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Solve the equations 
(x+a)×[(x+a)² - x²] - x [x(x+a) - x²] + x [ x²- x(x+a)] =0
(x+a) ×[ x²+2ax+a² - x²] - x[ x² + ax -x²] + x [ x² - x² - ax] = 0
(x+a) [2ax + a²] - ax² -ax² =0
2ax² + a²x + 2a²x + a³ -2ax² = 0
3a²x +a³ = 0
a² (3x + a) = 0
a != 0, ( 3x + a)= 0
3x = -a
Prove that the determinant 
is independent of θ.
Hence, Δ is independent of θ.
Without expanding the determinant, prove that
Hence, the given result is proved.
If a, b and c are real numbers, and
,
Show that either a + b + c = 0 or a = b = c.
Expanding along R1, we have:
Hence, if Δ = 0, then either a + b + c = 0 or a = b = c.
Prove that 
Expanding along R3, we have:
Hence, the given result is proved.
Using properties of determinants, prove that:
Expanding along R3, we have:
Hence, the given result is proved.
Using properties of determinants, prove that:
Expanding along R3, we have:
Hence, the given result is proved.
Using properties of determinants, prove that:
Expanding along C1, we have:
Hence, the given result is proved.
Using properties of determinants, prove that:
Expanding along C1, we have:
Hence, the given result is proved.
Using properties of determinants, prove that:
Hence, the given result is proved.
Solve the system of the following equations
Let 
Then the given system of equations is as follows:
This system can be written in the form of AX = B, where
A
Thus, A is non-singular. Therefore, its inverse exists.
Now,
A11 = 75, A12 = 110, A13 = 72
A21 = 150, A22 = −100, A23 = 0
A31 = 75, A32 = 30, A33 = − 24
Choose the correct answer.
If a, b, c, are in A.P., then the determinant
A. 0 B. 1 C. x D. 2x
Here, all the elements of the first row (R1) are zero.
Hence, we have Δ = 0.
The correct answer is A.
Choose the correct answer.
If x, y, z are nonzero real numbers, then the inverse of matrix 
is
A. 
B. 
The correct answer is A.
Choose the correct answer.
Let
, where 0 ≤ θ≤ 2π, then
A. Det (A) = 0
B. Det (A) ∈ (2, ∞)
C. Det (A) ∈ (2, 4)
D. Det (A)∈ [2, 4]
C. 
D. 
Answer: D.png)
Now,
0≤θ≤2π
⇒−1≤sinθ≤1.png)
The correct answer is D.
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