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Three-dimensional Geometry - ISC Class 12 Mathematics Questions with Answers, Page 2

63 past-paper questions on Three-dimensional Geometry from ISC Class 12 Mathematics papers (2027-2017), newest first, in full. Questions 21-40 are on this page, 20 to a page. Tap "Show answer" under a question to see its answer.

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2025 · 4 marks · Short answerOpen: Imagine you are at a point A, a café you visit often. Your friend is at the…
Imagine you are at a point A, a café you visit often. Your friend is at the point B, a bookstore a few blocks away on a straight road. You want to meet your friend at a point on the line joining the café and the bookstore. Another friend, who is at home on the other side of the same road represented by point P, also wants to join. You decide to determine the exact meeting point by finding the foot of the perpendicular from P on the line joining the café and the bookstore. Given that co-ordinates of the café (A) are $(1, 2, 4)$, of the bookstore (B) are $(3, 4, 5)$ and the home are $(2, 1, 3)$, find the location of the meeting point.

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2025 · 4 marks · Short answerOpen: There are two solar panels, A and B placed parallel to each other on the…
There are two solar panels, A and B placed parallel to each other on the terrace of a building. The equation of the surface of the panel A is $2x + 2y - z + 9 = 0$.
(a)[1.0]
The direction cosines of the plane are:
  • (1)$\left(\frac{2}{3}, \frac{2}{3}, -\frac{1}{3}\right)$
  • (2)$\left(\frac{2}{3}, \frac{2}{3}, \frac{1}{3}\right)$
  • (3)$\left(-\frac{2}{3}, -\frac{2}{3}, \frac{1}{3}\right)$
  • (4)$\left(-\frac{2}{3}, \frac{2}{3}, \frac{1}{3}\right)$
(b)[1.0]
A fly is sitting on panel A. The position of the fly is $(2, m, 3)$. Find the value of '$m$'.
(c)[1.0]
Find the equation of the panel B if the point $(3, 2, -4)$ lies on it.
(d)[1.0]
Find the distance between the panels.

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2025 · 5 marks · Short answerOpen: In subparts (i) and (ii) choose the correct options and in subparts (iii) to…
In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed.
(i)[1.0]
Consider the following statements and choose the correct option: Statement 1: If $\vec{a}$ and $\vec{b}$ represents two adjacent sides of a parallelogram then the diagonals are represented by $\vec{a} + \vec{b}$ and $\vec{a} - \vec{b}$. Statement 2: If $\vec{a}$ and $\vec{b}$ represents two diagonals of a parallelogram then the adjacent sides are represented by $2(\vec{a} + \vec{b})$ and $2(\vec{a} - \vec{b})$. Which of the following is correct? [Recall]
  • (a)Only Statement 1
  • (b)Only Statement 2
  • (c)Both Statements 1 and 2
  • (d)Neither Statement 1 nor Statement 2
(ii)[1.0]
The distance of the plane through $(1, 1, 1)$ and perpendicular to the line $\frac{x-1}{3} = \frac{y-1}{0} = \frac{z-1}{4}$ from the origin is [Application]
  • (a)$\frac{3}{4}$
  • (b)$\frac{4}{3}$
  • (c)$\frac{7}{5}$
  • (d)1
(iii)[1.0]
If the direction cosines of a line are $\left\langle \frac{1}{c}, \frac{1}{c}, \frac{1}{c} \right\rangle$ then [Understanding]
  • (a)c > 0
  • (b)0 < c < 1
  • (c)c = ±3
  • (d)c > 2
(iv)[1.0]
If $\vec{a}$ is a unit vector perpendicular to $\vec{b}$ and $(\vec{a} + 2\vec{b}) \cdot (3\vec{a} - \vec{b}) = -5$, find $|\vec{b}|$. [Understanding]
(v)[1.0]
Shown below is a cuboid. Find $\vec{BA} \cdot \vec{BC}$. [Analysis]
Figure for part (v)

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2025 · 1 mark · MCQOpen: In the picture given above, take a look at the double-arrowed lines drawn on…
In the picture given above, take a look at the double-arrowed lines drawn on the overpass. This is an example of skew lines in the real world. Based on this, which of these statements is INCORRECT?
  • (a)These lines are not parallel.
  • (b)These lines are intersecting.
  • (c)These lines are not coplanar.
  • (d)These lines can only exist in 3 or higher dimensional space.
Figure for this question

No answer yet.

