WBCHSE Solutions For Class 12 Maths Algebra

Algebra

Algebra Exercise 1A Review Of Facts And Formulae

1. Results on transposition of matrices

(A +B)’=(A’+B’)

(AB)’= (BA)’

(KB)’= (K.A’)

(A’)’= (A)

2.

A is symmetric ⇔  A’= A

A is skew-symmetric ⇔ A’=-A

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3.

A is idempotent ⇔ A²= A

A is nilpotent of order n = Aⁿ= 0

A is orthogonal ⇔ AA’ = A’A =I

A is involutory ⇔ A²=I

4.

A is singular ⇔ I A I = 0

A is non-singular ⇔ I A I # 0

A^-1 exists ⇔ I A I # 0

(AB)^-1 = B^-1 A^-1

(kA)^-11= \(\frac{1}{k}\)A^-1

(A’)^-1 = (A^-1)’

I A^-1I = \(\frac{1}{|A|}\)

5.

A(adj . A) = I AI I

I adj. A I = I AI^n-1

adj . AB = (adj . B) . (adj – A)

6. For square matrices A and B of the same order, we have

1. (A + B)²= A²+AB +BA + B²
2. (A- B)² = A²– AB- BA + B²
3. (A + B) . (A- B) = A²– AB +BA- B²
4. A and B anticommute AB= -BA

Class 12 Algebra Multiple Choice Questions

Question 1.  If A and B are 2 – 2-rowed square matrices such that   

Answer: 2. \(\left[\begin{array}{rr}
7 & -5 \\
1 & -5
\end{array}\right]\)

⇒ 2A= (A + B) + (A- B) and 2B = (A + B)- (A- B)

Question  2. \(\text { If }\left[\begin{array}{rr}
3 & -2 \\
5 & 6
\end{array}\right]+2 A=\left[\begin{array}{rr}
5 & 6 \\
-7 & 10
\end{array}\right], \text { then } A=?\)

1. \(\left[\begin{array}{rr}
1 & 3 \\
-5 & 4
\end{array}\right]\)

2. \(\left[\begin{array}{rr}
-1 & 5 \\
-3 & 4
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
1 & 4 \\
-6 & 2
\end{array}\right]\)

4. None of these

Answer: 3. \(\left[\begin{array}{rr}
1 & 4 \\
-6 & 2
\end{array}\right]\)

⇒ 2A= \(\left[\begin{array}{rr}
5 & 6 \\
-7 & 10
\end{array}\right]\)– \(\left[\begin{array}{rr}
3 & -2 \\
5 & 6
\end{array}\right]\)= \(\left[\begin{array}{rr}
2 & 8 \\
-12 & 4
\end{array}\right]\)

Question 3. \(A=\left[\begin{array}{rr}
2 & 0 \\
-3 & 1
\end{array}\right] \text { and } B=\left[\begin{array}{rr}
4 & -3 \\
-6 & 2
\end{array}\right] \text { are such that } 4 A+3 X=5 B \text {, then } X=\text { ? }\)

1. \(\left[\begin{array}{rr}
4 & -5\\
-6 & 2
\end{array}\right]\)

2. \(\left[\begin{array}{rr}
4 & 5 \\
-6 & -2
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
-4 & 5 \\
-6 & -2
\end{array}\right]\)

4. None of these

Answer: 1. \(\left[\begin{array}{rr}
4 & -5\\
-6 & 2
\end{array}\right]\)

⇒ 4A + 3X= 5B => 3X= (5B-4A) => X = \(\frac{1}{3}\)(5B-4A).

Question 4. \(\text { If }(A-2 B)=\left[\begin{array}{rr}
1 & -2 \\
3 & 0
\end{array}\right] \text { and }(2 A-3 B)=\left[\begin{array}{rr}
-2 & 2 \\
3 & -3
\end{array}\right] \text {, then } B=\text { ? }\)

1. \(\left[\begin{array}{rr}
6 & -4 \\
-3 & 3
\end{array}\right]\)

2. \(\left[\begin{array}{rr}
-4 & 6 \\
-3 & -3
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
4 & -6 \\
-3 & -3
\end{array}\right]\)

4. None of these

Answer: 2. \(\left[\begin{array}{rr}
-4 & 6 \\
-3 & -3
\end{array}\right]\)

⇒ B = (2A-3B)-2(A-2B).

Question 5. \(\text { If }(2 A-B)=\left[\begin{array}{rrr}
6 & -6 & 0 \\
-4 & 2 & 1
\end{array}\right] \text { and }(2 B+A)=\left[\begin{array}{rrr}
3 & 2 & 5 \\
-2 & 1 & -7
\end{array}\right] \text {, then } A=\text { ? }\)

1. \(\left[\begin{array}{rrr}
-3 & 2 & 1 \\
2 & 1 & -1
\end{array}\right]\)

2. \(\left[\begin{array}{rrr}
3 & 2 & -1 \\
2 & -1 & -1
\end{array}\right]\)

3. \(\left[\begin{array}{rrr}
3 & -2 & 1 \\
-2 & 1 & -1
\end{array}\right]\)

4. None of these

Answer: 3. \(\left[\begin{array}{rrr}
3 & -2 & 1 \\
-2 & 1 & -1
\end{array}\right]\)

⇒ 5A = 2(2A- B) + (2B + A). Then, A = \(\frac{1}{5}\)(5A)

Question 6. \(\text { If } 2\left[\begin{array}{ll}
3 & 4 \\
5 & x
\end{array}\right]+\left[\begin{array}{ll}
1 & y \\
0 & 1
\end{array}\right]=\left[\begin{array}{rr}
7 & 0 \\
10 & 5
\end{array}\right] \text {, then }\)

1. (x = -2,y = 8)

2. x =  2y = 8

3. x = 3,y = -6

4. x =-3,y = 6

Answer: 2. x =  2y = 8

⇒ [2x+1 = 5 ⇒ x= 2] and [8 + y= 0 => y= -8].

Question 7. \(\text { If }\left[\begin{array}{cc}
x-y & 2 x-y \\
2 x+z & 3 z+w
\end{array}\right]=\left[\begin{array}{rr}
-1 & 0 \\
5 & 13
\end{array}\right] \text {, then }\)

1. z = 3,w = 4

2. z = 4,w = 3

3. z =1,w = 2

2. z = 2,w =-1

Answer: 1. z = 3,w = 4

⇒ (x- y=-1 and 2x- y= 0) => (x=1, y= 2)

⇒  (2x+ z = 5 => z = 3) and (3z+ w= 13 => w= 4).

