Math Assignment Class XII Ch-11: Three Dimensional Geometry
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Math. Assignment Class XII Chapter 11
Question 1: A line makes angle 𝜶, 𝛃, 𝛾 with x-axis, y-axis, z-axis. Then find cos2𝜶, cos2𝛃, cos2𝛾 ? Ans. 1 -1 |
Question 2:
Write vector
equation of the line: \[\frac{2x-1}{\sqrt{3}} =\frac{y+2}{2} =\frac{z-3}{3}\] |
Question 3: Find the Cartesian equation of the line which passes through the point (-2, 4, -5) and is parallel to the line \[\frac{x+3}{3}=\frac{4-y}{5}=\frac{z+8}{6}\] Hint: Two parallel lines have same direction ratio. Ans. 3 \[\frac{x+2}{3}=\frac{y-4}{-5}=
\frac{z+5}{6}\] |
Question 4: Find angle between pair of lines \[\frac{2-x}{-2}=\frac{y-1}{7}=\frac{z+3}{-3}\; and\]\[\frac{x+2}{-1}=\frac{2y-8}{4}=\frac{z-5}{4}\] Ans 4: 90^{o} |
Question 5: Find the point on the given line at
a distance 3√2 from the point (1, 2, 3). \[\frac{x+2}{3}=\frac{y+1}{2}=\frac{z-3}{2}\] Ans 5: \[\lambda =0, \lambda =\frac{30}{17}\;\; and\; Point\; is\;
A(-2,-1,3)\; or\; A\left (\frac{56}{17},\frac{43}{17},\frac{111}{17} \right
)\] |
Question 6:
Find the length and
foot of perpendicular drawn from the point (2, -1, 5) on the line
\[\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}\] Ans 6: \[\lambda
=-1, Q \left ( 1,2,3 \right ), \; \; \left | PQ \right |=\sqrt{14}\] |
Question 7:
Find the shortest
distance between the lines :-
\[\vec{r}=(1+\lambda)\hat{i}+(2-\lambda)\hat{j}+(\lambda
+1)\hat{k}\]\[\vec{r}=(2\hat{i}-\hat{j}-\hat{k})+\mu (2\hat{i}+\hat{j}+2\hat{k})\] Ans 7: \[\vec{b_{1}}\times \vec{b_{2}}=-3\hat{i}+3\hat{k},\; Distance\;
=\frac{3\sqrt{2}}{2}\] |
Question 8: Find the equation of a line passing through the point P(2,-1,3) and perpendicular to the lines:\[\vec{r}=(\hat{i}+\hat{j}+\hat{k})+\lambda (2\hat{i}-2\hat{j}+\hat{k})\]\[\vec{r}=(2\hat{i}-\hat{j}-3\hat{k})+\mu(\hat{i}+2\hat{j}+2\hat{k})\] Ans 8: \[\vec{r}=\left ( 2\hat{i}-\hat{j}+3\hat{k} \right )+\lambda \left ( 2\hat{i}+\hat{j}-2\hat{k} \right )\] |
Question 9: Show that the following lines are intersecting, also find their point of intersection. \[\vec{r}=3\hat{i}+2\hat{j}-4\hat{k}+\lambda(\hat{i}+2\hat{j}+2\hat{k})\]\[\vec{r}=5\hat{i}-2\hat{j}+\mu (3\hat{i}+2\hat{j}+6\hat{k})\]Ans 9 (-1, -6, -12) |
Question 10:
Show that the following
lines are coplaner. \[\frac{5-x}{-4}=\frac{y-7}{4}=\frac{z+3}{-5} \;
and\]\[\frac{x-8}{7}=\frac{2y-8}{2}=\frac{z-5}{3}\] |
Question 11: Find the distance of the given plane from the origin: \[\vec{r}.(2\hat{i}+3\hat{j}-6\hat{k})+2=0\] Ans 11 : 2/7 |
Question 12: Find the angle between the planes \[\vec{r}.(\hat{i}-2\hat{j}-2\hat{k})=1 \;\; and\]\[\vec{r}.(3\hat{i}-6\hat{j}+2\hat{k})=0\]Ans 12 \[cos^{-1}\left ( \frac{11}{21} \right )\] |
Question 13:
Find the equation of
the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which
contains the line of intersection of planes x + 2y + 3z - 4 = 0 and 2x + y -
z + 5 = 0 Ans 13 51x + 15y - 50z + 173 = 0 , λ = -29/7 |
Question 14:
Find the equation of
the plane passing through the point (-1, 3, 2) and perpendicular to each of
the plane x + 2y + 3z = 5 and 3x + 3y + z = 0. Ans 14 7x - 8y + 3z + 25 = 0 |
Question 15:
Find the equation of
the plane through (2, 1, -1) and (-1, 3, 4) and perpendicular to the
plane x -2y + 4z = 10. Ans 15 18x + 17y + 4z -49 = 0 |
Question 16:
\[ Find \; image\;
p'\; of\; point\; p\; having\; position\; vector\; \;
\hat{i}+3\hat{j}+4\hat{k}\;\; in\; the\;
plane\]\[\vec{r}.(2\hat{i}-\hat{j}+\hat{k})+3=0\; and\; also\; find\; \;
\left | pp' \right |\] Ans. 16 Image (-3,5,2), Length = 2√6 |
Question 17: Find the distance of the point (1, -2,
3) from the plane x - y + z = 5 measured parallel to the line
\[\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}\] Ans 17 λ = 1/7, AB = 1 |
Question 18:
Find the distance of
the point (-2, 3, -4) from the given line measured ∥ to the plane 4x + 12y - 3z + 1 = 0 \[\frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}\] Ans 18 λ = 2/3 , Point is (4, 5/2, 2), Distance = 17/2 |
Question 19: Find the equation of the plane passing through the point P(4, 6, 2) and the point of intersection of the plane x + y - z = 8 and the line \[\frac{x-1}{3}=\frac{y}{2}= \;\frac{z+1}{7}\]Ans 19 \[\frac{x-4}{1}=\frac{y-6}{1}=\frac{z-2}{2}\] |
Question 20: Find the vector equation of the plane through the line of intersection of planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0. Hence find whether plane thus obtained contains the line \[\frac{x+2}{5}=\frac{y-3}{4}= \;\frac{z}{5}\]Ans 20 \[\vec{r}\left ( \hat{i}-\hat{k} \right )+2=0,\; \; \;Yes\] |
Question 21: Find the angle between the following line and plane : \[\vec{r}=\left (\hat{i}-\hat{j}+\hat{k} \right )+\lambda (2\hat{i}-\hat{j}+3\hat{k})\]\[and\; plane\; \; \vec{r}.\left (2\hat{i}+\hat{j}-\hat{k} \right )=4\] Ans 21 0^{o} |
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