Determinants of order two and three — practice questions

Question 1 · difficulty L2 · understanding

For $$\alpha, \beta \in \mathbb{R}$$ and a natural number $$n$$, let $$A_r=\left|\begin{array}{ccc}r & 1 & \frac{n^2}{2}+\alpha \\ 2 r & 2 & n^2-\beta \\ 3 r-2 & 3 & \frac{n(3 n-1)}{2}\end{array}\right|$$. Then $$2 A_{10}-A_8$$ is

• A. $$4 \alpha+2 \beta$$
• B. 0
• C. $$2 n$$
• D. $$2 \alpha+4 \beta$$

Question 2 · difficulty L2 · understanding

Let \begin{aligned}A = \left( {\matrix{ {[x + 1]} & {[x + 2]} & {[x + 3]} \cr {[x]} & {[x + 3]} & {[x + 3]} \cr {[x]} & {[x + 2]} & {[x + 4]} \cr } } \right)\end{aligned}, where [t] denotes the greatest integer less than or equal to t. If det(A) = 192, then the set of values of x is the interval :

• A. [68, 69)
• B. [62, 63)
• C. [65, 66)
• D. [60, 61)

Question 3 · difficulty L2 · understanding

The value of \begin{aligned}\left| {\matrix{ {(a + 1)(a + 2)} & {a + 2} & 1 \cr {(a + 2)(a + 3)} & {a + 3} & 1 \cr {(a + 3)(a + 4)} & {a + 4} & 1 \cr } } \right|\end{aligned} is :

• A. $$-$$2
• B. 0
• C. (a + 2)(a + 3)(a + 4)
• D. (a + 1)(a + 2)(a + 3)

Question 4 · difficulty L3 · understanding

Let $A=\left[$$\begin{array}{ccc}1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7\end{array}$$\right]$and$\operatorname{det}(A-\alpha I)=0$, where$\alpha$is a real number. If the largest possible value of$\alpha$is$p$, then the circle$(x-p)^2+(y-2 p)^2=320$, intersects the co-ordinate axes at

• A. 1 point
• B. 2 points
• C. 3 points
• D. 4 points

Question 5 · difficulty L3 · understanding

Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:

• A. $\frac{5}{8}$
• B. $\frac{1}{4}$
• C. $\frac{3}{4}$
• D. $\frac{3}{8}$

Question 6 · difficulty L3 · understanding

Let $ A = $$\begin{bmatrix} a_{ij} \end{bmatrix}$$ = $$\begin{bmatrix} \log_5 128 & \log_4 5 \\ \log_5 8 & \log_4 25 \end{bmatrix}$$ $. If$ A_{ij} $is the cofactor of$ a_{ij} $,$ C_{ij} = \sum\limits_{k=1}^{2} a_{ik} A_{jk} , 1 \leq i, j \leq 2 $, and$ C=[C_{ij}] $, then$ 8|C| $ is equal to :

• A. 288
• B. 262
• C. 222
• D. 242

Question 7 · difficulty L3 · understanding

Let M and m respectively be the maximum and the minimum values of $f(x)=\left|$$\begin{array}{ccc}1+\sin ^2 x & \cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & 1+\cos ^2 x & 4 \sin 4 x \\ \sin ^2 x & \cos ^2 x & 1+4 \sin 4 x\end{array}$$\right|, x \in R$Then$ M^4 - m^4 $ is equal to :

• A. 1280
• B. 1040
• C. 1215
• D. 1295

Question 8 · difficulty L3 · understanding

For some $a, b,$ let $f(x)=\left|$$\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \\ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}$$\right|, x \neq 0, \lim \limits_{x \rightarrow 0} f(x)=\lambda+\mu \mathrm{a}+\nu \mathrm{b}.$Then$(\lambda+\mu+v)^2$ is equal to :

• A. 25
• B. 16
• C. 9
• D. 36

Question 9 · difficulty L3 · understanding

$$\text { Let } A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }$$

• A. 9
• B. 17
• C. 3
• D. 5

Question 10 · difficulty L3 · understanding

The values of $$\alpha$$, for which $$\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0$$, lie in the interval

• A. $$(-2,1)$$
• B. $$\left(-\frac{3}{2}, \frac{3}{2}\right)$$
• C. $$(-3,0)$$
• D. $$(0,3)$$