# Mathematics Questions and Answers – Determinant – 3

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This set of Mathematics Quiz for Class 12 focuses on “Determinant – 3”.

1. What will be the value of $$\begin{vmatrix}0 & p-q & a – b\\q – p & 0 & x – y\\b – a & y – x & 0 \end {vmatrix}$$?
a) 0
b) a + b
c) x + y
d) p + q

Explanation: The above matrix is a skew symmetric matrix and its order is odd
And we know that for any skew symmetric matrix with odd order has determinant = 0
Therefore, the value of the given determinant = 0.

2. What will be the value of f(x) if $$\begin{vmatrix}x & b & c\\a & y & c\\a & b & z \end {vmatrix}$$?
a) (x – a)(y – b)(z – c)($$\frac{x}{x-a} + \frac{b}{y – b} – \frac{c}{z-c}$$ – 2)
b) (x – a)(y – b)(z – c)($$\frac{x}{x-a} – \frac{b}{y – b} – \frac{c}{z-c}$$ – 2)
c) (x – a)(y – b)(z – c)($$\frac{x}{x-a} + \frac{b}{y – b} + \frac{c}{z-c}$$ – 2)
d) (x – a)(y – b)(z – c)($$\frac{x}{x-a} + \frac{b}{y – b} + \frac{c}{z-c}$$ + 2)

Explanation: Given, $$\begin{vmatrix}x & b & c\\a & y & c\\a & b & z \end {vmatrix}$$ = $$\begin{vmatrix}x & b & c\\a – x & y – b & 0\\0 & b – y & z – c \end {vmatrix}$$

Applying the operation R2 = R2 – R1 and R3 = R3 – R2
= (x – a)(y – b)(z – c)$$\begin{vmatrix}x/(x – a) & b/(y – b) & c/(z – c)\\-1 & y 1 & 0\\0 & -1 & 1 \end {vmatrix}$$
Now, expanding the determinant we get,
= (x – a)(y – b)(z – c)($$\frac{x}{x-a} + \frac{b}{y – b} + \frac{c}{z-c}$$)
= (x – a)(y – b)(z – c)($$\frac{x}{x-a} + \frac{b}{y – b} + \frac{c}{z-c}$$ – 2)
This is because,
$$\frac{b}{y – b} + \frac{c}{z-c} = \frac{y-(y-b)}{y-b} + \frac{z-(z-c)}{z-c} = \frac{y}{y-b}$$ – 1 + $$\frac{z}{z-c}$$ – 1 = $$\frac{y}{y-b} + \frac{z}{z-c}$$ – 2

3. What will be the value of $$\begin{vmatrix}cos^2⁡ θ & cosθ \, sinθ & -sinθ \\cosθ\, sinθ & sin^2⁡θ & cosθ \\sinθ & -cosθ & 0 \end {vmatrix}$$?
a) -1
b) 0
c) 1
d) 2

Explanation: The given matrix is, $$\begin{vmatrix}cos^2 θ & cosθ\, sinθ & -sinθ \\cosθ\, sinθ & sin^2⁡ θ & cosθ \\sinθ & -cosθ & 0 \end {vmatrix}$$
Now, performing the row operations R1 = R1 + sinθR3 and R2 = R2 – cosθR3
=$$\begin{vmatrix}cos^2⁡ θ + sin^2⁡ θ & cosθ\, sinθ – cosθ sinθ & -sinθ \\cosθ\, sinθ – cosθ sinθ & cos^2⁡ θ + sin^2 ⁡θ & cosθ \\sinθ & -cosθ & 0 \end {vmatrix}$$
Solving further,
= $$\begin{vmatrix}1 & 0 & -sinθ \\0 & 1 & cosθ \\sinθ & -cosθ & 0 \end {vmatrix}$$
Breaking the determinant, we get,
= 1(0 + cos2θ) – sinθ(0 – sinθ)
= 1

4. What is the value of $$\begin{vmatrix}sin^2 ⁡a & sina\, cosa & cos^2 ⁡a \\sin^2 ⁡b & sinb\, cosb & cos^2 ⁡b \\sin^2⁡ c & sinc\, cosc & cos^2⁡ c \end {vmatrix}$$?
a) -sin(a – b) sin(b – c) sin(c – a)
b) sin(a – b) sin(b – c) sin(c – a)
c) -sin(a + b) sin(b + c) sin(c + a)
d) sin(a + b) sin(b + c) sin(c + a)