Question 8. \(\text { If }\left[\begin{array}{cc}
x & y \\
3 y & x
\end{array}\right]\left[\begin{array}{l}
1 \\
2
\end{array}\right]=\left[\begin{array}{l}
3 \\
5
\end{array}\right] \text {, then }\)

1. x = 1,y = 2

2. x = 2,y = 1

3. x =1,y = 1

4. None of these

Answer: 3. x =1,y = 1

⇒  Solve x+ 2y= 3 and 3y+ 2x = 5.

Question 9. \(\text { If the matrix } A=\left[\begin{array}{cc}
3-2 x & x+1 \\
2 & 4
\end{array}\right] \text { is singular, then } x=\text { ? }\)

1. 0

2. 1

3. -1

4. -2

Answer:  2. 1

⇒  A is singular ⇔ IAI = 0.

Question 10. \(\text { If } A_\alpha=\left[\begin{array}{cc}
\cos \alpha & \sin \alpha \\
-\sin \alpha & \cos \alpha
\end{array}\right], \text { then }\left(A_\alpha\right)^2=\text { ? }\)

1. \(\left[\begin{array}{cc}
\cos ^2 \alpha & \sin ^2 \alpha \\
-\sin ^2 \alpha & \cos ^2 \alpha
\end{array}\right]\)

2. \(\left[\begin{array}{cc}
\cos 2 \alpha & \sin 2 \alpha \\
-\sin 2 \alpha & \cos 2 \alpha
\end{array}\right]\)

3. \(\left[\begin{array}{cc}
2 \cos \alpha & 2 \sin \alpha \\
-\sin \alpha & 2 \cos \alpha
\end{array}\right]\)

4. None of these

Answer: 1. \(\left[\begin{array}{cc}
\cos ^2 \alpha & \sin ^2 \alpha \\
-\sin ^2 \alpha & \cos ^2 \alpha
\end{array}\right]\)

Question 11. \(\text { If } A=\left[\begin{array}{cc}
\cos \alpha & \sin \alpha \\
-\sin \alpha & \cos \alpha
\end{array}\right] \text { be such that } A+A^{\prime}=I \text {, then } \alpha=\text { ? }\)

1. π

2. \(\frac{\pi}{3}\)

3. 2π

4. \(\frac{2 \pi}{3}\)

Answer: 2. \(\frac{\pi}{3}\)

Question 12. \(\text { If } A=\left[\begin{array}{rrr}
1 & k & 3 \\
3 & k & -2 \\
2 & 3 & -4
\end{array}\right] \text { is singular, then } k=\text { ? }\)

1. \(\frac{16}{3}\)

2. \(\frac{34}{5}\)

3. \(\frac{33}{2}\)

4. None of these

Answer: 3. \(\frac{33}{2}\)

⇒  A is singular ⇔ IAI = 0.

Question 13. \(\text { If } A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\), then adj. A= ?

1. \(\left[\begin{array}{ll}
d & -c \\
-b & a
\end{array}\right]\)

2. \(\left[\begin{array}{ll}
-d & b \\
c & -a
\end{array}\right]\)

3. \(\left[\begin{array}{ll}
d & -b \\
-c & a
\end{array}\right]\)

4. \(\left[\begin{array}{ll}
-d & -b \\
c & a
\end{array}\right]\)

Answer: 3. \(\left[\begin{array}{ll}
d & -b \\
-c & a
\end{array}\right]\)

Question 14. \(\text { If } A=\left[\begin{array}{cc}
2 x & 0 \\
x & x
\end{array}\right] \text { and } A^{-1}=\left[\begin{array}{rr}
1 & 0 \\
-1 & 2
\end{array}\right], \text { then } x=?\)

1. 1

2. 2

3. \(\frac{1}{2}\)

4. -2

Answer: 3. \(\frac{1}{2}\)

Use AA^-1 =I.

Question 15. If A and B are square matrices of the same order, then (A + B)(A — B) = ?

1. (A²– B²)

2. A² + AB + BA + B²

3. A²-AB + BA-B²

4. None of these

Answer: 3. A²-AB + BA-B²

⇒ Using distributive law, we have

(A +B)-(A-B) = A(A-B) +B(A-B) = (A²-AB +BA-B²).

Question 16. If A and B are square matrices of the same order, then (A + B)²=?

1. A²+ 2AB + B²

2.  A²-AB-BA + B²

3. A² + 2BA + B²

4. None of these

Answer: 2.  A²-AB-BA + B²

⇒ (A + B)²= (A + B)-(A + B)=A(A + B) + B(A + B) = (A²+ AB + BA + B²).

Question 17. If A and B are square matrices of the same order, then (A- B)² = ?

1. A²– 2AB + B²

2. A²+ AB-BA-B²

3. A²– 2BA + B²

4. None of these

Answer: 2. A²+ AB-BA-B²

⇒ (A- B)² = (A-B) . (A-B)= A(A- B)- B(A- B) = (A2- AB- BA + B2).

Question 18. If A and B are symmetric matrices of the same order, then (AB-BA) is
always

1. A symmetric matrix

2. A skew-symmetric matrix

3. A zero matrix

4. An identity matrix

Answer: 2. A skew-symmetric matrix

Given A’= A and B’ = B.

∴ (AB- BA)’= (AB)’- (BA)’= (B’A’- A’B’) = (BA- AB) = -(AB- BA)

∴ (AB- BA) is skew-symmetric.

Question 19. Matrices A and B are inverses of each other only when

1. AB = BA

2. AB = BA = O

3. AB = O,BA =I

4. AB = BA = I

Answer: 4. AB = BA = I

A and B are inverses of each other only when AB = BA =I.

Question 20. For square matrices A and B of the same order, we have adj (AB) = ?

1. (adj A)(adj B)

2.  (adj A)(adj B)

3. I ABI

4. None of these

Answer: 2.  (adj A)(adj B)

⇒ adj (AB)= (adj B) (adj A).

Question 21. If A is a 3-rowed square matrix and I A I =4, then adj (adj A) =?

1. 4A

2. 16A

3. 64A

4. None of these

Answer: 1. 4A

adj(adj A) = IAI^(n-1)=  IAI^(3-2).A= IAI -A = 4A.

Question 22. If A is a 3-rowed square matrix and I A I =5, then I adj AI =?

1. 5

2. 25

3. 125

4. None of these

Answer: 2. 25

I adj AI = IA^(n-1)I= I A 1²= 5²= 25.

