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Question 158 of 162

Q.(a) Find the non-parametric form of Vector equation, and the Cartesian equation of the plane passing through the point (0,1,−5)(0, 1, -5) and parallel to the straight lines r⃗=(i^+2j^−4k^)+s(2i^+3j^+6k^)\vec{r}=\left(\hat{i}+2\hat{j}-4\hat{k}\right)+s\left(2\hat{i}+3\hat{j}+6\hat{k}\right) and r⃗=(i^−3j^+5k^)+t(i^+j^−k^)\vec{r}=\left(\hat{i}-3\hat{j}+5\hat{k}\right)+t\left(\hat{i}+\hat{j}-\hat{k}\right) OR

(b) The growth of a population is proportional to the number present. If the population of a colony doubles in 50 years, in how many years will the population become triple ?
Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2025Subjective· 5mImportance★★★★★
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(a) Builds the plane's normal as the cross product of the two given line directions and writes both the vector and Cartesian equations through the given point; (b) solves the exponential-growth model using the doubling time to find the tripling time. Both alternatives answered below.

(a) Plane through (0,1,−5)(0,1,-5) parallel to two given lines

1. Data. The plane must be parallel to both line directions: d⃗1=2i^+3j^+6k^\vec d_1=2\hat i+3\hat j+6\hat k (from the first line) and d⃗2=i^+j^−k^\vec d_2=\hat i+\hat j-\hat k (from the second), and it must pass through A(0,1,−5)A(0,1,-5).

2. Normal of the plane (perpendicular to both directions):

n⃗=d⃗1×d⃗2=∣i^j^k^23611−1∣=i^(3(−1)−6(1))−j^(2(−1)−6(1))+k^(2(1)−3(1))\vec n=\vec d_1\times\vec d_2=\begin{vmatrix}\hat i&\hat j&\hat k\\2&3&6\\1&1&-1\end{vmatrix}=\hat i(3(-1)-6(1))-\hat j(2(-1)-6(1))+\hat k(2(1)-3(1))

=i^(−3−6)−j^(−2−6)+k^(2−3)=−9i^+8j^−k^=\hat i(-3-6)-\hat j(-2-6)+\hat k(2-3)=-9\hat i+8\hat j-\hat k

So n⃗=(−9,8,−1)\vec n=(-9,8,-1).

3. Non-parametric vector equation. r⃗⋅n⃗=a⃗⋅n⃗\vec r\cdot\vec n=\vec a\cdot\vec n, where a⃗=0i^+j^−5k^\vec a=0\hat i+\hat j-5\hat k:

a⃗⋅n⃗=0(−9)+1(8)+(−5)(−1)=8+5=13\vec a\cdot\vec n=0(-9)+1(8)+(-5)(-1)=8+5=13

⇒ r⃗⋅(−9i^+8j^−k^)=13\Rightarrow\ \vec r\cdot(-9\hat i+8\hat j-\hat k)=13

4. Cartesian equation. With normal (−9,8,−1)(-9,8,-1) through (0,1,−5)(0,1,-5):

−9(x−0)+8(y−1)−1(z+5)=0 ⇒ −9x+8y−8−z−5=0 ⇒ −9x+8y−z−13=0-9(x-0)+8(y-1)-1(z+5)=0\ \Rightarrow\ -9x+8y-8-z-5=0\ \Rightarrow\ -9x+8y-z-13=0

i.e. 9x−8y+z+13=09x-8y+z+13=0.

5. Check. Point (0,1,−5)(0,1,-5): 9(0)−8(1)+(−5)+13=−8−5+13=09(0)-8(1)+(-5)+13=-8-5+13=0 ✓.

(b) Time for the population to triple, given it doubles in 5050 years

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