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) \[^{n+4}C_{r+4}\]
Explanation:
