x³ - 1 = x³ + 0x² + 0x - 1 

The coefficient form of the polynomial x³ - 1 is (1, 0, 0, −1).

It is true that every non zero number has a multiplicative inverse as an integer.

The multiplicative inverse of a fraction is called the reciprocal.

–16 ÷ [8 ÷ (–2)] 

= (-16) ÷ (-4)

 = 4

Vector Algebra class 12 Formulas:

A quantity which has both magnitudes, as well as direction, is said to be vector quantity. It is denoted by an arrow pointing direction and length of its tail as the magnitude. Thus, It's symbol is \(\overrightarrow{A}\) and its magnitude is given as |A|. Thus, read vector algebra formulas further.

Position Vector and Magnitude:

We write position vector of any point P(x, y, z) as \(\overrightarrow{OP}\) (=\(\overrightarrow{r}\) ) = x\(\hat{k}\) + y\(\hat{k}\) + z\(\hat{k}\).

Also, its magnitude as \(\sqrt{x^{2} + y^{2} + z^{2}}\).

Direction Ratio:

Scalar components of any vector are its direction ratios and represent its projections along the respective axes. Thus, direction ratio and direction ratios are considered in vector algebra formula.

Relation between Direction Ratio and Direction Cosine:

The magnitude (r), direction ratios (a, b, c) and direction cosines (l, m, n) of any vector are related as:
l = \(\frac{a}{r}\), m = \(\frac{b}{r}\), n = \(\frac{c}{r}\)

Triangle Formula:

The vector algebra formulas are absolute for triangles. The vector sum of the three sides of a triangle taken in order is \(\overrightarrow{0}\).

Dividing the line segment in given ratio:

1. Internally:

We write the position vector of a point R dividing a line segment in the ratio m : n joining the points P and Q whose position vectors are a and b respectively as:

\(\frac{n\overrightarrow{a} + m\overrightarrow{b}}{m + n}\)

2. Externally:

Similarly we can find position vector of R as

\(\frac{n\overrightarrow{a} - m\overrightarrow{b}}{m - n}\)

Scalar product:

For two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) also having angle θ between them, the scalar product is:

\(\overrightarrow{a}.\overrightarrow{b}\) = |\(\overrightarrow{a}\)||\(\overrightarrow{b}\)|cosθ

Thus, Also written as cosθ = \(\frac{\overrightarrow{a}.\overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}\)

Vector product:

The cross product of two vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) also having angle θ between them is:

\(\overrightarrow{a}\) x \(\overrightarrow{b}\) = |\(\overrightarrow{a}\)||\(\overrightarrow{b}\)| sinθ \(\hat{n}\)

Here, \(\hat{n}\) is a unit vector. Also, lies perpenducular the plane of \(\overrightarrow{a}\) and \(\overrightarrow{b}\). Then, we see that it is simple to denote these at same extent.

Properties of vector Algebra:

If we write vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) in its component form as

\(\overrightarrow{a}\) = a₁\(\hat{i}\) + a₂\(\hat{j}\) + a₃\(\hat{k}\)

\(\overrightarrow{b}\) = b₁\(\hat{i}\) + b₂\(\hat{j}\) + b₃\(\hat{k}\) also λ be any scalar.

Then vector algebra formulas are:

1. \(\overrightarrow{a}\) + \(\overrightarrow{b}\) = (a₁ + b₁)\(\hat{i}\) + (a₂ + b₂)\(\hat{j}\) + (a₃ + b₃)\(\hat{k}\)
2. λ \(\overrightarrow{a}\) = (λa₁)\(\hat{i}\) + (λa₂)\(\hat{j}\) + (λa₃)\(\hat{k}\)
3. \(\overrightarrow{a}\) . \(\overrightarrow{b}\) = a₁b₁ + a₂b₂ + a₃b₃
4. \(\vec{a} \times \vec{b} = \begin{vmatrix} i & j & k \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{vmatrix}\)

Thus, All the formulas will help you to deal with problem solving assesment. In vector algebra class 12 Everything is there with full details.

