Matrices & Determinants · NDA previous year questions

What is the value of following? $\cot \left[{\sin }^{-1}\frac{3}{5}+{\cot }^{-1}\frac{3}{2}\right]$

1. A.$\frac{6}{17}$
2. B.$\frac{7}{16}$
3. C.$\frac{16}{7}$
4. D.$\frac{17}{6}$

Let $4{\sin }^{2}x=3$, where $0\le x\le \pi$. What is $\tan 3x$ equal to?

1. A.-2
2. B.-1
3. C.0
4. D.1

What is the value of the following determinant? $$\left|\begin{array}{ccc}\cos C& \tan A& 0\\ \sin B& 0& -\tan A\\ 0& \sin B& \cos C\end{array}\right|$$

1. A.-1
2. B.0
3. C.$2\tan A\sin B\sin C$
4. D.$-2\tan A\sin B\sin C$

How many distinct matrices exist with all four entries taken from (1, 2)?

1. A.16
2. B.24
3. C.32
4. D.48

If $i=\sqrt{-1}$, then how many values does $i^{-2n}$ have for different $n\in \mathbb{Z}$?

1. A.One
2. B.Two
3. C.Four
4. D.Infinite

If $x=\frac{a}{b-c}$, $y=\frac{b}{c-a}$, $z=\frac{c}{a-b}$ then what is the value of the following? $$\left|\begin{array}{ccc}1& -x& x\\ 1& 1& -y\\ 1& z& 1\end{array}\right|$$

1. A.0
2. B.1
3. C.abc
4. D.$ab+bc+ca$

1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3

The element in the $i^{\text{th}}$ row and the $j^{\text{th}}$ column of a determinant of third order is equal to $2(i+j)$. What is the value of the determinant?

1. A.0
2. B.2
3. C.4
4. D.6

With the numbers 2, 4, 6, 8, all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?

1. A.128
2. B.64
3. C.32
4. D.0

1. A.02 or 2
2. B.-3 or 3
3. C.-1 or 1
4. D.3 or 4

If $\Delta$ is the value of the determinant $$\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$$ then what is the value of the following determinant? $$\left|\begin{array}{ccc}p{a}_1& b_1& q{c}_1\\ p{a}_2& b_2& q{c}_2\\ p{a}_3& b_3& q{c}_3\end{array}\right|$$ ($p\ne 0$ or 1, $q\ne 0$ or 1)

1. A.$p\Delta$
2. B.$q\Delta$
3. C.$(p+q)\Delta$
4. D.$pq\Delta$

If $a+b+c=4$ and $ab+bc+ca=0$, then what is the value of the following determinant? $$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$$

1. A.32
2. B.-64
3. C.-128
4. D.64

The number of integer values of k, for which the equation 2sinx = 2k + 1 has a solution, is

1. A.zero
2. B.one
3. C.two
4. D.four

If $a_1,a_2,a_3,\dots ,a_9$ are in GP, then what is the value of the following determinant? $$\left|\begin{array}{ccc}\ln a_1& \ln a_2& \ln a_3\\ \ln a_4& \ln a_5& \ln a_6\\ \ln a_7& \ln a_8& \ln a_9\end{array}\right|$$

1. A.0
2. B.1
3. C.2
4. D.4

If A and B are two matrices such that AB is of order n × n, then which one of the following is correct?

1. A.A and B should be square matrices of same order.
2. B.Either A or B should be a square matrix.
3. C.Both A and B should be of same order
4. D.Orders of A and B need not be the same.

How many matrices of different orders are possible with elements comprising all prime numbers less than 30?

1. A.4
2. B.3
3. C.8
4. D.6

Let $A=\left[\begin{array}{cc}p& q\\ r& s\end{array}\right]$ where p, q, r and s are any four different prime numbers less than 20. What is the maximum value of the determinant?

1. A.215
2. B.311
3. C.317
4. D.323

If A and B are square matrices of order 2 such that det(AB) = det(BA), then which one of the following is correct?

1. A.A must be a unit matrix.
2. B.B must be a unit matrix.
3. C.Both A and B must be unit matrices.
4. D.A and B need not be unit matrices.

If matrix $A=\left[\begin{array}{cc}1-i& i\\ -i& 1-i\end{array}\right]$ where $i=\sqrt{-1}$, then which one of the following is correct?

1. A.$A$ is hermitian
2. B.$A$ is skew-hermitian
3. C.$(\bar{A}{)}^T+A$ is hermitian
4. D.$(\bar{A}{)}^T+A$ is skew-hermitian

1. A.Only one
2. B.Only two
3. C.Only four
4. D.Infinite

Let $A=\left[\begin{array}{cc}x+y& y\\ 2x& x-y\end{array}\right]$, $B=\left[\begin{array}{c}2\\ -1\end{array}\right]$ and $C=\left[\begin{array}{c}3\\ 2\end{array}\right]$. If $AB=C$, then what is the value of the determinant of the matrix $A$?

1. A.$-10$
2. B.$-14$
3. C.$-24$
4. D.$-34$

If $1.5\le x\le 4.5$, then which one of the following is correct?

1. A.$(2x-3)(2x-9)>0$
2. B.$(2x-3)(2x-9)<0$
3. C.$(2x-3)(2x-9)\ge 0$
4. D.$(2x-3)(2x-9)\le 0$

1. A.0
2. B.$-2$
3. C.$4i$
4. D.$-4i$

1. A.$\left[\begin{array}{c}ax+hy+gz\\ y\\ z\end{array}\right]$
2. B.$\left[\begin{array}{c}ax+hy+gz\\ hx+by+fz\\ z\end{array}\right]$
3. C.$\left[\begin{array}{c}ax+hy+gz\end{array}\right]$
4. D.$\left[\begin{array}{ccc}ax+hy+gz& hx+by+fz& gx+fy+cz\end{array}\right]$

If A is a matrix of order $3\times 5$ and B is a matrix of order $5\times 3$, then the order of AB and BA will respectively be

1. A.$3\times 3$ and $3\times 3$
2. B.$3\times 5$ and $5\times 3$
3. C.$3\times 3$ and $5\times 5$
4. D.$5\times 3$ and $3\times 5$