# 2nd PUC Maths Question Bank Chapter 4 Determinants Miscellaneous Exercise

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## karnataka 2nd PUC Maths Question Bank Chapter 4 Determinants Miscellaneous Exercise

Question 1.
Prove that the determinant
$$\left| \begin{array}{rrr} { x } & { \sin \theta } & { \cos \theta } \\ { -\sin \theta } & { -x } & { 1 } \\ { 1 } & { 1 } & x \end{array} \right| is\quad independent\quad of\quad \theta$$

Question 2.
With out expanding, prove that
$$\left|\begin{array}{lll}{\mathbf{a}} & {\mathbf{a}^{2}} & {\mathbf{b} \mathbf{c}} \\{\mathbf{b}} & {\mathbf{b}^{2}} & {\mathbf{c} \mathbf{a}} \\{\mathbf{c}} & {\mathbf{c}^{2}} & {\mathbf{a} \mathbf{b}}\end{array}\right|=\left|\begin{array}{ccc}{\mathbf{1}} & {\mathbf{a}^{2}} & {\mathbf{a}^{3}} \\{\mathbf{1}} & {\mathbf{b}^{2}} & {\mathbf{b}^{3}} \\{\mathbf{1}} & {\mathbf{c}^{2}} & {\mathbf{c}^{3}}\end{array}\right|$$

Question 3.
Evaluate
$$\left|\begin{array}{ccc}{\cos \alpha \cos \beta} & {\cos \alpha \sin \beta} & {-\sin \alpha} \\{\sin \alpha \cos \beta} & {\sin \alpha \sin \beta} & {\cos \alpha}\end{array}\right|$$
expand along c3
-sin α (-sin α sin2β – sin α cos2β) + cosa (cosα cos2 β+cosα sin2β)
= sin2 α (sin2 β + cos2 β )+cos2 α (cos2 β + sin2 β)
= sin2α + cos2α= 1

Question 4.
It a, b, c are real, find the factors of the determination
$$\Delta=\left|\begin{array}{lll}{\mathbf{b}+\mathbf{c}} & {\mathbf{c}+\mathbf{a}} & {\mathbf{a}+\mathbf{b}} \\{\mathbf{c}+\mathbf{a}} & {\mathbf{a}+\mathbf{b}} & {\mathbf{b}+\mathbf{c}} \\{\mathbf{a}+\mathbf{b}} & {\mathbf{b}+\mathbf{c}} & {\mathbf{c}+\mathbf{a}}\end{array}\right|$$

Question 5.
Solve if a ≠ 0 and
$$\left|\begin{array}{ccc}{\mathbf{x}+\mathbf{a}} & {\mathbf{x}} & {\mathbf{x}} \\{\mathbf{x}} & {\mathbf{x}+\mathbf{a}} & {\mathbf{x}} \\{\mathbf{x}} & {\mathbf{x}} & {\mathbf{x}+\mathbf{a}}\end{array}\right|=\mathbf{0}$$

Question 6.
Prove that
$$\left|\begin{array}{ccc}{\mathbf{a}^{2}} & {\mathbf{b} \mathbf{c}} & {\mathbf{a c}+\mathbf{c}^{2}} \\{\mathbf{a}^{2}+\mathbf{a} \mathbf{b}} & {\mathbf{b}^{2}} & {\mathbf{a c}} \\{\mathbf{a b}} & {\mathbf{b}^{2}+\mathbf{b c}} & {\mathbf{c}^{2}}\end{array}\right|=4 \mathbf{a}^{2} \mathbf{b}^{2} \mathbf{c}^{2}$$

Question 7.
\begin{aligned}&\text { If } \mathbf{A}^{-1}=\left[\begin{array}{rrr}{3} & {-1} & {1} \\{-15} & {6} & {-5} \\{5} & {-2} & {2}\end{array}\right], \mathbf{B}=\left[\begin{array}{rrr}{1} & {2} & {-2} \\{-1} & {3} & {0} \\{0} & {-2} & {1}\end{array}\right] \text { find }\\&(\mathbf{A} \mathbf{B})^{-1}\end{aligned}

Question 8.
$$\text { Let } A=\left[\begin{array}{ccc}{1} & {-2} & {1} \\{-2} & {3} & {1} \\{1} & {1} & {5}\end{array}\right], \text { verify that }$$
(ii) (A1)1 = A

Question 9.
Evaluate
$$\left[\begin{array}{ccc}{\mathbf{x}} & {\mathbf{y}} & {\mathbf{x}+\mathbf{y}} \\{\mathbf{y}} & {\mathbf{x}+\mathbf{y}} & {\mathbf{x}} \\{\mathbf{x}+\mathbf{y}} & {\mathbf{x}} & {\mathbf{y}}\end{array}\right]$$

