Matrices & Determinants · NDA previous year questions

What is the value of the determinant $$\left|\begin{array}{ccc}1& 1& 1\\ 1& 1+xyz& 1\\ 1& 1& 1+xyz\end{array}\right|$$ ?

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

1. A.$\frac{x}{y}$ is one of the cube roots of unity
2. B.$x$ is one of the cube roots of unity
3. C.$y$ is one of the cube roots of unity
4. D.$\frac{x}{y}$ is one of the cube roots of $-1$

If $\sin A=\frac{3}{5}$, where $450^{\circ }<A<540^{\circ }$, then $\cos \frac{A}{2}$ is equal to

1. A.$\frac{1}{\sqrt{10}}$
2. B.$-\sqrt{\frac{3}{10}}$
3. C.$\frac{\sqrt{3}}{\sqrt{10}}$
4. D.None of the above

What is $\frac{1}{\sin 10^{\circ }}-\frac{\sqrt{3}}{\cos 10^{\circ }}$ equal to?

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

If $A=\left[\begin{array}{cc}1& -1\\ 2& 3\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 3\\ -1& -2\end{array}\right]$, then which of the following is/are correct ? 1. $AB(A^{-1}{B}^{-1})$ is a unit matrix. 2. $(AB{)}^{-1}=A^{-1}{B}^{-1}$. 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

Consider the following for the next two (02) items that follow : For the system of linear equations $2x+3y+5z=9$, $7x+3y-2z=8$ and $2x+3y+\lambda z=\mu$ Under what condition does the above system of equations have infinitely many solutions ?

1. A.$\lambda =5$ and $\mu \ne 9$
2. B.$\lambda =5$ and $\mu =9$
3. C.$\lambda =9$ and $\mu =5$
4. D.$\lambda =9$ and $\mu \ne 5$

Consider the following for the next two (02) items that follow : For the system of linear equations $2x+3y+5z=9$, $7x+3y-2z=8$ and $2x+3y+\lambda z=\mu$ Under what condition does the above system of equations have unique solutions ?

1. A.$\lambda =5$ and $\mu =9$
2. B.$\lambda \ne 5$ and $\mu =7$ only
3. C.$\lambda \ne 5$ and $\mu$ has any real value
4. D.$\lambda$ has any real value and $\mu \ne 9$

If A is a square matrix of order 3 and $\det A=5$, then what is $\det [(2A{)}^{-1}]$ equal to ?

1. A.1/10
2. B.2/5
3. C.8/5
4. D.1/40

Consider the following for the next two (02) items that follow : Let $a{x}^3+b{x}^2+cx+d=\left|\begin{array}{ccc}x+1& 2x& 3x\\ 2x+3& x+1& x\\ 2-x& 3x+4& 5x-1\end{array}\right|$, then What is the value of c ?

1. A.$-1$
2. B.34
3. C.35
4. D.50

Consider the following for the next two (02) items that follow : Let $a{x}^3+b{x}^2+cx+d=\left|\begin{array}{ccc}x+1& 2x& 3x\\ 2x+3& x+1& x\\ 2-x& 3x+4& 5x-1\end{array}\right|$, then What is the value of a + b + c + d ?

1. A.62
2. B.63
3. C.65
4. D.68

If $m=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $n=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, then what is the value of the determinant of $m\cos \theta -n\sin \theta$ ?

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

If $f(x)=\left[\begin{array}{ccc}\sin x& \cos x& 0\\ 0& 0& 1\end{array}\right]$, then which of the following are correct ? 1. $f(\theta )\times f(\varphi )=f(\theta +\varphi )$. 2. The value of the determinant of the matrix $f(\theta )\times f(\varphi )$ is 1. 3. The determinant of $f(x)$ is an even function. 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

Which of the following are correct in respect of the system of equations $x+y+z=8$, $x-y+2z=6$ and $3x-y+5z=k$ ? 1. They have no solution, if $k=15$. 2. They have infinitely many solutions, if $k=20$. 3. They have unique solution, if $k=25$. 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

$\sin A+2\sin 2A+\sin 3A$ is equal to which of the following? 1. $4\sin 2A{\cos }^2\left(\frac{A}{2}\right)$ 2. $2\sin 2A{\left(\sin \frac{A}{2}+\cos \frac{A}{2}\right)}^2$ 3. $8\sin A\cos A{\cos }^2\left(\frac{A}{2}\right)$ 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

If $x=\sin 70^{\circ }\cdot \sin 50^{\circ }$ and $y=\cos 60^{\circ }\cdot \cos 80^{\circ }$, then what is xy equal to?

1. A.1/16
2. B.1/8
3. C.1/4
4. D.1/2

If $\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3+\sin {\theta }_4=4$, then what is the value of $\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3+\cos {\theta }_4$?

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

What is the value of $(1+\cos \frac{\pi }{8})(1+\cos \frac{3\pi }{8})(1+\cos \frac{5\pi }{8})(1+\cos \frac{7\pi }{8})$?

1. A.$\frac{1}{2}$
2. B.$\frac{1}{2}+\frac{1}{2\sqrt{2}}$
3. C.$\frac{1}{2}-\frac{1}{2\sqrt{2}}$
4. D.$\frac{1}{8}$

If $x\cos \theta +y\sin \theta =z$, then what is the value of $(x\sin \theta -y\cos \theta {)}^2$?

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

If $\sin 18^{\circ }=\frac{\sqrt{5}-1}{4}$, then what is the value of $\sin 81^{\circ }$?

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

What is $\frac{1-\tan 2^{\circ }\cot 62^{\circ }}{\tan 152^{\circ }-\cot 88^{\circ }}$ equal to?

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

What is the value of $\cos (2{\cos }^{-1}0.8)$?

1. A.0.81
2. B.0.56
3. C.0.48
4. D.0.28

The top of a hill when observed from the top and bottom of a building of height h is at angles of elevation p and q respectively. What is the height of the hill?

1. A.$\frac{h\cot q}{\cot q-\cot p}$
2. B.$\frac{h\cot p}{\cot p-\cot q}$
3. C.$\frac{2h\tan p}{\tan p-\tan q}$
4. D.$\frac{2h\tan q}{\tan q-\tan p}$

Consider the following in respect of the matrix $A=\left(\begin{array}{cc}-1& 1\\ 1& -1\end{array}\right)$ : 1. $A^2=-A$ 2. $A^3=4A$ 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

Which of the following determinants have value 'zero'? 1. $\left|\begin{array}{ccc}41& 1& 5\\ 79& 7& 9\\ 29& 5& 3\end{array}\right|$ 2. $\left|\begin{array}{ccc}1& a& b+c\\ 1& b& c+a\\ 1& c& a+b\end{array}\right|$ 3. $\left|\begin{array}{ccc}0& c& b\\ -c& 0& a\\ -b& -a& 0\end{array}\right|$ 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

The system of linear equations $kx+y+z=1,x+ky+z=1$ and $x+y+kz=1$ has a unique solution under which one of the following conditions?

1. A.$k\ne 1$ and $k\ne -2$
2. B.$k\ne 1$ and $k\ne 2$
3. C.$k\ne -1$ and $k\ne -2$
4. D.$k\ne -1$ and $k\ne 2$