Matrices & Determinants · NDA previous year questions

If in a triangle $ABC$, ${\sin }^{3}A+{\sin }^{3}B+{\sin }^{3}C=3\sin A\sin B\sin C$, then what is the value of the determinant $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$, where $a,b,c$ are sides of the triangle?

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

If ${\cos }^{-1}x={\sin }^{-1}x$, then which one of the following is correct?

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

What is the number of solutions of $(\sin \theta -\cos \theta {)}^2=2$ where $-\pi <\theta <\pi$?

1. A.Only one
2. B.Only two
3. C.Four
4. D.No solution

$ABC$ is a triangle such that angle $C=60^{\circ }$, then what is $\frac{\cos A+\cos B}{\cos \left(\frac{A-B}{2}\right)}$ equal to?

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

What is $\sqrt{15+{\cot }^2\left(\frac{\pi }{4}-2{\cot }^{-1}3\right)}$ equal to?

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

What is the value of $\sin 10^{\circ }\cdot \sin 50^{\circ }+\sin 50^{\circ }\cdot \sin 250^{\circ }+\sin 250^{\circ }\cdot \sin 10^{\circ }$ equal to?

1. A.$-\frac{1}{4}$
2. B.$-\frac{3}{4}$
3. C.$3\sin 10^{\circ }/4$
4. D.$3\cos 10^{\circ }/4$

What is ${\tan }^{-1}\left(\frac{a}{b}\right)-{\tan }^{-1}\left(\frac{a-b}{a+b}\right)$ equal to?

1. A.$-\frac{\pi }{4}$
2. B.$\frac{\pi }{4}$
3. C.${\tan }^{-1}\left(\frac{a^2-b^2}{a^2+b^2}\right)$
4. D.${\tan }^{-1}\left(\frac{2ab}{a^2+b^2}\right)$

Under which one of the following conditions does the equation $(\cos \beta -1)x^2+(\cos \beta )x+\sin \beta =0$ in $x$ have a real root for $\beta \in [0,\pi ]$?

1. A.$1-\cos \beta <0$
2. B.$1-\cos \beta \le 0$
3. C.$1-\cos \beta >0$
4. D.$1-\cos \beta \ge 0$

In a triangle $ABC$, $AB=16\mathrm{cm}$, $BC=63\mathrm{cm}$ and $AC=65\mathrm{cm}$. What is the value of $\cos 2A+\cos 2B+\cos 2C$?

1. A.-1
2. B.0
3. C.1
4. D.$\frac{76}{65}$

If $f(\theta )=\frac{1}{1+\tan \theta }$ and $\alpha +\beta =\frac{5\pi }{4}$, then what is the value of $f(\alpha )f(\beta )$?

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

1. A.15
2. B.6
3. C.0
4. D.-1

If A, B and C are square matrices of order 3 and $\det (BC)=2\det (A)$, then what is the value of $\det (2A^{-1}BC)$?

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

If the $n^{\text{th}}$ term of a sequence is $\frac{2n+5}{7}$, then what is the sum of its first 140 terms?

1. A.2840
2. B.2780
3. C.2920
4. D.5700

Let A be a skew-symmetric matrix of order 3. What is the value of $\det (4A^4)-\det (3A^3)+\det (2A^2)-\det (A)+\det (-I)$ where I is the identity matrix of order 3?

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

1. A.$A^2$ is symmetric matrix with $\det (A^2)=0$.
2. B.$A^2$ is symmetric matrix with $\det (A^2)\ne 0$.
3. C.$A^2$ is skew-symmetric matrix with $\det (A^2)=0$.
4. D.$A^2$ is skew-symmetric matrix with $\det (A^2)\ne 0$.

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 all elements of a third order determinant are equal to 1 or -1, then the value of the determinant is:

1. A.0 only
2. B.an even number but not necessarily 0
3. C.an odd number
4. D.0, 1 or -1

1. A.5
2. B.25
3. C.125
4. D.625

1. A.Null matrix of order 3
2. B.Identity matrix of order 3
3. C.$\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 0\\ 0& 0& 2\end{array}\right]$
4. D.$\left[\begin{array}{ccc}7& 0& 0\\ 0& 7& 0\end{array}\right]$

The value of the determinant of a matrix A of order 3 is 3. If C is the matrix of cofactors of the matrix A, then what is the value of determinant of $C^2$?

1. A.3
2. B.9
3. C.81
4. D.729

If $A_k=\left[\begin{array}{cc}k-1& k\\ k-2& k+1\end{array}\right]$, then what is $\det (A_1)+\det (A_2)+\det (A_3)+\cdots +\det (A_{100})$ equal to ?

1. A.100
2. B.1000
3. C.9900
4. D.10000

Let A be a matrix of order $3\times 3$ and $|A|=4$. If $|2\mathrm{adj}(3A)|=2^{\alpha }{3}^{\beta }$, then what is the value of $(\alpha +\beta )$?

1. A.12
2. B.13
3. C.17
4. D.24

If $\alpha$ and $\beta$ are the distinct roots of equation $x^2-x+1=0$, then what is the value of $\left|\frac{{\alpha }^{100}+{\beta }^{100}}{{\alpha }^{100}-{\beta }^{100}}\right|$ ?

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

Let A and B be symmetric matrices of same order, then which one of the following is correct regarding (AB - BA) ? 1. Its diagonal entries are equal but nonzero 2. The sum of its non-diagonal entries is zero 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 statements in respect of square matrices A, B, C each of same order n : 1. AB = AC ⇒ B = C if A is non-singular 2. If BX = CX for every column matrix X having n rows then B = C Which of the statements given above is/are correct ?

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