions, then $| A | = 0 and (adj A) B = 0$

Here $A = [\begin{matrix} 1 & 1 & k \\ 2 & 3 & - 1 \\ 3 & 4 & 2 \end{matrix}]$

$\Rightarrow | A | = 0 \Rightarrow 1 (6 + 4) - 1 (4 + 3) + k (8 - 9) = 0$

$\Rightarrow 3 - k = 0, k = 3$

For $k = 3$ , $(adj A) B = 0$

Now, new system will be $4 x + 5 y = 7, 7 x + 8 y = 10$

After solving these two equations, we get $x = - 2, y = 3$

$\Rightarrow x - y = - 2 - (3) = - 5 and x + y = - 2 + 3 = 1$

Q 18 :

For $α, β \in R,$ suppose the system of linear equations

$x - y + z = 5$

$2 x + 2 y + α z = 8$

$3 x - y + 4 z = β$

has infinitely many solutions. Then $α$ and $β$ are the roots of [2023]

$x^{2} - 18 x + 56 = 0$
$x^{2} + 14 x + 24 = 0$
$x^{2} - 10 x + 16 = 0$
$x^{2} + 18 x + 56 = 0$

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(1)

Given, $x - y + z = 5, 2 x + 2 y + α z = 8, 3 x - y + 4 z = β$

It can be written as $A X = B$

Where, $A = [\begin{matrix} 1 & - 1 & 1 \\ 2 & 2 & α \\ 3 & - 1 & 4 \end{matrix}], B = [\begin{matrix} 5 \\ 8 \\ β \end{matrix}], X = [\begin{matrix} x \\ y \\ z \end{matrix}]$

Condition for infinite solutions $| A | = 0 and (Adj A) B = 0$

$∴$ $| A | = 8 + α + 8 - 3 α - 8 = 0 \Rightarrow α = 4$

$Adj A = [\begin{matrix} 12 & 3 & - 6 \\ 4 & 1 & - 2 \\ - 8 & - 2 & 4 \end{matrix}]$

$(Adj A) B = [\begin{matrix} 12 & 3 & - 6 \\ 4 & 1 & - 2 \\ - 8 & - 2 & 4 \end{matrix}] [\begin{matrix} 5 \\ 8 \\ β \end{matrix}] = 0$

$\Rightarrow 60 + 24 - 6 β = 0 \Rightarrow β = 14$ $∴$ $α + β = 4 + 14 = 18, α β = 4 \times 14 = 56$

$∴$ $Required equation is given by x^{2} - 18 x + 56 = 0$

Q 19 :

For the system of linear equations $x + y + z = 6, α x + β y + 7 z = 3, x + 2 y + 3 z = 14,$ which of the following is NOT true? [2023]

If $α = β$ and $α \neq 7$ , then the system has a unique solution
There is a unique point $(α, β)$ on the line $x + 2 y + 18 = 0$ for which the system has infinitely many solutions
For every point $(α, β) \neq (7, 7)$ on the line $x - 2 y + 7 = 0$ , the system has infinitely many solutions
If $α = β = 7$ , then the system has no solution

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(3)

We have, $x + y + z = 6, α x + β y + 7 z = 3, x + 2 y + 3 z = 14$

Now, let
$A = [\begin{matrix} 1 & 1 & 1 \\ α & β & 7 \\ 1 & 2 & 3 \end{matrix}], B = [\begin{matrix} 6 \\ 3 \\ 14 \end{matrix}]$

$\Rightarrow$ $| A |$ $= 1 (3 β - 14) - 1 (3 α - 7) + 1 (2 α - β) = 2 β - α - 7$

If $α = β$ and $α \neq 7$ , then $| A | \neq 0$ , so the system has unique solutions.

Hence (1) is true.

If $α = β = 7$ , then $| A | = 0$ and the system of equations is:

$x + y + z = 6 ...(i)$

$7 x + 7 y + 7 z = 3 ...(ii)$

$x + 2 y + 3 z = 14 ...(iii)$

From equations (i) and (ii), two rows in matrix $A$ will get identical; therefore, the system has no solutions.

