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

Q 1 :

If the system of equations

$x + (\sqrt{2} \sin α) y + (\sqrt{2} \cos α) z = 0$

$x + (\cos α) y + (\sin α) z = 0$

$x + (\sin α) y - (\cos α) z = 0$

has a non-trivial solution, then $α \in (0, \frac{π}{2})$ is equal to [2024]

$\frac{7 π}{24}$
$\frac{3 π}{4}$
$\frac{11 π}{24}$
$\frac{5 π}{24}$

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(4)

Given, $x + (\sqrt{2} \sin α) y + (\sqrt{2} \cos α) z = 0$

$x + (\cos α) y + (\sin α) z = 0$

$x + (\sin α) y - (\cos α) z = 0$

For non-trivial solution,

$| \begin{matrix} 1 & \sqrt{2} \sin α & \sqrt{2} \cos α \\ 1 & \cos α & \sin α \\ 1 & \sin α & - \cos α \end{matrix} |$ $= 0$

$\Rightarrow 1 [- \cos^{2} α - \sin^{2} α] - 1 [- \sqrt{2} \sin α \cos α - \sqrt{2} \sin α \cos α] + 1 [\sqrt{2} \sin^{2} α - \sqrt{2} \cos^{2} α] = 0$

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

$\Rightarrow \sqrt{2} \sin 2 α - \sqrt{2} \cos 2 α = 1$

$\Rightarrow \sin (2 α - \frac{π}{4}) = \sin \frac{π}{6} \Rightarrow 2 α - \frac{π}{4} = n π + {(- 1)}^{n} \frac{π}{6}$

for $n = 0 \Rightarrow α = \frac{5 π}{24}$

Q 2 :

If the system of equations

$11 x + y + λ z = - 5$

$2 x + 3 y + 5 z = 3$

$8 x - 19 y - 39 z = μ$

has infinitely many solutions, then $λ^{4} - μ$ is equal to: [2024]

49
51
47
45

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4

(3)

We have, $11 x + y + λ z = - 5$

$2 x + 3 y + 5 z = 3$

$8 x - 19 y - 39 z = μ$

has infinitely many solutions

$∴$ $Δ =$ $| \begin{matrix} 11 & 1 & λ \\ 2 & 3 & 5 \\ 8 & - 19 & - 39 \end{matrix} |$ $= 0$

$\Rightarrow 11 (- 22) - 1 (- 118) + λ (- 62) = 0$

$\Rightarrow - 124 - 62 λ = 0$

$\Rightarrow λ = - 2$

Also, $Δ_{3} =$ $| \begin{matrix} 11 & 1 & - 5 \\ 2 & 3 & 3 \\ 8 & - 19 & μ \end{matrix} |$ $= 0$

$\Rightarrow 11 (3 μ + 57) - 1 (2 μ - 24) - 5 (- 38 - 24)$

$\Rightarrow 31 μ + 961 = 0 \Rightarrow μ = - 31$

So, $λ^{4} - μ = {(- 2)}^{4} + 31 = 16 + 31 = 47$

Q 3 :

The values of $m, n$ for which the system of equations $x + y + z = 4, 2 x + 5 y + 5 z = 17, x + 2 y + m z = n$ has infinitely many solutions, satisfy the equation: [2024]

$m^{2} + n^{2} + m n = 68$
$m^{2} + n^{2} - m - n = 46$
$m^{2} + n^{2} - m n = 39$
$m^{2} + n^{2} + m + n = 64$

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(3)

Since the system of equations has infinitely many solutions

$∴$ $Δ = 0 and Δ_{3} = 0$

$\Rightarrow$ $| \begin{matrix} 1 & 1 & 1 \\ 2 & 5 & 5 \\ 1 & 2 & m \end{matrix} |$ $= 0$ and $| \begin{matrix} 1 & 1 & 4 \\ 2 & 5 & 17 \\ 1 & 2 & n \end{matrix} |$ $= 0$

$\Rightarrow (5 m - 10) - (2 m - 5) + (- 1) = 0$ and $(5 n - 34) - (2 n - 17) + 4 (- 1) = 0$

$\Rightarrow 3 m - 6 = 0$ and $3 n - 21 = 0$

$\Rightarrow m = 2$ and $n = 7$

Clearly $m^{2} + n^{2} - m n = 4 + 49 - 14 = 39$

Q 4 :

If the system of equations $x + 4 y - z = λ,$ $7 x + 9 y + μ z = - 3,$ $5 x + y + 2 z = - 1$ has infinitely many solutions, then $(2 μ + 3 λ)$ is equal to: [2024]

$- 2$
$3$
$- 3$
$2$

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(3)

We have, $x + 4 y - z = λ$

$7 x + 9 y + μ z = - 3; 5 x + y + 2 z = - 1$

$[\begin{matrix} 1 & 4 & - 1 \\ 7 & 9 & μ \\ 5 & 1 & 2 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} λ \\ - 3 \\ - 1 \end{matrix}]$

$A = [\begin{matrix} 1 & 4 & - 1 \\ 7 & 9 & μ \\ 5 & 1 & 2 \end{matrix}], B = [\begin{matrix} λ \\ - 3 \\ - 1 \end{matrix}]$

$A X = B \Rightarrow X = A^{- 1} B = \frac{adj A}{| A |} B$

Since, the system has infinitely many solutions,

$∴$ $| A | = 0 and (adj A) \cdot B = 0 .$

Now, $| A |$ $= 0 \Rightarrow 18 - μ - 4 (14 - 5 μ) - 1 (7 - 45) = 0$

$\Rightarrow 18 - μ - 56 + 20 μ + 38 = 0$

$\Rightarrow 19 μ = 0 \Rightarrow μ = 0$

Also, $adj A = [\begin{matrix} 18 & - 9 & 9 \\ - 14 & 7 & - 7 \\ - 38 & 19 & - 19 \end{matrix}]$

$(adj A \cdot B) = 0$

$\Rightarrow [\begin{matrix} 18 & - 9 & 9 \\ - 14 & 7 & - 7 \\ - 38 & 19 & - 19 \end{matrix}] [\begin{matrix} λ \\ - 3 \\ - 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

$18 λ + 27 - 9 = 0 \Rightarrow 18 λ = - 18 \Rightarrow λ = - 1$

Hence, $2 μ + 3 λ = 2 (0) + 3 (- 1) = - 3$

Q 5 :

Let $λ, μ \in R .$ If the system of equations

$3 x + 5 y + λ z = 3$

$7 x + 11 y - 9 z = 2$

$97 x + 155 y - 189 z = μ$

has infinitely many solutions, then $μ + 2 λ$ is equal to: [2024]

24
27
25
22

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4

(3)

$3 x + 5 y + λ z = 3$

$7 x + 11 y - 9 z = 2$

$97 x + 155 y - 189 z = μ$

$[\begin{matrix} 3 & 5 & λ \\ 7 & 11 & - 9 \\ 97 & 155 & - 189 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 3 \\ 2 \\ μ \end{matrix}]$

$A X = B$

For infinitely many solutions,

$| A |$ $= 0 and (adj A) B = 0$

$\Rightarrow$ $| \begin{matrix} 3 & 5 & λ \\ 7 & 11 & - 9 \\ 97 & 155 & - 189 \end{matrix} |$ $= 0$

$\Rightarrow - 2052 + 2250 + 18 λ = 0$

$\Rightarrow λ = - 11$

$adj A = [\begin{matrix} - 684 & - 760 & 76 \\ 450 & 500 & - 50 \\ 18 & 20 & - 2 \end{matrix}]$

$(adj A) B = [\begin{matrix} - 684 & - 760 & 76 \\ 450 & 500 & - 50 \\ 18 & 20 & - 2 \end{matrix}] [\begin{matrix} 3 \\ 2 \\ μ \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

$\Rightarrow 54 + 40 - 2 μ = 0$

$\Rightarrow 2 μ = 94 \Rightarrow μ = 47$

Hence, $μ + 2 λ = 25$

Q 6 :

If the system of equations $2 x + 3 y - z = 5,$ $x + α y + 3 z = - 4,$ $3 x - y + β z = 7$ has infinitely many solutions, then $13 α β$ is equal to _____ . [2024]

1210
1110
1220
1120

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(4)

