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Worksheet: Solving by Using Properties of Determinants

In this worksheet, we will practice using the properties of determinants to solve problems.

Q1:

Find the value of

Q2:

Find the value of | | | | − 5 2 − 4 0 5 0 3 0 0 | | | | .

Q3:

Using the properties of determinants, find the value of

Q4:

Use the properties of determinants to evaluate

Q5:

Without expanding, find the value of the determinant

Q6:

Using the properties of determinants, find the value of

Q7:

Evaluate

Q8:

Evaluate

Q9:

Which of the following is equal to the determinant

  • A | | | | 9 𝑏 − 1 8 𝑏 7 𝑐 − 7 𝑐 9 𝑐 − 6 𝑑 6 𝑎 − 7 𝑎 − 3 𝑏 | | | |
  • B | | | | 9 𝑏 − 9 𝑏 7 𝑐 − 7 𝑐 2 𝑐 − 6 𝑑 6 𝑎 − 𝑎 − 3 𝑏 | | | |
  • C | | | | − 9 𝑏 − 1 8 𝑏 7 𝑐 2 𝑐 9 𝑐 − 6 𝑑 − 𝑎 − 7 𝑎 − 3 𝑏 | | | |
  • D | | | | − 1 8 𝑏 9 𝑏 7 𝑐 9 𝑐 − 7 𝑐 − 6 𝑑 − 7 𝑎 6 𝑎 − 3 𝑏 | | | |

Q10:

Consider the equation

Evaluate

Q11:

Consider the equation

Determine all the possible values of 𝑥 given that 0 ≤ 𝑥 ≤ 3 6 0 ∘ ∘ .

  • A 3 1 5 ∘ , 2 2 5 ∘ , 1 8 0 ∘ , 3 6 0 ∘
  • B 3 1 5 ∘ , 4 5 ∘ , 9 0 ∘ , 2 7 0 ∘
  • C 3 1 5 ∘ , 4 5 ∘ , 1 8 0 ∘ , 3 6 0 ∘
  • D 3 1 5 ∘ , 2 2 5 ∘ , 9 0 ∘ , 2 7 0 ∘

Q12:

Use the properties of determinants to find the value of 𝑘 which makes 𝑥 a factor of the determinant

  • A36
  • B − 4
  • C 4 3
  • D4

Q13:

Consider the equation Given that 𝑎 + 𝑏 + 𝑐 = − 1 , find its solution set.

  • A − 5 , √ ( 𝑎 − 𝑐 ) ( 5 𝑎 − 5 𝑏 ) , − √ ( 𝑎 − 𝑐 ) ( 5 𝑎 − 5 𝑏 )
  • B − 1 , 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 ) , − 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 )
  • C − 1 , √ ( 𝑎 − 𝑐 ) ( 5 𝑎 − 5 𝑏 ) , − √ ( 𝑎 − 𝑐 ) ( 5 𝑎 − 5 𝑏 )
  • D − 5 , 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 ) , − 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 )
  • E − 5 , 2 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 ) , − 2 5 √ ( 𝑎 − 𝑏 ) ( 𝑎 − 𝑐 )

Q14:

Find, without expanding, the value of the determinant

Q15:

Given that 𝑛 = | | | | 6 − 8 9 1 5 − 9 − 1 1 − 7 2 − 4 | | | | and 𝑚 = | | | | 1 8 − 2 4 2 7 9 0 − 5 4 − 6 6 − 3 5 1 0 − 2 0 | | | | , find a relation between 𝑚 and 𝑛 without expanding either determinant.

  • A 1 5 𝑛 = 𝑚
  • B 1 8 𝑛 = 𝑚
  • C 𝑛 = 𝑚
  • D 9 0 𝑛 = 𝑚

Q16:

Select the determinant that is equal to

  • A 7 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • B 2 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • C 1 4 | | | | 1 4 9 3 2 2 1 2 5 3 2 2 2 3 | | | |
  • D 1 4 | | | | 1 3 2 4 9 2 1 3 2 2 5 2 3 2 | | | |
  • E 2 | | | | 1 4 9 3 2 2 1 2 5 3 2 2 2 3 | | | |

Q17:

Find, in its simplest form, an expression for the determinant

  • A − 9 ( 9 + 1 8 𝑚 − 7 𝑛 + 2 𝑘 )
  • B − 9 ( 9 + 1 8 𝑚 + 7 𝑛 + 2 𝑘 )
  • C − 9 ( 9 + 1 8 𝑚 − 7 𝑛 − 2 𝑘 )
  • D − 9 ( 9 + 1 8 𝑚 + 7 𝑛 − 2 𝑘 )
  • E − 8 1 − 1 8 𝑚 − 7 𝑛 + 1 8 𝑘

Q18:

Select a factor of the determinant

  • A 𝑥 + 8
  • B 𝑥 + 6
  • C 𝑥 + 9
  • D 𝑥

Q19:

Consider the equation

Find, without expanding, the value of the determinant

Q20:

Put the determinant | | | | − 4 − 5 7 8 − 4 1 6 5 2 2 3 8 | | | | in upper triangular form, and find its value.

  • A | | | | − 4 0 0 − 5 − 1 4 0 7 3 0 − 9 | | | | , − 5 5 4 4
  • B | | | | − 4 − 5 7 0 1 4 − 3 0 0 0 9 | | | | , − 5 0 4
  • C | | | | − 4 0 0 − 5 − 1 4 0 7 3 0 9 9 | | | | , 5 544
  • D | | | | − 4 − 5 7 0 − 1 4 3 0 0 0 9 | | | | , 504

Q21:

Which of the following pairs of points does the straight line represented by the equation pass through?

  • A ( 0 , − 6 ) , ( − 3 , − 4 )
  • B ( − 4 , − 3 ) , ( − 6 , 0 )
  • C ( − 3 , − 4 ) , ( − 6 , 0 )
  • D ( − 4 , − 3 ) , ( 0 , − 6 )

Q22:

Use the properties of determinants to find

Q23:

Consider

Find the value of

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