Explanation: We have, $$\begin{vmatrix}sin^2 a & sina \,cosa & cos^2 a\\sin^2 b & sinb\, cosb & cos^2 b\\sin^2 c & sinc\, cosc & cos^2 c \end {vmatrix}$$
Now, multiplying by 2 in both numerator and denominator of column 2 and C1 = C1 + C3 we get,
1/2 $$\begin{vmatrix}sin^2 a + cos^2 a & 2sina\, cosa & cos^2 a\\sin^2 b + cos^2 b & 2sinb\, cosb & cos^2 b\\sin^2 c + cos^2 c & 2sinc\, cosc & cos^2 c \end {vmatrix}$$
= 1/2 $$\begin{vmatrix}sin^2 a + cos^2 a & sin2a & cos^2 a\\sin^2 b + cos^2 b & sin2b & cos^2 b\\sin^2 c + cos^2 c & sin2c & cos^2 c \end {vmatrix}$$
= 1/2 $$\begin{vmatrix}1 & sin2a & cos^2 a\\1 & sin2b & cos^2 b\\1 & sin2c & cos^2 c \end {vmatrix}$$
Solving further,
= 1/2 $$\begin{vmatrix}1 & sin2a & cos^2⁡a \\0 & sin2b-sin2a & cos^2⁡ b-cos^2 ⁡a \\0 & sin2c-sin2a & cos^2 ⁡c-cos^2 ⁡a \end {vmatrix}$$
= 1/2 [(sin2b – sin2a)(cos2⁡c – cos2⁡a) – (cos2 b – cos2a)(sin2c – sin2a)]
Now, since, [cos2 ⁡A + cos2 B = sin(A + B) * sin(B – A)]
So, 1/2 [2 cos(a + b) sin(b – a) * sin(c + a)sin(a – c) – sin(a + b)sin(a – b) * 2 cos(a + c)sin(c – a)]
= sin(a – b)sin(c – a)[sin(c + a)cos(a + b) – cos(c + a) sin(a + b)]
= sin(a – b) sin(c – a) sin(c + a – a – b)
= -sin(a – b) sin(b – c) sin(c – a)

5. What will be the value of f(x) if $$\begin{vmatrix}2ab & a^2 & b^2 \\a^2 & b^2 & 2ab \\b^2 & 2ab & a^2 \end {vmatrix}$$?
a) a2 + b2
b) -(a2 + b2)
c) -(a2 + b2)3
d) -(a3 + b3)2

Explanation: Given,$$\begin{vmatrix}2ab & a^2 & b^2 \\a^2 & b^2 & 2ab \\b^2 & 2ab & a^2 \end {vmatrix}$$
Using C1 = C1 + C2 + C3
= $$\begin{vmatrix}a^2 + b^2 + 2ab & a^2 & b^2 \\a^2 + b^2 + 2ab & b^2 & 2ab \\a^2 + b^2 + 2ab & 2ab & a^2 \end {vmatrix}$$
= (a + b)2$$\begin{vmatrix}1 & a^2 & b^2 \\1 & b^2 & 2ab \\1 & 2ab & a^2 \end {vmatrix}$$
= (a + b)2$$\begin{vmatrix}1 & a^2 & b^2 \\1 & b^2 – a^2 & 2ab – b^2 \\0 & 2ab – a^2 & a^2 – b^2 \end {vmatrix}$$
= (a + b)2[(b2 – a2)(a2 – b2) – (2ab – b2)( 2ab – a2)]
= -(a + b)2[(a2 – b2)2 + 4a2b2 – 2ab(a2 + b2) + a2 b2)]
= –(a + b)2[(a2+b2)2 – 2(a2+b2) (ab)+(ab)2]
= –(a + b)2(a2 + b2 – ab)2
= –[(a + b)2(a2 + b2 – ab)2]2
= –(a3 + b3)2
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6. What will be the value of f(x) if $$\begin{vmatrix}1 & ab & (\frac{1}{a} + \frac{1}{b}) \\1 & bc & (\frac{1}{b} + \frac{1}{c}) \\1 & ca & (\frac{1}{c} + \frac{1}{a})\end {vmatrix}$$?
a) -1
b) 0
c) 1
d) Can’t be predicted

