## MP Board Class 12th Maths Important Questions Chapter 4 Determinants

### Determinants Structure Important Questions

Determinants Structure Objective Type Questions:

Question 1.

Question 1.
If A is a square matrix of order 3 × 3, then value of |Adj. A| will be:
(a) |A|
(b) |A|2
(c) |A|3
(d) 3|A|
(b) |A|2

Question 2.
If a, b, c are in Arithematic series, then value of determinant $$\left|\begin{array}{ccc} {x+2} & {x+3} & {x+2 a} \\ {x+3} & {x+4} & {x+2 b} \\ {x+4} & {x+5} & {x+2 c} \end{array}\right|$$ will be:
(a) 0
(b) 1
(c) x
(d) 2x
(a) 0

Question 3.
Matrix A = $$\begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$$, then A-1 will be:
(a) A-1 = $$\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$$
(b) A-1 = $$\begin{bmatrix} 2 & -3 \\ 1 & -2 \end{bmatrix}$$
(c) A-1 = $$\begin{bmatrix} -2 & 3 \\ -1 & 2 \end{bmatrix}$$
(d) A-1 = $$\begin{bmatrix} -2 & 3 \\ 1 & -2 \end{bmatrix}$$
(a) A-1 = $$\begin{bmatrix} 2 & -3 \\ -1 & 2 \end{bmatrix}$$

Question 4.
If ω is the cube roots of unitary, then $$\left|\begin{array}{ccc} {\mathbf{1}} & {\omega} & {\omega^{2}} \\ {\omega} & {\omega^{2}} & {1} \\ {\omega^{2}} & {1} & {\omega} \end{array}\right|$$ =
(a) 1
(b) 0
(c) ω
(d) ω2
(b) 0

Question 5.
Determinat $$\left|\begin{array}{ccc} {a+b} & {a+2 b} & {a+3 b} \\ {a+2 b} & {a+3 b} & {a+4 b} \\ {a+4 b} & {a+5 b} & {a+6 b} \end{array}\right|$$ =
(a) a2 + b2 + c2 – 3abc
(b) 0
(c) a3 + b3 + c3
(d) None of these
(b) 0

Question 2.
Fill in the blanks:

1. If $$\begin{vmatrix} 3 & m \\ 4 & 5 \end{vmatrix}$$ = 3, then m = ………………………..
2. In determinant $$\begin{vmatrix} 2 & -3 \\ 1 & -2 \end{vmatrix}$$, the cofactor of element – 3 is ………………………..
3. If A = $$\left|\begin{array}{lll}{1} & {0} & {1} \\ {0} & {1} & {2} \\ {0} & {0} & {4}\end{array}\right|$$, then the value of |3A| is …………………………..
4. The value of determinant $$\begin{vmatrix} 1 & log_{ b }a \\ log_{ a }b & 1 \end{vmatrix}$$ will be ……………………………
5. The value of determinant $$\begin{vmatrix} cos70^{ \circ }\quad & sin20^{ \circ } \\ sin70^{ \circ } & cos20^{ \circ } \end{vmatrix}$$ will be ………………………

1. 3
2. -1
3. 27A
4. 0
5. 0

Question 3.
Write True/False

1. The value of determinant $$\left|\begin{array}{ccc}{0} & {a} & {-b} \\ {-a} & {0} & {-c} \\ {b} & {c} & {0}\end{array}\right|$$ is abc?
2. The maximum value of determinant
$$\left|\begin{array}{ccc} {1} & {1} & {1} \\ {1} & {1+\sin \theta} & {1} \\ {1} & {1} & {1+\cos \theta} \end{array}\right|$$ is $$\frac{1}{2}$$?
3. If A is the matrix of order 3 × 3, then find the value of |kA| will be k2|A|2?
4. If $$\begin{vmatrix} x\quad & 2 \\ 18 & x \end{vmatrix}$$ = $$\begin{vmatrix} 6\quad & 2 \\ 18 & 6 \end{vmatrix}$$, then find the value of x is ± 3?
5. The value of the determinant $$\begin{vmatrix} 1\quad & \omega \\ \omega & -\omega \end{vmatrix}$$ is 1?