Question 23. For any two matrices A and B

1. AB = BA is always true

2. AB = BA is never true

3. Sometimes AB = BA and sometimes AB BA

4. Whenever AB exists, then BA exists

Answer: 3. Sometimes AB = BA and sometimes AB BA

Question 24.  \(\text { If } A \cdot\left[\begin{array}{rr}
3 & 2 \\
1 & -1
\end{array}\right]=\left[\begin{array}{ll}
4 & 1 \\
2 & 3
\end{array}\right] \text {, then } A=\text { ? }\)

1. \(\left[\begin{array}{rr}
1 & -1 \\
1 & 1
\end{array}\right]\)

2. \(\left[\begin{array}{rr}
1 & 1 \\
-1 & 1
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
1 & 1 \\
1 & -1
\end{array}\right]\)

4. None of these

Answer: 3. \(\left[\begin{array}{rr}
1 & 1 \\
1 & -1
\end{array}\right]\)

⇒ Let \(\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]\left[\begin{array}{rr}
3 & 2 \\
1 & -1
\end{array}\right]=\left[\begin{array}{ll}
4 & 1 \\
2 & 3
\end{array}\right]\), find a,b,c and d

Question 25. If A is an invertible square matrix, then 1A^-1I =?

1. |A|

2. \(\frac{1}{|A|}\)

3. 1

4. 0

Answer: 2. \(\frac{1}{|A|}\)

AA^-1 = ⇒  IAA^-1 I = III =1

⇒  IAI. I A^-1 I  =1 = \(\frac{1}{|A|}\)

Question 26. If A and B are invertible matrices of the same order, then (AB)^-1 =?

1. (A^-1 x B^-1)

2. (A x B^-1)

3. (A^-1x B)

4. (B^-1x A^-1)

Answer: 4. (B^-1x A^-1)

⇒  (AB)^-1= B^-1A^-1

Question 27. If A and B are two non-zero square matrices of the same order such that AB = 0, then

1.  IAI =0 or IBI =0

2. IAI = 0 and IBI =0

3.  IAI ≠ 0and IB I ≠ 0

4. None of these

Answer: 2. IAI = 0 and IBI =0

⇒ [AB= 0 and A ≠ 0, B≠ 0] => IAI =0 and IBI =0.

Question 28. If A is a square matrix such that IAI ≠ 0 and A²– A + 2I = 0, then A^-1 =?

1. (I-A)

2. (I+A)

3. \(\frac{1}{2}\)(I-A)

4. \(\frac{1}{2}\)(I+A)

Answer:  3. \(\frac{1}{2}\)(I-A)

⇒ 21=(A- A² ) => 2A^-1 = A^-1 A-A^-1AA = I-IA = (I-A)

A^-1 = \(\frac{1}{2}\) (I-A)

Question 29.  \(
\text { If } A=\left[\begin{array}{lll}
1 & \lambda & 2 \\
1 & 2 & 5 \\
2 & 1 & 1
\end{array}\right] \text { is not invertible, then } \lambda=\text { ? }\)

1. 2

2. 1

3. -1

4. 0

Answer: 2. 1

⇒ A is not invertible ⇒ I A I = 0.

Question 30. \(\text { If } A=\left[\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right] \text {, then } A^{-1}=\text { ? }\)

1. A

2. -A

3. adj A-

4. -adj A

Answer: 3. adj A-

IAI =1 ⇒ A^-1 =\(
\frac{1}{|A|}
\) adj A= (adj A).

Question 31. \(
\text { The matrix } A=\left[\begin{array}{cc}
a b & b^2 \\
-a^2 & -a b
\end{array}\right] \text { is }\)

1. Idempotent

2. Orthogonal

3. Nilpotent

4. None of these

Answer:  3. Nilpotent

A²= 0  ⇒ A is nilpotent.

Question 32. \(
\text { The matrix } A=\left[\begin{array}{rrr}
2 & -2 & -4 \\
-1 & 3 & 4 \\
1 & -2 & -3
\end{array}\right] \text { is }\)

1. Non-singular

2. Idempotent

3. Nilpotent

4. Orthogonal

Answer: 2. Idempotent

A²= A ⇒ A is idempotent.

Question 33. If A is singular, then A (adj A) =?

1. A unit matrix

2. A null matrix

3. A symmetric matrix

4. None of these

Answer: 2. A null matrix

Given IAI = 0. So, A(adj A) = IAI.I = 0-I = 0.

∴ A(adj A) is a null matrix.

Question 34. \(
\text { For any 2-rowed square matrix } A, \text { if } A \cdot({adj} A)=\left[\begin{array}{ll}
8 & 0 \\
0 & 8
\end{array}\right] \text {, then the value of }\)

1. 0

2. 8

3. 64

4. 4

Answer: 2. 8

A . adj  A = IAI. I= 8 \(\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]\) = 8I= |A|

Question 35. \(\text { If } A=\left[\begin{array}{rr}
-2 & 3 \\
1 & 1
\end{array}\right] \text {, then }\left|A^{-1}\right|=\text { ? }\)

1. -5

2. \(\frac{-1}{5}\)

3. \(\frac{1}{25}\)

4. 25

Answer: 2. \(\frac{-1}{5}\)

AA^-1  = I ⇒  I AA^-1 I = 1I1 => IAI .I A^-1 I =1 ⇒ IA^-1 I = \(\frac{1}{|A|}\)

IAI = \(\left|\begin{array}{rr}
-2 & 3 \\
1 & 1
\end{array}\right|\)=(-2- 3) = (-5)⇒I A^-1I= \(\frac{-1}{5}\)

Question 36. \(
\text { If } A=\left[\begin{array}{ll}
3 & 1 \\
7 & 5
\end{array}\right] \text { and } A^2+x I=y A \text {, then the values of } x \text { and } y \text { are }\)

1.  x = 6, y = 6

2.  x = 8, y = 8

3. x = 5, y – 8

4. x = 6, y = 8

Answer: 2.  x = 8, y = 8

⇒ \(\left[\begin{array}{ll}
3 & 1 \\
7 & 5
\end{array}\right]\left[\begin{array}{ll}
3 & 1 \\
7 & 5
\end{array}\right]+x\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]=y\left[\begin{array}{ll}
3 & 1 \\
7 & 5
\end{array}\right]\)

⇒ \(\left[\begin{array}{rr}
16 & 8 \\
56 & 32
\end{array}\right]+\left[\begin{array}{ll}
x & 0 \\
0 & x
\end{array}\right]=\left[\begin{array}{rr}
3 y & y \\
7 y & 5 y
\end{array}\right]\)

⇒ \(\left[\begin{array}{cc}
16+x & 8 \\
56 & 32+x
\end{array}\right]=\left[\begin{array}{cc}
3 y & y \\
7 y & 5 y
\end{array}\right]\)

y= 8 and (16 + x= 3y= 3 x 8= 24) => x= 8.

Question 37. If matrices A and B anticommuting, then

1. AB = BA

2. AB = -BA

3. (AB) = (BA)^-1

4. None of these

Answer: 2. AB = -BA

A and B anticommute ⇔ AB= -BA.

