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
Read and Learn More WBCHSE Solutions For Class 12 Maths
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 AIn-1
adj . AB = (adj . B) . (adj – A)
6. For square matrices A and B of the same order, we have
- (A + B)2= A2+AB +BA + B2
- (A- B)2 = A2– AB- BA + B2
- (A + B) . (A- B) = A2– AB +BA- B2
- 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. \(\left|\begin{array}{lll}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{array}\right|\)
= \(a_{11} \cdot\left|\begin{array}{ll}
a_{22} & a_{23} \\
a_{32} & a_{33}
\end{array}\right|-a_{12} \cdot\left|\begin{array}{ll}
a_{21} & a_{23} \\
a_{31} & a_{33}
\end{array}\right|+a_{13} \cdot\left|\begin{array}{ll}
a_{21} & a_{22} \\
a_{31} & a_{32}
\end{array}\right|\)
2.
1. Minor of aij is given by
Mij= det. obtained after deleting ith row and jth column.
2. Co-factor of aij is given by Cij= (-1)i+j.Mij
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.
⇒ Ri —>Rj+ kRj 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
- 0
- cos 50°
- 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
- \(\frac{1}{2}\)
- \(\frac{\sqrt{3}}{2}\)
- 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. \(\left|\begin{array}{rr}
\sin 23^{\circ} & -\sin 7^{\circ} \\
\cos 23^{\circ} & \cos 7^{\circ}
\end{array}\right|=?\)
- \(\frac{\sqrt{3}}{2}\)
- \(\frac{1}{2}\)
- sin 16
- cos 16
Answer: 2. \(\frac{1}{2}\)
Question 4. \(\left|\begin{array}{cc}
a+i b & c+i d \\
-c+i d & a-i b
\end{array}\right|\) = ?
- (a2+ b2– c2– d22
- (a2-b2+ c2-d2)
- (a2+ b2+ c2+ d2)
- 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. \(\left|\begin{array}{lll}
a-b & b-c & c-a \\
b-c & c-a & a-b \\
c-a & a-b & b-c
\end{array}\right|\)=?
- (a + b + c)
- 3[a+ b+ c)
- 3abc
- 0
Answer: 4. 0
R1 -+ (R1 + R2+ R3) gives all zeros in R1 and this gives Δ= 0
Question 6. \(\left|\begin{array}{ccc}
1 & 1+p & 1+p+q \\
2 & 3+2 p & 1+3 p+2 q \\
3 & 6+3 p & 1+6 p+3 q
\end{array}\right|\) =?
- 0
- 1
- -1
- None of these
Answer: 2. 1
Applying R2→ (R2– 2R1) and R3 -+ (R3– 3R1) and expandingby C1 we get Δ=1.
Question 7. \(\left|\begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^3 & b^3 & c^3
\end{array}\right|\) =?
- (a-b)(b-c)(c-a)
- -(a-b)(b-c)(c-a)
- (a- b)(b- c)(c- a)(a + b + c)
- 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 + a2+ b2)
- (1 + a2+ b2)2
- (1 + a2 + b2)3
- None of these
Answer: 3. (1 + a2 + b2)3
Apply R1 → R1 + bR3 and R2 → R2 — aR3
Question 9. \(\left|\begin{array}{lll}
\sin \alpha & \cos \alpha & \sin (\alpha+\delta) \\
\sin \beta & \cos \beta & \sin (\beta+\delta) \\
\sin \gamma & \cos \gamma & \sin (\gamma+\delta)
\end{array}\right|\)= ?
- 0
- 1
- sin (a+ 8) + sin (P + 8) + sin (Y+ 8)
- None of these
Answer: 1. 0
Apply C3→ C3– (cos δ)C1– (sinδ )C2
Question 10. \(\left|\begin{array}{ccc}
x+y & x & x \\
5 x+4 y & 4 x & 2 x \\
10 x+8 y & 8 x & 3 x
\end{array}\right|\)
- 0
- x3
- y3
- None of these
Answer: 2. x3
Take x common from C2 and x common from C3.
Apply R3→R3– 2R2.
Question 11. \(\left|\begin{array}{lll}
b+c & c+a & a+b \\
c+a & a+b & b+c \\
a+b & b+c & c+a
\end{array}\right|=?\)
- (a + b + c)
- (a + b + c)2
- 0
- None of these
Answer: 4. None of these
Apply C1 → (C1 + C12+ C3).