1. True
2. True
3. Flase
4. Flase
5. True

Question 4.
Write the answer in one word/sentence:

1. How many number of value k for which the linear equation 4x+ ky + 2z = 0, kx + 4y + z = 0, 2x + 2y + z = 0 passes a non – zero solution?
2. If α, β are the roots of equation 2x2 + 3x + 5 = 0, then find the value of $$\left|\begin{array}{lll} {0} & {\beta} & {\beta} \\ {\alpha} & {0} & {\alpha} \\ {\beta} & {\alpha} & {0} \end{array}\right|$$?
3. If area of the traingle with vertices (2, -6), (5, 4) and (k, 4) be 35 square units, then find the value of k?
4. If x ∈ N and A = $$\begin{vmatrix} x+3\quad & -2 \\ -3x & 2x \end{vmatrix}$$ = 8, then find the value of x?
5. Find the value of determinant $$\left|\begin{array}{ccc} {1^{2}} & {2^{2}} & {3^{2}} \\ {2^{2}} & {3^{2}} & {4^{2}} \\ {3^{2}} & {4^{2}} & {5^{2}} \end{array}\right|$$?

1. 2
2. $$\frac{-15}{4}$$
3. 12
4. 2
5. -8

Determinants Structure Very Short Answer Type Questions

Question 1.
Find the value of $$\begin{vmatrix} 2\quad & 20 \\ 1 & 6 \end{vmatrix}$$?
-8.

Question 2.
Find the value of y if $$\begin{vmatrix} -6\quad & 2 \\ 3 & y \end{vmatrix}$$ = 24?
-5.

Question 3.
Find x if $$\begin{vmatrix} 2\quad & 4 \\ x & 0 \end{vmatrix}$$ = -16?
4.

Question 4.
If $$\begin{vmatrix} a\quad & b \\ c & d \end{vmatrix}$$ = 5, then find the value of $$\begin{vmatrix} 3a\quad & 3b \\ 3c & 3d \end{vmatrix}$$?
45.

Question 5.
If $$\begin{vmatrix} a\quad & ω \\ ω & -ω \end{vmatrix}$$ = 1, then value of a will be?
1.

Question 6.
If $$\begin{vmatrix} 3\quad & m \\ 4 & 5 \end{vmatrix}$$ = 3, then find the value of m?
3.

Question 7.
If $$\begin{vmatrix} 2\quad & x \\ 4 & 9 \end{vmatrix}$$ = 30, then find the value of x?
– 3.

Question 8.
If $$\begin{vmatrix} 4\quad & -3 \\ m & m \end{vmatrix}$$ = 21, then find the value of x?
3.

Question 9.
If $$\begin{vmatrix} 2\quad & 4 \\ 3 & x \end{vmatrix}$$ = 0, then find the value of x?
6.

Question 10.
If $$\begin{vmatrix} 4\quad & -3 \\ -m & m \end{vmatrix}$$, then find the value of m?
21.

Question 11.
If $$\begin{vmatrix} -6\quad & 2 \\ 3 & m \end{vmatrix}$$ = 12, then find the value of m?
An.swer:
-3.

Question 12.
If $$\begin{vmatrix} 4\quad & -6 \\ -2 & x \end{vmatrix}$$ = 20, then find the value of x?
8.

Question 13.
If ω, ω2 are the cube root of unity, then find the value of $$\begin{vmatrix} 1\quad & \omega \\ \omega & -\omega \end{vmatrix}$$?
1.

Question 14.
Find the value of $$\left|\begin{array}{ccc} {224} & {777} & {32} \\ {735} & {888} & {105} \\ {812} & {999} & {116} \end{array}\right|$$?
0.

Question 15.
If $$\begin{vmatrix} x\quad & 4 \\ 3 & 3 \end{vmatrix}$$ = 0 then find the value of x?
4.

Question 16.
If $$\begin{vmatrix} 2+3i\quad & 4 \\ 1 & 2-3i \end{vmatrix}$$ find its value?
9.

Question 17.
In determinants $$\begin{vmatrix} 2\quad & -3 \\ 1 & -2 \end{vmatrix}$$ then find the co – factor of element – 3?
1.