Question 10.
$$\left[\begin{array}{ccc}{\mathbf{1}} & {\mathbf{x}} & {\mathbf{y}} \\{\mathbf{1}} & {\mathbf{x}+\mathbf{y}} & {\mathbf{y}} \\{\mathbf{1}} & {\mathbf{x}} & {\mathbf{x}+\mathbf{y}}\end{array}\right]$$

Prove the following using properties

Question 11.
$$\left[\begin{array}{lll}{\alpha} & {\alpha^{2}} & {\beta+\alpha} \\{\beta} & {\beta^{2}} & {\gamma+\alpha} \\{\gamma} & {\gamma^{2}} & {\alpha+\beta}\end{array}\right]=(\beta-\alpha)(\gamma-\alpha)(\alpha-\beta)$$

Question 12.
\begin{aligned}&\left[\begin{array}{lll}{\mathbf{x}} & {\mathbf{x}^{2}} & {\mathbf{1}+\mathbf{p} \mathbf{x}^{3}} \\{\mathbf{y}} & {\mathbf{y}^{2}} & {\mathbf{1}+\mathbf{p} \mathbf{y}^{3}} \\{\mathbf{z}} & {\mathbf{z}^{2}} & {\mathbf{1}+\mathbf{p}\mathbf{z}^{3}}\end{array}\right]=(\mathbf{x}-\mathbf{y})(\mathbf{y}-\mathbf{z})(\mathbf{z}-\mathbf{x})\\&(1+p x y z)\end{aligned}

Question 13.
$$\left[\begin{array}{rrr}{3 a} & {-a+b} & {-a+c} \\{-b+a} & {3 b} & {-b+c} \\{-c+a} & {-c+b} & {3 c}\end{array}\right]=3(a+b+c)(a b+b c+c a)$$

Question 14.
$$\left|\begin{array}{ccc}{1} & {1+p} & {1+p+q} \\{2} & {3+2 p} & {4+3 p+2 q} \\{3} & {6+3 p} & {10+6 p+3 q}\end{array}\right|=1$$

Question 15.
$$\left|\begin{array}{ccc}{\sin \alpha} & {\cos \alpha} & {\cos (\alpha+\delta)} \\{\sin \beta} & {\cos \beta} & {\cos (\beta+\delta)} \\{\sin \gamma} & {\cos \gamma} & {\cos (\gamma+\delta)}\end{array}\right|=0$$

Question 16.
Solve
$$\frac{2}{x}+\frac{3}{y}+\frac{10}{z}=4$$
$$\frac{4}{x}-\frac{6}{y}+\frac{5}{z}=1$$
$$\frac{6}{x}+\frac{9}{y}-\frac{20}{z}=2$$

Question 17.
It a, b, c are in AP, then the determinant
$$\left|\begin{array}{ccc}{x+2} & {x+3} & {x+2 a} \\{x+3} & {x+4} & {x+2 b} \\{x+4} & {x+5} & {x+2 c}\end{array}\right|$$

Question 18.
If x, y, z are real, then the inverse of
$$\mathbf{A}=\left|\begin{array}{lll}{\mathbf{x}} & {\mathbf{0}} & {\mathbf{0}} \\{\mathbf{0}} & {\mathbf{y}} & {\mathbf{0}} \\{\mathbf{0}} & {\mathbf{0}} & {\mathbf{z}}\end{array}\right|$$

Question 19.
Let
$${ A }=\left[ \begin{array}{ccc} { { 1 } } & { \sin \theta } & { { 1 } } \\ { -\sin \theta } & { { 1 } } & { \sin \theta } \\ { -1 } & { -\sin \theta } & { { 1 } } \end{array} \right] { 0 }\leq \theta \leq 2\pi$$

### 2nd PUC Maths Chapter 4 Determinants Miscellaneous Exercise Additional Questions and Answers

Try These Questions

Question 1.
$$\text { If } \mathbf{A}=\left[\begin{array}{lll}{1} & {2} & {2} \\{2} & {1} & {2} \\{2} & {2} & {1}\end{array}\right] \text { find } A^{-1}$$
and hence prove that A2 – 4A – 5 I = 0

Question 2.
use product
$$\left[\begin{array}{ccc}{1} & {-1} & {2} \\{0} & {2} & {-3} \\{3} & {-2} & {4}\end{array}\right]\left[\begin{array}{ccc}{-2} & {0} & {1} \\{9} & {2} & {-3} \\{6} & {1} & {-2}\end{array}\right]$$
solve x – y + 2z = 1
2y – 3z = 1
3x – 2y + 4z = 2

Question 3.
If the point (x, y), (a, 0), (0, b) are collinear, prove that
$$\frac{x}{a}+\frac{y}{b}=1$$