Hence (1) is true.

For every point $(α, β) \neq (7, 7)$ on the line $x - 2 y + 7 = 0$ ,

$| A | \neq 0$ . Hence, the system has unique solutions. Therefore, (3) is false.

$(α, β) \equiv (3, 3) does not satisfy x + 2 y + 18 = 0, then$

$Δ_{1} =$ $| \begin{matrix} 6 & 1 & 1 \\ 3 & β & 7 \\ 14 & 2 & 3 \end{matrix} |$ $= 6 (3 β - 14) - 1 (9 - 98) + 1 (6 - 14 β)$

$= 4 β + 11$

$Δ_{2} =$ $| \begin{matrix} 1 & 6 & 1 \\ α & 3 & 7 \\ 1 & 14 & 3 \end{matrix} |$ $= 1 (9 - 98) - 6 (3 α - 7) + 1 (14 α - 3)$

$= - 4 α - 50$

$Δ_{3} =$ $| \begin{matrix} 1 & 1 & 6 \\ α & β & 3 \\ 1 & 2 & 14 \end{matrix} |$ $= 1 (14 β - 6) - 1 (14 α - 3) + 6 (2 α - β)$

$= 8 β - 2 α - 3$

For infinite solution, $| A | = 0$ and $Δ_{1} = Δ_{2} = Δ_{3} = 0$ , then

$2 β - α = 7, β = \frac{- 11}{4}, α = \frac{- 25}{2}, and 8 β - 2 α = 3$

Hence, there is a unique $(α, β)$ on the line $x + 2 y + 18 = 0$ for which the system has infinitely many solutions.

Hence (2) is true.

Q 20 :

Let S be the set of values of $λ$ , for which the system of equations

$6 λ x - 3 y + 3 z = 4 λ^{2},$

$2 x + 6 λ y + 4 z = 1,$

$3 x + 2 y + 3 λ z = λ$ has no solution. Then $12 \sum_{λ \in S} | λ |$ is equal to ___________ . [2023]

(24)

For no solution, $Δ = 0$

$\Rightarrow$ $| \begin{matrix} 6 λ & - 3 & 3 \\ 2 & 6 λ & 4 \\ 3 & 2 & 3 λ \end{matrix} |$ $= 0$

$\Rightarrow 6 λ (18 λ^{2} - 8) + 3 (6 λ - 12) + 3 (4 - 18 λ) = 0$

$\Rightarrow 18 λ^{3} - 14 λ - 4 = 0 \Rightarrow λ = 1, \frac{- 1}{3}, \frac{- 2}{3}$

$For each λ, Δ_{1} =$ $| \begin{matrix} 6 λ & - 3 & 4 λ^{2} \\ 2 & 6 λ & 1 \\ 3 & 2 & λ \end{matrix} |$ $\neq 0$

$Then, 12 \sum_{λ \in S} | λ | = 12 (1 + \frac{1}{3} + \frac{2}{3}) = 24$

Q 21 :

System of linear equations $5 x + 2 y = 4,$ $7 x + 3 y = 5$ has:

unique solution
infinitely many solutions
no solution
none of the above

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(1)

Ans. Unique solution

Explanation:

Given,

$5 x + 2 y = 4$

$7 x + 3 y = 5$

Here,

$\frac{a_{1}}{a_{2}} = \frac{5}{7},$ $\frac{b_{1}}{b_{2}} = \frac{2}{3}$

Since, $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$

Thus, given system of linear equations has a unique solution.

Q 22 :

The following system of linear equations

$3 x + 5 y = 7$

$2 x - 5 y = 9$

is:

consistent
inconsistent
data incomplete
can't be said anything

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(1)

Ans. Consistent

Explanation:

Given,

$3 x + 5 y = 7$

$2 x - 5 y = 9$

The coefficient matrix is

$A = [\begin{matrix} 3 & 5 \\ 2 & - 5 \end{matrix}]$

Now, $| A |$ $= 3 (- 5) - 2 (5) = - 15 - 10 = - 25 \neq 0$

$∴$ $A^{- 1}$ can be calculated

$∴$ System of linear equations is consistent.

