Maths

Vector $\lambda \overrightarrow{a}$ is a unit vector if $\left|\lambda \overrightarrow{a}\right|=1$

Now,

$\begin{array}{l}\left|\lambda \overrightarrow{a}\right|=1\\ \left|\lambda \right|\left|\overrightarrow{a}\right|=1\\ \left|\overrightarrow{a}\right|=\frac{1}{\left|\lambda \right|}\left[\lambda \ne 0\right]\\ a=\frac{1}{\left|\lambda \right|}\left[\left|\overrightarrow{a}\right|=a\right]\end{array}$

$\therefore$ Therefore, vectar $\lambda \overrightarrow{a}$ is a unit vector if a= $\frac{1}{\left|\lambda \right|}$ .

Option (D)is correct.

Let vector $2\widehat{i}-\widehat{j}+\widehat{k},\widehat{i}-3\widehat{j}-5\widehat{k}$ and $3\widehat{i}-4\widehat{j}-4\widehat{k}$ be position vector of point A, B, C respectively.

So,

$\begin{array}{l}O\overrightarrow{A}=2\widehat{i}-\widehat{j}+\widehat{k}\\ O\overrightarrow{B}=\widehat{i}-3\widehat{j}-5\widehat{k}\\ O\overrightarrow{C}=3\widehat{i}-4\widehat{j}-4\widehat{k}\end{array}$

Now, vectors $A\overrightarrow{B},B\overrightarrow{C}$ and $A\overrightarrow{C}$ represents the sides of $?ABC$ .

Hence,

Given, point are

51. The given system of inequality is

x+2y≤ 10- (1)

x+y≥ 1 - (2)

x – y ≤ 0 - (3)

x≥ 0 and y≥ 0 - (4)

The corresponding equation of (1), (2) and (3) are

x + 2y = 10

and x + y =1

and x – y = 0

Putting (2,0)= (x, y) in inequality (1), (2) and (3),

2+2 * 0 ≤ 10 =>  2≤ 10 is true.

and 2+0 ≥ 1 =>  2 ≥ 1 is true.

and 2 – 0 ≤ 0  => 2 ≤ 0 is false.

So, the solution of inequality (1) and (2) is the plane that includes point (2,0) whereas the solution of inequality (3) is the plane which includes point (2, 0)

∴ The shaded region represents the solution of the given system of inequality.

Vertices of $?ABC$ are given as

$\begin{array}{l}A\left(1,2,3\right),B\left(-1,0,0\right),C\left(0,1,2\right)\\ \end{array}$

$\therefore ?ABC$ is the angle between the vectors $B\overrightarrow{A}$ and $B\overrightarrow{C}$

Consider

$\begin{array}{l}\overrightarrow{a}=2\widehat{i}+4\widehat{j}+3\widehat{k}\\ \overrightarrow{b}=3\widehat{i}+3\widehat{j}-6\widehat{k}\end{array}$ and

Then,

$\begin{array}{l}\overrightarrow{a}.\overrightarrow{b}=2.3+4.3+3.(-6)\\ =6+12-18=0\end{array}$

Therefore, the converse of the given statement need not be true.

50.The given system of inequality is

3x+2y≤ 150- (1)

x+4y≤ 80- (2)

x≤ 15 - (3)

y≥ 0 and x≥ 0 - (4)

The corresponding equation of (1) and (2) are

3x + 2y = 150

and x + 4y =80

Putting (0,0)= (x, y) in inequality (1) and (2) we get,

3 * 0+2 * 0 ≤ 150  => 0 ≤ 150 is true.

and 0+4 * 0 ≤ 80  => 0 ≤ 80 is true.

So, the solution plane of both inequality (1) and (2) includes the origin (0,0).

∴ The shaded region is the solution of the given system of inequality.

$\left|\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right|=\left(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right).\left(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\right)$

$\begin{array}{l}=\overrightarrow{a}.\overrightarrow{a}+\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{a}.\overrightarrow{c}+\overrightarrow{b}.\overrightarrow{a}+\overrightarrow{b}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}+\overrightarrow{c}.\overrightarrow{b}+\overrightarrow{c}.\overrightarrow{c}\\ ={\left|\overrightarrow{a}\right|}^2+{\left|\overrightarrow{b}\right|}^2+{\left|\overrightarrow{c}\right|}^2+2\left(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}\right)\\ =1+1+1+2\left(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}\right)\\ =3+2\left(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}\right)\\ \Rightarrow \overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}=\frac{-3}{2}\end{array}$

2. Given, f (x) = 2x2 1

At x = 3

Lim f (x) = $\frac{dy}{dx}=\frac{-\left(3x^2+2xy+y^2\right)}{\left(x^2+2xy+3y^2\right).}$ 2 (3)2 1 = 18 1 = 17.

So, f is continuous at x = 3.