Mathematics Test

If the lines \(\vec r = \overrightarrow {{a_1}} + \lambda \overrightarrow {{b_1}} \;and\;\vec r = \overrightarrow {{a_2}} + \mu \overrightarrow {{b_2}} \) are coplanar then which of the following is correct ?

\(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right) \cdot \left( {\overrightarrow {{b_1}} \times \;\overrightarrow {{b_2}} } \right) = 0\)

\(\left( {\overrightarrow {{a_2}} + \overrightarrow {{a_1}} } \right) \cdot \left( {\overrightarrow {{b_1}} \times \;\overrightarrow {{b_2}} } \right) = 0\)

\(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right) \times \left( {\overrightarrow {{b_1}} \cdot \;\overrightarrow {{b_2}} } \right) = 0\)

Question 1/10

x + z = 1

y = 1

x = z = 1

x = z = 0

Question 2/10

\( {1 \over \sqrt{14}}, {2 \over \sqrt{14}}, {3 \over \sqrt{14}}\)

\(\pm {1 \over \sqrt{14}}, \pm{2 \over \sqrt{14}}, \pm{3 \over \sqrt{14}}\)

\(\pm {1 \over \sqrt{14}}, \mp{2 \over \sqrt{14}}, \pm {3 \over \sqrt{14}}\)

\( {1 \over \sqrt{14}}, -{2 \over \sqrt{14}}, - {3 \over \sqrt{14}}\)

Question 3/10

\(\rm \dfrac{x-a}{1}=\dfrac{y-b}{0}=\dfrac{z-c}{0}\)

\(\rm \dfrac{x-a}{0}=\dfrac{y-b}{0}=\dfrac{z-c}{1}\)

\(\rm \dfrac{x-a}{0}=\dfrac{y-b}{1}=\dfrac{z-c}{0}\)

\(\rm \dfrac{x-a}{0}=\dfrac{y-b}{1}=\dfrac{z-c}{1}\)

Question 4/10

1

7

-7

2

Question 5/10

2 ∶ 1 internally

3 ∶ 2 externally

5 ∶ 3 internally

5 ∶ 3 externally

Question 6/10

6

5

4

2

Question 7/10

\({\sin ^{ - 1}}\left( {\frac{8}{21}} \right)\)

\({\sin ^{ - 1}}\left( {\frac{2}{21}} \right)\)

\({\sin ^{ - 1}}\left( {\frac{5}{21}} \right)\)

None of these

Question 8/10

line parallel to x - axis

line parallel to y - axis

line parallel to z - axis

plane parallel to xy - plane

Question 9/10

〈1, 2, 3〉

〈2, 1, 3〉

〈3, 2, 1〉

〈1, 3, 2〉

Question 10/10

\(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right) \cdot \left( {\overrightarrow {{b_1}} \times \;\overrightarrow {{b_2}} } \right) = 0\)

\(\left( {\overrightarrow {{a_2}} + \overrightarrow {{a_1}} } \right) \cdot \left( {\overrightarrow {{b_1}} \times \;\overrightarrow {{b_2}} } \right) = 0\)

\(\left( {\overrightarrow {{a_2}} - \overrightarrow {{a_1}} } \right) \times \left( {\overrightarrow {{b_1}} \cdot \;\overrightarrow {{b_2}} } \right) = 0\)

None of these