Q 23 :

The value of $x$ and $y$ , if the system of linear equations is $2 x - y = 2$ and $3 x + 4 y = 3$ is:

$x = 1, y = 0$
$x = 1, y = - 1$
$x = 0, y = 1$
$x = - 1, y = - 1$

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(1)

Ans. $x = 1, y = 0$

Explanation:

Given,

$2 x - y = 2$

$3 x + 4 y = 3$

The coefficient matrix is

$A = [\begin{matrix} 2 & - 1 \\ 3 & 4 \end{matrix}]$

$B = [\begin{matrix} 2 \\ 3 \end{matrix}], X = [\begin{matrix} x \\ y \end{matrix}]$

Now, $| A |$ $= 2 (4) - (- 1) (3) = 8 + 3 = 11$

$adj (A) = [\begin{matrix} 4 & 1 \\ - 3 & 2 \end{matrix}]$

Therefore, $A^{- 1} = \frac{1}{11} [\begin{matrix} 4 & 1 \\ - 3 & 2 \end{matrix}]$

Hence, $X = A^{- 1} B$

$[\begin{matrix} x \\ y \end{matrix}] = \frac{1}{11} [\begin{matrix} 4 & 1 \\ - 3 & 2 \end{matrix}] [\begin{matrix} 2 \\ 3 \end{matrix}] = \frac{1}{11} [\begin{matrix} 8 + 3 \\ - 6 + 6 \end{matrix}]$

Thus, $x = \frac{11}{11} = 1, y = \frac{0}{11} = 0$

Hence, $x = 1, y = 0$

Q 24 :

The number of distinct real values of $' K'$ such that the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = K, x + 5 y + 10 z = K^{2}$ has infinitely many solutions is:

0
4
2
3

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(3)

$D =$ $| \begin{matrix} 1 & 2 & 1 \\ 1 & 3 & 4 \\ 1 & 5 & 10 \end{matrix} |$ $= 0$

$D_{1} =$ $| \begin{matrix} 1 & 2 & 1 \\ K & 3 & 4 \\ K^{2} & 5 & 10 \end{matrix} |$ $= 5 (K^{2} - 3 K + 2) = 5 (K - 1) (K - 2)$

$D_{2} =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & K & 4 \\ 1 & K^{2} & 10 \end{matrix} |$ $= - 3 (K^{2} - 3 K + 2) = - 3 (K - 2) (K - 1)$

$D_{3} =$ $| \begin{matrix} 1 & 2 & 1 \\ 1 & 3 & K \\ 1 & 5 & K^{2} \end{matrix} |$ $= K^{2} - 3 K + 2 = (K - 2) (K - 1)$

Q 25 :

The system of equations

$(1 - λ) x + 3 y - 4 z = 0$

$x - (3 + λ) y + 5 z = 0$

$3 x + y - λ z = 0$

possesses non-trivial solutions for

exactly three distinct real values of $λ$
exactly two distinct real values of $λ$
exactly one real value of $λ$
infinite number of distinct real values of $λ$

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(2)

$Put$ $| \begin{matrix} 1 - λ & 3 & - 4 \\ 1 & - (3 + λ) & 5 \\ 3 & 1 & - λ \end{matrix} |$ $= 0$

$\Rightarrow λ {(λ + 1)}^{2} = 0 \Rightarrow λ = - 1, 0$

Q 26 :

$If the system of equations$

$x + y + z = 16$

$2 x + 5 y + α z = β$

$x + 2 y + 3 z = 14$

$has infinitely many solutions, then α + β is equal to$

(34)

It has infinitely many solutions.