Question 38. \(
\text { If } A=\left[\begin{array}{ll}
2 & 5 \\
1 & 3
\end{array}\right], \text { then adj } A=\text { ? }\)

1. \(\left[\begin{array}{rr}
3 & -5 \\
-1 & 2
\end{array}\right]\)

2. \(\left[\begin{array}{rr}
3 & -1 \\
-5 & 2
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
-1 & 2 \\
3 & -5
\end{array}\right]\)

4. None of these

Answer: 1. \(\left[\begin{array}{rr}
3 & -5 \\
-1 & 2
\end{array}\right]\)

Question 39. \(
\text { If } A=\left[\begin{array}{rr}
3 & -4 \\
-1 & 2
\end{array}\right] \text { and } B \text { is a square matrix of order } 2 \text { such that } A B=I \text {, then }\) B= ?

1. \(\left[\begin{array}{ll}
1 & 2 \\
2 & 3
\end{array}\right]\)

2. \(\left[\begin{array}{ll}
1 & 2 \\
2 & 3
\end{array}\right]\)

3. \(\left[\begin{array}{ll}
1 & 2 \\
2 & 3
\end{array}\right]\)

4. None of these

Answer: 3. \(\left[\begin{array}{ll}
1 & 2 \\
2 & 3
\end{array}\right]\)

AB =7 ⇒ B = A^-1

Question 40. If A and B are invertible square matrices of the same order, then (AB)^-1 = ?

1. AB^-1

2. A^-1B

3. A^-1B^-1

4. B^-1A^-1

Answer: 4. B^-1A^-1

(AB)^-1 = B^-1 A^-1

Question 41. \(
\text { If } A=\left[\begin{array}{rr}
2 & -1 \\
1 & 3
\end{array}\right] \text {, then } A^{-1}=\text { ? }\)

1. \(\left[\begin{array}{cc}
\frac{3}{7} & \frac{-1}{7} \\
\frac{1}{7} & \frac{2}{7}
\end{array}\right]\)

2. \(\left[\begin{array}{cc}
\frac{3}{7} & \frac{-1}{7} \\
\frac{1}{7} & \frac{2}{7}
\end{array}\right]\)

3. \(\left[\begin{array}{cc}
\frac{3}{7} & \frac{-1}{7} \\
\frac{1}{7} & \frac{2}{7}
\end{array}\right]\)

4. None of these

Answer: 2. \(\left[\begin{array}{cc}
\frac{3}{7} & \frac{-1}{7} \\
\frac{1}{7} & \frac{2}{7}
\end{array}\right]\)

⇒  \(|A|=\left|\begin{array}{rr}
2 & -1 \\
1 & 3
\end{array}\right|=(6+1)=7 \neq 0\)

M₁₁ = 3, M₁₂= 1, M₂₁ = -1 and M₂₂= 2

∴ C₁₁= 3, C₁₂= -1, C₂₁ =1 and C₂₂= 2

⇒ A \(=\left[\begin{array}{rr}
3 & -1 \\
1 & 2
\end{array}\right]^{\prime}=\left[\begin{array}{rr}
3 & 1 \\
-1 & 2
\end{array}\right.]\)

Question 42. \(
\text { If }|A|=3 \text { and } A^{-1}=\left[\begin{array}{rr}
3 & -1 \\
\frac{-5}{3} & \frac{2}{3}
\end{array}\right], \text { then adj } A=?\)

1. \(\left[\begin{array}{rr}
9 & 3 \\
-5 & -2
\end{array}\right]\)

2. \(\left[\begin{array}{rr}
9 & -3 \\
-5 & 2
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
-9 & 3 \\
5 & -2
\end{array}\right]\)

4. \(\left[\begin{array}{rr}
9 & -3 \\
5 & -2
\end{array}\right]\)

Answer: 2. \(\left[\begin{array}{rr}
9 & -3 \\
-5 & 2
\end{array}\right]\)

⇒ \(
A^{-1}=\frac{1}{|A|} \cdot {adj} A \Rightarrow {adj} A=|A| \cdot A^{-1}=3 A^{-1}=\left[\begin{array}{rr}
9 & -3 \\
-5 & 2
\end{array}\right]\)

Question 43. \(
\text { If } A \text { is an invertible matrix and } A^{-1}=\left[\begin{array}{ll}
3 & 4 \\
5 & 6
\end{array}\right] \text {, then } A=\text { ? }\)

1. \(\left[\begin{array}{rr}
6 & -4 \\
-5 & 3
\end{array}\right]\)

2. \(\left[\begin{array}{ll}
\frac{1}{3} & \frac{1}{4} \\
\frac{1}{5} & \frac{1}{6}
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
-3 & 2 \\
\frac{5}{2} & \frac{-3}{2}
\end{array}\right]\)

4. None of these

Answer: 3. \(\left[\begin{array}{rr}
-3 & 2 \\
\frac{5}{2} & \frac{-3}{2}
\end{array}\right]\)

⇒ A = (A^-1)^-1. So, find the inverse of A^-1.

Question 44. \(
\text { If } A=\left[\begin{array}{rr}
1 & 2 \\
4 & -3
\end{array}\right] \text { and } f(x)=2 x^2-4 x+5, \text { then } f(A)=?\)

1. \(\left[\begin{array}{rr}
19 & 2 \\
4 & -3
\end{array}\right]\)

2. \(\left[\begin{array}{rr}
19 & -16 \\
-32 & 51
\end{array}\right]\)

3. \(\left[\begin{array}{rr}
19 & -11 \\
-27 & 51
\end{array}\right]\)

4. None of these

Answer: 2. \(\left[\begin{array}{rr}
19 & -16 \\
-32 & 51
\end{array}\right]\)

⇒ (A) = 2A²-4A+5I.

Question 45. \(
\text { If } A=\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right], \text { then } A^2-4 A=\text { ? }\)

1. I

2. 5I

3. 3I

4. 0

Answer: 2. 5I

⇒ \(\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]\left[\begin{array}{ll}
1 & 4 \\
2 & 3
\end{array}\right]-\left[\begin{array}{ll}
4 & 16 \\
8 & 12
\end{array}\right]=\left[\begin{array}{ll}
9 & 16 \\
8 & 17
\end{array}\right]-\left[\begin{array}{ll}
4 & 16 \\
8 & 12
\end{array}\right]=\left[\begin{array}{ll}
5 & 0 \\
0 & 5
\end{array}\right]\)=5I

Question 46. If A is a 2-rowed square matrix and I A I =6, then A . adj A = ?