Question 18.
If $$\begin{vmatrix} 3\quad & -2 \\ -4 & x \end{vmatrix}$$ = 16, then find the value of x?
8.

Question 19.
The value of $$\begin{vmatrix} 1\quad & log_{ b }a \\ log_{ a }b & 1 \end{vmatrix}$$?
0.

Question 20.
Find the value of 2 from determinant $$\begin{vmatrix} 1\quad & 3 \\ 2 & 4 \end{vmatrix}$$?
3.

Question 21.
Find the value of $$\begin{vmatrix} 2+5i\quad & 5 \\ 4 & 2-5i \end{vmatrix}$$?
9.

Question 22.
Find the value of $$\begin{vmatrix} cot x\quad & cosec x \\ cosec x & cot x \end{vmatrix}$$?
-1.

Question 23.
Find the value of $$\begin{vmatrix} cos70^{ \circ }\quad & sin20^{ \circ } \\ sin70^{ \circ } & cos20^{ \circ } \end{vmatrix}$$?
0.

Determinants Long Answer Type Questions – I

Question 1.
Prove that:
$$\left|\begin{array}{ccc} {a+b+2 c} & {a} & {b} \\ {c} & {b+c+2 a} & {b} \\ {c} & {a} & {c+a+2 b} \end{array}\right|$$ = 2 (a + b + c)3?
Solution:
Let ∆ =

= 2 (a + b + c) . 1. [(a + b + c)2 – 0]
= 2 (a + b + c)3. Proved.

Question 2.
Prove that:
$$\left|\begin{array}{ccc} {a^{2}+1} & {a b} & {a c} \\ {a b} & {b^{2}+1} & {b c} \\ {a c} & {b c} & {c^{2}+1} \end{array}\right|$$ = 1 + a2 + b2 + c2? (NCERT, CBSE 2016)
Solution:
Let ∆ =

= (1 + a2 + b2 + c2) . 1 $$\begin{vmatrix} 1\quad & 0 \\ 0 & 1 \end{vmatrix}$$
∆ = (1 + a2 + b2 + c2). Proved.

Question 3.
Prove that $$\left|\begin{array}{ccc} {a^{2}} & {b c} & {a c+c^{2}} \\ {a^{2}+a b} & {b^{2}} & {a c} \\ {a b} & {b^{2}+b c} & {c^{2}} \end{array}\right|$$ = 4a2 b2 c2 ?(NCERT, CBSE 2015)
Solution:
Let ∆ =

= abc.2c. $$\begin{vmatrix} a-b\quad & -a \\ a+b & -a \end{vmatrix}$$
= 2abc2 [-a(a – b) + a (a + b)]
= 22bc2[-a + b + a + b]
= 2a2bc2.2b = 4a2b2c2. Proved.

Question 4.
Solve the following determinant:
$$\left|\begin{array}{ccc} {x+1} & {3} & {5} \\ {2} & {x+2} & {5} \\ {2} & {3} & {x+4} \end{array}\right|$$ = 0?
Solution:

⇒ (x + 9) × 1 $$\begin{vmatrix} x-1\quad & 0 \\ 0 & x-1 \end{vmatrix}$$ = 0
⇒ (x + 9) (x – 1)2 = 0
∴ (x + 9) = 0 or (x – 1)2 = 0
⇒ x = -9 or x = 1, 1.

Question 5.
Prove that:
$$\left|\begin{array}{ccc} {\alpha} & {\beta} & {\lambda} \\ {\alpha^{2}} & {\beta^{2}} & {\lambda^{2}} \\ {\beta+\lambda} & {\lambda+\alpha} & {\alpha+\beta} \end{array}\right|$$ = (α – β) (β – λ) (λ – α) (α + β + λ)?
Solution:
Let ∆ =

⇒ ∆ = (α + β + λ) (α – β) (β – λ) (β + λ – α – β)
⇒ ∆ = (α – β) (β – λ) (λ – α) (α + β + λ)
= R.H.S. Proved.