Given, system of equations can be written as $A X = B,$

where $A = [\begin{matrix} 2 & 3 & - 1 \\ 1 & α & 3 \\ 3 & - 1 & β \end{matrix}], B = [\begin{matrix} 5 \\ - 4 \\ 7 \end{matrix}]$ and $X = [\begin{matrix} x \\ y \\ z \end{matrix}]$

Using Cramer's rule for infinite solutions,

$D = D_{1} = D_{2} = D_{3} = 0$

$D_{3} =$ $| \begin{matrix} 2 & 3 & 5 \\ 1 & α & - 4 \\ 3 & - 1 & 7 \end{matrix} |$ $= 2 (7 α - 4) - 3 (7 + 12) + 5 (- 1 - 3 α) = 0$

$\Rightarrow α = - 70$

Similarly,

$D_{2} =$ $| \begin{matrix} 2 & 5 & - 1 \\ 1 & - 4 & 3 \\ 3 & 7 & β \end{matrix} |$ $= 2 (- 4 β - 21) - 5 (β - 9) - 1 (7 + 12) = 0$

$\Rightarrow β = - \frac{16}{13}$

$∴$ $13 α β = 13 \times (- 70) \times (\frac{- 16}{13}) = 1120$

Q 7 :

Let the system of equations $x + 2 y + 3 z = 5,$ $2 x + 3 y + z = 9,$ $4 x + 3 y + λ z = μ$ have infinite number of solutions. Then $λ + 2 μ$ is equal to: [2024]

22
17
15
28

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(2)

Given system of equations can be written as, $A X = B$

Where $A = [\begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 4 & 3 & λ \end{matrix}], X = [\begin{matrix} x \\ y \\ z \end{matrix}],$ and $B = [\begin{matrix} 5 \\ 9 \\ μ \end{matrix}]$

For infinitely many solutions, $D = 0 \Rightarrow$ $| \begin{matrix} 1 & 2 & 3 \\ 2 & 3 & 1 \\ 4 & 3 & λ \end{matrix} |$ $= 0$

$\Rightarrow λ = - 13$

Also, $D_{1} = 0 \Rightarrow$ $| \begin{matrix} 5 & 2 & 3 \\ 9 & 3 & 1 \\ μ & 3 & - 13 \end{matrix} |$ $= 0 \Rightarrow μ = 15$

So, $λ + 2 μ = - 13 + 30 = 17$

Q 8 :

Consider the system of linear equations $x + y + z = 4 μ,$ $x + 2 y + 2 λ z = 10 μ,$ $x + 3 y + 4 λ^{2} z = μ^{2} + 15,$ where $λ, μ \in R .$ Which one of the following statements is NOT correct? [2024]

The system is inconsistent if $λ = \frac{1}{2}$ and $μ \neq 1$
The system has unique solution if $λ \neq \frac{1}{2}$ and $μ \neq 1, 15$
The system has infinite number of solutions if $λ = \frac{1}{2}$ and $μ = 15$
The system is consistent if $λ \neq \frac{1}{2}$

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(1)

We have, $[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 2 λ \\ 1 & 3 & 4 λ^{2} \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 4 μ \\ 10 μ \\ μ^{2} + 15 \end{matrix}]$

For $λ = \frac{1}{2}$ and $μ = 15,$ we get $[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 3 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 60 \\ 150 \\ 240 \end{matrix}]$

System is consistent but does not have unique solution as matrix have zero determinant because two columns are same.

Also, let $λ \neq \frac{1}{2}$ and $λ = μ = 1,$ then we have

$[\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 2 \\ 1 & 3 & 4 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 4 \\ 10 \\ 16 \end{matrix}]$

$\Rightarrow x + y + z = 4; x + 2 y + 2 z = 10; x + 3 y + 4 z = 16$

On solving these equations, we get $(- 2, 6, 0)$ as unique solution.

Clearly, for $λ = \frac{1}{2}$ and $μ \neq 1$ i.e., $μ = 15$ the system is consistent and have infinite solution.

So, statement (a) is not correct.

Q 9 :

Consider the system of linear equations $x + y + z = 5,$ $x + 2 y + λ^{2} z = 9,$ $x + 3 y + λ z = μ,$ where $λ, μ \in R .$ Then which of the following statement is NOT correct? [2024]

System has unique solution if $λ \neq 1$ and $μ \neq 13$ .
System is inconsistent if $λ = 1$ and $μ \neq 13 .$
System is consistent if $λ \neq 1$ and $μ = 13 .$
System has infinite number of solutions if $λ = 1$ and $μ = 13 .$

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(1)

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

$\Rightarrow λ = 1, - \frac{1}{2} and$ $| \begin{matrix} 1 & 1 & 5 \\ 2 & λ^{2} & 9 \\ 3 & λ & μ \end{matrix} |$ $= 0 \Rightarrow μ = 13$

For infinite solution, $λ = 1$ and $μ = 13$

For unique solution $λ \neq 1$

For no solution, $λ = 1$ and $μ \neq 13$

If $λ \neq 1$ and $μ \neq 13$

Considering the case when $λ = - \frac{1}{2}$ and $μ \neq 13$ this will generate no solution case.

Q 10 :

If the system of linear equations $x - 2 y + z = - 4,$ $2 x + α y + 3 z = 5,$ $3 x - y + β z = 3$ has infinitely many solutions, then $12 α + 13 β$ is equal to [2024]

54
64
58
60

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(3)

Given, $x - 2 y + z = - 4$

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

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

$\Rightarrow α β - 18 - 2 - 3 α + 3 + 4 β = 0 = α β - 3 α + 4 β - 17 = 0$

$Δ y =$ $| \begin{matrix} 1 & - 4 & 1 \\ 2 & 5 & 3 \\ 3 & 3 & β \end{matrix} |$ $= 0$

$Δ y = 1 (5 β - 9) + 4 (2 β - 9) + 1 (6 - 15)$

$= 5 β - 9 + 8 β - 36 + 6 - 15$

$\Rightarrow 13 β - 54 = 0 \Rightarrow 13 β = 54 \Rightarrow β = \frac{54}{13}$

$Δ z =$ $| \begin{matrix} 1 & - 2 & - 4 \\ 2 & α & 5 \\ 3 & - 1 & 3 \end{matrix} |$ $= 0$

$Δ z = 1 (3 α + 5) + 2 (6 - 15) - 4 (- 2 - 3 α)$

$\Rightarrow 3 α + 5 + 12 - 30 + 8 + 12 α$

$\Rightarrow 15 α - 5 = 0 \Rightarrow 15 α = 5 \Rightarrow α = \frac{5}{15} = \frac{1}{3}$

So, $12 α + 13 β = 12 \times \frac{1}{3} + 54 \Rightarrow 4 + 54 = 58$

Q 11 :

Let $α β γ = 45; α, β, γ \in R .$ If $x (α, 1, 2) + y (1, β, 2) + z (2, 3, γ) = (0, 0, 0)$ for some $x, y, z \in R, x y z \neq 0,$ then $6 α + 4 β + γ$ is equal to _______. [2024]

(55)

We have, $x (α, 1, 2) + y (1, β, 2) + z (2, 3, γ) = (0, 0, 0)$

$\Rightarrow α x + y + 2 z = 0; x + β y + 3 z = 0; 2 x + 2 y + γ z = 0$

Since, $x y z \neq 0$ so the system of equations has non-trivial solution.

Now, $| \begin{matrix} α & 1 & 2 \\ 1 & β & 3 \\ 2 & 2 & γ \end{matrix} |$ $= 0$

$\Rightarrow α (β γ - 6) - 1 (γ - 6) + 2 (2 - 2 β) = 0$

$\Rightarrow α β γ - 6 α - γ + 6 + 4 - 4 β = 0$

$\Rightarrow 45 + 10 - 6 α - γ - 4 β = 0 \Rightarrow 6 α + 4 β + γ = 55$

Q 12 :

If the system of equations, $2 x + 7 y + λ z = 3,$ $3 x + 2 y + 5 z = 4,$ $x + μ y + 32 z = - 1$ has infinitely many solutions, then $(λ - μ)$ is equal to _______. [2024]

(38)

Given, $2 x + 7 y + λ z = 3$

$3 x + 2 y + 5 z = 4$

$x + μ y + 32 z = - 1$ has infinitely many solutions.

$\Rightarrow Δ = 0, Δ_{1} = 0, Δ_{2} = 0$ and $Δ_{3} = 0$

$Δ_{2} =$ $| \begin{matrix} 2 & λ & 3 \\ 3 & 5 & 4 \\ 1 & 32 & - 1 \end{matrix} |$ $= 0$ and $Δ_{3} =$ $| \begin{matrix} 2 & 7 & 3 \\ 3 & 2 & 4 \\ 1 & μ & - 1 \end{matrix} |$ $= 0$

$\Rightarrow - 266 + λ (7) + 273 = 0$ and $- 4 - 8 μ + 49 + 9 μ - 6 = 0$

$\Rightarrow 7 λ + 7 = 0$ and $μ + 39 = 0$

$\Rightarrow λ = - 1$ and $μ = - 39$

So, $λ - μ = - 1 + 39 = 38$

Q 13 :

If the system of Linear equations

$3 x + y + β z = 3$

$2 x + α y - z = - 3$

$x + 2 y + z = 4$

has infinitely many solutions, then the value of $22 β - 9 α$ is : [2025]

31
49
37
43

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4

(1)

Since, the given system of equations has infinitely many solutions, $△ = △_{1} = △_{2} = △_{3} = 0$

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

$\Rightarrow 3 α + 4 β - α β + 3 = 0$ ...(i)

$△_{3} =$ $| \begin{matrix} 3 & 1 & 3 \\ 2 & α & - 3 \\ 1 & 2 & 4 \end{matrix} |$ $= 0$

$\Rightarrow 9 α + 19 = 0 \Rightarrow α = \frac{- 19}{9}$

From (i), we get

$β = \frac{6}{11}$

$∴ 22 β - 9 α = 22 \times \frac{6}{11} - 9 \times \frac{- 19}{9} = 12 + 19 = 31$ .

Q 14 :

If the system of equations

$2 x + λ y + 3 z = 5$

$3 x + 2 y - z = 7$

$4 x + 5 y + μ z = 9$

has infinitely many solutions, then $(λ^{2} + μ^{2})$ is equal to: [2025]

22
18
26
30

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2

3

4

(3)

For infinitely man solutions, we have $△ = △_{1} = △_{2} = △_{3} = 0$

Now, $△ =$ $| \begin{matrix} 2 & λ & 3 \\ 3 & 2 & - 1 \\ 4 & 5 & μ \end{matrix} |$ $= 0$

$\Rightarrow 2 (2 μ + 5) - λ (3 μ + 4) + 3 (15 - 8) = 0$

$\Rightarrow 4 μ + 10 - 3 λ μ - 4 λ + 21 = 0$

$\Rightarrow 4 μ - 4 λ - 3 λ μ + 31 = 0$ ... (i)

Now, $△_{2} =$ $| \begin{matrix} 2 & 5 & 3 \\ 3 & 7 & - 1 \\ 4 & 9 & μ \end{matrix} |$ $= 0$

$\Rightarrow 2 (7 μ + 9) - 5 (3 μ + 4) + 3 (27 - 28)$

$\Rightarrow 14 μ + 18 - 15 μ - 20 - 3 = 0$

$\Rightarrow - μ + 18 - 23 = 0 \Rightarrow μ = - 5$

Substituting the value of $μ$ in (i), we get

$- 20 - 4 λ + 15 λ + 31 = 0$

$\Rightarrow 11 λ + 11 = 0 \Rightarrow λ = - 1$

$∴ λ^{2} + μ^{2} = {(- 1)}^{2} + {(- 5)}^{2} = 26$ .

Q 15 :

Let the system of equations :

$2 x + 3 y + 5 z = 9$

$7 x + 3 y - 2 z = 8$

$12 x + 3 y - (4 + λ) z = 16 - μ$

have infinitely many solutions. Then the radius of the circle centred at $(λ, μ)$ and touching the line 4x = 3y is [2025]

$\frac{17}{5}$
$\frac{7}{5}$
$\frac{21}{5}$
7

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(2)

Since, the given system of equation has infinitely many solutions.

$△ = △_{1} = △_{2} = △_{3} = 0$

$△ =$ $| \begin{matrix} 2 & 3 & 5 \\ 7 & 3 & - 2 \\ 12 & 3 & - (λ + 4) \end{matrix} |$ $= 0$

$\Rightarrow 12 (- 21) - 3 (- 39) - (λ + 4) (- 15) = 0$

$\Rightarrow - 252 + 117 + 15 (λ + 4) = 0$

$\Rightarrow 15 λ - 75 = 0 \Rightarrow λ = 5$

$△_{1} =$ $| \begin{matrix} 9 & 3 & 5 \\ 8 & 3 & - 2 \\ 16 - μ & 3 & - 9 \end{matrix} |$ $= 0$

$\Rightarrow 9 (- 21) - 3 (- 40 - 2 μ) + 5 (- 24 + 3 μ) = 0$

$\Rightarrow μ = 9$

So, centre of circle (5, 9)

Also, radius = length of $⊥$ from centre (5, 9) to 4x = 3y

$∴ r =$ $| \frac{4 (5) - 3 (9)}{\sqrt{{(4)}^{2} + {(3)}^{2}}} |$ $=$ $| \frac{20 - 27}{5} |$ $= \frac{7}{5}$ .

Q 16 :

Let the system of equations

$x + 5 y - z = 1$

$4 x + 3 y - 3 z = 7$

$24 x + y + λ z = μ$

$λ, μ \in R$ , have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy $7 \leq x + y + z \leq 77$ , is : [2025]

3
5
6
4

1

2

3

4

(1)

For infinitely many solutions, we have D = 0

$\Rightarrow$ $| \begin{matrix} 1 & 5 & - 1 \\ 4 & 3 & - 3 \\ 24 & 1 & λ \end{matrix} |$ $= 0$

$\Rightarrow 1 (3 λ + 3) - 5 (4 λ + 72) - 1 (4 - 72) = 0$

$\Rightarrow 17 λ = - 289 \Rightarrow λ = - 17$ ... (i)

Now, $D_{1} = 0$

$\Rightarrow$ $| \begin{matrix} 1 & 5 & - 1 \\ 7 & 3 & - 3 \\ μ & 1 & - 17 \end{matrix} |$ $= 0$

$\Rightarrow 1 (- 51 + 3) - 5 (- 119 + 3 μ) - 1 (7 - 3 μ) = 0$

$\Rightarrow - 48 + 595 - 15 μ - 7 + 3 μ = 0$

$\Rightarrow 12 μ = 540$

$\Rightarrow μ = 45$ ... (ii)

Using (i) and (ii), we get

x + 5y – z = 1, 4x + 3y – 3z =7, 24x + y –17z = 45

$\Rightarrow z = x + 5 y - 1$

$∴ 4 x + 3 y - 3 x - 15 y + 3 = 7$

$\Rightarrow x - 12 y = 4$

$\Rightarrow x = 4 + 12 y and z = 4 + 12 y + 5 y - 1 = 3 + 17 y$

$∴ (x, y, z) = (4 + 12 k, k, 3 + 17 k)$ ( $∵$ Assume y = k)

Also, $7 \leq 7 + 30 k \leq 77$

$\Rightarrow 0 \leq 30 k < 70$

$\Rightarrow 0 \leq k \leq 2.3$

$\Rightarrow$ k = 0, 1, 2

Thus, there are three possible solutions.

Q 17 :

If the system of linear equations:

x + y + 2z = 6

2x + 3y + az = a + 1

– x – 3y + bz = 2b

where $a, b \in R$ , has infinitely many solutions, then $7 a + 3 b$ is equal to : [2025]

16
22
9
12

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4

(1)

For infinitely many solutions, we have

$△ =$ $| \begin{matrix} 1 & 1 & 2 \\ 2 & 3 & a \\ - 1 & - 3 & b \end{matrix} |$ $= 0 \Rightarrow 2 a + b - 6 = 0$ ...(i)

Also, $△_{3} =$ $| \begin{matrix} 1 & 1 & 6 \\ 2 & 3 & a + 1 \\ - 1 & - 3 & 2 b \end{matrix} |$ $= 0 \Rightarrow a + b - 8 = 0$ ... (ii)

Solving (i) and (ii), we get a = –2, b = 10

$∴$ 7a + 3b = 7(–2) + 3(10) =16.

