## Matrices and Determinants Questions and Answers Part-10

1. Suppose a, b, c are distinct non-zero real numbers and
$\triangle_{1}=\begin{bmatrix}a^{2} & b^{2} & c^{2} \\b^{2}+c^{2} & c^{2}+a^{2} & a^{2}+b^{2} \\bc & ca & ab\end{bmatrix}$
and $\triangle_{2}=\begin{bmatrix}1 & a & a^{3} \\1 & b & b^{3} \\1 & c & c^{3}\end{bmatrix}$
then $\frac{\triangle_{2}-\triangle_{1}}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}$         is equal to
a) a + b + c
b) $a^{2}+b^{2}+c^{2}$
c) $\left(a+b+c\right)\left(1+a^{2}+b^{2}+c^{2}\right)$
d) $4\left(abc\right)^{2}$

Explanation: Using R2 $\rightarrow$ R2 + R1, we get

2. Let $\omega=-\frac{1}{2}+\frac{i\sqrt{3}}{2}$     . Then the value of the determinant $\triangle=\begin{bmatrix}1 & 1 & 1 \\1 & -1-\omega^{2} & \omega^{2} \\1 &\omega^{2} & \omega^{4}\end{bmatrix}$
is
a) $3\omega$
b) $3\omega\left(\omega-1\right)$
c) $3\omega^{2}$
d) $3\omega\left(1-\omega\right)$

Explanation:

3. if $n\epsilon N$ and $A=\begin{bmatrix}n! & \left(n+1\right)! & \left(n+2\right)! \\\left(n+1\right)! & \left(n+2\right)! & \left(n+3\right)! \\\left(n+2\right)! & \left(n+3\right)! & \left(n+4\right)!\end{bmatrix}$
then $\lim_{n \rightarrow \infty}\frac{\left(3n^{3}-5\right)\triangle_{n}}{\triangle_{n+1}}$     equal
a) $\frac{3}{2}$
b) $\frac{5}{2}$
c) $-\frac{5}{2}$
d) 3

Explanation: Taking n! common from R1, (n + 1)! from R2 and (n + 2)! from R3, we obtain and

4. If $f(x)=\begin{bmatrix}1 & x & x+1 \\2x & x\left(x-1\right) & \left(x+1\right)x \\3x\left(x-1\right) & x\left(x-1\right)\left(x-2\right) & \left(x+1\right)x\left(x-1\right)\end{bmatrix}$
then f(2018) is equal to
a) 0
b) 1
c) 2018
d) 2019

Explanation: Taking x common from R2 and x(x – 1) common from R3, we get

5. The number of positive integral solutions of the equation Δ = 12, where $\triangle=\begin{bmatrix}y+z & z & y \\z & z+x & x \\y & x & x+y\end{bmatrix}$
is
a) 3
b) 15
c) 4
d) $2^{3}3^{2}$

Explanation: Using C1 $\rightarrow$ C1 + C2 + C3, we get

6. If $a_{1},a_{2},a_{3},.......,a_{9}$     are in H.P. and $a_{4}=5,a_{5}=4$   , then value of the determinant
$\triangle=\begin{bmatrix}a_{1} & a_{2} & a_{3} \\a_{4} & a_{5} & a_{6} \\a_{7} & a_{8} & a_{9}\end{bmatrix}$
equals
a) $\frac{1}{20}$
b) $\frac{50}{21}$
c) $\frac{3}{20}$
d) $\frac{7}{90}$

Explanation:

7. Let $\omega$ be the complex number $\cos\left(\frac{2\pi}{3}\right)+i\sin\left(\frac{2\pi}{3}\right)$     Then the number of distinct complex number z satisfying
$\triangle=\begin{bmatrix}z+1 & \omega & \omega^{2} \\\omega & z+\omega^{2} & 1 \\\omega^{2} & 1 & z+\omega\end{bmatrix} = 0$
is
a) 1
b) 2
c) 3
d) infinte

Explanation:

8. If a, b, c are three complex number such that $a^{2}+b^{2}+c^{2}=0$     and
$\triangle=\begin{bmatrix}b^{2}+c^{2} & ab & ac \\ab & c^{2}+a^{2} & bc \\ac & bc & a^{2}+b^{2}\end{bmatrix}=ka^{2}b^{2}c^{2} ,$
then value of k is
a) 1
b) 2
c) -2
d) 4

Explanation:

9. The determinant $\triangle=\begin{bmatrix}a^{2} & a & 1 \\\cos\left(nx\right) & \cos\left(n+1\right) x &\cos\left(n+2\right)x \\sin\left(nx\right) & \sin\left(n+1\right)x & \sin\left(n+2\right)x \end{bmatrix}$
is independent of
a) n
b) x
c) a
d) none of these

Explanation:

10. Suppose $n\epsilon N$  and $1\leq r\leq n-3$    and
$\triangle\left(n,r\right)=\begin{bmatrix}^{n}C_{r-1} & ^{n}C_{r} & \left(r+1\right) \left(^{n+2}C_{r+1}\right)\\^{n}C_{r} & ^{n}C_{r+1} &\left(r+2\right) \left(^{n+2}C_{r+2}\right) \\^{n}C_{r+1} & ^{n}C_{r+2} & \left(r+3\right) \left(^{n+2}C_{r+3}\right) \end{bmatrix}$
then $\triangle\left(n,r\right)$   is equal to
a) 0
b) n! – r!
c) $^{n+4}C_{r+3}$
d) $^{n+4}C_{r+4}$