Question 6.
Prove that:
$$\left|\begin{array}{ccc} {1+a} & {1} & {1} \\ {1} & {1+b} & {1} \\ {1} & {1} & {1+c} \end{array}\right|$$ = (abc) (1 + $$\frac{1}{a}$$ + $$\frac{1}{b}$$ + $$\frac{1}{c}$$ )? (NCERT; CBSE 2012, 14)
Solution:
Let

R.H.S. Proved.

Question 7.
Prove that:
$$\left|\begin{array}{ccc} {-a^{2}} & {a b} & {a c} \\ {a b} & {-b^{2}} & {b c} \\ {a c} & {b c} & {-c^{2}} \end{array}\right|$$ = 4a2b2
c2?
Solution:

= a2b2c2[2 (1 + 1)]
= a2b2c2.4 = 4a2b2c2. Proved.

Question 8.
Solve 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?
Solution:

⇒ (3x – 2) (3x – 11) [1. (1 – 0)] = 0
⇒ (3x – 2) (3x – 11) = 0
∴x = $$\frac{2}{3}$$, $$\frac{11}{3}$$.

Question 9.
Solve 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?
Solution:

⇒ (3a – x).1.(4x2 – 0) = 0
⇒ 3a – x = 0, 4x2 = 0
⇒ x = 3a, 0.

Question 10.
Prove that:
$$\left|\begin{array}{ccc} {a} & {a+b} & {a+b+c} \\ {2 a} & {3 a+2 b} & {4 a+3 b+2 c} \\ {3 a} & {6 a+3 b} & {10 a+6 b+3 c} \end{array}\right|$$ = 0?
Solution:
Let ∆ =

= a2 [7a + 3b – 6a – 3b]
= a2(a)
= a3. Proved.

Question 11.
Prove that:
$$\left|\begin{array}{ccc} {x} & {x+y} & {x+2 y} \\ {x+2 y} & {x} & {x+y} \\ {x+y} & {x+2 y} & {x} \end{array}\right|$$ = 9y2 (x + y)? (CBSE 2017)
Solution:
Let ∆

⇒ ∆ = 9y(x + y) [-x – y + x+ 2y]
⇒ ∆ = 9y2 (x + y). Proved.

Question 12.
Prove that:
$$\left|\begin{array}{ccc} {x+4} & {2 x} & {2 x} \\ {2 x} & {x+4} & {2 x} \\ {2 x} & {2 x} & {x+4} \end{array}\right|$$ = (5x + 4) (4 – x)2? (NCERT)
Solution:
Let

= (5x + 4) (x – 4) (2x – x – 4)
= (5x + 4) (x – 4) (x – 4)
= (5x + 4) (x – 4)2. [∵(a – b)2 = (b – a)2]
⇒ ∆ = (5x + 4) (4 – x)2. Proved.

Question 13.
Prove that:
$$\left|\begin{array}{ccc} {x} & {y} & {x+y} \\ {y} & {x+y} & {x} \\ {x+y} & {x} & {y} \end{array}\right|$$ = -2(x3 + y3). (NCERT)
Solution:
Let ∆ =

= 2 (x + y) (x2 – xy + y2)
⇒ ∆ = – 2(x2 + y2). Proved.

Question 14.
Prove that:
$$\left|\begin{array}{ccc} {a^{2}+2 a} & {2 a+1} & {1} \\ {2 a+1} & {a+2} & {1} \\ {3} & {3} & {1} \end{array}\right|$$ = (a – 1)3? (CBSE 2017)
Solution:
Let ∆ =

⇒ ∆ = (a – 1)2 (a + 1 – 2)
⇒ ∆ = (a – 1)2 (a – 1)
⇒ ∆ = (a – 1)3. Proved.

Question 15.
Prove that:
$$\left|\begin{array}{ccc} {1} & {1} & {1+3 x} \\ {1+3 y} & {1} & {1} \\ {1} & {1+3 z} & {1} \end{array}\right|$$ = 9(3 xyz + xy + yz + zx)? (CBSE 2018)
Solution:
Let ∆ =

= 9[x{(1 + y) (1 + 3z) – (1 + z)} – z{(x – y) – 0}]
= 9[x{1 + y + 3z + 3yz – 1 – z} – zx + zy]
= 9 [xy + 3xz + 3xyz – xz – zx + zy]
= 9 [3xyz + xy + yz + zx]. Proved.