Mathematics - Class 12 Practice Test 8

1. Find the shortest distance between the lines given below.
\(\vec{r}=(1-p) \hat{i}+(p-3) \hat{j}+(1+p) \hat{k}\)
\(\vec{r}=(q-1) \hat{i}-(2q+3) \hat{j}+(1+q)\hat{k}\)

\(\frac{6}{\sqrt{24}}\)

\(\frac{12}{\sqrt{14}}\)

\(\frac{32}{\sqrt{14}}\)

\(\frac{6}{\sqrt{14}}\)

2. If a, b, c are the direction ratios of the line and l, m, n are the direction cosines of the line, then which of the following is true?

\(\frac{l}{a}=\frac{n+1}{b}=\frac{n}{c}=μ\)

\(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ-1\)

\(\frac{l}{a}=\frac{m}{c}=\frac{n}{b}=μ\)

\(\frac{l}{a}=\frac{m}{b}=\frac{n}{c}=μ\)

3. Which of the given set of planes are perpendicular to each other?

\(\vec{r}.(\hat{i}-2\hat{j}+\hat{k})\)=7 and \(\vec{r}.(\hat{i}+\hat{j}+2\hat{k})\)=2

\(\vec{r}.(2\hat{i}-2\hat{j}+\hat{k})\)=4 and \(\vec{r}.(\hat{i}+2\hat{j}+2\hat{k})\)=5

\(\vec{r}.(3\hat{i}-2\hat{j}+\hat{k})\)=2 and \(\vec{r}.(\hat{i}+2\hat{j}+8\hat{k})\)=8

\(\vec{r}.(2\hat{i}+2\hat{j}+\hat{k})\)=5 and \(\vec{r}.(\hat{i}+2\hat{j}+2\hat{k})\)=5

4. Find the equation of the plane passing through three points (1,2,-1), (0,-1,2) and (3,1,1).

\((\vec{r}-(\hat{i}+2\hat{j}-\hat{k})).[(\hat{i}-3\hat{j}+3\hat{k})×(2\hat{i}-\hat{j}+2\hat{k})]\)=0

\((\vec{r}-(\hat{i}+2\hat{j}-\hat{k})).[(3\hat{i}+\hat{j}+\hat{k})×(2\hat{i}-\hat{j}+2\hat{k})]\)=0

\((\vec{r}-(\hat{i}+2\hat{j}-\hat{k})).[(-\hat{j}+2\hat{k})×(2\hat{i}-\hat{j}+2\hat{k})]\)=0

\((\vec{r}-(\hat{i}+2\hat{j})).[(-\hat{i}-3\hat{j}+3\hat{k})×(2\hat{i}-\hat{j}+2\hat{k})]\)=0

5. Which trigonometric function is used to find the angle between two planes?

Cosecant

Sine

Tangent

Secant

6. Which of the following is not the correct formula for representing a plane?

lx+my+nz=d

al+mb+cn=d²

\(\vec{r}.\hat{n}=d\)

ax+by+cz=d

7. If a line is passing through two points \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\) then which of the following is the vector equation of the line?

\(\vec{r}=\vec{a}+λ(\vec{b}-\vec{a})\)

\(\vec{r}=\vec{a}+λ(\vec{a}-\vec{b})\)

\(\vec{r}=λ\vec{a}+(\vec{b}-\vec{a})\)

\(\vec{r}=\vec{a}+λ(\vec{b}+\vec{a})\)

8. If a line has direction ratios 2, -3, 7 then find the direction cosines.

l=\(\frac{2}{\sqrt{6}},m=-\frac{3}{\sqrt{6}},n=\frac{7}{\sqrt{6}}\)

l=\(\frac{2}{\sqrt{62}},m=-\frac{3}{\sqrt{62}},n=\frac{7}{\sqrt{62}}\)

l=\(\frac{2}{\sqrt{62}},m=-\frac{7}{\sqrt{62}},n=\frac{7}{\sqrt{62}}\)

l=-\(\frac{2}{\sqrt{62}},m=-\frac{3}{\sqrt{62}},n=-\frac{7}{\sqrt{62}}\)

9. Find the angle between the lines \(\vec{r}=2\hat{i}+6\hat{j}-\hat{k}+λ(\hat{i}-2\hat{j}+3\hat{k})\) and \(\vec{r}=4\hat{i}-7\hat{j}+3\hat{k}+μ(5\hat{i}-3\hat{j}+3\hat{k})\).

θ=\(cos^{-1}\frac{⁡8}{\sqrt{602}}\)

θ=\(cos^{-1}⁡\frac{14}{\sqrt{598}}\)

θ=\(cos^{-1}\frac{⁡20}{\sqrt{682}}\)

θ=\(cos^{-1}\frac{⁡20}{\sqrt{602}}\)

10. Find the Cartesian equation of the plane \(\vec{r}.[(λ+2μ) \hat{i}+(2λ-μ) \hat{j}+(3λ-2μ)\hat{k}]\)=12.

(λ+2μ)x-2λy+(3λ-2μ)z=12

(λ-μ)x+y+(3λ-2μ)z=12

(λ+2μ)x+(2λ-μ)y+(3λ-2μ)z=12

(λ+3μ)x+(2+μ)y+(3λ-2μ)z=12