Question 12. \(\left|\begin{array}{ccc}
a & b & c \\
a-b & b-c & c-a \\
b+c & c+a & a+b
\end{array}\right|=?\)
- (a3 + b3 + c3)
- (a+b+c)3
- 3abc(a + b + c)
- (a3 + b3 + c3– 3abc
Answer: 4. (a3 + b3 + c3– 3abc
Apply C1 →(C1 + C2+ C3)
Question 13. \(\left|\begin{array}{ccc}
0 & a-b & a-c \\
b-a & 0 & b-c \\
c-a & c-b & 0
\end{array}\right|=?\)
- (a+b+c)
- (a+b-c)
- 1
- 0
Answer: 4. 0
Apply R1 → (R1 − R12) and → (R3−R2)
Question 14. \(\left|\begin{array}{ccc}
a+b & a & b \\
a & a+c & c \\
b & c & b+c
\end{array}\right|=?\)
- a+b+c
- abc
- 4abc
- a2b2c2
Answer: 3. 4abc
Apply R1 → R1– (R2+ R3).
Question 15. \(\left|\begin{array}{lll}
a & 1 & b+c \\
b & 1 & c+a \\
c & 1 & a+b
\end{array}\right|=?\)
- (a+b+c)
- 2(a+b+c)
- 0
- None of these
Answer: 3. 0
Apply C1 → (C1 + C3) and take (a+ b+ c) common from C1
Question 16. \(\left|\begin{array}{ccc}
a-b-c & 2 b & 2 c \\
2 a & b-c-a & 2 c \\
2 a & 2 b & c-a-b
\end{array}\right|=?\)
- (a+b+c)
- (a+b+c)2
- (a+b+c)3
- None of these
Answer: 3. (a+b+c)3
Apply Cx1→ C1+ C2+ C3
Question 17. \(\left|\begin{array}{ccc}
a^2+1 & a b & a c \\
a b & b^2+1 & b c \\
c a & c b & c^2+1
\end{array}\right|=?\)
- (a2+b2+c2)
- ( 1+a2+b2+c2)
- ( 3+a2+b2+c2)
- 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. \(\left|\begin{array}{lll}
265 & 240 & 219 \\
240 & 225 & 198 \\
219 & 198 & 181
\end{array}\right|=?\)
- 0
- 78
- -39
- 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. \(\left|\begin{array}{lll}
1^2 & 2^2 & 3^2 \\
2^2 & 3^2 & 4^2 \\
3^2 & 4^2 & 5^2
\end{array}\right|=?\)
- 8
- -8
- 16
- 142
Answer: 2. -8
Question 20.\(\left|\begin{array}{lll}
1 ! & 2 ! & 3 ! \\
2 ! & 3 ! & 4 ! \\
3 ! & 4 ! & 5 !
\end{array}\right|=?\)
- 2
- 6
- 24
- 120
Answer: 3. 24
Question 21.\(\left|\begin{array}{ccc}
1 & 1 & 1 \\
1 & 1+x & 1 \\
1 & 1 & 1+y
\end{array}\right|\)
- (x+y)
- (x-y)
- xy
- None of these
Answer: 3. xy
Question 22. \(\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|=?\)
- abc (a+b+c)
- (a3 -b3-c3 + 3abc)
- (-a3 -b3-c3 + 3abc)
- 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
- Positive
- Negative
- A perfect square
- 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. \(\left|\begin{array}{ccc}
-a^2 & a b & a c \\
a b & -b^2 & b c \\
a c & b c & -c^2
\end{array}\right|=?\)
- 0
- abc
- 4 a2b2c2
- None of these
Answer: 3. 4 a2b2c2
Question 25. \(\left|\begin{array}{lll}
b c & b+c & 1 \\
c a & c+a & 1 \\
a b & a+b & 1
\end{array}\right|=?\)
- (a-b)(b-c)(c-a)
- – (a-b)(b-c)(c-a)
- (a+b)(b+c)(c+a)
- None of these
Answer: 1. (a-b)(b-c)(c-a)
Question 26. \(\left|\begin{array}{ccc}
a^2+2 a & 2 a+1 & 1 \\
2 a+1 & a+2 & 1 \\
3 & 3 & 1
\end{array}\right|=?\)
- (a-1)2
- (a-1)3
- (a-1)
- None of these
Answer: 2. (a-1)3
Apply R1→ (R1 — R2) and R3 → (R3 — R2)
Question 27. \(\left|\begin{array}{lll}
\frac{1}{a} & a^2 & b c \\
\frac{1}{b} & b^2 & a c \\
\frac{1}{c} & c^2 & a b
\end{array}\right|=?\)
- 0
- 1
- -1
- 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.\(\left|\begin{array}{ccc}
x+1 & x+2 & x+4 \\
x+3 & x+5 & x+8 \\
x+7 & x+10 & x+14
\end{array}\right|=?\)
- -2
- 2
- x2– 2
- x + 2
Answer: 1. -2
Apply C2→(C2– C1) and C3 → (C3 — C1).