1. \(\left[\begin{array}{ll}
6 & 0 \\
0 & 6
\end{array}\right]\)

2. \(\left[\begin{array}{ll}
3 & 0 \\
0 & 3
\end{array}\right]\)

3. \(\left[\begin{array}{ll}
\frac{1}{6} & 0 \\
0 & \frac{1}{6}
\end{array}\right]\)

4. None of these

Answer: 1. \(\left[\begin{array}{ll}
6 & 0 \\
0 & 6
\end{array}\right]\)

⇒ A.(adj A) = IAI.

⇒ \(
I=6 \cdot\left[\begin{array}{ll}
1 & 0 \\
0 & 1
\end{array}\right]=\left[\begin{array}{ll}
6 & 0 \\
0 & 6
\end{array}\right]\)

Question 47. If A is an invertible square matrix and A:is a non-negative real number, (kA^-1) = ?

1. k. A^-1

2. \(\frac{1}{k} \cdot A^{-1}\)

3. – K. A^-1

4. None of these

Answer: 2. \(\frac{1}{k} \cdot A^{-1}\)

⇒ \((k A)^{-1}=\frac{1}{k} \cdot A^{-1}\) is true.

Question 48. \(
\text { If } A=\left[\begin{array}{rrr}
3 & 4 & 1 \\
1 & 0 & -2 \\
-2 & -1 & 2
\end{array}\right], \text { then } A^{-1}=\text { ? }\)

1. \(\left[\begin{array}{rrr}
2 & 9 & -8 \\
-2 & 8 & 7 \\
-1 & 5 & -4
\end{array}\right]\)

2. \(\left[\begin{array}{rrr}
2 & 9 & -8 \\
2 & 8 & 7 \\
-1 & -5 & -4
\end{array}\right]\)

3. \(\left[\begin{array}{rrr}
2 & -9 & -8 \\
2 & 8 & 7 \\
-1 & -5 & -4
\end{array}\right]\)

4. None of these

Answer: 3. \(\left[\begin{array}{rrr}
2 & -9 & -8 \\
2 & 8 & 7 \\
-1 & -5 & -4
\end{array}\right]\)

⇒ \(A^{-1}=\frac{1}{|A|}\).adj A.

Question 49. If A is a square matrix, then (A + A’) is

1. Anull matrix

2. An identity matrix

3. A symmetric matrix

4. A skew-symmetric matrix

Answer: 3. A symmetric matrix

⇒ A is a square matrix ⇒ (A + A’) is symmetric.

Question 50. If A is a square matrix, then (A- A’) is

1. Anull matrix

2. An identity matrix

3. A symmetric matrix

4. A skew-symmetric matrix

Answer: 4. A skew-symmetric matrix

A is a square matrix ⇒(A -A’) is skew-symmetric.

Question 51. If A is a 3-rowed square matrix and I 3 AI = k I  A I, then K=?

1. 3

2. 9

3. 27

4. 1

Answer: 3. 27

⇒ I3AI =( 3 x 3 x 3) IAI =27. I A I.

Question 52. Which one of the following is a scalar matrix?

1. \(\left[\begin{array}{ll}
1 & 1 \\
1 & 1
\end{array}\right]\)

2. \(\left[\begin{array}{ll}
6 & 0 \\
0 & 3
\end{array}\right]\)

3. \(\left[\begin{array}{ll}
8 & 0 \\
0 & -8
\end{array}\right]\)

4. None of these

Answer: 3. \(\left[\begin{array}{ll}
8 & 0 \\
0 & -8
\end{array}\right]\)

⇒ A scalar matrix is a square matrix each of whose non-diagonal elements is 0 and all diagonal elements are equal.

Question 53. \(
\text { If } A=\left[\begin{array}{ll}
1 & -1 \\
2 & -1
\end{array}\right] \text { and } B=\left[\begin{array}{rr}
a & 1 \\
b & -1
\end{array}\right] \text { and }(A+B)^2=\left(A^2+B^2\right) \text {, then }\)

1. a= 2, b= -3

2. a= -2, b= 3

3. a= 1, b= 4

4. None of these

Answer:  3. a= 1, b= 4

A+ B)² = (A²+ B²) ⇒ A²+ B²+AB+ BA = (A²+ B²) ⇒ AB =-BA

⇒ \(\left[\begin{array}{cc}
a-b & 2 \\
2 a-b & 3
\end{array}\right]=\left[\begin{array}{ll}
-a-2 & a+1 \\
-b+2 & b-1
\end{array}\right]\)

Now, (a +1 = 2 and b-1 = 3)⇒ (a =1 and 6 = 4).

Algebra Exercise 1B Review Of Facts And Formulae

1.

1.\(\left|\begin{array}{ll}
a_{11} & a_{12} \\
a_{21} & a_{22}
\end{array}\right|=\left(a_{11} a_{22}-a_{12} a_{21}\right)\)

2. \(\left|\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|\)

= \(a_{11} \cdot\left|\begin{array}{ll}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}\right|-a_{12} \cdot\left|\begin{array}{ll}
a_{21} & a_{23} \\
a_{31} & a_{33}
\end{array}\right|+a_{13} \cdot\left|\begin{array}{ll}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}\right|\)

\(=a_{11}\left(a_{22} a_{33}-a_{32} a_{23}\right)-a_{12}\left(a_{21} a_{33}-a_{31} a_{23}\right)+a_{13}\left(a_{21} a_{32}-a_{31} a_{22}\right)\)

2. 

1. Minor of a_ij is given by

M_ij= det. obtained after deleting ith row and jth column.

2. Co-factor of a_ij is given by C_ij= (-1)^i+j.M_ij

Δ= sum of the products of the elements of any row or column with their
corresponding co-factors.

Δ= a₁₁C₁₁+ a₁₂C₁₂ + a₁₃C₁₃

= a₁₁C₁₁+ a₂₁C₂₁ + a₃₁C₃₁

3. Properties of determinants

If the rows and columns of a determinant are interchange the value of the determinant remains unchanged.

⇒ R₁ ↔ R₃ shows the interchange of first and 3rd rows

If any two adjacent rows (or columns) of a determinant are interchanged, the value of the new determinant is the negative of the value of original determinant.

If all the elements of one row (or column) of a determinant are multiplied by k, the value of the original determinant is multiplied by k.

⇒ R₂—> R₂ shows the multiplication of each element of the second rowby k.

If the elements of a row (or a column) of a determinant are added k times the corresponding elements of another row (or column), the value of the determinant remains unchanged.

⇒ R_i —>R_j+ kR_j shows that k times the elements of jth row are added to the
corresponding elements of ith row.

If two rows (or columns) of a determinant are identical, the value of the
the determinant is zero.

If each element of a row (or a column) of a determinant is 0, the value of
the determinant is 0.