Q 18 :

If the system of equations

$(λ - 1) x + (λ - 4) y + λ z = 5$

$λ x + (λ - 1) y + (λ - 4) z = 7$

$(λ + 1) x + (λ + 2) y - (λ + 2) z = 9$

has infinitely many solutions, then $λ^{2} + λ$ is equal to [2025]

20
12
6
10

1

2

3

4

(2)

For infinitely many solutions,

$D =$ $| \begin{matrix} (λ - 1) & (λ - 4) & λ \\ λ & (λ - 1) & (λ - 4) \\ (λ + 1) & (λ + 2) & - (λ + 2) \end{matrix} |$ $= 0$ ... (i)

On solving (i) along column (1), we get

$2 λ^{2} - 5 λ - 3 = 0$

$\Rightarrow (2 λ + 1) (λ - 3) = 0$

$\Rightarrow λ = 3 or - \frac{1}{2}$ ... (ii)

$D_{2} = 0 \Rightarrow$ $| \begin{matrix} (λ - 1) & 5 & λ \\ λ & 7 & (λ - 4) \\ (λ + 1) & 9 & - (λ + 2) \end{matrix} |$ $= 0$

$\Rightarrow 2 λ^{2} - 13 λ + 21 = 0$

$\Rightarrow (2 λ - 7) (λ - 3) = 0$

$\Rightarrow λ = 7 / 2 or λ = 3$ ... (iii)

Thus, $λ = 3$ (From (ii) and (iii))

$∴ λ^{2} + λ = {(3)}^{2} + 3 = 9 + 3 = 12$ .

Q 19 :

The system of equations

x + y + z = 6

x + 2y + 5x = 9

$x + 5 y + λ z = μ$

has no solution if [2025]

$λ = 15, μ \neq 17$
$λ \neq 17, μ \neq 18$
$λ = 17, μ = 18$
$λ = 17, μ \neq 18$

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(4)

Let $A = [\begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 1 & 5 & λ \end{matrix}] ∴$ $| A |$ $=$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 5 \\ 1 & 5 & λ \end{matrix} |$ $= 0$

$\Rightarrow 1 (2 λ - 25) - (λ - 5) + 1 (5 - 2) = 0 \Rightarrow λ = 17$

$D_{z} =$ $| \begin{matrix} 1 & 1 & 6 \\ 1 & 2 & 9 \\ 1 & 5 & μ \end{matrix} |$ $\neq 0$ $[\begin{matrix} ∵ If | A | = 0 then atleast one D_{x}, D_{y}, D_{z} \\ is non zero for inconsistent system \end{matrix}]$

$\Rightarrow μ \neq 18$ .

Q 20 :

If the system of equations

2x – y + z = 4

$5 x + λ y + 3 z = 12$

$100 x - 47 y + μ z = 212$

has infinitely many solutions, then $μ - 2 λ$ is equal to [2025]

55
59
56
57

1

2

3

4

(4)

As the given system of equations has infinitely many solutions, then

$△ = △_{1} = △_{2} = △_{3} = 0$

$△_{2} =$ $| \begin{matrix} 2 & 4 & 1 \\ 5 & 12 & 3 \\ 100 & 212 & μ \end{matrix} |$ $= 0$

$\Rightarrow 2 (12 μ - 636) - 4 (5 μ - 300) + 1 (1060 - 1200) = 0$

$\Rightarrow 4 μ - 212 = 0$

$\Rightarrow μ = 53$

$△_{3} =$ $| \begin{matrix} 2 & - 1 & 4 \\ 5 & λ & 12 \\ 100 & - 47 & 212 \end{matrix} |$ $= 0$

$\Rightarrow 2 (212 λ + 564) + 1 (1060 - 1200) + 4 (- 235 - 100 λ) = 0$

$\Rightarrow 24 λ + 48 = 0 \Rightarrow λ = - 2$

$∴ μ - 2 λ = 53 - 2 (- 2) = 57$ .

Q 21 :

If the system of equations

x + 2y –3z = 2

$2 x + λ y + 5 z = 5$

$14 x + 3 y + μ z = 33$

has infinitely many solutions, then $λ + μ$ is equal to : [2025]

13
11
12
10

1

2

3

4

(3)

Given : x + 2y – 3z = 2

$2 x + λ y + 5 z = 5$

$14 x + 3 y + μ z = 33$

For infinitely many solutions, we have $∆ = 0$

$\Rightarrow$ $| \begin{matrix} 1 & 2 & - 3 \\ 2 & λ & 5 \\ 14 & 3 & μ \end{matrix} |$ $= 0$

$\Rightarrow 1 (λ μ - 15) - 2 (2 μ - 70) - 3 (6 - 14 λ) = 0$

$\Rightarrow λ μ + 42 λ - 4 μ + 107 = 0$

We also have, $△_{1} = 2 λ μ + 99 λ - 10 μ + 225$

$△_{2} = μ - 13, △_{3} = 5 λ + 5$

Now, $△_{2} = 0 and △_{3} = 0$

$∴ 13 - μ = 0 and 5 λ + 5 = 0 \Rightarrow μ = 13 \Rightarrow λ = - 1$

Thus, we get $λ + μ = 13 + (- 1) = 13 - 1 = 12$

Q 22 :

Let $α, β (α \neq β)$ be the values of m, for which the equations x + y + z = 1; x + 2y +4z = m and x + 4y + 10z = $m^{2}$ have infinitely many solutions. Then the value of $\sum_{n = 1}^{10} (n^{α} + n^{β})$ is equal to : [2025]

3080
560
3410
440

1

2

3

4

(4)

We have, x + y + z = 1

x + 2y +4z = m and x + 4y + 10z = $m^{2}$

$∴ △ =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 4 & 10 \end{matrix} |$ $= 1 (4) - 1 (6) + 1 (2) = 0$

For infinite many solutions, $△ x = △ y = △ z = 0$

Now, $△ x = 0 \Rightarrow$ $| \begin{matrix} 1 & 1 & 1 \\ m & 2 & 4 \\ m^{2} & 4 & 10 \end{matrix} |$ $= 0$

$\Rightarrow 1 (4) - 1 (10 m - 4 m^{2}) + 1 (4 m - 2 m^{2}) = 0$

$\Rightarrow 4 - 10 m + 4 m^{2} + 4 m - 2 m^{2} = 0$

$\Rightarrow 2 m^{2} - 6 m + 4 = 0$

$\Rightarrow m^{2} - 3 m + 2 = 0 \Rightarrow m = 1, 2$

$∴ α = 1, β = 2$

Now, $\sum_{n = 1}^{10} (n^{α} + n^{β}) = \sum_{n = 1}^{10} n + \sum_{n = 1}^{10} n^{2}$

$= \frac{10 (11)}{2} + \frac{10 (11) (21)}{6}$ = 55 + 385 = 440.

Q 23 :

If the system of equations

$x + y + a z = b$

$2 x + 5 y + 2 z = 6$

$x + 2 y + 3 z = 3$

has infinitely many solutions, then $2 a + 3 b$ is equal to [2023]

20
25
28
23

1

2

3

4

(4)

Given system of equations are

$x + y + a z = b$

$2 x + 5 y + 2 z = 6$

$x + 2 y + 3 z = 3$

For infinitely many solutions, $Δ = 0, Δ_{1} = 0, Δ_{2} = 0, Δ_{3} = 0$

$Δ =$ $| \begin{matrix} 1 & 1 & a \\ 2 & 5 & 2 \\ 1 & 2 & 3 \end{matrix} |$ $= 0 \Rightarrow 1 (15 - 4) - 1 (6 - 2) + a (4 - 5) = 0$

$\Rightarrow 11 - 4 - a = 0 \Rightarrow a = 7$

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

$\Rightarrow b (15 - 4) - 1 (18 - 6) + a (12 - 15) = 0$

$\Rightarrow 11 b - 12 + 7 (- 3) = 0 \Rightarrow b = \frac{33}{11} = 3$

$∴$ $2 a + 3 b = 2 (7) + 3 (3) = 23$

Q 24 :

For the system of equations $x + y + z = 6, x + 2 y + α z = 10, x + 3 y + 5 z = β,$ which one of the following is not true? [2023]