Q 27 :

Let $α, β \in R$ be such that the system of linear equations x + 2y + z = 5; $2 x + y + α z = 5$ ; $8 x + 4 y + β z = 18$ has no solution. Then $\frac{β}{α}$ is equal to : [2026]

–4
4
8
–8

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(2)

Given system of linear equations: x + 2y + z = 5, $2 x + y + α z = 5$ , $8 x + 4 y + β z = 18$

Since the given system of linear equations has no solution, so, $△ = 0$

$\Rightarrow △ = | \begin{matrix} 1 & 2 & 1 \\ 2 & 1 & α \\ 8 & 4 & β \end{matrix} | = 12 α - 3 β = 0$

$\Rightarrow β = 4 α$

$△_{z} = | \begin{matrix} 1 & 2 & 5 \\ 2 & 1 & 5 \\ 8 & 4 & 18 \end{matrix} | = 6 \neq 0$

$\Rightarrow z = \frac{△_{z}}{△} \Rightarrow z = \frac{6}{0}$ = not defined

Hence, $\frac{β}{α} = 4$

Q 28 :

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 [2026]

y – x = 3
x – y = 3
x + y = 11
x + y = 12

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(2)

$△ = | \begin{matrix} 1 & 5 & 6 \\ 2 & 3 & 4 \\ 1 & 6 & a \end{matrix} | = 0$

$\Rightarrow (3 a - 24) - 5 (2 a - 4) + 6 (9) = 0$

$\Rightarrow - 7 a + 50 = 0 \Rightarrow a = \frac{50}{7}$

Now, $△_{z} = = 0 \Rightarrow | \begin{matrix} 1 & 5 & 4 \\ 2 & 3 & 7 \\ 1 & 6 & b \end{matrix} | = 0$

$\Rightarrow (3 b - 42) - 5 (2 b - 7) + 4 (9) = 0$

$\Rightarrow - 7 b + 29 = 0 \Rightarrow b = \frac{29}{7}$

Now, a – b = 3

$∴$ x – y = 3

Q 29 :

If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 2y + $λ$ z = $μ$ has infinitely many solutions, then the value of $λ + μ$ is: [2026]

16
18
19
21

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(2)

Since, given system equations have infinitely many solutions.

$∴ | \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & λ \end{matrix} | = 0$

$\Rightarrow 1 (2 λ - 9) - 1 (λ - 3) + 1 (3 - 2) = 0$

$\Rightarrow 2 λ - 9 - (λ - 3) + 1 = 0 \Rightarrow λ - 5 = 0$

$\Rightarrow λ = 5$

$∴ △_{3} = | \begin{matrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & μ \end{matrix} | = 0$

$\Rightarrow 1 (2 μ - 27) - 1 (μ - 9) + 5 (3 - 2) = 0$

$\Rightarrow (2 μ - 27) - (μ - 9) + 5 (1) = 0$

$\Rightarrow μ - 13 = 0 \Rightarrow μ = 13$

So, $λ + μ = 5 + 13 = 18$

Q 30 :

The sum of all possible values of $θ \in [0, 2 π]$ , for which the system of equations:

$x \cos 3 θ - 8 y - 12 z = 0$ ;

$x \cos 2 θ + 3 y + 3 z = 0$ ;

x + y + 3z = 0

has a non-trivial solution, is equal to: [2026]

$π$
$2 π$
$3 π$
$4 π$

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(4)

Since the given system of equations has non-trivial solution,

$△ = | \begin{matrix} \cos 3 θ & - 8 & - 12 \\ \cos 2 θ & 3 & 3 \\ 1 & 1 & 3 \end{matrix} |$

Expanding along $C_{1}$

$\Rightarrow \cos 3 θ | \begin{matrix} 3 & 3 \\ 1 & 3 \end{matrix} | - \cos 2 θ | \begin{matrix} - 8 & - 12 \\ 1 & 3 \end{matrix} | + 1 | \begin{matrix} - 8 & - 12 \\ 3 & 3 \end{matrix} | = 0$