4.  Area of Δ A ABC with vertices A(x₁,y₁) B(x₂, y₂) and C(x₃, y₃) is given by

⇒ \(\Delta=\frac{1}{2}\left|\begin{array}{lll}
x_1 & y_1 & 1 \\
x_2 & y_2 & 1 \\
x_3 & y_3 & 1
\end{array}\right|\)

Points A(x₁,y₁) B(x₂, y₂) and C(x₃, y₃) are collinear ⇔ ar(Δ ABC) = 0 ⇔ Δ = 0.

Algebra Exercise 1B Multiple Choice Questions

Question 1. \(\left|\begin{array}{ll}
\cos 70^{\circ} & \sin 20^{\circ} \\
\sin 70^{\circ} & \cos 20^{\circ}
\end{array}\right|\)

1.  1
2.  0
3.  cos 50°
4. sin 50°

Answer: 2. 0

Question 2. \(\left|\begin{array}{ll}
\cos 15^{\circ} & \sin 15^{\circ} \\
\sin 15^{\circ} & \cos 15^{\circ}
\end{array}\right|\)

1. 1
2. \(\frac{1}{2}\)
3. \(\frac{\sqrt{3}}{2}\)
4. None of these

Answer:  3. \(\frac{\sqrt{3}}{2}\)

Δ = \(\left(\cos ^2 15^{\circ}-\sin ^2 15^{\circ}\right)\)

= cos (2×15° )= cos 30°

= \(\frac{\sqrt{3}}{2}\)

Question 3. \(\left|\begin{array}{rr}
\sin 23^{\circ} & -\sin 7^{\circ} \\
\cos 23^{\circ} & \cos 7^{\circ}
\end{array}\right|=?\)

1. \(\frac{\sqrt{3}}{2}\)
2. \(\frac{1}{2}\)
3. sin 16
4. cos 16

Answer: 2. \(\frac{1}{2}\)

Question 4. \(\left|\begin{array}{cc}
a+i b & c+i d \\
-c+i d & a-i b
\end{array}\right|\) = ?

1.  (a²+ b²– c²– d2²
2. (a²-b²+ c²-d²)
3. (a²+ b²+ c²+ d²)
4.  None of these

Answer: 3.  (a²+ b²+ c²+ d²)

Δ = \(\left|\begin{array}{cc}
a+i b & c+i d \\
-(c-i d) & a-i b
\end{array}\right|=(a+i b)(a-i b)+(c-i d)(c+i d)\)

= (a²+b²+c²+d²)

Question 5. \(\left|\begin{array}{lll}
a-b & b-c & c-a \\
b-c & c-a & a-b \\
c-a & a-b & b-c
\end{array}\right|\)=?

1.  (a + b + c)
2.  3[a+ b+ c)
3.  3abc
4.  0

Answer: 4. 0

R₁ -+ (R₁ + R₂+ R₃) gives all zeros in R₁ and this gives Δ= 0

Question 6. \(\left|\begin{array}{ccc}
1 & 1+p & 1+p+q \\
2 & 3+2 p & 1+3 p+2 q \\
3 & 6+3 p & 1+6 p+3 q
\end{array}\right|\) =?

1. 0
2. 1
3. -1
4. None of these

Answer: 2. 1

Applying R₂→ (R₂– 2R₁) and R₃ -+ (R₃– 3R₁) and expandingby C₁ we get Δ=1.

Question 7.  \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^3 & b^3 & c^3
\end{array}\right|\) =?

1. (a-b)(b-c)(c-a)
2.  -(a-b)(b-c)(c-a)
3. (a- b)(b- c)(c- a)(a + b + c)
4. abc (a- b)(b- c){c- a)

Answer: 3. (a- b)(b- c)(c- a)(a + b + c)

Question 8. \(\left|\begin{array}{ccc}
1+a^2-b^2 & 2 a b & -2 b \\
2 a b & 1-a^2+b^2 & 2 a \\
2 b & -2 a & 1-a^2-b^2
\end{array}\right|=?\)

1. (1 + a²+ b²)
2. (1 + a2+ b²)²
3. (1 + a² + b²)³
4. None of these

Answer:  3. (1 + a² + b²)³

Apply R₁ → R₁ + bR₃ and R₂ → R₂ — aR₃

Question 9. \(\left|\begin{array}{lll}
\sin \alpha & \cos \alpha & \sin (\alpha+\delta) \\
\sin \beta & \cos \beta & \sin (\beta+\delta) \\
\sin \gamma & \cos \gamma & \sin (\gamma+\delta)
\end{array}\right|\)= ?

1. 0
2. 1
3. sin (a+ 8) + sin (P + 8) + sin (Y+ 8)
4.  None of these

Answer: 1. 0

Apply C₃→ C₃– (cos δ)C₁– (sinδ )C₂

Question 10. \(\left|\begin{array}{ccc}
x+y & x & x \\
5 x+4 y & 4 x & 2 x \\
10 x+8 y & 8 x & 3 x
\end{array}\right|\)

1. 0
2.  x³
3. y³
4. None of these

Answer: 2. x³

Take x common from C₂ and x common from C₃.

Apply R₃→R₃– 2R₂.

Question 11. \(\left|\begin{array}{lll}
b+c & c+a & a+b \\
c+a & a+b & b+c \\
a+b & b+c & c+a
\end{array}\right|=?\)

1. (a + b + c)
2. (a + b + c)²
3.  0
4. None of these

Answer: 4. None of these

Apply C₁ → (C₁ + C₁₂+ C₃).

Question 12. \(\left|\begin{array}{ccc}
a & b & c \\
a-b & b-c & c-a \\
b+c & c+a & a+b
\end{array}\right|=?\)

1. (a³ + b³ + c³)
2.  (a+b+c)³
3. 3abc(a + b + c)
4. (a³ + b³ + c³– 3abc

Answer: 4. (a³ + b³ + c³– 3abc

Apply C₁ →(C₁ + C₂+ C₃)

Question 13. \(\left|\begin{array}{ccc}
0 & a-b & a-c \\
b-a & 0 & b-c \\
c-a & c-b & 0
\end{array}\right|=?\)

1. (a+b+c)
2. (a+b-c)
3. 1
4. 0

Answer:  4. 0

Apply R₁ → (R₁ − R₁₂) and → (R₃−R₂)

Question 14. \(\left|\begin{array}{ccc}
a+b & a & b \\
a & a+c & c \\
b & c & b+c
\end{array}\right|=?\)

1. a+b+c
2. abc
3. 4abc
4. a²b²c²

Answer: 3. 4abc

Apply R₁ →  R₁– (R₂+ R₃).

