Let R= ( lll a & 3 & ~b c & 2 & ~d 0 & 5 & 0 ): a, b, c, d 0,3,5,… - Vidyadip

MCQ

Let $R=\left\{\left(\begin{array}{lll}\mathrm{a} & 3 & \mathrm{~b} \\ \mathrm{c} & 2 & \mathrm{~d} \\ 0 & 5 & 0\end{array}\right): \mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d} \in\{0,3,5,7,11,13,17,19\}\right\}$. Then the number of invertible matrices in $\mathrm{R}$ is

A
$500$
✓
$3780$
C
$515$
D
$520$

✓

Answer

Correct option: B.

$3780$

b
Let us calculate when $|R|=0$

Case-I $a d=b c=0$

Now $\mathrm{ad}=0$

$\Rightarrow$ Total - (When none of a & $d$ is 0 )

$=8^2-1=15$ ways

Similarly bc $=0 \Rightarrow 15$ ways

$\therefore 15 \times 15=225$ ways of $a d=b c=0$

Case-II $a d=b c \neq 0$

either $a=d=b=c \quad$ OR $\quad a \neq d, b \neq d$ but $a d=b c$

${ }^7 \mathrm{C}_1=7$ ways

${ }^7 \mathrm{C}_2 \times 2 \times 2=84$ ways

Total 91 ways

$\therefore|\mathbb{R}|=0 \text { in } 225+91=316 \text { ways }$

$|\mathbb{R}| \neq 0 \text { in } 8^4-316=3780$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\tan \theta = - \frac{1}{{\sqrt {10} }}$ and $\theta $ lies in the fourth quadrant, then $\cos \theta = $

Let $x _{1}, x _{2}, x _{3}, \ldots ., x _{20}$ be in geometric progression with $x_{1}=3$ and the common ration $\frac{1}{2}$. A new data is constructed replacing each $x_{i}$ by $\left(x_{i}-i\right)^{2}$. If $\bar{x}$ is the mean of new data, then the greatest integer less than or equal to $\bar{x}$ is $.....$

If $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,then

Let $f:[0, \infty) \rightarrow[0,3]$ be a function defined by

$f(x)=\max \{\sin t: 0 \leq t \leq x\}, \quad 0 \leq x \leq \pi$

$\quad \quad \quad \quad \quad \quad 2+\cos x,\quad \quad \quad \quad x>\pi$

Then which of the following is true?

A closed conical vessel is filled with water fully and is placed with its vertex down. The water is let out at a constant speed. After $21\,min$, it was found that the height of the water column is half of the original height. How much more time in minutes does it require to empty the vessel?

If $0 \le x < 2\pi $ , then the number of real values of $x,$ which satisfy the equation $\cos x + \cos 2x + \cos 3x + \cos 4x = 0$ is . . .

The mean of a data set consisting of $20$ observations is $40$. If one observation $53$ was wrongly recorded as $33$, then the correct mean will be

If the sum of the two roots of the equation $4{x^3} + 16{x^2} - 9x - 36 = 0$ is zero, then the roots are

The $A.M.$ of a $50$ set of numbers is $38$. If two numbers of the set, namely $55$ and $45$ are discarded, the $A.M.$ of the remaining set of numbers is

$\int_0^{1/\sqrt 2 } {\frac{{{{\sin }^{ - 1}}x}}{{{{(1 - {x^2})}^{3/2}}}}dx = } $