Mathematics - Online Test

In the word 'MATHAEMATICS'  there are   7 consonants which are M-2 ,,H-1,C-1,S-1, T-2 .Since they have to occur together we treat them as a single unit.

Now this single unit together with the remaining 4 vowels which are A-2,E-1,and I-1  will account for 5 letters.

Now in these 5 letters we have A is repeating twice  these can be aarranged in 5!2!  different ways.

Corresponding to each of these arrangements the consonents can be arranged in 7!2!.2! different ways.

Hence the number of ways it can be arranged is 7!5!2!2!2! =75600

As we know that if a lines makes angles a , b and c with X-axis , Y-axis and Z-axis respectively then direction cosines are given by  < cos a , cos b ,cos c >

In our case line is X-axis itself which we know makes angle of 0∘${}^{\circ }$ , 90∘${}^{\circ }$ , 90∘${}^{\circ }$ with  X-axis , Y-axis and Z-axis respectively then direction cosine will be

<cos 0∘${}^{\circ }$ , cos 90∘${}^{\circ }$, cos 90∘${}^{\circ }$>

= < 1 0 0 > 

23x3=29=512.

The number of elements in a 3 X 3 matrix is the product 3 X 3=9.

Each element can either be a 0 or a 1.

Given this, the total possible matrices that can be selected is 29=512