TEFA: GCE A/L

March 18, 2021, 9:07 p.m.
KENG ElSON
revision

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March 5, 2021, 7 p.m.
Wanyu
integrate 1/(x+2)^(1/2)

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Feb. 20, 2021, 1:04 p.m.
Zozo
please who knows a good furthermaths text book for beginners???

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Feb. 14, 2021, 6:17 p.m.
Ndifon
Solution

$\vec{a}X\vec{b}=\begin{vmatrix} \hat{i }&\hat{j} &\hat{k} \\ \alpha &-1 &-4 \\ 3 & 2 &(1+2\beta ) \end{vmatrix}=(7-2\beta )\hat{i}-(12+\alpha +2\alpha \beta )\hat{j}+(3+2\alpha )\hat{k}$

(i) Finding $\alpha$ and $\beta$

from the question, $\vec{a}X\vec{b}=3\hat{i}-\frac{41}{2}\hat{j}+6\hat{k}$

So $3\hat{i}-\frac{41}{2}\hat{j}+6\hat{k}=(7-2\beta )\hat{i}-(12+\alpha +2\alpha \beta )\hat{j}+(3+2\alpha )\hat{k}$

$\therefore \left\{\begin{matrix} 3=7-2\beta \Rightarrow \beta =2\\ 6=3+2\alpha \Rightarrow \alpha =\frac{3}{2}\\ -\frac{41}{2}=-(12+\alpha +2\alpha \beta )\\ \end{matrix}\right.$

Substituting the values of $\alpha$ and $\beta$ in the third equation verfifies.

$\therefore \left\{\begin{matrix} \alpha =\frac{3}{2}\\ \beta =2 \end{matrix}\right.$

(ii) Stating the vectors a and b

$\left\{\begin{matrix} \vec{a}=\frac{3}{2}\hat{i}-\hat{j}-4\hat{k}\\ \vec{b}=3\hat{i}+2\hat{j}+5\hat{k} \end{matrix}\right.$

(iii) Showing that a and b are linearly independent

$\vec{a}X\vec{b}=3\hat{i}-\frac{41}{2}\hat{j}+6\hat{k}\neq \vec{0}$

Therefore a and b are linearly independent.

(iv) Finding the cartesian equation of the plane containing a and b

Normal to the plane

$\vec{n}=\vec{a}X\vec{b}=3\hat{i}-\frac{41}{2}\hat{j}+6\hat{k}$

Let P be a general point on the plane with coordinates:

$P(x,y,z)$

Using vector a

$\vec{OP}\cdot \vec{n}=\vec{a}\cdot\vec{n}$

$\Rightarrow (x\hat{i}+y\hat{j}+z\hat{k})\cdot (3\hat{i}-\frac{41}{2}\hat{j}+6\hat{k})=(\frac{3}{2}\hat{i}-\hat{j}-4\hat{k})\cdot(3\hat{i}-\frac{41}{2}\hat{j}+6\hat{k})$

$3x-\frac{41}{2}y+6z=1\Rightarrow 6x-41y+12z=2$

So the plane has equation

$6x-41y+12z=2$

N/B: One could equally use vector b or any other vector on the plane to get the equation. Note also that any vector parallel to vector n is equally a normal to the plane and can be used to get the equation of the plane.

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Feb. 14, 2021, 3:05 p.m.
Wanyu
Given the two vectors

$a=\alpha i-j-4k$ and $b=3i+2j+(1+2\beta )k, \alpha ,\beta \epsilon \mathbb{Z},$ $aXb=3i-21j+6k,$

(i) Calculate the values of $\alpha$ and $\beta$ (3mks)

(ii) By using the values of $\alpha$ and $\beta$ , state the vectors a and b (1mk)

(iii) Show that a and b are linearly independent (2mks)

(iv)Find the cartesian equation of th plane containing a and b. (2mks)

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Feb. 14, 2021, 2:42 p.m.
Wanyu
June 2019 question 8

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Feb. 14, 2021, 2:42 p.m.
Wanyu
June 2019 question 8

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