Matrices & Determinants · NDA previous year questions

Consider the following statements : 1. The matrix is singular. 2. The matrix is non-singular. 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

The cofactor of the element 4 in the determinant $$\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 5& 8& 9\end{array}\right|$$ is

1. A.2
2. B.4
3. C.6
4. D.$-6$

If $A$ is a square matrix of order 3 with $|A|\ne 0$, then which one of the following is correct?

1. A.$|\mathrm{adj}A|=|A|$
2. B.$|\mathrm{adj}A|=|A{|}^2$
3. C.$|\mathrm{adj}A|=|A{|}^3$
4. D.$|\mathrm{adj}A{|}^2=|A|$

Let $A=\{a,b,c,d\}$ and $B=\{x,y,z\}$. What is the number of elements in $A\times B$?

1. A.6
2. B.7
3. C.12
4. D.64

If $C(28,2r)=C(28,2r-4)$, then what is $r$ equal to?

1. A.7
2. B.8
3. C.12
4. D.16

If $A=\left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right),B=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right),C=\left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)$ where $i=\sqrt{-1}$, then which one of the following is correct?

1. A.$AB=-C$
2. B.$AB=C$
3. C.$A^2=B^2=C^2=I$, where $I$ is the identity matrix
4. D.$BA\ne C$

What is the value of the determinant $$\left|\begin{array}{ccc}1& bc& a(b+c)\\ 1& ca& b(c+a)\\ 1& ab& c(a+b)\end{array}\right|$$ ?

1. A.0
2. B.$abc$
3. C.$ab+bc+ca$
4. D.$abc(a+b+c)$

Consider the following : 1. $A\cup (B\cap C)=(A\cap B)\cup (A\cap C)$ 2. $A\cap (B\cup C)=(A\cup B)\cap (A\cup C)$ Which of the above is/are correct?

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

Let $n$ be a positive integer and $(1+x{)}^n=a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n$ What is $a_0+a_1+a_2+\cdots +a_n$ equal to?

1. A.1
2. B.$2^n$
3. C.$2^{n-1}$
4. D.$2^{n+1}$

If A and B are two non-singular square matrices such that $AB=A$, then which one of the following is correct ?

1. A.B is an identity matrix
2. B.$B=A^{-1}$
3. C.$B=A^2$
4. D.Determinant of B is zero

What is the value of the minor of the element 9 in the determinant $\left|\begin{array}{ccc}10& 19& 2\\ 0& 13& 1\\ 9& 24& 2\end{array}\right|$ ?

1. A.$-9$
2. B.$-7$
3. C.7
4. D.0

The value of the determinant $\left|\begin{array}{ccc}p& m& n\\ m& n& p\\ n& p& m\end{array}\right|$

1. A.is a perfect cube
2. B.is a perfect square
3. C.has linear factor
4. D.is zero

The determinant of an orthogonal matrix is

1. A.$\pm 1$
2. B.2
3. C.0
4. D.$\pm 2$

If D is a determinant of order 3 and $D^{\prime }$ is the determinant obtained by replacing the elements of D by their cofactors, then which one of the following is correct ?

1. A.$D^{\prime }=D^2$
2. B.$D^{\prime }=D^3$
3. C.$D^{\prime }=3D^2$
4. D.$D^{\prime }=3D^3$

Consider the following statements : 1. A matrix is not a number. 2. Two determinants of different orders may have the same value. 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

What is the value of $\tan 105^{\circ }$ ?

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

If $\tan A=x+1$ and $\tan B=x-1$, then $x^2\tan (A-B)$ has the value

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

What is the value of $({\sin }^4\theta -{\cos }^4\theta +1){\csc }^2\theta$?

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

If d is the number of degrees contained in an angle, m is the number of minutes and s is the number of seconds, then the value of $(s-m)/(m-d)$ is

1. A.1
2. B.60
3. C.1/60
4. D.None of the above

The expression $\frac{\cot x+\csc x-1}{\cot x-\csc x+1}$ is equal to

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

What is $\frac{1-{\tan }^2\frac{x}{2}}{1+{\tan }^2\frac{x}{2}}$ equal to?

1. A.$\sin x\cdot \cos x$
2. B.$\tan x$
3. C.$\sin x$
4. D.$\cos x$

If each element in a row of a determinant is multiplied by the same factor r, then the value of the determinant:

1. A.is multiplied by $r^3$
2. B.is increased by $3r$
3. C.remains unchanged
4. D.is multiplied by r

The inverse of a diagonal matrix is a:

1. A.symmetric matrix
2. B.skew-symmetric matrix
3. C.diagonal matrix
4. D.None of the above

1. A.B is the inverse of A
2. B.B is the adjoint of A
3. C.B is the transpose of A
4. D.None of the above

1. A.$-5$
2. B.0
3. C.5
4. D.10