Matrices & Determinants · NDA previous year questions

If $A$ is a square matrix, then what is $\mathrm{adj}(A^{-1})-(\mathrm{adj}A{)}^{-1}$ equal to?

1. A.$2|A|$
2. B.Null matrix
3. C.Unit matrix
4. D.None of the above

The matrix $A=\left[\begin{array}{ccc}1& 3& 2\\ 1& x-1& 1\\ 2& 7& x-3\end{array}\right]$ will have inverse for every real number $x$ \textit{except} for

1. A.$x=\frac{11\pm \sqrt{5}}{2}$
2. B.$x=\frac{9\pm \sqrt{5}}{2}$
3. C.$x=\frac{11\pm \sqrt{3}}{2}$
4. D.$x=\frac{9\pm \sqrt{3}}{2}$

If $A=\left[\begin{array}{ccc}1& 0& -2\\ 2& -3& 4\end{array}\right]$, then the matrix $X$ for which $2X+3A=0$ holds true is

1. A.$\left[\begin{array}{ccc}-\frac{3}{2}& 0& -3\\ -3& -\frac{9}{2}& -6\end{array}\right]$
2. B.$\left[\begin{array}{ccc}\frac{3}{2}& 0& -3\\ 3& -\frac{9}{2}& -6\end{array}\right]$
3. C.$\left[\begin{array}{ccc}\frac{3}{2}& 0& 3\\ 3& \frac{9}{2}& -6\end{array}\right]$
4. D.$\left[\begin{array}{ccc}-\frac{3}{2}& 0& 3\\ -3& \frac{9}{2}& -6\end{array}\right]$

If $A=\left[\begin{array}{ccc}1& 1& -1\\ 2& -3& 4\\ 3& -2& 3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-1& -2& -1\\ 6& 12& 6\\ 5& 10& 5\end{array}\right]$, then which of the following is/are correct? 1. $A$ and $B$ commute. 2. $AB$ is a null matrix. 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

If $A$ is an orthogonal matrix of order 3 and $B=\left[\begin{array}{ccc}1& 2& 3\\ -3& 0& 2\\ 2& 5& 0\end{array}\right]$, then which of the following is/are correct? 1. $|AB|=\pm 47$ 2. $AB=BA$ 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

If $A$ is an invertible matrix of order $n$ and $k$ is any positive real number, then the value of $[\det (kA){]}^{-1}\det A$ is

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

If the value of the determinant $\left|\begin{array}{ccc}a& 1& 1\\ 1& b& 1\\ 1& 1& c\end{array}\right|$ is positive, where $a\ne b\ne c$, then the value of $abc$

1. A.cannot be less than 1
2. B.is greater than $-8$
3. C.is less than $-8$
4. D.must be greater than 8

Consider the following statements in respect of the determinant $\left|\begin{array}{cc}{\cos }^2\frac{\alpha }{2}& {\sin }^2\frac{\alpha }{2}\\ {\sin }^2\frac{\beta }{2}& {\cos }^2\frac{\beta }{2}\end{array}\right|$ where $\alpha ,\beta$ are complementary angles 1. The value of the determinant is $\frac{1}{\sqrt{2}}\cos \left(\frac{\alpha -\beta }{2}\right)$. 2. The maximum value of the determinant is $\frac{1}{\sqrt{2}}$. 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 $a,b,c$ are real numbers, then the value of the determinant $\left|\begin{array}{ccc}1-a& a-b-c& b+c\\ 1-b& b-c-a& c+a\\ 1-c& c-a-b& a+b\end{array}\right|$ is

1. A.0
2. B.$(a-b)(b-c)(c-a)$
3. C.$(a+b+c{)}^2$
4. D.$(a+b+c{)}^3$

The value of ${\sin }^2{5}^{\circ }+{\sin }^{2}10^{\circ }+{\sin }^{2}15^{\circ }+{\sin }^{2}20^{\circ }+\cdots +{\sin }^{2}90^{\circ }$ is

1. A.7
2. B.8
3. C.9
4. D.$\frac{19}{2}$

On simplifying $\frac{{\sin }^{3}A+\sin 3A}{\sin A}+\frac{{\cos }^{3}A-\cos 3A}{\cos A}$, we get