System has infinitely many solutions for $α = 3, β = 14 .$
System has no solution for $α = 3, β = 24 .$
System has a unique solution for $α = 3, β \neq 14 .$
System has a unique solution for $α = - 3, β = 14 .$

1

2

3

4

(3)

Let $Δ =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & 2 & α \\ 1 & 3 & 5 \end{matrix} |$ $= 1 (10 - 3 α) - 1 (5 - α) + 1$ $= 6 - 2 α$

For a unique solution, $Δ \neq 0 \Rightarrow α \neq 3$

Now, if $α = 3$ , then

$Δ_{1} =$ $| \begin{matrix} 6 & 1 & 1 \\ 10 & 2 & 3 \\ β & 3 & 5 \end{matrix} |$ $= 6 (10 - 9) - 1 (50 - 3 β) + 1 (30 - 2 β)$ $= β - 14$

$Δ_{2} =$ $| \begin{matrix} 1 & 6 & 1 \\ 1 & 10 & 3 \\ 1 & β & 5 \end{matrix} |$ $= 1 (50 - 3 β) - 6 (5 - 3) + 1 (β - 10)$ $= 28 - 2 β = - 2 (β - 14)$

$Δ_{3} =$ $| \begin{matrix} 1 & 1 & 6 \\ 1 & 2 & 10 \\ 1 & 3 & β \end{matrix} |$ $= 1 (2 β - 30) - 1 (β - 10) + 6 (3 - 2)$ $= β - 14$

Clearly, at $β = 14$ , $Δ_{i} = 0 ∀ i = 1, 2, 3$

$∴$ $At α = 3, β = 14, the system of equations has infinitely many solutions.$

Q 25 :

Let $S$ be the set of all values of $θ \in [- π, π]$ for which the system of linear equations

$x + y + \sqrt{3} z = 0,$

$- x + (\tan θ) y + \sqrt{7} z = 0,$

$x + y + (\tan θ) z = 0$

has a non-trivial solution. Then $\frac{120}{π} \sum_{θ \in S} θ$ is equal to [2023]

30
10
40
20

1

2

3

4

(4)

For non-trivial solution, we have $| \begin{matrix} 1 & 1 & \sqrt{3} \\ - 1 & \tan θ & \sqrt{7} \\ 1 & 1 & \tan θ \end{matrix} |$ $= 0$

$\Rightarrow 1 (\tan^{2} θ - \sqrt{7}) - 1 (- \tan θ - \sqrt{7}) + \sqrt{3} (- 1 - \tan θ) = 0$

$\Rightarrow \tan^{2} θ - \sqrt{7} + \tan θ + \sqrt{7} - \sqrt{3} - \sqrt{3} \tan θ = 0$

$\Rightarrow \tan^{2} θ + \tan θ - \sqrt{3} \tan θ - \sqrt{3} = 0$

$\Rightarrow \tan θ (\tan θ + 1) - \sqrt{3} (\tan θ + 1) = 0$

$\Rightarrow \tan θ = - 1 or \tan θ = \sqrt{3} \Rightarrow θ = \frac{- π}{4}, \frac{3 π}{4}, \frac{π}{3}, \frac{- 2 π}{3}$

So, $\frac{120}{π} \sum_{θ \in S} θ = \frac{120}{π} [\frac{- π}{4} + \frac{3 π}{4} + \frac{π}{3} - \frac{2 π}{3}] = \frac{120}{π} \times \frac{π}{6} = 20$

Q 26 :

For the system of linear equations $2 x - y + 3 z = 5; 3 x + 2 y - z = 7; 4 x + 5 y + α z = β$ , which of the following is NOT correct? [2023]

The system has infinitely many solutions for $α = - 6$ and $β = 9$
The system has infinitely many solutions for $α = - 5$ and $β = 9$
The system has a unique solution for $α \neq - 5$ and $β = 8$
The system is inconsistent for $α = - 5$ and $β = 8$

1

2

3

4

(1)

We have,

$2 x - y + 3 z = 5$

$3 x + 2 y - z = 7$

$4 x + 5 y + α z = β$

By Cramer's rule,

$Δ =$ $| \begin{matrix} 2 & - 1 & 3 \\ 3 & 2 & - 1 \\ 4 & 5 & α \end{matrix} |$ $= 7 α + 35$

$Δ_{1} =$ $| \begin{matrix} 5 & - 1 & 3 \\ 7 & 2 & - 1 \\ β & 5 & α \end{matrix} |$ $= 17 α - 5 β + 130$

$Δ_{2} =$ $| \begin{matrix} 2 & 5 & 3 \\ 3 & 7 & - 1 \\ 4 & β & α \end{matrix} |$ $= 11 β - α - 104$

$Δ_{3} =$ $| \begin{matrix} 2 & - 1 & 5 \\ 3 & 2 & 7 \\ 4 & 5 & β \end{matrix} |$ $= 7 (β - 9)$

For infinite many solutions, $Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0$

$\Rightarrow α = - 5 and β = 9$

So, option (a) is incorrect and option (b) is correct.

For unique solution, $Δ \neq 0 \Rightarrow α \neq - 5$ and $β$ can be any value.

$∴ Option (c) is correct.$

At $α = - 5$ and $β = 8$

$Δ = 0 and Δ_{1} = 5 \neq 0$ $∴ Solution is inconsistent.$

$So, option (4) is correct.$

Q 27 :

If the system of linear equations $7 x + 11 y + α z = 13, 5 x + 4 y + 7 z = β, 175 x + 194 y + 57 z = 361$

has infinitely many solutions, then $α + β + 2$ is equal to: [2023]

4
3
6
5

1

2

3

4

(1)

For infinitely many solutions, $Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0$

$Δ =$ $| \begin{matrix} 7 & 11 & α \\ 5 & 4 & 7 \\ 175 & 194 & 57 \end{matrix} |$

$= 7 (228 - 1358) - 11 (285 - 1225) + α (970 - 700)$

$= - 7910 + 10340 + 270 α = 0 \Rightarrow - 2430 + 270 α = 0 \Rightarrow α = - 9$

$Δ_{1} =$ $| \begin{matrix} 13 & 11 & - 9 \\ β & 4 & 7 \\ 361 & 194 & 57 \end{matrix} |$ $= 13 (228 - 1358) - 11 (57 β - 2527) - 9 (194 β - 1444)$

$= - 14690 - 627 β + 27797 - 1746 β + 12996 = 26103 - 2373 β = 0$

$\Rightarrow β = 11$ $∴ α + β + 2 = - 9 + 11 + 2 = 4$

Q 28 :

For the system of linear equations

$2 x + 4 y + 2 a z = b,$

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

$2 x - 5 y + 2 z = 8$

which of the following is NOT correct? [2023]

It has infinitely many solutions if $a = 3, b = 8$
It has infinitely many solutions if $a = 3, b = 6$
It has unique solution if $a = b = 8$
It has unique solution if $a = b = 6$

1

2

3

4

(2)

We have, system of linear equations

$2 x + 4 y + 2 a z = b$

$x + 2 y + 3 z = 4$

$2 x - 5 y + 2 z = 8$

It can be written as $A X = B$ ,

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

$| A |$ $= 2 (4 + 15) - 4 (2 - 6) + 2 a (- 5 - 4) = 54 - 18 a$

If $| A |$ $= 0$ , then $a = 3$ and the system has infinitely many solutions.

If $| A |$ $\neq 0$ , then $a \neq 3$ and the system has a unique solution.