Explanation: We have,$$\begin{vmatrix}1 & ab & (\frac{1}{a} + \frac{1}{b}) \\1 & bc & (\frac{1}{b} + \frac{1}{c}) \\1 & ca & (\frac{1}{c} + \frac{1}{a})\end {vmatrix}$$
= (1/abc)$$\begin{vmatrix}1 & ab & \frac{b + a}{ab} * abc \\1 & bc & \frac{b + c}{bc} * abc \\1 & ca & \frac{c + a}{ac} * abc \end {vmatrix}$$
= (1/abc)$$\begin{vmatrix}1 & ab & bc + ac \\1 & bc & ac + ab \\1 & ca & ab + bc \end {vmatrix}$$
Operating, C3 = C3 + C2
= (1/abc)$$\begin{vmatrix}1 & ab & ab + bc + ac \\1 & bc & ab + bc + ac \\1 & ca & ab + bc + ac \end {vmatrix}$$
= ((ab + bc + ac)/abc)$$\begin{vmatrix}1 & ab & 1 \\1 & bc & 1 \\1 & ca & 1 \end {vmatrix}$$
= 0

7. What is the value of $$\begin{vmatrix}1 & cosx-sinx & cosx + sinx \\1 & cosy-siny & cosy + siny \\1 & cosz-sinz & cosz + sinz \end {vmatrix}$$?
a) 3$$\begin{vmatrix}1 & cosx & sinx \\1 & cosy & siny \\1 & cosz & sinz \end {vmatrix}$$
b) $$\begin{vmatrix}1 & cosx & sinx \\1 & cosy & siny \\1 & cosz & sinz \end {vmatrix}$$
c) 2$$\begin{vmatrix}1 & cosx & sinx \\1 & cosy & siny \\1 & cosz & sinz \end {vmatrix}$$
d) 4$$\begin{vmatrix}1 & cosx & sinx \\1 & cosy & siny \\1 & cosz & sinz \end {vmatrix}$$

Explanation: Let, a = cosx, b = cosy, c = cosz, p =sinx, q = siny and r = sinz
So, $$\begin{vmatrix}1 & a – p & a + p \\1 & b – q & b + q \\1 & c – r & c + r \end {vmatrix}$$
Making C3 = C3 + C2
= $$\begin{vmatrix}1 & a – p & 2a \\1 & b – q & 2b \\1 & c – r & 2c \end {vmatrix}$$

= 2$$\begin{vmatrix}1 & a – p & a \\1 & b – q & b \\1 & c – r & c \end {vmatrix}$$
Making C2 = C2 – C3
= -2$$\begin{vmatrix}1 & p & a \\1 & q & b \\1 & r & c \end {vmatrix}$$
Interchanging 2nd and 3rd column, we get,
2$$\begin{vmatrix}1 & a & p \\1 & b & q \\1 & c & r \end {vmatrix}$$
= 2$$\begin{vmatrix}1 & cosx & sinx \\1 & cosy & siny \\1 & cosz & sinz \end {vmatrix}$$

8. What will be the value of $$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}$$?
a) (a3 + b3 + c3 + 3abc)
b) –(a3 + b3 + c3 + 3abc)
c) (a3 + b3 + c3 – 3abc)
d) –(a3 + b3 + c3 – 3abc)

Explanation: Given, $$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}$$
Replacing R1 = R1 + R2 + R3
$$\begin{vmatrix}a + b + c & a + b + c & a + b + c \\b & c & a \\c & a & b \end {vmatrix}$$
= (a + b + c)$$\begin{vmatrix}1 & 1 & 1 \\b & c & a \\c & a & b \end {vmatrix}$$
Replacing 2nd column by C2 – C1 and 3rd column by C3 – C1
= (a + b + c)$$\begin{vmatrix}1 & 0 & 0 \\b & c-b & a-b \\c & a-c & b-c \end {vmatrix}$$
= (a + b + c)[(c – b)(b – c) – (a – b)(a – c)]
= (a + b + c)(bc – b2 – c2 + bc + a2 + ac + ab – bc)
= -(a + b + c)(a2 + b2 + c2 – ab – bc – ac)
= -(a3 + b3 + c3 – 3abc)