2025 · 4 marks · Long answerOpen: Two friends are planning a road trip. One friend stays in City A represented by…
Two friends are planning a road trip. One friend stays in City A represented by the position vector $(-2\hat{i} + 3\hat{j} + 5\hat{k})$. The trip will start from City A and proceed towards the City B represented by the position vector $(\hat{i} + 2\hat{j} + 3\hat{k})$. The friend living in City C represented by the position vector $7\hat{i} - \hat{k}$ will join when the first friend passes through her city.
(a)
Find the vector equation for the straight path between the cities A and B.
(b)
Hence, find out whether the three cities lie on the same straight path.
(c)
If the two friends now plan to travel $\sqrt{126}$ units along the vector $\vec{AB}$ from the City C, find the position vector of the destination point.

No answer yet.

2025 · 4 marks · DerivationOpen: Show that the line whose vector equation is is parallel to the plane whose…
Show that the line whose vector equation is $\vec{r} = (2\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k})$ is parallel to the plane whose vector equation is $\vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5$. Also find the distance between them. [Application]

Given: $\vec{r} = (2\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k}), \quad \vec{r} \cdot (\hat{i} + 5\hat{j} + \hat{k}) = 5$

To show: $\vec{r} \text{ is parallel to the plane}$

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2025 · 2 marks · Short answerOpen: Find the shortest distance between the lines:
Find the shortest distance between the lines: $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ $\vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(4\hat{i} + 6\hat{j} + 8\hat{k})$

No answer yet.

2025 · 4 marks · Long answerOpen: In the beautiful town of Darjeeling in the Himalayan foothills, the city…
In the beautiful town of Darjeeling in the Himalayan foothills, the city planning committee wants to construct two major roads to connect the various neighbourhoods. The two roads are represented by the equations $\frac{x+1}{3} = \frac{y+3}{5} = \frac{z+5}{7}$ and $\frac{x-2}{1} = \frac{y-4}{k} = \frac{z-6}{7}$. As an in charge of the planning committee, ensure that these roads lie on the same plane to facilitate efficient urban planning and infrastructure development.
(a)
For what value of k, will the construction meet the requirement?
(b)
Hence, find the equation of the plane containing these lines.

No answer yet.

2024 · 4 marks · Short answerOpen: A mobile tower is situated at the top of a hill. Consider the surface on which…
A mobile tower is situated at the top of a hill. Consider the surface on which the tower stands as a plane having points $A(1, 0, 2)$, $B(3, -1, 1)$ and $C(1, 2, 1)$ on it. The mobile tower is tied with three cables from the points $A$, $B$ and $C$ such that it stands vertically on the ground. The top of the tower is at point $P(2, 3, 1)$ as shown in the figure below. The foot of the perpendicular from the point $P$ on the plane is at the point $Q\left(\frac{43}{29}, \frac{77}{29}, \frac{9}{29}\right)$. Answer the following questions:
Figure for this question
(i)
Find the equation of the plane containing the points $A$, $B$ and $C$.
(ii)
Find the equation of the line $PQ$.
(iii)
Calculate the height of the tower.

No answer yet.

2023 · 1 mark · MCQOpen: The distance of the point from the plane will be:
The distance of the point $2\hat{i} + \hat{j} - \hat{k}$ from the plane $\vec{r} \cdot (\hat{i} - 2\hat{j} + 4\hat{k}) = 9$ will be:
  • (a)$13$
  • (b)$\frac{13}{\sqrt{21}}$
  • (c)$21$
  • (d)$\frac{21}{\sqrt{13}}$

No answer yet.

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