Question 29.\(\left|\begin{array}{ccc}
b+c & a & a \\
b & c+a & b \\
c & c & a+b
\end{array}\right|=?\)
- 4abc
- 2(a + b + c)
- (ab + bc + ca)
- None of these
Answer: 1. 4abc
Apply R1→R1– (R2+R3) and then apply R2 → (R2-R1) and R3 → (R3-R1).
Question 30. \(\left|\begin{array}{ccc}
a & a+2 b & a+2 b+3 c \\
3 a & 4 a+6 b & 5 a+7 b+9 c \\
6 a & 9 a+12 b & 11 a+15 b+18 c
\end{array}\right|=?\)
- a3
- -a3
- 0
- None of these
Answer: 2. -a3
Apply R2 → R2 − 3R1 and R3→ R3 − 6R1.
Question 31. \(\left|\begin{array}{lll}
1 & b c & b c(b+c) \\
1 & c a & c a(c+a) \\
1 & a b & a b(a+b)
\end{array}\right|=?\)
- abc
- 2 abc
- abc(a+b+c)
- 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
- Independent of o only
- Independent of o only
- Independent of o only
- 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. \(\left|\begin{array}{lll}
b+c & a & b \\
c+a & c & a \\
a+b & b & c
\end{array}\right|=?\)
- (a + b+ c)(a- c)
- (a + b + c)(b- c)
- (a + b + c)(a-cf
- (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
- 0
- 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 \(\left|\begin{array}{ccc}
1 & \omega & 1+\omega \\
1+\omega & 1 & \omega \\
\omega & 1+\omega & 1
\end{array}\right|\)
- 2
- 4
- 0
- -3
Answer: 2. 4
1 +ω+ ω2 => (1 + ω) = – ω2. Put (1 + ω) = -ω2 and expand.
Question 36. If \(\left|\begin{array}{lll}
a+b & b+c & c+a \\
b+c & c+a & a+b \\
c+a & a+b & b+c
\end{array}\right|=k\left|\begin{array}{lll}
a & b & c \\
b & c & a \\
c & a & b
\end{array}\right|, \text { then } k=?\)
- 0
- 1
- 2
- -2
Answer: 3. 2
Question 37. \(\text { The solution set of the equation }\left|\begin{array}{lll}
x & 3 & 7 \\
2 & x & 2 \\
7 & 6 & x
\end{array}\right|=0 \text { is }\)
- {2,-3,7}
- {2,7,-9}
- {-2,3,-7}
- 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
- {4}
- {2,4}
- {2,8}
- {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
- {1,0}
- {3a,0}
- {a,3a}
- 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
- {0}
- {6}
- {-6}
- {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
- \(\left\{\frac{2}{3}, \frac{8}{3}\right\}\)
- \(\left\{\frac{2}{3}, \frac{11}{3}\right\}\)
- \(\left\{\frac{3}{2}, \frac{8}{3}\right\}\)
- 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
- 17.5 sq units
- 35 sq units
- 32 sq units
- 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
- 2
- -3
- 5
- -4
Answer: 3. 5
⇒ \(\text { If } \Delta=\left|\begin{array}{rrr}
3 & -2 & 1 \\
k & 2 & 1 \\
8 & 8 & 1
\end{array}\right| \text {, then we must have } \Delta=0 \text {. }\)