9. What will be the value of $$\begin{vmatrix}2bc – a^2 & c^2 & b^2 \\c^2 & 2ac – b^2 & a^2 \\b^2 & a^2 & 2ab – c^2 \end {vmatrix}$$ if given another determinant $$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}$$?
a) (a3 + b3 + c3 + 3abc)2
b) –(a3 + b3 + c3 + 3abc)2
c) (a3 + b3 + c3 – 3abc)2
d) –(a3 + b3 + c3 – 3abc)2

Explanation: Now, $$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}$$
Interchanging 2nd and 3rd columns,
= –$$\begin{vmatrix}a & c & b \\b & a & c \\c & b & a \end {vmatrix}$$
= $$\begin{vmatrix}-a & c & b \\ -b & a & c \\-c & b & a \end {vmatrix}$$
So, $$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}^2$$ = $$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}$$$$\begin{vmatrix} -a & c & b \\ -b & a & c \\-c & b & a \end {vmatrix}$$
= {–(a3 + b3 + c3 – 3abc)}2 = $$\begin{vmatrix}-a^2 + bc + bc & -ab + ab + c^2 & -ac + b^2 + ca \\-ab + ab + c^2 & -ac – b^2 + ca & -a^2 + bc + bc \\-ac + b^2 + ca & -a^2 + bc + bc & -ab + ab – c^2 \end {vmatrix}$$
=> $$\begin{vmatrix}2bc – a^2 & c^2 & b^2 \\c^2 & 2ac – b^2 & a^2 \\b^2 & a^2 & 2ab – c^2 \end {vmatrix}$$ = (a3 + b3 + c3 – 3abc)2

10. What will be the value of f(x) if $$\begin{vmatrix}1 & 1 & 1 \\x & y & z \\x^3 & y^3 & z^3 \end {vmatrix}$$?
a) -1
b) 0
c) 1
d) 2

Explanation: Given, $$\begin{vmatrix}1 & 1 & 1 \\x & y & z \\x^3 & y^3 & z^3 \end {vmatrix}$$
Operating, C1 = C1 – C2 and C2 = C2 – C3
= $$\begin{vmatrix}1 & 1 & 1 \\x – y & y – z & y \\x^3 – y^3 & y^3 – z^3 & z^3 \end {vmatrix}$$
Expanding by the 1st row,
= (x – y)(y3 – z3) – (y – z)(x3 – y3)
= (x – y)(y – z)[(y2 + yz + z2) – (x2 + xy + y2)]
= (x – y)(y – z)(z – x)(x + y + z)
As, x + y + z = 0
= 0

11. If, x3 = 1, then, what will be the value of$$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}$$?
a) -(a + bx + cx2)$$\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}$$
b) (a + bx + cx2)$$\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}$$
c) (a – bx – cx2)$$\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}$$
d) (a + bx – cx2)$$\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}$$

Explanation: We have, $$\begin{vmatrix}a & b & c \\b & c & a \\c & a & b \end {vmatrix}$$
As, x3 = 1,
= $$\begin{vmatrix}a & bx & cx^2 \\b & cx & ax^2 \\c & ax & bx^2 \end {vmatrix}$$
Replacing the 1st column by C1 + C2 + C3 we get,
= $$\begin{vmatrix}a + bx + cx^2 & bx & cx^2 \\ b + cx + ax^2 & cx & ax^2 \\c + ax + bx^2 & ax & bx^2 \end {vmatrix}$$
As, x3 = 1 so, x4 = x3 * x = x
= $$\begin{vmatrix}a + bx + cx^2 & b & c \\x^2 (a + bx + cx^2) & c & a \\x(a + bx + cx^2) & a & b \end {vmatrix}$$
= (a + bx + cx2)$$\begin{vmatrix}1 & b & c \\x^2 & c & a \\x & a & b \end {vmatrix}$$

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