2nd PUC Maths Question Bank Chapter 4 Determinants Ex 4.2

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Karnataka 2nd PUC Maths Question Bank Chapter 4 Determinants Ex 4.2

2nd PUC Maths Determinants NCERT Text Book Questions and Answers Ex 4.2

Using die property of determinants and without prove the following Q. 1-5

Question 1.
\(\left|\begin{array}{lll}{\mathbf{x}} & {\mathbf{a}} & {\mathbf{x}+\mathbf{a}} \\{\mathbf{y}} & {\mathbf{b}} & {\mathbf{y}+\mathbf{b}} \\{\mathbf{x}} & {\mathbf{c}} & {\mathbf{x}+\mathbf{c}}\end{array}\right|=\mathbf{0}\)
Answer:

Question 2.
\(\left|\begin{array}{ccc}{\mathbf{a}-\mathbf{b}} & {\mathbf{b}-\mathbf{c}} & {\mathbf{c}-\mathbf{a}} \\{\mathbf{b}-\mathbf{c}} & {\mathbf{c}-\mathbf{a}} & {\mathbf{a}-\mathbf{b}} \\{\mathbf{c}-\mathbf{a}} & {\mathbf{a}-\mathbf{b}} & {\mathbf{b}-\mathbf{c}}\end{array}\right|=\mathbf{0}\)
Answer:

Question 3.
\(\left|\begin{array}{lll}{2} & {7} & {65} \\{3} & {8} & {75} \\{5} & {9} &{86}\end{array}\right|=0\)
Answer:

Question 4.
\(\left|\begin{array}{lll}{\mathbf{1}} & {\mathbf{b} \mathbf{c}} & {\mathbf{a}(\mathbf{b}+\mathbf{c})} \\{\mathbf{1}} & {\mathbf{c} \mathbf{a}} & {\mathbf{b}(\mathbf{c}+\mathbf{a})} \\{\mathbf{1}} & {\mathbf{a} \mathbf{b}} & {\mathbf{c}(\mathbf{a}+\mathbf{b})}\end{array}\right|=\mathbf{0}\)
Answer:

Question 5.
\(\left|\begin{array}{ccc}{\mathbf{b}+\mathbf{c}} & {\mathbf{q}+\mathbf{r}} & {\mathbf{y}+\mathbf{z}} \\{\mathbf{c}+\mathbf{a}} & {\mathbf{r}+\mathbf{p}} & {\mathbf{z}+\mathbf{x}} \\{\mathbf{a}+\mathbf{b}} & {\mathbf{p}+\mathbf{q}} & {\mathbf{x}+\mathbf{y}}\end{array}\right|=\mathbf{2}\left|\begin{array}{ccc}{\mathbf{a}} & {\mathbf{p}} & {\mathbf{x}} \\{\mathbf{b}} & {\mathbf{q}} & {\mathbf{y}} \\{\mathbf{c}} & {\mathbf{r}} & {\mathbf{z}}\end{array}\right|\)
Answer:

By using properties of determination .show that

Question 6.
\(\left|\begin{array}{ccc}{\mathbf{0}} & {\mathbf{a}} & {\mathbf{-} \mathbf{b}} \\{-\mathbf{a}} & {\mathbf{o}} & {\mathbf{-c}} \\{\mathbf{b}} & {\mathbf{c}} & {\mathbf{0}}\end{array}\right|=\mathbf{0}\)
Answer:

Question 7.
\(\left|\begin{array}{ccc}{-a^{2}} & {a b} & {a c} \\{b a} & {-b^{2}} & {b c} \\{c a} & {c b} & {-c^{2}}\end{array}\right|=4 a^{2} b^{2} c^{2}\)
Answer:

Question 8.
(i)
\(\left| \begin{array}{ccc} { 1 } & { a } & { a^{ 2 } } \\ 1 & b & { b }^{ 2 } \\ { 1 } & { c } & { c^{ 2 } } \end{array} \right| =(a-b)(b-c)(c-a)\)
Answer:

(ii)
\(\left| \begin{array}{lll} { { l } } & { { 1 } } & { { 1 } } \\ { { a } } & { { b } } & { { c } } \\ { { a }^{ { 3 } } } & { { b }^{ { 3 } } } & { c }^{ { 3 } } \end{array} \right| \)
Answer:

Question 9.
\(\left|\begin{array}{lll}{\mathbf{x}} & {\mathbf{x}^{2}} & {\mathbf{y} \mathbf{z}} \\{\mathbf{y}} & {\mathbf{y}^{2}} & {\mathbf{z} \mathbf{x}} \\{\mathbf{z}} & {\mathbf{z}^{2}} & {\mathbf{x} \mathbf{y}}\end{array}\right|\)
Answer:

Question 10.
(i)
\(\left|\begin{array}{ccc}{x+4} & {2 x} & {2 x} \\{2 x} & {x+4} & {2 x} \\{2 x} & {2 x} & {x+4}\end{array}\right|=(5 x+4)(4-x)^{2}\)
Answer:

(ii)
\(\left|\begin{array}{ccc}{\mathbf{y}+\mathbf{K}} & {\mathbf{y}} & {\mathbf{y}} \\{\mathbf{y}} & {\mathbf{y}+\mathbf{k}} & {\mathbf{y}} \\{\mathbf{y}} & {\mathbf{y}} & {\mathbf{y}+\mathbf{k}}\end{array}\right|=\mathbf{k}^{2}(5 \mathbf{y}+\mathbf{k})\)
Answer:
R₁⇒ R₁ + R₂ + R₃

Question 11.
(i)
\(\left|\begin{array}{ccc}{\mathbf{a}-\mathbf{b}-\mathbf{c}} & {\mathbf{2 a}} & {\mathbf{2} \mathbf{a}} \\{\mathbf{2} \mathbf{b}} & {\mathbf{b}-\mathbf{c}-\mathbf{a}} & {\mathbf{2} \mathbf{b}} \\{\mathbf{2 c}} & {\mathbf{2 c}} & {\mathbf{c}-\mathbf{a}-\mathbf{b}}\end{array}\right|=(\mathbf{a}+\mathbf{b}+\mathbf{c})^{3}\)
Answer:

(ii)
\(\left|\begin{array}{ccc}{\mathbf{x}+\mathbf{y}+2 \mathbf{z}} & {\mathbf{x}} & {\mathbf{y}} \\{\mathbf{z}} & {\mathbf{y}+\mathbf{z}+2 \mathbf{x}} & {\mathbf{y}} \\{\mathbf{z}} & {\mathbf{x}} & {\mathbf{z}+\mathbf{x}+2 \mathbf{y}}\end{array}\right|=2(\mathbf{x}+\mathbf{y}+\mathbf{z})^{3}\)
Answer:

Question 12.
\(\left|\begin{array}{ccc}{\mathbf{1}} & {\mathbf{x}} & {\mathbf{x}^{2}} \\{\mathbf{x}^{2}} & {\mathbf{1}} & {\mathbf{x}} \\{\mathbf{x}} & {\mathbf{x}^{2}} & {\mathbf{1}}\end{array}\right|=\left(1-\mathbf{x}^{3}\right)^{2}\)
Answer:

Question 13.
\(\left|\begin{array}{ccc}{1+a^{2}-b^{2}} & {2 a b} & {-2 b} \\{2 a b} & {1-a^{2}+b^{2}} & {2 a} \\{2 b} & {-2 a} & {1-a^{2}-b^{2}}\end{array}\right|=\left(1+a^{2}+b^{2}\right)^{3}\)
Answer:

Question 14.
\(\left|\begin{array}{ccc}{\mathbf{a}^{2}+\mathbf{1}} & {\mathbf{a} \mathbf{b}} & {\mathbf{a c}} \\{\mathbf{a b}} & {\mathbf{b}^{2}+\mathbf{1}} & {\mathbf{b c}} \\{\mathbf{c a}} & {\mathbf{c b}} &{\mathbf{c}^{2}+\mathbf{1}}\end{array}\right|=\mathbf{1}+\mathbf{a}^{2}+\mathbf{b}^{2}+\mathbf{c}^{2}\)
Answer:

Question 15.
Let A be a square matrix of order 3 x 3 then |KA| is equal to
(A) K |A|
(B) K² |A|
(C) K³ |A|
(D) 3K |A|
Answer:
|KA| = K³ |A|
hence (C) is the correct answer.

Question 16.
Which of the following is true
(A) Determinant is a square matrix
(B) Determinant is a member associated to a matrix
(C) Determinant is a number associated with a square matrix.
(D) None of these
Answer:
Determinant is a number associated with . a square matrix hence the answer is (C).