NCERT Solutions, Determinants, Part -7, Class 12, Maths Notes - Class 12

Class 12: NCERT Solutions, Determinants, Part -7, Class 12, Maths Notes - Class 12

The document NCERT Solutions, Determinants, Part -7, Class 12, Maths Notes - Class 12 is a part of Class 12 category.
All you need of Class 12 at this link: Class 12

Determinants


Miscellaneous Solutions


Question 1:
Prove that the determinant 
determinants, mathematics,Question and Answer (Q & A)  is independent of θ.


Answer

determinants, mathematics,Question and Answer (Q & A)
Hence, Δ is independent of θ.


Question 2: Without expanding the determinant, prove that

determinants, mathematics,Question and Answer (Q & A)


Answer

determinants, mathematics,Question and Answer (Q & A)

Hence, the given result is proved.


Question 3: Evaluate

determinants, mathematics,Question and Answer (Q & A)
 

Answer

determinants, mathematics,Question and Answer (Q & A)
Expanding along C3, we have:
  determinants, mathematics,Question and Answer (Q & A)

 
Question 4: If a, b and c are real numbers, and
determinants, mathematics,Question and Answer (Q & A) ,
Show that either a + b + c = 0 or a = b = c.


Answer

determinants, mathematics,Question and Answer (Q & A)
Expanding along R1, we have:

determinants, mathematics,Question and Answer (Q & A)
Hence, if Δ = 0, then either a + b + c = 0 or a = b = c.


Question 5: Solve the equations

determinants, mathematics,Question and Answer (Q & A)


Answer

determinants, mathematics,Question and Answer (Q & A)


Question 6: Prove that

determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)

determinants, mathematics,Question and Answer (Q & A)
Expanding along R3, we have:

determinants, mathematics,Question and Answer (Q & A)
Hence, the given result is proved.


Question 8: Let verify that
determinants, mathematics,Question and Answer (Q & A)

determinants, mathematics,Question and Answer (Q & A)

determinants, mathematics,Question and Answer (Q & A)

determinants, mathematics,Question and Answer (Q & A)

determinants, mathematics,Question and Answer (Q & A)
 

Question 9: Evaluate
determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)
Expanding along R1, we have:

determinants, mathematics,Question and Answer (Q & A)


Question 10: Evaluate

determinants, mathematics,Question and Answer (Q & A)


Answer

determinants, mathematics,Question and Answer (Q & A)


Question 11: Using properties of determinants, prove that:

determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)
determinants, mathematics,Question and Answer (Q & A)
Hence, the given result is proved.


Question 12: Using properties of determinants, prove that:

determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)
Expanding along R3, we have:

determinants, mathematics,Question and Answer (Q & A)
Hence, the given result is proved.

Question 13: Using properties of determinants, prove that:

determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)
Hence, the given result is proved.


Question 14: Using properties of determinants, prove that:
determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)
Hence, the given result is proved.


Question 15: Using properties of determinants, prove that:

determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)
Hence, the given result is proved.


Question 16: Solve the system of the following equations

determinants, mathematics,Question and Answer (Q & A)


Answer
determinants, mathematics,Question and Answer (Q & A)
This system can be written in the form of AX = B, where
determinants, mathematics,Question and Answer (Q & A)
 determinants, mathematics,Question and Answer (Q & A)

 

Question 17: Choose the correct answer. If a, b, c, are in A.P., then the determinant

determinants, mathematics,Question and Answer (Q & A)
A. 0

B. 1

C. x

D. 2x


Answer
Answer: A

determinants, mathematics,Question and Answer (Q & A)
Here, all the elements of the first row (R1) are zero.
Hence, we have Δ = 0.
The correct answer is A.


Question 18: Choose the correct answer. If x, y, z are nonzero real numbers, then the inverse of matrix    

determinants, mathematics,Question and Answer (Q & A)  is
A.
determinants, mathematics,Question and Answer (Q & A)

B. determinants, mathematics,Question and Answer (Q & A)
C.  determinants, mathematics,Question and Answer (Q & A)

D. determinants, mathematics,Question and Answer (Q & A)


Answer: A

determinants, mathematics,Question and Answer (Q & A)
determinants, mathematics,Question and Answer (Q & A)

The correct answer is A.


Question 19: Choose the correct answer.
Let
determinants, mathematics,Question and Answer (Q & A) , where 0 ≤ θ≤ 2π, then
A). Det (A) = 0
B). Det (A) (2, ∞)
C). Det (A) (2, 4)
D). Det (A) [2, 4]


Answer: D

determinants, mathematics,Question and Answer (Q & A)

The correct answer is D.

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