$\Rightarrow 6 \cos 3 θ - (- 12) \cos 2 θ + 12 = 0$

$\Rightarrow \cos 3 θ + 2 \cos 2 θ + 2 = 0$

Since, $\cos 2 θ = 2 \cos^{2} θ - 1 and \cos 3 θ = 4 \cos^{3} θ - 3 \cos θ$

$∴ (4 \cos^{3} θ - 3 \cos θ) + 4 \cos^{2} θ - 2 + 2 = 0$

$\Rightarrow 4 \cos^{3} θ + 4 \cos^{2} θ - 3 \cos θ = 0$

$\Rightarrow \cos θ (4 \cos^{2} θ + 4 \cos θ - 3) = 0$

$\Rightarrow \cos θ (2 \cos θ - 1) (2 \cos θ + 3) = 0$

$\Rightarrow \cos θ = 0, \cos θ = \frac{1}{2}$ and $\cos θ = \frac{- 3}{2}$ (no real soluion)

If $\cos θ = 0, θ = \frac{π}{2}, \frac{3 π}{2}$

If $\cos θ = \frac{1}{2}, θ = \frac{π}{3}, \frac{3 π}{3}$

$∴ Required sum = \frac{π}{2} + \frac{3 π}{2} + \frac{π}{3} + \frac{5 π}{3}$

$= 2 π + 2 π = 4 π$

Q 31 :

If the system of linear equations:

x + y + z = 6

x + 2y + 5z = 10

2x + 3y + $λ$ z = $μ$

has infinitely many solutions, then the value of $λ + μ$ equals: [2026]

12
16
22
28

1

2

3

4

(3)

x + y + z = 6

x + 2y + 3z = 10

2x + 3y + $λ$ z = $μ$

For infinitely many solutions, $△ = 0$

$\Rightarrow | \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 2 & 3 & λ \end{matrix} | = 0$

$\Rightarrow (2 λ - 15) - (λ - 10) + (3 - 4) = 0$

$\Rightarrow λ = 6$

Also, $△_{1} = 0$

$\Rightarrow | \begin{matrix} 6 & 1 & 1 \\ 10 & 2 & 5 \\ μ & 3 & 6 \end{matrix} | = 0$

$\Rightarrow 6 (12 - 15) - (60 - 5 μ) + (30 - 2 μ) = 0$

$\Rightarrow - 18 - 60 + 5 μ + 30 - 2 μ = 0$

$\Rightarrow 3 μ = 48 \Rightarrow μ = 16$

$∴ λ + μ = 22$

Q 32 :

If the system of equations $x + y + z = 5, x + 2 y + 3 z = 9, x + 3 y + α z = β$ has infinitely many solutions, then $β - 2 α$ equals:

21
3
18
5

1

2

3

4

(2)

$x + y + z = 5,$ $x + 2 y + 3 z = 9,$ $x + 3 y + α z = β$

$D =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & α \end{matrix} |$ $= 0$

$\Rightarrow (2 α - 9) + (3 - α) + (3 - 2) = 0$

$\Rightarrow α = 5$

$Now,$ $D_{3} =$ $| \begin{matrix} 1 & 1 & 5 \\ 1 & 2 & 9 \\ 1 & 3 & β \end{matrix} |$ $= 0$

$\Rightarrow 2 β - 27 + 9 - β + 5 (3 - 2) = 0$

$\Rightarrow β = 13$

$\Rightarrow at α = 5, β = 13$

Q 33 :

If A and B are the two real values of $k$ for which the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = k, x + 5 y + 10 z = k^{2}$ is consistent,

then $A^{2} + B^{2}$ is _______

(5)

Here $Δ = 0$ and $Δ_{1} = Δ_{2} = Δ_{3} = 0 \Rightarrow k = 1, 2$ then $A^{2} + B^{2} = 1 + 4 = 5$

Topic Question Set

Q 11 :

Let S denote the set of all real values of $λ$ such that the system of equations $λ x + y + z = 1, x + λ y + z = 1, x + y + λ z = 1$ is inconsistent. Then $\sum_{λ \in S} (| λ |^{2} + | λ |)$ is equal to [2023]

Q 12 :

For the system of linear equations $α x + y + z = 1, x + α y + z = 1, x + y + α z = β,$ which one of the following statements is NOT correct? [2023]

Q 14 :

If the system of equations

$x + 2 y + 3 z = 3$

$4 x + 3 y - 4 z = 4$

$8 x + 4 y - λ z = 9 + μ$

has infinitely many solutions, then the ordered pair $(λ, μ)$ is equal to [2023]