1. A.$\sin 3A$
2. B.$\cos 3A$
3. C.$\sin A+\cos A$
4. D.3

If $\sin x+\sin y=a$ and $\cos x+\cos y=b$, then ${\tan }^2\left(\frac{x+y}{2}\right)+{\tan }^2\left(\frac{x-y}{2}\right)$ is equal to

1. A.$\frac{a^4+b^4+4b^2}{a^2{b}^2+b^4}$
2. B.$\frac{a^4-b^4+4b^2}{a^2{b}^2+b^4}$
3. C.$\frac{a^4-b^4+4a^2}{a^2{b}^2+a^4}$
4. D.None of the above

If $p=\tan \left(-\frac{11\pi }{6}\right)$, $q=\tan \left(\frac{21\pi }{4}\right)$ and $r=\cot \left(\frac{283\pi }{6}\right)$, then which of the following is/are correct? 1. The value of $p\times r$ is 2. 2. $p,q$ and $r$ are in G.P. 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

The value of $\tan \left(2{\tan }^{-1}\frac{1}{5}-\frac{\pi }{4}\right)$ is

1. A.$-\frac{7}{17}$
2. B.$\frac{5}{16}$
3. C.$\frac{5}{4}$
4. D.$\frac{7}{17}$

Consider the following: 1. ${\sin }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{3}{5}=\frac{\pi }{2}$ 2. ${\tan }^{-1}\sqrt{3}+{\tan }^{-1}1=-{\tan }^{-1}\left(2+\sqrt{3}\right)$ 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

Two poles are 10 m and 20 m high. The line joining their tops makes an angle of $15^{\circ }$ with the horizontal. The distance between the poles is approximately equal to

1. A.36.3 m
2. B.37.3 m
3. C.38.3 m
4. D.39.3 m

A vertical tower standing on a levelled field is mounted with a vertical flag staff of length 3 m. From a point on the field, the angles of elevation of the bottom and tip of the flag staff are $30^{\circ }$ and $45^{\circ }$ respectively. Which one of the following gives the best approximation to the height of the tower?

1. A.3.90 m
2. B.4.00 m
3. C.4.10 m
4. D.4.25 m

For the next two (02) items that follow : Consider a triangle $ABC$ satisfying $2a{\sin }^2\left(\frac{C}{2}\right)+2c{\sin }^2\left(\frac{A}{2}\right)=2a+2c-3b$ The sides of the triangle are in

1. A.G.P.
2. B.A.P.
3. C.H.P.
4. D.Neither in G.P. nor in A.P. nor in H.P.

Consider a triangle $ABC$ satisfying $2a{\sin }^2\left(\frac{C}{2}\right)+2c{\sin }^2\left(\frac{A}{2}\right)=2a+2c-3b$ $\sin A$, $\sin B$, $\sin C$ are in

1. A.G.P.
2. B.A.P.
3. C.H.P.
4. D.Neither in G.P. nor in A.P. nor in H.P.

The area of the figure formed by the lines $ax+by+c=0$, $ax-by+c=0$, $ax+by-c=0$ and $ax-by-c=0$ is

1. A.$\frac{c^2}{ab}$
2. B.$\frac{2c^2}{ab}$
3. C.$\frac{c^2}{2ab}$
4. D.$\frac{c^2}{4ab}$

If a line is perpendicular to the line $5x-y=0$ and forms a triangle of area 5 square units with coordinate axes, then its equation is

1. A.$x+5y\pm 5\sqrt{2}=0$
2. B.$x-5y\pm 5\sqrt{2}=0$
3. C.$5x+y\pm 5\sqrt{2}=0$
4. D.$5x-y\pm 5\sqrt{2}=0$

The three lines $4x+4y=1$, $8x-3y=2$, $y=0$ are

1. A.the sides of an isosceles triangle
2. B.concurrent
3. C.mutually perpendicular
4. D.the sides of an equilateral triangle

Which one of the following matrices is an elementary matrix?

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

If $$A=\left[\begin{array}{cc}2& 7\\ 1& 5\end{array}\right]$$ then what is $A+3A^{-1}$ equal to? where $I$ is the identity matrix of order 2.

1. A.$3I$
2. B.$5I$
3. C.$7I$
4. D.None of the above

The matrix $$\left[\begin{array}{cc}0& -4+i\\ 4+i& 0\end{array}\right]$$ is

1. A.symmetric
2. B.skew-symmetric
3. C.Hermitian
4. D.skew-Hermitian