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Worksheet: Solving by Using Properties of Determinants

Q1:

Find the value of

Q2:

Find the value of | | | | βˆ’ 5 2 βˆ’ 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 βˆ’ 1 8 𝑏 7 𝑐 βˆ’ 7 𝑐 9 𝑐 βˆ’ 6 𝑑 6 π‘Ž βˆ’ 7 π‘Ž βˆ’ 3 𝑏 | | | |
  • B | | | | 9 𝑏 βˆ’ 9 𝑏 7 𝑐 βˆ’ 7 𝑐 2 𝑐 βˆ’ 6 𝑑 6 π‘Ž βˆ’ π‘Ž βˆ’ 3 𝑏 | | | |
  • C | | | | βˆ’ 9 𝑏 βˆ’ 1 8 𝑏 7 𝑐 2 𝑐 9 𝑐 βˆ’ 6 𝑑 βˆ’ π‘Ž βˆ’ 7 π‘Ž βˆ’ 3 𝑏 | | | |
  • D | | | | βˆ’ 1 8 𝑏 9 𝑏 7 𝑐 9 𝑐 βˆ’ 7 𝑐 βˆ’ 6 𝑑 βˆ’ 7 π‘Ž 6 π‘Ž βˆ’ 3 𝑏 | | | |

Q10:

Consider the equation

Evaluate

Q11:

Consider the equation

Determine all the possible values of π‘₯ given that 0 ≀ π‘₯ ≀ 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 π‘˜ which makes π‘₯ a factor of the determinant

  • A36
  • B βˆ’ 4
  • C 4 3
  • D4

Q13:

Consider the equation Given that π‘Ž + 𝑏 + 𝑐 = βˆ’ 1 , find its solution set.

  • A  βˆ’ 5 , √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) , βˆ’ √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) 
  • B  βˆ’ 1 , 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) , βˆ’ 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) 
  • C  βˆ’ 1 , √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) , βˆ’ √ ( π‘Ž βˆ’ 𝑐 ) ( 5 π‘Ž βˆ’ 5 𝑏 ) 
  • D  βˆ’ 5 , 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) , βˆ’ 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) 
  • E  βˆ’ 5 , 2 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) , βˆ’ 2 5 √ ( π‘Ž βˆ’ 𝑏 ) ( π‘Ž βˆ’ 𝑐 ) 

Q14:

Find, without expanding, the value of the determinant

Q15:

Given that 𝑛 = | | | | 6 βˆ’ 8 9 1 5 βˆ’ 9 βˆ’ 1 1 βˆ’ 7 2 βˆ’ 4 | | | | and π‘š = | | | | 1 8 βˆ’ 2 4 2 7 9 0 βˆ’ 5 4 βˆ’ 6 6 βˆ’ 3 5 1 0 βˆ’ 2 0 | | | | , find a relation between π‘š and 𝑛 without expanding either determinant.

  • A 1 5 𝑛 = π‘š
  • B 1 8 𝑛 = π‘š
  • C 𝑛 = π‘š
  • D 9 0 𝑛 = π‘š

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 βˆ’ 9 ( 9 + 1 8 π‘š βˆ’ 7 𝑛 + 2 π‘˜ )
  • B βˆ’ 9 ( 9 + 1 8 π‘š + 7 𝑛 + 2 π‘˜ )
  • C βˆ’ 9 ( 9 + 1 8 π‘š βˆ’ 7 𝑛 βˆ’ 2 π‘˜ )
  • D βˆ’ 9 ( 9 + 1 8 π‘š + 7 𝑛 βˆ’ 2 π‘˜ )
  • E βˆ’ 8 1 βˆ’ 1 8 π‘š βˆ’ 7 𝑛 + 1 8 π‘˜

Q18:

Select a factor of the determinant

  • A π‘₯ + 8
  • B π‘₯ + 6
  • C π‘₯ + 9
  • D π‘₯

Q19:

Consider the equation

Find, without expanding, the value of the determinant

Q20:

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

  • A | | | | βˆ’ 4 0 0 βˆ’ 5 βˆ’ 1 4 0 7 3 0 βˆ’ 9 | | | | , βˆ’ 5 5 4 4
  • B | | | | βˆ’ 4 βˆ’ 5 7 0 1 4 βˆ’ 3 0 0 0 9 | | | | , βˆ’ 5 0 4
  • C | | | | βˆ’ 4 0 0 βˆ’ 5 βˆ’ 1 4 0 7 3 0 9 9 | | | | , 5 544
  • D | | | | βˆ’ 4 βˆ’ 5 7 0 βˆ’ 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 , βˆ’ 6 ) , ( βˆ’ 3 , βˆ’ 4 )
  • B ( βˆ’ 4 , βˆ’ 3 ) , ( βˆ’ 6 , 0 )
  • C ( βˆ’ 3 , βˆ’ 4 ) , ( βˆ’ 6 , 0 )
  • D ( βˆ’ 4 , βˆ’ 3 ) , ( 0 , βˆ’ 6 )

Q22:

Use the properties of determinants to find

Q23:

Consider

Find the value of