Mathematics - Class 12 Mock Test 8

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Question 1 of 10

1. Find the vector equation of the line which is passing through the point (2,-3,5) and parallel to the vector \(3\hat{i}+4\hat{j}-2\hat{k}\).

\((9+3λ) \hat{i}+(λ-3) \hat{j}+(5-2λ)\hat{k}\)

\((2+3λ) \hat{i}+(4λ-3) \hat{j}+(5-2λ)\hat{k}\)

\((7+λ) \hat{i}+(4λ+3) \hat{j}+(5-2λ)\hat{k}\)

\((2+3λ) \hat{i}+(4λ+3) \hat{j}+(5-λ)\hat{k}\)

Question 1 of 10

Question 2 of 10

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

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

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

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

Question 2 of 10

Question 3 of 10

3. Find the angle between 2x + 3y - 2z + 4 = 0 and 4x + 3y + 2z + 2 = 0.

38.2

19.64

89.21

54.54

Question 3 of 10

Question 4 of 10

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

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

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

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

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

Question 4 of 10

Question 5 of 10

5. If L₁ and L₂ have the direction ratios \(a_1,b_1,c_1 \,and \,a_2,b_2,c_2\) respectively then what is the angle between the lines?

\(θ=2 \,cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)

\(θ=tan^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)

\(θ=cos^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)

\(θ=2tan^{-1}⁡\left|\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}}\right |\)

Question 5 of 10

Question 6 of 10

6. Find the angle between the vectors if \(|\vec{a}|=|\vec{b}|=3\sqrt{2}\) and \(\vec{a}.\vec{b}=9\sqrt{3}\).

\(\frac{π}{3}\)

\(\frac{π}{2}\)

\(\frac{π}{5}\)

\(\frac{π}{6}\)

Question 6 of 10

Question 7 of 10

7. Find the projection of vector \(\vec{b}=2\hat{i}+2\sqrt{2} \,\hat{j}-2\hat{k}\) on the vector \(\vec{a}=\hat{i}-\hat{j}-\sqrt{2} \,\hat{k}\).

\(2\sqrt{2}\)

\(\sqrt{2}\)

Question 7 of 10

Question 8 of 10

8. If the plane passes through three collinear points \((x_1,y_1,z_1),(x_2,y_2,z_2),(x_3,y_3,z_3)\) then which of the following is true?

\(\begin{vmatrix}x_1\\y_2\\z_3\end{vmatrix}\)=0

\(x_1 y_1 z_1+x_2 y_2 z_2+x_3 y_3 z_3\)=0

\(\begin{vmatrix}x_1&y_1&z_1\\x_2&y_2&z_2\\x_3&y_3&z_3\end{vmatrix}\)=0

\(x_1 x_2 x_3+y_1 y_2 y_3+z_1 z_2 z_3=0\)

Question 8 of 10

Question 9 of 10

9. Find the cartesian equation of a line passing through two points (1,-9,8) and (4,-1,6).

\(\frac{x-1}{3}=\frac{y+9}{8}=\frac{z-8}{-2}\)

\(\frac{2x-1}{3}=\frac{6y+9}{8}=\frac{4z-8}{-2}\)

\(\frac{x-1}{7}=\frac{y+9}{-2}=\frac{z-8}{5}\)

\(\frac{x+1}{3}=\frac{y-9}{8}=\frac{-z-8}{2}\)

Question 9 of 10

Question 10 of 10

10. Find the shortest distance between two lines l₁ and l₂ whose vector equations is given below.
\(\vec{r}=3\hat{i}-4\hat{j}+2\hat{k}+λ(4\hat{i}+\hat{j}+\hat{k})\)
\(\vec{r}=5\hat{i}+\hat{j}-\hat{k}+μ(2\hat{i}-\hat{j}-3\hat{k})\)

\(\frac{10}{\sqrt{11}}\)

\(\frac{18}{\sqrt{10}}\)

\(\frac{23}{\sqrt{10}}\)

\(\frac{11}{\sqrt{12}}\)

Question 10 of 10

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