Question 15.  \(\left|\begin{array}{lll}
a & 1 & b+c \\
b & 1 & c+a \\
c & 1 & a+b
\end{array}\right|=?\)

1. (a+b+c)
2. 2(a+b+c)
3. 0
4. None of these

Answer: 3. 0

Apply C₁ →  (C₁ + C₃) and take (a+ b+ c) common from C₁

Question 16.  \(\left|\begin{array}{ccc}
a-b-c & 2 b & 2 c \\
2 a & b-c-a & 2 c \\
2 a & 2 b & c-a-b
\end{array}\right|=?\)

1. (a+b+c)
2. (a+b+c)²
3. (a+b+c)³
4. None of these

Answer: 3. (a+b+c)³

Apply Cx₁→  C₁+ C₂+ C₃

Question 17. \(\left|\begin{array}{ccc}
a^2+1 & a b & a c \\
a b & b^2+1 & b c \\
c a & c b & c^2+1
\end{array}\right|=?\)

1.  (a²+b²+c²)
2. ( 1+a²+b²+c²)
3. ( 3+a²+b²+c²)
4. None of these

Answer: 2. ( 1+a²+b²+c²)

⇒ \(\Delta=\left|\begin{array}{ccc}
a\left(a+\frac{1}{a}\right) & a b & a c \\
a b & \left(b+\frac{1}{b}\right) b & b c \\
a c & b c & \left(c+\frac{1}{c}\right) c
\end{array}\right|=(a b c) \cdot\left|\begin{array}{ccc}
a+\frac{a}{a} & a & a \\
b & b+\frac{1}{b} & b \\
c & c & c+\frac{1}{c}
\end{array}\right|\)

⇒ \(=(a b c)\left|\begin{array}{ccc}
\frac{\left(a^2+1\right)}{a} & a & a \\
b & \frac{b^2+1}{b} & b \\
c & c & \frac{c^2+1}{c}
\end{array}\right|=\left(\frac{a b c}{a b c}\right) \cdot\left|\begin{array}{ccc}
a^2+1 & a^2 & a^2 \\
b^2 & b^2+1 & b^2 \\
c^2 & c^2 & c^2+1
\end{array}\right|\)

Now apply, R₁→ R₁+R₂+R₃

Question 18.  \(\left|\begin{array}{lll}
265 & 240 & 219 \\
240 & 225 & 198 \\
219 & 198 & 181
\end{array}\right|=?\)

1.  0
2.  78
3. -39
4. 108

Answer: 1. 0

Applying C₁→(C₁-C₃) and C₂→(C₂-C₃) we get:

⇒ \(\Delta=\left|\begin{array}{lll}
46 & 21 & 219 \\
42 & 27 & 198 \\
38 & 17 & 181
\end{array}\right|=\left|\begin{array}{rrr}
4 & 21 & 9 \\
-12 & 27 & -72 \\
4 & 17 & 11
\end{array}\right|\)

C₁→(C₁-2C₂)

C₃→(C₃-10C₂)

Now , apply R₁→(R₁-R₃) and R₂→(R₂-3R₃)

Question 19. \(\left|\begin{array}{lll}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2
\end{array}\right|=?\)

1. 8
2. -8
3. 16
4. 142

Answer: 2. -8

Question 20.\(\left|\begin{array}{lll}
1 ! & 2 ! & 3 ! \\
2 ! & 3 ! & 4 ! \\
3 ! & 4 ! & 5 !
\end{array}\right|=?\)

1. 2
2. 6
3. 24
4. 120

Answer: 3. 24

Question 21.\(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1+x & 1 \\
1 & 1 & 1+y
\end{array}\right|\)

1.  (x+y)
2. (x-y)
3. xy
4.  None of these

Answer: 3. xy

Question 22. \(\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|=?\)

1. abc (a+b+c)
2. (a³ -b³-c³ + 3abc)
3. (-a³ -b³-c³ + 3abc)
4. None of these

Answer: 3. (-a³ -b³-c³ + 3abc)

Question 23. If a, b, cbe distinct positive real numbers, then the value of \(\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|\) is

1. Positive
2. Negative
3. A perfect square
4. 0

Answer: 2. Negative

Δ= -(a+ b+ c)(a²+ b²+ c²-ab-bc-ca)

=- \(\frac{1}{2}\) (a+ b + c)[(a- b)²+ (b- c)²+ (c- a)²], which is negative.

Question 24. \(\left|\begin{array}{ccc}
-a^2 & a b & a c \\
a b & -b^2 & b c \\
a c & b c & -c^2
\end{array}\right|=?\)

1. 0
2. abc
3. 4 a²b²c²
4. None of these

Answer: 3. 4 a²b²c²

Question 25. \(\left|\begin{array}{lll}
b c & b+c & 1 \\
c a & c+a & 1 \\
a b & a+b & 1
\end{array}\right|=?\)

1. (a-b)(b-c)(c-a)
2. – (a-b)(b-c)(c-a)
3. (a+b)(b+c)(c+a)
4.  None of these

Answer: 1. (a-b)(b-c)(c-a)

Question 26. \(\left|\begin{array}{ccc}
a^2+2 a & 2 a+1 & 1 \\
2 a+1 & a+2 & 1 \\
3 & 3 & 1
\end{array}\right|=?\)

1. (a-1)²
2. (a-1)³
3. (a-1)
4. None of these

Answer: 2. (a-1)³

Apply R₁→ (R₁ — R₂) and R₃ → (R₃ — R₂)

Question 27. \(\left|\begin{array}{lll}
\frac{1}{a} & a^2 & b c \\
\frac{1}{b} & b^2 & a c \\
\frac{1}{c} & c^2 & a b
\end{array}\right|=?\)

1. 0
2. 1
3. -1
4. None of these

Answer: 1.0

⇒ \(\Delta=\frac{1}{a b c}\left|\begin{array}{ccc}
1 & a^3 & a b c \\
1 & b^3 & a b c \\
1 & c^3 & a b c
\end{array}\right|=\left(\frac{a b c}{a b c}\right) \cdot\left|\begin{array}{ccc}
1 & a^3 & 1 \\
1 & b^3 & 1 \\
1 & c^3 & 1
\end{array}\right|=0\)

Question 28.\(\left|\begin{array}{ccc}
x+1 & x+2 & x+4 \\
x+3 & x+5 & x+8 \\
x+7 & x+10 & x+14
\end{array}\right|=?\)

1.  -2
2. 2
3. x²– 2
4. x + 2

Answer: 1. -2

Apply C₂→(C₂– C₁) and C₃ → (C₃ — C₁).

