Matrices & Determinants · NDA previous year questions

Consider the following for the next 02 (two) items: It is given that $\cos (\theta -\alpha )=a,\cos (\theta -\beta )=b$. What is ${\sin }^2(\alpha -\beta )+2ab\cos (\alpha -\beta )$ equal to?

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

If $\sin \alpha +\cos \alpha =p$, then what is ${\cos }^2(2\alpha )$ equal to?

1. A.$p^2$
2. B.$p^2-1$
3. C.$p^2(2-p^2)$
4. D.$p^2+1$

If $\tan \theta =\frac{1}{2}$ and $\tan \phi =\frac{1}{3}$, then what is the value of $(\theta +\phi )$?

1. A.0
2. B.$\frac{\pi }{6}$
3. C.$\frac{\pi }{4}$
4. D.$\frac{\pi }{2}$

If $\cos A=\frac{3}{4}$, then what is the value of $\sin \left(\frac{A}{2}\right)\sin \left(\frac{3A}{2}\right)$?

1. A.$\frac{5}{8}$
2. B.$\frac{5}{16}$
3. C.$\frac{5}{24}$
4. D.$\frac{7}{32}$

What is the value of $\tan 75^{\circ }+\cot 75^{\circ }$?

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

What is the value of $\cos 46^{\circ }\cos 47^{\circ }\cos 48^{\circ }\cos 49^{\circ }\cos 50^{\circ }\dots$ $\cos 135^{\circ }$?

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

If $\sin 2\theta =\cos 3\theta$, where $0<\theta <\frac{\pi }{2}$, then what is $\sin \theta$ equal to?

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

If A is a square matrix of order $n>1$, then which one of the following is correct?

1. A.$\det (-A)=\det A$
2. B.$\det (-A)=(-1{)}^n\det A$
3. C.$\det (-A)=-\det A$
4. D.$\det (-A)=n\det A$

Consider the following for the next 02 (two) items: Let A and B be $(3\times 3)$ matrices with $\det A=4$ and $\det B=3$. What is $\det (2AB)$ equal to?

1. A.96
2. B.72
3. C.48
4. D.36

Consider the following for the next 02 (two) items: Let A and B be $(3\times 3)$ matrices with $\det A=4$ and $\det B=3$. What is $\det (3A{B}^{-1})$ equal to?

1. A.12
2. B.18
3. C.36
4. D.48

A square matrix A is called orthogonal if

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

If A and B are two invertible square matrices of same order, then what is $(AB{)}^{-1}$ equal to?

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

Consider the following in respect of matrices A, B and C of same order: 1. $(A+B+C{)}^{\prime }=A^{\prime }+B^{\prime }+C^{\prime }$ 2. $(AB{)}^{\prime }=A^{\prime }{B}^{\prime }$ 3. $(ABC{)}^{\prime }=C^{\prime }{B}^{\prime }{A}^{\prime }$ where $A^{\prime }$ is the transpose of the matrix A. Which of the 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

Consider the following in respect of matrices A and B of same order: 1. $A^2-B^2=(A+B)(A-B)$ 2. $(A-I)(I+A)=O\iff A^2=I$ where I is the identity matrix and O is the null matrix. 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

What is the adjoint of the matrix $\left(\begin{array}{cc}\cos (-\theta )& -\sin (-\theta )\\ -\sin (-\theta )& \cos (-\theta )\end{array}\right)$?

1. A.$\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)$
2. B.$\left(\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$
3. C.$\left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right)$
4. D.$\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$

What should be the value of $x$ so that the matrix $\left(\begin{array}{cc}2& 4\\ -8& x\end{array}\right)$ does not have an inverse?

1. A.16
2. B.$-16$
3. C.8
4. D.$-8$

For a square matrix A, which of the following properties hold? 1. $(A^{-1}{)}^{-1}=A$ 2. $\det (A^{-1})=\frac{1}{\det A}$ 3. $(\lambda A{)}^{-1}=\lambda A^{-1}$ where $\lambda$ is a scalar Select the correct answer using the code given below:

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

1. A.$x-3$
2. B.$x-y$
3. C.$y-3$
4. D.$x-3y$

If $a+b+c=0$, then one of the solutions of $\left|\begin{array}{ccc}a-x& c& b\\ c& b-x& a\\ b& a& c-x\end{array}\right|=0$ is

1. A.$x=a$
2. B.$x=\sqrt{\frac{3(a^2+b^2+c^2)}{2}}$
3. C.$x=\sqrt{\frac{2(a^2+b^2+c^2)}{3}}$
4. D.$x=0$

If $u,v$ and $w$ (all positive) are the $p^{th},q^{th}$ and $r^{th}$ terms of a GP, then the determinant of the matrix $\left(\begin{array}{ccc}\ln u& p& 1\\ \ln v& q& 1\\ \ln w& r& 1\end{array}\right)$ is

1. A.0
2. B.1
3. C.$(p-q)(q-r)(r-p)$
4. D.$\ln u\times \ln v\times \ln w$

Let matrix B be the adjoint of a square matrix A, $l$ be the identity matrix of same order as A. If $k(\ne 0)$ is the determinant of the matrix A, then what is AB equal to?

1. A.$l$
2. B.$kl$
3. C.$k^{2}l$
4. D.$(1/k)l$

1. A.$(x-y)(y-z)(z-x)$
2. B.$(x-y)(y-z)$
3. C.$(y-z)(z-x)$
4. D.$(z-x{)}^2(x+y+z)$

If $A=\left(\begin{array}{cc}1& 2\\ 2& 3\end{array}\right)$ and $A^2-kA-I_2=O$, where $I_2$ is the $2\times 2$ identity matrix, then what is the value of $k$ ?

1. A.4
2. B.$-4$
3. C.8
4. D.$-8$

What is the inverse of the matrix ?

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

What is $\frac{\sin 5x-\sin 3x}{\cos 5x+\cos 3x}$ equal to ?

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