$A_{11} = 19, A_{12} = 4, A_{13} = - 9, A_{21} = - 38, A_{22} = - 8, A_{23} = 18, A_{31} = 0, A_{32} = 0, A_{33} = 0$

$adj A = [\begin{matrix} A_{11} & A_{21} & A_{31} \\ A_{12} & A_{22} & A_{32} \\ A_{13} & A_{23} & A_{33} \end{matrix}] = [\begin{matrix} 19 & - 38 & 0 \\ 4 & - 8 & 0 \\ - 9 & 18 & 0 \end{matrix}]$

For infinitely many solutions:

$(adj A) B = O \Rightarrow [\begin{matrix} 19 & - 38 & 0 \\ 4 & - 8 & 0 \\ - 9 & 18 & 0 \end{matrix}] [\begin{matrix} b \\ 4 \\ 8 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

$\Rightarrow 19 b - 152 = 0 \Rightarrow b = 8$

Q 29 :

If the system of equations $2 x + y - z = 5, 2 x - 5 y + λ z = μ, x + 2 y - 5 z = 7$

has infinitely many solutions, then ${(λ + μ)}^{2} + {(λ - μ)}^{2}$ is equal to: [2023]

912
916
904
920

1

2

3

4

(2)

$2 x + y - z = 5$

$2 x - 5 y + λ z = μ$

$x + 2 y - 5 z = 7$

The system of equations has infinitely many solutions when $Δ = Δ_{1} = Δ_{2} = Δ_{3} = 0$

$Δ = 0$

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

$\Rightarrow 2 (25 - 2 λ) - 1 (- 10 - λ) - 1 (4 + 5) = 0$

$\Rightarrow 50 - 4 λ + 10 + λ - 9 = 0 \Rightarrow - 3 λ = - 51 \Rightarrow λ = 17$

$Δ_{3} = 0$

$| \begin{matrix} 2 & 1 & 5 \\ 2 & - 5 & μ \\ 1 & 2 & 7 \end{matrix} |$ $= 0$ $\Rightarrow 2 (- 35 - 2 μ) - 1 (14 - μ) + 5 (4 + 5) = 0$

$\Rightarrow - 70 - 4 μ - 14 + μ + 45 = 0 \Rightarrow - 3 μ = 39 \Rightarrow μ = - 13$

$∴$ ${(λ + μ)}^{2} + {(λ - μ)}^{2} = {(17 - 13)}^{2} + {(17 + 13)}^{2} = 4^{2} + 30^{2} = 16 + 900 = 916$

Q 30 :

Let the system of linear equations

$- x + 2 y - 9 z = 7$

$- x + 3 y + 7 z = 9$

$- 2 x + y + 5 z = 8$

$- 3 x + y + 13 z = λ$

has a unique solution $x = α, y = β, z = γ$ . Then the distance of the point $(α, β, γ)$ from the plane $2 x - 2 y + z = λ$ is [2023]

9
7
13
11

1

2

3

4

(2)

$We have, - x + 2 y - 9 z = 7 ... (i)$

$- x + 3 y + 7 z = 9 ... (ii)$

$- 2 x + y + 5 z = 8 ... (iii)$

$- 3 x + y + 13 z = λ ... (iv)$

From (i), we have $x = 2 y - 9 z - 7 ... (v)$

Substituting the value of $x$ in (ii) and (iii), we get

$y + 16 z = 2$ and $- 3 y + 23 z = - 6$

Solving these two equations, we get $y = 2, z = 0$ .

$\Rightarrow x = 2 \times 2 - 9 \times 0 - 7 = 4 - 7 = - 3 [Using (v)]$

Now, substituting the value of $x, y, z$ in equation (iv), we get

$- 3 \times (- 3) + 2 + 13 \times (0) = λ \Rightarrow λ = 11$

$Distance of (- 3, 2, 0) from the plane 2 x - 2 y + z = 11 is$

$d =$ $| \frac{- 6 - 4 - 11}{\sqrt{4 + 4 + 1}} |$ $= \frac{21}{3} = 7 units.$

Topic Question Set

Q 1 :

If the system of equations

$x + (\sqrt{2} \sin α) y + (\sqrt{2} \cos α) z = 0$

$x + (\cos α) y + (\sin α) z = 0$

$x + (\sin α) y - (\cos α) z = 0$

has a non-trivial solution, then $α \in (0, \frac{π}{2})$ is equal to [2024]

Q 2 :

If the system of equations

$11 x + y + λ z = - 5$

$2 x + 3 y + 5 z = 3$

$8 x - 19 y - 39 z = μ$

has infinitely many solutions, then $λ^{4} - μ$ is equal to: [2024]

Q 3 :

The values of $m, n$ for which the system of equations $x + y + z = 4, 2 x + 5 y + 5 z = 17, x + 2 y + m z = n$ has infinitely many solutions, satisfy the equation: [2024]

Q 4 :

If the system of equations $x + 4 y - z = λ,$ $7 x + 9 y + μ z = - 3,$ $5 x + y + 2 z = - 1$ has infinitely many solutions, then $(2 μ + 3 λ)$ is equal to: [2024]

Q 5 :

Let $λ, μ \in R .$ If the system of equations

$3 x + 5 y + λ z = 3$

$7 x + 11 y - 9 z = 2$

$97 x + 155 y - 189 z = μ$

has infinitely many solutions, then $μ + 2 λ$ is equal to: [2024]

Q 6 :

If the system of equations $2 x + 3 y - z = 5,$ $x + α y + 3 z = - 4,$ $3 x - y + β z = 7$ has infinitely many solutions, then $13 α β$ is equal to _____ . [2024]

Q 7 :

Let the system of equations $x + 2 y + 3 z = 5,$ $2 x + 3 y + z = 9,$ $4 x + 3 y + λ z = μ$ have infinite number of solutions. Then $λ + 2 μ$ is equal to: [2024]

Q 8 :

Consider the system of linear equations $x + y + z = 4 μ,$ $x + 2 y + 2 λ z = 10 μ,$ $x + 3 y + 4 λ^{2} z = μ^{2} + 15,$ where $λ, μ \in R .$ Which one of the following statements is NOT correct? [2024]

Q 9 :

Consider the system of linear equations $x + y + z = 5,$ $x + 2 y + λ^{2} z = 9,$ $x + 3 y + λ z = μ,$ where $λ, μ \in R .$ Then which of the following statement is NOT correct? [2024]

Q 10 :

If the system of linear equations $x - 2 y + z = - 4,$ $2 x + α y + 3 z = 5,$ $3 x - y + β z = 3$ has infinitely many solutions, then $12 α + 13 β$ is equal to [2024]

Q 11 :

Let $α β γ = 45; α, β, γ \in R .$ If $x (α, 1, 2) + y (1, β, 2) + z (2, 3, γ) = (0, 0, 0)$ for some $x, y, z \in R, x y z \neq 0,$ then $6 α + 4 β + γ$ is equal to _______. [2024]

Q 12 :

If the system of equations, $2 x + 7 y + λ z = 3,$ $3 x + 2 y + 5 z = 4,$ $x + μ y + 32 z = - 1$ has infinitely many solutions, then $(λ - μ)$ is equal to _______. [2024]

Q 13 :

If the system of Linear equations

$3 x + y + β z = 3$

$2 x + α y - z = - 3$

$x + 2 y + z = 4$

has infinitely many solutions, then the value of $22 β - 9 α$ is : [2025]

Q 14 :

If the system of equations

$2 x + λ y + 3 z = 5$

$3 x + 2 y - z = 7$

$4 x + 5 y + μ z = 9$

has infinitely many solutions, then $(λ^{2} + μ^{2})$ is equal to: [2025]

Q 15 :

Let the system of equations :

$2 x + 3 y + 5 z = 9$

$7 x + 3 y - 2 z = 8$

$12 x + 3 y - (4 + λ) z = 16 - μ$

have infinitely many solutions. Then the radius of the circle centred at $(λ, μ)$ and touching the line 4x = 3y is [2025]

Q 16 :

Let the system of equations

$x + 5 y - z = 1$

$4 x + 3 y - 3 z = 7$

$24 x + y + λ z = μ$

$λ, μ \in R$ , have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy $7 \leq x + y + z \leq 77$ , is : [2025]

Q 17 :

If the system of linear equations:

x + y + 2z = 6

2x + 3y + az = a + 1

– x – 3y + bz = 2b

where $a, b \in R$ , has infinitely many solutions, then $7 a + 3 b$ is equal to : [2025]

Q 18 :

If the system of equations

$(λ - 1) x + (λ - 4) y + λ z = 5$

$λ x + (λ - 1) y + (λ - 4) z = 7$

$(λ + 1) x + (λ + 2) y - (λ + 2) z = 9$

has infinitely many solutions, then $λ^{2} + λ$ is equal to [2025]

Q 19 :

The system of equations

x + y + z = 6

x + 2y + 5x = 9

$x + 5 y + λ z = μ$

has no solution if [2025]

Q 20 :