Question 29.\(\left|\begin{array}{ccc}
b+c & a & a \\
b & c+a & b \\
c & c & a+b
\end{array}\right|=?\)

1.  4abc
2.  2(a + b + c)
3.  (ab + bc + ca)
4.  None of these

Answer: 1. 4abc

Apply R₁→R₁– (R₂+R₃) and then apply R₂ → (R₂-R₁) and R₃ → (R₃-R₁).

Question 30. \(\left|\begin{array}{ccc}
a & a+2 b & a+2 b+3 c \\
3 a & 4 a+6 b & 5 a+7 b+9 c \\
6 a & 9 a+12 b & 11 a+15 b+18 c
\end{array}\right|=?\)

1.  a³
2. -a³
3. 0
4. None of these

Answer: 2. -a³

Apply R₂ → R₂ − 3R₁ and R₃→ R₃ − 6R₁.

Question 31. \(\left|\begin{array}{lll}
1 & b c & b c(b+c) \\
1 & c a & c a(c+a) \\
1 & a b & a b(a+b)
\end{array}\right|=?\)

1.  abc
2.  2 abc
3. abc(a+b+c)
4. 0

Answer: 4. 0

Apply R₂ → R₂– R₁ and R₃ → R₃-R₁.

Question 32. The value of \(\left|\begin{array}{ccc}
\cos (\theta+\phi) & -\sin (\theta+\phi) & \cos 2 \phi \\
\sin \theta & \cos \theta & \sin \phi \\
-\cos \theta & \sin \theta & \cos \phi
\end{array}\right|\) is

1.  Independent of o only
2.  Independent of o only
3.  Independent of o only
4.  Dependent of o only

Answer: 1. Independent of o only

Apply R₁ → R₁+ (sin Φ)R₂ − (cos Φ)R₃.

Take (2cosΦ) common from R₁. Now, apply R₁ →4 (R₁ +R₃).

Question 33. \(\left|\begin{array}{lll}
b+c & a & b \\
c+a & c & a \\
a+b & b & c
\end{array}\right|=?\)

1. (a + b+ c)(a- c)
2.  (a + b + c)(b- c)
3.  (a + b + c)(a-cf
4.  (a + b + c)(b- c)

Answer: 3. (a + b + c)(a-cf)

Express A as the sum of two determinants and simplify each.

Question 34. If co is a complex root of unity, then \(\left|\begin{array}{ccc}
1 & \omega & \omega^2 \\
\omega & \omega^2 & 1 \\
\omega^2 & 1 & \omega
\end{array}\right|=?\)

1. 1
2. -1
3.  0
4. None of these

Answer: 3. 0

Apply R₁→R₁+ R₂+ R₃ and use the result (1 + ω+ ω²) = 0.

Question 35. If ω is a complex cube root of unity, then the value of \(\left|\begin{array}{ccc}
1 & \omega & 1+\omega \\
1+\omega & 1 & \omega \\
\omega & 1+\omega & 1
\end{array}\right|\)

1. 2
2.  4
3.  0
4. -3

Answer: 2. 4

1 +ω+ ω² => (1 + ω) = – ω². Put (1 + ω) = -ω² and expand.

Question 36. If \(\left|\begin{array}{lll}
a+b & b+c & c+a \\
b+c & c+a & a+b \\
c+a & a+b & b+c
\end{array}\right|=k\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|, \text { then } k=?\)

1. 0
2.  1
3.  2
4. -2

Answer:  3. 2

Question 37. \(\text { The solution set of the equation }\left|\begin{array}{lll}
x & 3 & 7 \\
2 & x & 2 \\
7 & 6 & x
\end{array}\right|=0 \text { is }\)

1. {2,-3,7}
2.  {2,7,-9}
3. {-2,3,-7}
4.  None of these

Answer: 2. {2,7,-9}

Apply C₁ → C₁  −C₃ and take (x-7) common from C₁

Question 38. The solution set of the equation \(\left|\begin{array}{ccc}
x-2 & 2 x-3 & 3 x-4 \\
x-4 & 2 x-9 & 3 x-16 \\
x-8 & 2 x-27 & 3 x-64
\end{array}\right|\) is

1.  {4}
2.  {2,4}
3.  {2,8}
4.  {4,8}

Answer: 1. {4}

Apply C₂ → C₂ – 2C₁ and C₃→ C₃– 3C₁

Then, apply R₂ → R₂− R₁ and R₃→ R₃− R₁

Question 39. The solution set of the equation \(\left|\begin{array}{lll}
a+x & a-x & a-x \\
a-x & a+x & a-x \\
a-x & a-x & a+x
\end{array}\right|=0 \text { is }\) = 0 is

1. {1,0}
2.  {3a,0}
3.  {a,3a}
4.  None of these

Answer: 2. {3a,0}

Apply C₁ → (C₁ + C₂+ C₃) and take (3a- x) common from C₁

Question 40. The solution set of the equation \(\left|\begin{array}{rrr}
5 & 3 & -1 \\
-7 & x & 2 \\
9 & 6 & -2
\end{array}\right|=\) =0 is

1.  {0}
2.  {6}
3.  {-6}
4.  {0,9}

Answer: 3. {-6}

Question 41. The solution set of the equation \(\left|\begin{array}{ccc}
3 x-8 & 3 & 3 \\
3 & 3 x-8 & 3 \\
3 & 3 & 3 x-8
\end{array}\right|\) = 0 is

1.  \(\left\{\frac{2}{3}, \frac{8}{3}\right\}\)
2.  \(\left\{\frac{2}{3}, \frac{11}{3}\right\}\)
3.  \(\left\{\frac{3}{2}, \frac{8}{3}\right\}\)
4. None of these

Answer: 2. \(\left\{\frac{2}{3}, \frac{11}{3}\right\}\)

Question 42. The vertices of a A ABC are A(-2, 4), B(2, -6) and C(5, 4). The area of Δ ABC is

1.  17.5 sq units
2.  35 sq units
3. 32 sq units
4. 28 squats

Answer: 2. 35 sq units

⇒ \(\Delta=\frac{1}{2}\left|\begin{array}{rrr}
-2 & 4 & 1 \\
2 & -6 & 1 \\
5 & 4 & 1
\end{array}\right|=\frac{1}{2}\left|\begin{array}{rrr}
-2 & 4 & 1 \\
4 & -10 & 0 \\
7 & 0 & 0
\end{array}\right|\)

= 35 sq units

Question 43. If the points A(3, -2), B(k, 2) and C(8, 8) are collinear, then the value of k is

1. 2
2.  -3
3. 5
4. -4

Answer: 3. 5

⇒ \(\text { If } \Delta=\left|\begin{array}{rrr}
3 & -2 & 1 \\
k & 2 & 1 \\
8 & 8 & 1
\end{array}\right| \text {, then we must have } \Delta=0 \text {. }\)