Q 16 :

Consider the following system of equations

$α x + 2 y + z = 1$

$2 α x + 3 y + z = 1$

$3 x + α y + 2 z = β$

for some $α, β \in ℝ$ . Then which of the following is NOT correct? [2023]

Q 17 :

Let the system of linear equations

$x + y + k z = 2$

$2 x + 3 y - z = 1$

$3 x + 4 y + 2 z = k$

have infinitely many solutions. Then the system

$(k + 1) x + (2 k - 1) y = 7$

$(2 k + 1) x + (k + 5) y = 10$

has [2023]

Q 18 :

For $α, β \in R,$ suppose the system of linear equations

$x - y + z = 5$

$2 x + 2 y + α z = 8$

$3 x - y + 4 z = β$

has infinitely many solutions. Then $α$ and $β$ are the roots of [2023]

Q 19 :

For the system of linear equations $x + y + z = 6, α x + β y + 7 z = 3, x + 2 y + 3 z = 14,$ which of the following is NOT true? [2023]

Q 20 :

Let S be the set of values of $λ$ , for which the system of equations

$6 λ x - 3 y + 3 z = 4 λ^{2},$

$2 x + 6 λ y + 4 z = 1,$

$3 x + 2 y + 3 λ z = λ$ has no solution. Then $12 \sum_{λ \in S} | λ |$ is equal to ___________ . [2023]

Q 21 :

System of linear equations $5 x + 2 y = 4,$ $7 x + 3 y = 5$ has:

Q 22 :

The following system of linear equations

$3 x + 5 y = 7$

$2 x - 5 y = 9$

is:

Q 23 :

The value of $x$ and $y$ , if the system of linear equations is $2 x - y = 2$ and $3 x + 4 y = 3$ is:

Q 24 :

The number of distinct real values of $' K'$ such that the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = K, x + 5 y + 10 z = K^{2}$ has infinitely many solutions is:

Q 25 :

The system of equations

$(1 - λ) x + 3 y - 4 z = 0$

$x - (3 + λ) y + 5 z = 0$

$3 x + y - λ z = 0$

possesses non-trivial solutions for

Q 26 :

$If the system of equations$

$x + y + z = 16$

$2 x + 5 y + α z = β$

$x + 2 y + 3 z = 14$

$has infinitely many solutions, then α + β is equal to$

Q 27 :

Let $α, β \in R$ be such that the system of linear equations x + 2y + z = 5; $2 x + y + α z = 5$ ; $8 x + 4 y + β z = 18$ has no solution. Then $\frac{β}{α}$ is equal to : [2026]

Q 28 :

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 [2026]

Q 29 :

If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 2y + $λ$ z = $μ$ has infinitely many solutions, then the value of $λ + μ$ is: [2026]

Q 30 :

The sum of all possible values of $θ \in [0, 2 π]$ , for which the system of equations:

$x \cos 3 θ - 8 y - 12 z = 0$ ;

$x \cos 2 θ + 3 y + 3 z = 0$ ;

x + y + 3z = 0

has a non-trivial solution, is equal to: [2026]

Q 31 :

If the system of linear equations:

x + y + z = 6

x + 2y + 5z = 10

2x + 3y + $λ$ z = $μ$

has infinitely many solutions, then the value of $λ + μ$ equals: [2026]

Q 32 :

If the system of equations $x + y + z = 5, x + 2 y + 3 z = 9, x + 3 y + α z = β$ has infinitely many solutions, then $β - 2 α$ equals:

Q 33 :

If A and B are the two real values of $k$ for which the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = k, x + 5 y + 10 z = k^{2}$ is consistent,

then $A^{2} + B^{2}$ is _______

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Topic Question Set

Q 11 : Let S denote the set of all real values of λ such that the system of equations λx+y+z=1, x+λy+z=1, x+y+λz=1 is inconsistent. Then ∑λ∈S(|λ|2+|λ|) is equal to [2023]

Q 12 : For the system of linear equations αx+y+z=1, x+αy+z=1, x+y+αz=β, which one of the following statements is NOT correct? [2023]