Option : (a). {1, 5, 9}

\(\because |1-1|=0\) which is a multiple of 4

Therefore, \((1, 1) \in R\)

and \(|1-5|=4\) which is a multiple of 4

Therefore, \((1, 5) \in R\)

Also \(|1-9|=8\) which is a multiple of 4

Therefore, \((1, 9) \in R\)

Since, \((1, 1), (1, 5), (1, 9) \in R\)

Therefore, the equivalence class of 1 is [1] = {1, 5, 9}

Option : (d)

sin(tan⁻¹x) = sin(sin⁻¹(\(\frac{x}{\sqrt{1+x^2}}\)))

= \(\frac{x}{\sqrt{1+x^2}}\)

Correct option is (B) 9

\(\frac{x^2}9+\frac{y^2}9=1\) 

equation of tangent to the ellipse is

y = mx \(\pm\sqrt{a^2m^2+b^2}\)

y = mx \(\pm\sqrt{16m^2+9}\)----(i)

x² + y² = 12

equation of tangent to the circle is

y = mx \(\pm\sqrt{12}\sqrt{1+m^2}\)--(ii)

for common tangent equate eq. (i) and (ii)

⇒ 16m² + 9 = 12(1 + m²)

16m² - 12m² = 3

4m² = 3

12m² = 9

Below is a table of differences between Computer Network and Data Communication:

#include<stdio .h>

#include<conio . h>

void main() 

{

char text[100],ch;

int i=0;

clrscr();

printf("Enter text. Press at the end\n"); do

{ ch=getchar();

text[i]=ch;

i++;

}while(ch!='\n');

i=i-1;

text[i]='\0';

printf("The entered text is");

printf("\n%s\n",text);

getch();

}

⇒  Number of elements in set A=n(A)=5
⇒  Number of elements in set B=n(B)=6
As, the number of bijection from A into B can only be possible when provided n(A)≥n(B).
But here, we can see, n(A)<n(B)
So, the number of bijection i.e. one-one and onto mappings from A to B=0

Equivalent postfix expression: 

= ((A-B)*(D/E))/(F*G*H) 

= ((AB-)*(DE/))/(FG*H*) 

= AB – DE /* FG* H*

Correct option is B) delegates

Correct option is D) acquitted

The traffic light at one particular crossing change every 40 seconds.

The traffic light at the next crossing changes every 32 seconds.

The time required to change together is equal to the least common multiplier of both times.

Required time = LCM (40, 32) = 160 sec = 2 min. 40 sec.

Correct option is B) acting

Correct option is C) is having

Correct option is B) had paid / pushed

Correct option is B) build / shall move

Students is the meaning of the term "Taliban"

Correct option is B) in front of / in

Correct option is C) was given

Correct option is A) provided

1) In Alexander’s opinion Indian soldiers were as brave as Greek soldiers but they might not be as skilful as the Greeks. 

2) The Indian generals used the old methods of warfare. Their battle plans are faulty. They depend too much upon their elephants. 

3) The Indian generals were brave. They were wonderful fighters. So Alexander had highest respect for Indian soldiers. 

4) Puru was injured badly. So the Greeks were able to capture him. 

5) Puru continued to fight even when others had left the field. He killed at least a hundred of Greek soldiers with his sword. He injured many more. Thus I can agree with, Alexander’s statements that Puru was a brave and a great soldier.

Additional Answer: 

1) In the opinion of Greek generals the Indian generals were not so brave and skilful, as the Greek generals. 

2) The Indian generals followed the old methods of warfare. Their battle plans were faulty and they depended too much on their elephants. Thus we can say Indian generals were not skilful. 

3) The name of the battle was the battle of Jhelum. 

4) The battle was fought on the bank of the river Jhelum. So it was called the battle of Jhelum. 

5) Puru was a brave man and a great soldier. He continued to fight even when others had left the field. He killed hundreds of Greeks and injured many with his sword. Alexander was inspired With his bravery and wished if Puru were one of his generals.

Correct option is A) angry

Correct option is B) can

Correct option is D) behind

Correct option is C) have your house painted

Correct option is D) since I saw

Correct option is A) much

Correct option is C) any

1) Alexander was the king of Macedonia. 

2) The battle of Jhelum was fought between Alexander and Puru, a king of northern India. 

3) Alexander thought that the battle of Jhelum would go down in the history because he had never fought such a battle. The Indian soldiers fought very bravely. He could not believe that he would be able to defeat the Indians. The army of Puru fought better than the army of the Greeks. 

4) According to Alexander every Indian soldier fought like a tiger. 

5) All Indian soldiers fought bravely but because of the inefficiency of the Indian generals Indians lost their battle.

Correct option is A) Pat the book

Correct option is C) similar to

Correct option is  D) has given

Correct option is B) men’s

Correct option is B) as far as

Correct option is  B) have lost

Correct option is  D) have been painting

Correct option is D) her

Correct option is A) did she

Correct option is D) have been painting

Correct option is B) did

Correct option is B) on

Correct option is A) She happened to find

Correct option is D) which

Every seat in this bus has a number.