Matrices & Determinants · NDA previous year questions

If the matrix AB is a zero matrix, then which one of the following is correct?

1. A.A must be equal to zero matrix or B must be equal to zero matrix
2. B.A must be equal to zero matrix and B must be equal to zero matrix
3. C.It is not necessary that either A is zero matrix or B is zero matrix
4. D.None of the above

1. A.$\alpha =-5$
2. B.$\alpha =5$
3. C.$\alpha =0$
4. D.$\alpha =1$

1. A.0
2. B.$x^2+y^2+z^2$
3. C.$x^2+y^2+z^2+1$
4. D.None of the above

A square matrix $[a_{ij}]$ such that $a_{ij}=0$ for $i\ne j$ and $a_{ij}=k$ where k is a constant for $i=j$ is called:

1. A.diagonal matrix, but not scalar matrix
2. B.scalar matrix
3. C.unit matrix
4. D.None of the above

What is the value of $\sin 15^{\circ }$?

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

If $4{\sin }^2\theta =1$, where $0<\theta <2\pi$, how many values does $\theta$ take?

1. A.1
2. B.2
3. C.4
4. D.None of the above

What is the value of $\sin 18^{\circ }\cos 36^{\circ }$ equal to?

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

What is $\sin \left[{\sin }^{-1}\left(\frac{3}{5}\right)+{\sin }^{-1}\left(\frac{4}{5}\right)\right]$ equal to?

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

If $\sec \alpha =\frac{13}{5}$ where $270^{\circ }<\alpha <360^{\circ }$, then what is $\sin \alpha$ equal to?

1. A.5/13
2. B.12/13
3. C.$-12/13$
4. D.$-13/12$

What is $\tan (-585^{\circ })$ equal to?

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

Consider the following statements: 1. The value of $\cos 46^{\circ }-\sin 46^{\circ }$ is positive. 2. The value of $\cos 44^{\circ }-\sin 44^{\circ }$ is negative. Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

If $A=\left(\begin{array}{cc}1& 2\\ 2& 3\end{array}\right)$ and $B=\left(\begin{array}{cc}1& 0\\ 1& 0\end{array}\right)$ then what is determinant of AB?

1. A.0
2. B.1
3. C.10
4. D.20

1. A.$4abc$
2. B.$4a^{2}bc$
3. C.$4a^2{b}^2{c}^2$
4. D.$-4a^2{b}^2{c}^2$

What is the distance between the lines $3x+4y=9$ and $6x+8y=18$?

1. A.0
2. B.3 units
3. C.9 units
4. D.18 units

A and B are two matrices such that $AB=A$ and $BA=B$ then what is $B^2$ equal to? where I is the identity matrix

1. A.B
2. B.A
3. C.I
4. D.$-1$

If a matrix A has inverses B and C, then which one of the following is correct?

1. A.B may not be equal to C
2. B.B should be equal to C.
3. C.B and C should be unit matrices
4. D.None of the above

Consider the following statements: 1. Every zero matrix is a square matrix. 2. A matrix has a numerical value. 3. A unit matrix is a diagonal matrix. Which of the above statements is/are correct?

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

If A is a square matrix such that $A^2=I$ where I is the identity matrix, then what is $A^{-1}$ equal to?

1. A.A + I
2. B.Null matrix
3. C.A
4. D.Transpose of A

If two rows of a determinant are identical, then what is the value of the determinant?

1. A.0
2. B.1
3. C.-1
4. D.can be any real value

If A = {0, 1} and B = {1, 0}, then what is A × B equal to?

1. A.{(0, 1), (1, 0)}
2. B.{(0, 0), (1, 1)}
3. C.{(0, 1), (1, 0), (1, 1)}
4. D.A × A

The sum and product of matrices A and B exist. Which of the following implications are necessarily true? 1. A and B are square matrices of same order. 2. A and B are non-singular matrices. Select the correct answer using the code given below :

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

1. A.$a^3+b^3+c^3$
2. B.$3bc$
3. C.$a^3+b^3+c^3-3abc$
4. D.0

Consider the following statements in respect of the square matrices A and B of same order: 1. A and B are non-zero and $AB=0\Rightarrow$ either $|A|=0$ or $|B|=0$ 2. $AB=0\Rightarrow A=0$ or $B=0$ Which of the above statements is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

For what value of k, are the lines $x+2y-9=0$ and $kx+4y+5=0$ parallel?

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

If the matrix $A=\left[\begin{array}{cc}\alpha & \beta \\ \beta & \alpha \end{array}\right]$ is such that $A^2=I$, then which one of the following is correct?

1. A.$\alpha =0,\beta =1$ or $\alpha =1,\beta =0$
2. B.$\alpha =0,\beta \ne 1$ or $\alpha \ne 0,\beta =1$
3. C.$\alpha =1,\beta \ne 0$ or $\alpha \ne 1,\beta =0$
4. D.$\alpha \ne 0,\beta \ne 0$