Q 13 : Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations x+y+z=1 2x+Ny+2z=2 3x+3y+Nz=3 has a unique solution is k6, then the sum of the value of k and all possible values of N is [2023]

Q 14 : If the system of equations x+2y+3z=3 4x+3y-4z=4 8x+4y-λz=9+μ has infinitely many solutions, then the ordered pair (λ,μ) is equal to [2023]

Q 15 : Let S1 and S2 be respectively the sets of all a∈R-{0} for which the system of linear equations ax+2ay-3az=1 (2a+1)x+(2a+3)y+(a+1)z=2 (3a+5)x+(a+5)y+(a+2)z=3 has a unique solution and infinitely many solutions. Then [2023]

Q 16 : Consider the following system of equations αx+2y+z=1 2αx+3y+z=1 3x+αy+2z=β for some α,β∈ℝ. Then which of the following is NOT correct? [2023]

Q 17 : Let the system of linear equations x+y+kz=2 2x+3y-z=1 3x+4y+2z=k have infinitely many solutions. Then the system (k+1)x+(2k-1)y=7 (2k+1)x+(k+5)y=10 has [2023]

Q 18 : For α,β∈R, suppose the system of linear equations x-y+z=5 2x+2y+αz=8 3x-y+4z=β has infinitely many solutions. Then α and β are the roots of [2023]

Q 19 : For the system of linear equations x+y+z=6, αx+βy+7z=3, x+2y+3z=14, which of the following is NOT true? [2023]

Q 20 : Let S be the set of values of λ, for which the system of equations 6λx-3y+3z=4λ2, 2x+6λy+4z=1, 3x+2y+3λz=λ has no solution. Then 12∑λ∈S|λ| is equal to ___________ . [2023]

Q 21 : System of linear equations 5x+2y=4, 7x+3y=5 has:

Q 22 : The following system of linear equations 3x+5y=7 2x-5y=9 is:

Q 23 : The value of x and y, if the system of linear equations is 2x-y=2 and 3x+4y=3 is:

Q 24 : The number of distinct real values of 'K' such that the system of equations x+2y+z=1, x+3y+4z=K, x+5y+10z=K2 has infinitely many solutions is:

Q 25 : The system of equations (1-λ)x+3y-4z=0 x-(3+λ)y+5z=0 3x+y-λz=0 possesses non-trivial solutions for

Q 26 : If the system of equations x+y+z=16 2x+5y+αz=β x+2y+3z=14 has infinitely many solutions, then α+β is equal to

Q 27 : Let α,β∈R be such that the system of linear equations x + 2y + z = 5; 2x+y+αz=5; 8x+4y+βz=18 has no solution. Then βα is equal to : [2026]

Q 28 : 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 [2026]

Q 29 : If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 2y + λz = μ has infinitely many solutions, then the value of λ+μ is: [2026]

Q 30 : The sum of all possible values of θ∈[0,2π], for which the system of equations: xcos3θ–8y–12z=0; xcos2θ+3y+3z=0; x + y + 3z = 0 has a non-trivial solution, is equal to: [2026]

Q 31 : If the system of linear equations: x + y + z = 6 x + 2y + 5z = 10 2x + 3y + λz = μ has infinitely many solutions, then the value of λ+μ equals: [2026]

Q 32 : If the system of equations x+y+z=5, x+2y+3z=9, x+3y+αz=β has infinitely many solutions, then β-2α equals:

Q 33 : If A and B are the two real values of k for which the system of equations x+2y+z=1, x+3y+4z=k, x+5y+10z=k2 is consistent, then A2+B2 is _______

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Q 11 :

Let S denote the set of all real values of $λ$ such that the system of equations $λ x + y + z = 1, x + λ y + z = 1, x + y + λ z = 1$ is inconsistent. Then $\sum_{λ \in S} (| λ |^{2} + | λ |)$ is equal to [2023]

Q 12 :

For the system of linear equations $α x + y + z = 1, x + α y + z = 1, x + y + α z = β,$ which one of the following statements is NOT correct? [2023]

