Matrices & Determinants · NDA previous year questions

. Consider the following statements in respect of the determinant $$\Delta =\left|\begin{array}{ccc}k(k+2)& 2k+1& 1\\ 2k+1& k+2& 1\\ 3& 3& 1\end{array}\right|$$ I. $\Delta$ is positive if k>0. II. $\Delta$ is negative if k<0. III. $\Delta$ is zero if k=0. How many of the statements given above are correct?

1. A.None
2. B.One
3. C.Two
4. D.All three

1. A.-10
2. B.-6
3. C.0
4. D.6

If $A^2+B^2+C^2=0$, then what is the value of the following? $$\left|\begin{array}{ccc}1& \cos C& \cos B\\ \cos C& 1& \cos A\\ \cos B& \cos A& 1\end{array}\right|$$

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

If $A=\left[\begin{array}{ccc}y& z& x\\ z& x& y\\ x& y& z\end{array}\right]$ where $x,y,z$ are integers, is an orthogonal matrix, then what is the value of $x^2+y^2+z^2$?

1. A.0
2. B.1
3. C.4
4. D.14

Consider the following in respect of a non-singular matrix $M$: I. $|M^2|=|M{|}^2$ II. $|M|=|M^{-1}|$ III. $|M|=|M^T|$ How many of the above are correct?

1. A.None
2. B.One
3. C.Two
4. D.All three

1. A.$\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]$
2. B.$\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$
3. C.$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$
4. D.$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

1. A.$-5I_3$
2. B.$-I_3$
3. C.$I_3$
4. D.$5I_3$

If the number of selections of r as well as (n+r) things from 5n different things are equal, then what is the value of r?

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

What is the number of selections of at most 3 things from 6 different things?

1. A.20
2. B.22
3. C.41
4. D.42

1. A.Null matrix
2. B.Identity matrix
3. C.A
4. D.-A

Let X be a matrix of order 3 x 3, Y be a matrix of order 2 x 3 and Z be a matrix of order 3 × 2. Which of the following statements are correct? I. (ZY)X is defined and is a square matrix of order 3. II. Y(XZ) is defined and is a square matrix of order 2. III. X(YZ) is not defined. Select the answer using the code given below.

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

Let A and B be two square matrices of same order. If AB is a null matrix, then which one of the following is correct?

1. A.Both A and B are null matrices
2. B.Either A or B is a null matrix
3. C.B is a null matrix if A is a nonsingular matrix
4. D.Both A and B are singular matrices

Consider the following in respect of the matrices $P=\left[\begin{array}{ccc}-c& 0& a\\ b& -a& 0\end{array}\right]$ and $Q=\left[\begin{array}{ccc}ab& b^2& bc\\ ac& bc& c^2\end{array}\right]$ I. PQ is a null matrix. II. QP is an identity matrix of order 3. III. PQ = QP Which of the above is/are correct?

1. A.I only
2. B.II only
3. C.I and III
4. D.II and III

If P is a skew-symmetric matrix of order 3, then what is det(P) equal to?

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

Let $A$ and $B$ be matrices of order $3\times 3$. If $|A|=\frac{1}{2\sqrt{2}}$ and $|B|=\frac{1}{729}$, then what is the value of $|2B(\mathrm{adj}(3A))|$?

1. A.27
2. B.$\frac{-27}{2\sqrt{2}}$
3. C.$\frac{27}{2}$
4. D.1

Consider the following statements in respect of two non-singular matrices $A$ and $B$ of the same order $n$: 1. $1.adj(AB)=(adjA)(adjB)$ 2. $2.adj(AB)=adj(BA)$ 3. $3.(AB)adj(AB)-|AB|I_n$ is a null matrix of order $n$ How many of the above statements are correct?

1. A.None
2. B.Only one statement
3. C.Only two statements
4. D.All three statements

Consider the following statements in respect of a non-singular matrix $A$ of order $n$: 1. $A(\mathrm{adj}{A}^T)=A(\mathrm{adj}A{)}^T$ 2. If $A^2=A$, then $A$ is identity matrix of order $n$ 3. If $A^3=A$, then $A$ is identity matrix of order $n$ Which of the statements given above are correct?

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

How many four-digit natural numbers are there such that all of the digits are even?

1. A.625
2. B.500
3. C.400
4. D.256

Consider the following statements in respect of a skew-symmetric matrix $A$ of order 3: 1. All diagonal elements are zero. 2. The sum of all the diagonal elements of the matrix is zero. 3. $A$ is orthogonal matrix. Which of the statements given above are correct?

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

Four digit numbers are formed by using the digits 1, 2, 3, 5 without repetition of digits. How many of them are divisible by 4?

1. A.120
2. B.24
3. C.12
4. D.6

What is the remainder when $2^{120}$ is divided by 7?

1. A.1
2. B.3
3. C.5
4. D.6

For what value of $n$ is the determinant $$\left|\begin{array}{ccc}C(9,4)& C(9,3)& C(10,n-2)\\ C(11,6)& C(11,5)& C(12,n)\\ C(m,7)& C(m,6)& C(m+1,n+1)\end{array}\right|=0$$ for every $m>n$ ?

1. A.4
2. B.5
3. C.6
4. D.7

If $ABC$ is a triangle, then what is the value of the determinant $$\left|\begin{array}{ccc}\cos C& \sin B& 0\\ \tan A& 0& \sin B\\ 0& \tan (B+C)& \cos C\end{array}\right|?$$

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

What is the number of different matrices, each having 4 entries that can be formed using 1, 2, 3, 4 (repetition is allowed)?

1. A.72
2. B.216
3. C.254
4. D.768

If $A=\left[\begin{array}{ccc}\cos 2\theta & 2\sin \theta \cos \theta & 0\\ 0& 0& 1\end{array}\right]$, then which of the following statements is/are correct? 1. $A^{-1}=\mathrm{adj}A$ 2. A is skew-symmetric matrix 3. $A^{-1}=A^T$ Select the correct answer using the code given below :

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