If the system of equations

2x – y + z = 4

$5 x + λ y + 3 z = 12$

$100 x - 47 y + μ z = 212$

has infinitely many solutions, then $μ - 2 λ$ is equal to [2025]

Q 21 :

If the system of equations

x + 2y –3z = 2

$2 x + λ y + 5 z = 5$

$14 x + 3 y + μ z = 33$

has infinitely many solutions, then $λ + μ$ is equal to : [2025]

Q 22 :

Let $α, β (α \neq β)$ be the values of m, for which the equations x + y + z = 1; x + 2y +4z = m and x + 4y + 10z = $m^{2}$ have infinitely many solutions. Then the value of $\sum_{n = 1}^{10} (n^{α} + n^{β})$ is equal to : [2025]

Q 23 :

If the system of equations

$x + y + a z = b$

$2 x + 5 y + 2 z = 6$

$x + 2 y + 3 z = 3$

has infinitely many solutions, then $2 a + 3 b$ is equal to [2023]

Q 24 :

For the system of equations $x + y + z = 6, x + 2 y + α z = 10, x + 3 y + 5 z = β,$ which one of the following is not true? [2023]

Q 25 :

Let $S$ be the set of all values of $θ \in [- π, π]$ for which the system of linear equations

$x + y + \sqrt{3} z = 0,$

$- x + (\tan θ) y + \sqrt{7} z = 0,$

$x + y + (\tan θ) z = 0$

has a non-trivial solution. Then $\frac{120}{π} \sum_{θ \in S} θ$ is equal to [2023]

Q 26 :

For the system of linear equations $2 x - y + 3 z = 5; 3 x + 2 y - z = 7; 4 x + 5 y + α z = β$ , which of the following is NOT correct? [2023]

Q 27 :

If the system of linear equations $7 x + 11 y + α z = 13, 5 x + 4 y + 7 z = β, 175 x + 194 y + 57 z = 361$

has infinitely many solutions, then $α + β + 2$ is equal to: [2023]

Q 28 :

For the system of linear equations

$2 x + 4 y + 2 a z = b,$

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

$2 x - 5 y + 2 z = 8$

which of the following is NOT correct? [2023]

Q 29 :

If the system of equations $2 x + y - z = 5, 2 x - 5 y + λ z = μ, x + 2 y - 5 z = 7$

has infinitely many solutions, then ${(λ + μ)}^{2} + {(λ - μ)}^{2}$ is equal to: [2023]

Q 30 :

Let the system of linear equations

$- x + 2 y - 9 z = 7$

$- x + 3 y + 7 z = 9$

$- 2 x + y + 5 z = 8$

$- 3 x + y + 13 z = λ$

has a unique solution $x = α, y = β, z = γ$ . Then the distance of the point $(α, β, γ)$ from the plane $2 x - 2 y + z = λ$ is [2023]

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Topic Question Set

Q 1 : If the system of equations x+(2sinα)y+(2cosα)z=0 x+(cosα)y+(sinα)z=0 x+(sinα)y-(cosα)z=0 has a non-trivial solution, then α∈(0,π2) is equal to [2024]

Q 2 : If the system of equations 11x+y+λz=-5 2x+3y+5z=3 8x-19y-39z=μ has infinitely many solutions, then λ4-μ is equal to: [2024]

Q 3 : The values of m,n for which the system of equationsx+y+z=4, 2x+5y+5z=17, x+2y+mz=n has infinitely many solutions, satisfy the equation: [2024]

Q 4 : If the system of equations x+4y-z=λ,7x+9y+μz=-3,5x+y+2z=-1 has infinitely many solutions, then (2μ+3λ) is equal to: [2024]

Q 5 : Let λ,μ∈R. If the system of equations 3x+5y+λz=3 7x+11y-9z=2 97x+155y-189z=μ has infinitely many solutions, then μ+2λ is equal to: [2024]

Q 6 : If the system of equations 2x+3y-z=5,x+αy+3z=-4,3x-y+βz=7 has infinitely many solutions, then 13αβ is equal to _____ . [2024]

Q 7 : Let the system of equations x+2y+3z=5,2x+3y+z=9,4x+3y+λz=μ have infinite number of solutions. Then λ+2μ is equal to: [2024]

Q 8 : Consider the system of linear equations x+y+z=4μ,x+2y+2λz=10μ,x+3y+4λ2z=μ2+15, where λ,μ∈R. Which one of the following statements is NOT correct? [2024]

Q 9 : Consider the system of linear equations x+y+z=5,x+2y+λ2z=9,x+3y+λz=μ, where λ,μ∈R. Then which of the following statement is NOT correct? [2024]

Q 10 : If the system of linear equations x-2y+z=-4,2x+αy+3z=5,3x-y+βz=3 has infinitely many solutions, then 12α+13β is equal to [2024]

Q 11 : Let αβγ=45; α,β,γ∈R. If x(α,1,2)+y(1,β,2)+z(2,3,γ)=(0,0,0) for some x,y,z∈R, xyz≠0, then 6α+4β+γ is equal to _______. [2024]

Q 12 : If the system of equations, 2x+7y+λz=3,3x+2y+5z=4,x+μy+32z=-1 has infinitely many solutions, then (λ-μ) is equal to _______. [2024]

Q 13 : If the system of Linear equations 3x+y+βz=3 2x+αy–z=–3 x+2y+z=4 has infinitely many solutions, then the value of 22β-9α is : [2025]

Q 14 : If the system of equations 2x+λy+3z=5 3x+2y–z=7 4x+5y+μz=9 has infinitely many solutions, then (λ2+μ2) is equal to: [2025]

Q 15 : Let the system of equations : 2x+3y+5z=9 7x+3y–2z=8 12x+3y–(4+λ)z=16–μ have infinitely many solutions. Then the radius of the circle centred at (λ, μ) and touching the line 4x = 3y is [2025]

Q 16 : Let the system of equations x+5y–z=1 4x+3y–3z=7 24x+y+λz=μ λ,μ∈R, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy 7≤x+y+z≤77, is : [2025]

Q 17 : If the system of linear equations: x + y + 2z = 6 2x + 3y + az = a + 1 – x – 3y + bz = 2b where a,b∈R, has infinitely many solutions, then 7a+3b is equal to : [2025]

Q 18 : If the system of equations (λ–1)x+(λ–4)y+λz=5 λx+(λ–1)y+(λ–4)z=7 (λ+1)x+(λ+2)y–(λ+2)z=9 has infinitely many solutions, then λ2+λ is equal to [2025]

Q 19 : The system of equations x + y + z = 6 x + 2y + 5x = 9 x+5y+λz=μ has no solution if [2025]

Q 20 : If the system of equations 2x – y + z = 4 5x+λy+3z=12 100x–47y+μz=212 has infinitely many solutions, then μ–2λ is equal to [2025]

Q 21 : If the system of equations x + 2y –3z = 2 2x+λy+5z=5 14x+3y+μz=33 has infinitely many solutions, then λ+μ is equal to : [2025]

Q 22 : Let α,β(α≠β) be the values of m, for which the equations x + y + z = 1; x + 2y +4z = m and x + 4y + 10z = m2 have infinitely many solutions. Then the value of ∑n=110(nα+nβ) is equal to : [2025]

Q 23 : If the system of equations x+y+az=b 2x+5y+2z=6 x+2y+3z=3 has infinitely many solutions, then 2a+3b is equal to [2023]

Q 24 : For the system of equations x+y+z=6, x+2y+αz=10, x+3y+5z=β, which one of the following is not true? [2023]

Q 25 : Let S be the set of all values of θ∈[-π,π] for which the system of linear equations x+y+3z=0, -x+(tanθ)y+7z=0, x+y+(tanθ)z=0 has a non-trivial solution. Then 120π∑θ∈Sθ is equal to [2023]

Q 26 : For the system of linear equations 2x-y+3z=5; 3x+2y-z=7; 4x+5y+αz=β, which of the following is NOT correct? [2023]

Q 27 : If the system of linear equations 7x+11y+αz=13, 5x+4y+7z=β, 175x+194y+57z=361 has infinitely many solutions, then α+β+2 is equal to: [2023]

Q 28 : For the system of linear equations 2x+4y+2az=b, x+2y+3z=4, 2x-5y+2z=8 which of the following is NOT correct? [2023]

Q 29 : If the system of equations 2x+y-z=5, 2x-5y+λz=μ, x+2y-5z=7 has infinitely many solutions, then (λ+μ)2+(λ-μ)2 is equal to: [2023]

Q 30 : Let the system of linear equations -x+2y-9z=7 -x+3y+7z=9 -2x+y+5z=8 -3x+y+13z=λ has a unique solution x=α, y=β, z=γ. Then the distance of the point (α,β,γ) from the plane 2x-2y+z=λ is [2023]

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Q 1 :

If the system of equations

$x + (\sqrt{2} \sin α) y + (\sqrt{2} \cos α) z = 0$

$x + (\cos α) y + (\sin α) z = 0$

$x + (\sin α) y - (\cos α) z = 0$

has a non-trivial solution, then $α \in (0, \frac{π}{2})$ is equal to [2024]

Q 2 :

If the system of equations

$11 x + y + λ z = - 5$

$2 x + 3 y + 5 z = 3$

$8 x - 19 y - 39 z = μ$

has infinitely many solutions, then $λ^{4} - μ$ is equal to: [2024]

Q 3 :

The values of $m, n$ for which the system of equations $x + y + z = 4, 2 x + 5 y + 5 z = 17, x + 2 y + m z = n$ has infinitely many solutions, satisfy the equation: [2024]

Q 4 :

If the system of equations $x + 4 y - z = λ,$ $7 x + 9 y + μ z = - 3,$ $5 x + y + 2 z = - 1$ has infinitely many solutions, then $(2 μ + 3 λ)$ is equal to: [2024]

Q 5 :

Let $λ, μ \in R .$ If the system of equations

$3 x + 5 y + λ z = 3$

$7 x + 11 y - 9 z = 2$

$97 x + 155 y - 189 z = μ$

has infinitely many solutions, then $μ + 2 λ$ is equal to: [2024]

Q 6 :

If the system of equations $2 x + 3 y - z = 5,$ $x + α y + 3 z = - 4,$ $3 x - y + β z = 7$ has infinitely many solutions, then $13 α β$ is equal to _____ . [2024]

Q 7 :

Let the system of equations $x + 2 y + 3 z = 5,$ $2 x + 3 y + z = 9,$ $4 x + 3 y + λ z = μ$ have infinite number of solutions. Then $λ + 2 μ$ is equal to: [2024]

Q 8 :

Consider the system of linear equations $x + y + z = 4 μ,$ $x + 2 y + 2 λ z = 10 μ,$ $x + 3 y + 4 λ^{2} z = μ^{2} + 15,$ where $λ, μ \in R .$ Which one of the following statements is NOT correct? [2024]

Q 9 :

Consider the system of linear equations $x + y + z = 5,$ $x + 2 y + λ^{2} z = 9,$ $x + 3 y + λ z = μ,$ where $λ, μ \in R .$ Then which of the following statement is NOT correct? [2024]

Q 10 :

If the system of linear equations $x - 2 y + z = - 4,$ $2 x + α y + 3 z = 5,$ $3 x - y + β z = 3$ has infinitely many solutions, then $12 α + 13 β$ is equal to [2024]

Q 11 :

Let $α β γ = 45; α, β, γ \in R .$ If $x (α, 1, 2) + y (1, β, 2) + z (2, 3, γ) = (0, 0, 0)$ for some $x, y, z \in R, x y z \neq 0,$ then $6 α + 4 β + γ$ is equal to _______. [2024]

Q 12 :

If the system of equations, $2 x + 7 y + λ z = 3,$ $3 x + 2 y + 5 z = 4,$ $x + μ y + 32 z = - 1$ has infinitely many solutions, then $(λ - μ)$ is equal to _______. [2024]

Q 13 :

If the system of Linear equations

$3 x + y + β z = 3$

$2 x + α y - z = - 3$

$x + 2 y + z = 4$

has infinitely many solutions, then the value of $22 β - 9 α$ is : [2025]

Q 14 :

If the system of equations

$2 x + λ y + 3 z = 5$

$3 x + 2 y - z = 7$

$4 x + 5 y + μ z = 9$

has infinitely many solutions, then $(λ^{2} + μ^{2})$ is equal to: [2025]

Q 15 :

Let the system of equations :

$2 x + 3 y + 5 z = 9$

$7 x + 3 y - 2 z = 8$

$12 x + 3 y - (4 + λ) z = 16 - μ$

have infinitely many solutions. Then the radius of the circle centred at $(λ, μ)$ and touching the line 4x = 3y is [2025]

Q 16 :

Let the system of equations

$x + 5 y - z = 1$

$4 x + 3 y - 3 z = 7$

$24 x + y + λ z = μ$

$λ, μ \in R$ , have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy $7 \leq x + y + z \leq 77$ , is : [2025]

Q 17 :

If the system of linear equations:

x + y + 2z = 6

2x + 3y + az = a + 1

– x – 3y + bz = 2b

where $a, b \in R$ , has infinitely many solutions, then $7 a + 3 b$ is equal to : [2025]

Q 18 :

If the system of equations

$(λ - 1) x + (λ - 4) y + λ z = 5$

$λ x + (λ - 1) y + (λ - 4) z = 7$

$(λ + 1) x + (λ + 2) y - (λ + 2) z = 9$

has infinitely many solutions, then $λ^{2} + λ$ is equal to [2025]

Q 19 :

The system of equations

x + y + z = 6

x + 2y + 5x = 9

$x + 5 y + λ z = μ$

has no solution if [2025]

Q 20 :

If the system of equations

2x – y + z = 4

$5 x + λ y + 3 z = 12$

$100 x - 47 y + μ z = 212$

has infinitely many solutions, then $μ - 2 λ$ is equal to [2025]

Q 21 :

If the system of equations

x + 2y –3z = 2

$2 x + λ y + 5 z = 5$

$14 x + 3 y + μ z = 33$

has infinitely many solutions, then $λ + μ$ is equal to : [2025]

Q 22 :

Let $α, β (α \neq β)$ be the values of m, for which the equations x + y + z = 1; x + 2y +4z = m and x + 4y + 10z = $m^{2}$ have infinitely many solutions. Then the value of $\sum_{n = 1}^{10} (n^{α} + n^{β})$ is equal to : [2025]

Q 23 :

If the system of equations

$x + y + a z = b$

$2 x + 5 y + 2 z = 6$

$x + 2 y + 3 z = 3$

has infinitely many solutions, then $2 a + 3 b$ is equal to [2023]

Q 24 :

For the system of equations $x + y + z = 6, x + 2 y + α z = 10, x + 3 y + 5 z = β,$ which one of the following is not true? [2023]

Q 25 :

Let $S$ be the set of all values of $θ \in [- π, π]$ for which the system of linear equations

$x + y + \sqrt{3} z = 0,$

$- x + (\tan θ) y + \sqrt{7} z = 0,$

$x + y + (\tan θ) z = 0$

has a non-trivial solution. Then $\frac{120}{π} \sum_{θ \in S} θ$ is equal to [2023]

Q 26 :

For the system of linear equations $2 x - y + 3 z = 5; 3 x + 2 y - z = 7; 4 x + 5 y + α z = β$ , which of the following is NOT correct? [2023]

Q 27 :

If the system of linear equations $7 x + 11 y + α z = 13, 5 x + 4 y + 7 z = β, 175 x + 194 y + 57 z = 361$

has infinitely many solutions, then $α + β + 2$ is equal to: [2023]

Q 28 :

For the system of linear equations

$2 x + 4 y + 2 a z = b,$

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

$2 x - 5 y + 2 z = 8$

which of the following is NOT correct? [2023]

Q 29 :

If the system of equations $2 x + y - z = 5, 2 x - 5 y + λ z = μ, x + 2 y - 5 z = 7$

has infinitely many solutions, then ${(λ + μ)}^{2} + {(λ - μ)}^{2}$ is equal to: [2023]

Q 30 :

Let the system of linear equations

$- x + 2 y - 9 z = 7$

$- x + 3 y + 7 z = 9$

$- 2 x + y + 5 z = 8$

$- 3 x + y + 13 z = λ$

has a unique solution $x = α, y = β, z = γ$ . Then the distance of the point $(α, β, γ)$ from the plane $2 x - 2 y + z = λ$ is [2023]