Question 4.
$$\text { If } \Delta=\left|\begin{array}{ccc}{\mathbf{1}} & {\mathbf{1}} & {\mathbf{1}} \\{\mathbf{1}} & {\mathbf{1}+\mathbf{x}} & {\mathbf{1}} \\{\mathbf{1}} & {\mathbf{1}} & {\mathbf{1}+\mathbf{y}}\end{array}\right|, \text { then } \Delta \text { is }$$
(a) Divisible by neither x nor y
(b) Divisible by y but not x
(c) Divisible by x but not y
(d) Divisible by both x and y [AIEEE 2007]

Question 5.
If a, b, c are positive and unequal, show that the value of the determinant
$$\left|\begin{array}{lll}{\mathbf{a}} & {\mathbf{b}} & {\mathbf{c}} \\{\mathbf{b}} & {\mathbf{c}} & {\mathbf{a}} \\{\mathbf{c}} & {\mathbf{a}} & {\mathbf{b}}\end{array}\right|$$ is always negative [Kerala CET 09]

since a, b, c are positive a + b + c is positive and (a – b)2 is always positive,
Δ is always negative.

Question 6.
Using properties of matrices prove that 1+abc is a factor of [CET]
$$\left|\begin{array}{lll}{\mathbf{a}} & {\mathbf{a}^{2}} &{\mathbf{1}+\mathbf{a}^{3}} \\{\mathbf{b}} & {\mathbf{b}^{2}} & {\mathbf{1}+\mathbf{b}^{3}} \\{\mathbf{c}} &{\mathbf{c}^{2}} &{\mathbf{1}+\mathbf{c}^{3}}\end{array}\right|$$

Question 7.
For what values of y is the matrix  [CBSE 2011]
$$\mathbf{A}=\left|\begin{array}{rr}{\mathbf{y}^{2}+6} & {\mathbf{2} \mathbf{y}} \\{\mathbf{y}+3} & {\mathbf{2}} \end{array}\right|$$singular.

Question 8.
Find y
$$\left[\begin{array}{cc}{x} & {x-y} \\{2 x+y} & {7}\end{array}\right]=\left[\begin{array}{ll}{3} & {1} \\{8} & {7}\end{array}\right]$$

Question 9.
If A be a square matrix of order 3 x 3 then find |KA|
|KA| = kn |A| when n is the order [CBSE 2007]
|KA| = k3 |A|

Question 10.
Evaluate  [PUC 2009]
$$\left|\begin{array}{cccc}{\sin 30} & {\cos } & {30} \\{-\sin 60} & {\cos } & {60}\end{array}\right|$$

Question 11.
From the matrix equation AB = AC, we can conclude that B = C provided
(A) A is singular
(B) A is non singular
(C) A is symmetric
(D) A is skew symmetric [IIT]
AB = AC
A-1 (AB) = A-1 (AC) ⇒ (A-1 A) B = (A-1 A) C ⇒ IB = IC ⇒ B = C
AB = AC ⇒ B = C if A1 exist
ie,. A is non singular

Question 12.
A2 – A + I = 0, Find the inverse if A [IIT]
A2 – A + I = 0
Pre – multiply by A-1
A-1 A2 – A-1 A + A-1  I = 0
A -1 + A1 = 0
A-1= I – A

Question 13.
It is a triangle
$$\mathbf{A B C},\left|\begin{array}{lll}{\mathbf{1}} & {\mathbf{a}} & {\mathbf{b}} \\{\mathbf{1}} & {\mathbf{c}} & {\mathbf{a}} \\{\mathbf{1}} & {\mathbf{b}} & {\mathbf{c}}\end{array}\right|=\mathbf{0}$$
then find the value of sin2A+ sin2B+ sin2C [Karnataka CET 2002]

Question 14.
Let A, B be two square matrices such that A + B = AB, then  [Karnataka CET]
(A) AB = BA
(B) AB = -BA
(C) AB + 2BA = 0
(D) None of these
Ans:
A + B = AB A + B – AB = 0
A + B – AB + I = 0+1
(A – I)(B – I) = 1
⇒ A -1 inverse of B -1
(B -1) (A -1) = I
BA – B – A + = I
BA – (B + A) ⇒ BA = B + A
= A + B  ∴ AB = BA
$$\mathbf{A}(\alpha)=\left[\begin{array}{cc}{\cos \alpha} & {\sin \alpha} \\{-\sin \alpha} & {\cos \alpha}\end{array}\right]$$,
$$\mathbf{A}=\left[\begin{array}{ll}{\mathbf{1}} & {-\mathbf{1}} \\{\mathbf{2}} & {-\mathbf{1}}\end{array}\right], \mathbf{B}=\left[\begin{array}{ll}{\mathbf{a}} & {-\mathbf{1}} \\{\mathbf{b}} & {-\mathbf{1}}\end{array}\right] \text { and }(\mathbf{A}+\mathbf{B})^{2}$$