Q 14 :

If the system of equations

$x + 2 y + 3 z = 3$

$4 x + 3 y - 4 z = 4$

$8 x + 4 y - λ z = 9 + μ$

has infinitely many solutions, then the ordered pair $(λ, μ)$ is equal to [2023]

Q 16 :

Consider the following system of equations

$α x + 2 y + z = 1$

$2 α x + 3 y + z = 1$

$3 x + α y + 2 z = β$

for some $α, β \in ℝ$ . Then which of the following is NOT correct? [2023]

Q 17 :

Let the system of linear equations

$x + y + k z = 2$

$2 x + 3 y - z = 1$

$3 x + 4 y + 2 z = k$

have infinitely many solutions. Then the system

$(k + 1) x + (2 k - 1) y = 7$

$(2 k + 1) x + (k + 5) y = 10$

has [2023]

Q 18 :

For $α, β \in R,$ suppose the system of linear equations

$x - y + z = 5$

$2 x + 2 y + α z = 8$

$3 x - y + 4 z = β$

has infinitely many solutions. Then $α$ and $β$ are the roots of [2023]

Q 19 :

For the system of linear equations $x + y + z = 6, α x + β y + 7 z = 3, x + 2 y + 3 z = 14,$ which of the following is NOT true? [2023]

Q 20 :

Let S be the set of values of $λ$ , for which the system of equations

$6 λ x - 3 y + 3 z = 4 λ^{2},$

$2 x + 6 λ y + 4 z = 1,$

$3 x + 2 y + 3 λ z = λ$ has no solution. Then $12 \sum_{λ \in S} | λ |$ is equal to ___________ . [2023]

Q 21 :

System of linear equations $5 x + 2 y = 4,$ $7 x + 3 y = 5$ has:

Q 22 :

The following system of linear equations

$3 x + 5 y = 7$

$2 x - 5 y = 9$

is:

Q 23 :

The value of $x$ and $y$ , if the system of linear equations is $2 x - y = 2$ and $3 x + 4 y = 3$ is:

Q 24 :

The number of distinct real values of $' K'$ such that the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = K, x + 5 y + 10 z = K^{2}$ has infinitely many solutions is:

Q 25 :

The system of equations

$(1 - λ) x + 3 y - 4 z = 0$

$x - (3 + λ) y + 5 z = 0$

$3 x + y - λ z = 0$

possesses non-trivial solutions for

Q 26 :

$If the system of equations$

$x + y + z = 16$

$2 x + 5 y + α z = β$

$x + 2 y + 3 z = 14$

$has infinitely many solutions, then α + β is equal to$

Q 27 :

Let $α, β \in R$ be such that the system of linear equations x + 2y + z = 5; $2 x + y + α z = 5$ ; $8 x + 4 y + β z = 18$ has no solution. Then $\frac{β}{α}$ is equal to : [2026]

Q 28 :

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 [2026]

Q 29 :

If the system of equations: x + y + z = 5; x + 2y + 3z = 9; x + 2y + $λ$ z = $μ$ has infinitely many solutions, then the value of $λ + μ$ is: [2026]

Q 30 :

The sum of all possible values of $θ \in [0, 2 π]$ , for which the system of equations:

$x \cos 3 θ - 8 y - 12 z = 0$ ;

$x \cos 2 θ + 3 y + 3 z = 0$ ;

x + y + 3z = 0

has a non-trivial solution, is equal to: [2026]

Q 31 :

If the system of linear equations:

x + y + z = 6

x + 2y + 5z = 10

2x + 3y + $λ$ z = $μ$

has infinitely many solutions, then the value of $λ + μ$ equals: [2026]

Q 32 :

If the system of equations $x + y + z = 5, x + 2 y + 3 z = 9, x + 3 y + α z = β$ has infinitely many solutions, then $β - 2 α$ equals:

Q 33 :

If A and B are the two real values of $k$ for which the system of equations $x + 2 y + z = 1, x + 3 y + 4 z = k, x + 5 y + 10 z = k^{2}$ is consistent,

then $A^{2} + B^{2}$ is _______