Three-dimensional Geometry - ISC Class 12 Mathematics Questions with Answers
63 past-paper questions on Three-dimensional Geometry from ISC Class 12 Mathematics papers (2027-2017), newest first. Open one to see its answer.
- A vector $\vec{n}$ of magnitude 8 units is inclined to x-axis at $45^\circ$, y-axis at $60^\circ$ and at an acute angle with… 2027 · 3 marks · Short answer
- The planes $\pi_1$ and $\pi_2$ have equations $2x + 6y - 2z = 5$ and $3x + 9y + pz = -\frac{51}{2}$ respectively. Verify that the… 2027 · 3 marks · Short answer
- A straight line $L_\theta$ has vector equation $\vec{r} = 5\hat{i} + \lambda(5\hat{i} + \sin\theta \hat{j} + \cos\theta \hat{k})$… 2027 · 2 marks · Short answer
- The point of intersection of the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{3-z}{-4}$ and… 2027 · 1 mark · MCQ
- The direction cosines of the line passing through the points $P(2, 3, 5)$ and $Q(-1, 2, 4)$ are: 2026 · 1 mark · MCQ
- Find the equation of the plane with intercepts $3, -4$ and $2$ on $x, y$ and $z$ axes respectively. 2026 · 1 mark · Short answer
- The paths traced by two hot air balloons are: $\frac{x - 1}{2} = \frac{y - b}{3} = \frac{z - 3}{4}$ and… 2026 · 4 marks · Short answer
- Find the foot of perpendicular from the point $P(2, 3, -8)$ to the line $\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z+4}{1}$. Also… 2026 · 4 marks · Short answer
- Find the equation of the plane passing through the points $(2, -3, 1)$ and $(-1, 1, -7)$ and perpendicular to the plane… 2026 · 4 marks · Short answer
- Find the equation of the plane parallel to ZOX plane with Y intercept 3. 2026 · 1 mark · Short answer
- Find the angle between the line $\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-3}{-3}$ and the plane $2x + y - 3z + 4 = 0$. 2026 · 1 mark · Short answer
- A school is preparing the stage for its annual day function. They want to place a hanging mic and a hanging light on the stage. -… 2026 · 4 marks · Case based
- Given, B and C lie on the line $\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$ and $BC = 5$ units find the area of $\Delta ABC$… 2026 · 4 marks · Short answer
- Find the equation of the plane passing through the point $(2, -1, 3)$ and containing the line… 2025 · 1 mark · MCQ
- The angle between the two planes $x + y + 2z = 9$ and $2x - y + z = 15$ is: 2025 · 1 mark · MCQ
- The angle between the line $\frac{x+1}{1} = \frac{y-1}{2} = \frac{z-2}{2}$ and the plane $2x - y + \sqrt{k}z + 4 = 0$ is… 2025 · 1 mark · Short answer
- Find the angle between the normal to the planes: $\vec{r} \cdot (\hat{i} - \hat{j} + \hat{k}) = 3$ and… 2025 · 1 mark · Short answer
- From any point $P(2, 1, 2)$ perpendiculars PM and PN are drawn to ZX and XY planes. If O is the origin, find the equation of the… 2025 · 4 marks · Long answer
- Two drones are being used for soil analysis over an area of farmland. Drone A has been programmed to fly on the path given by… 2025 · 4 marks · Short answer
- Two honeybees are flying parallel to each other in the garden to collect the nectar. The path traced by the bees is given in the… 2025 · 2 marks · Short answer
- 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… 2025 · 4 marks · Short answer
- Show that points $P(-2, 3, 5)$, $Q(1, 2, 3)$ and $R(7, 0, -1)$ are collinear. 2025 · 1 mark · Short answer
- 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… 2025 · 4 marks · Short answer
- A variable plane at unit distance from the origin cuts the co-ordinate axes at A, B and C. The centroid of the triangle formed by… 2025 · 1 mark · Short answer
- Find the equation of the plane containing the line $\frac{x}{-2} = \frac{y-1}{3} = \frac{1-z}{1}$ and the point $(-1, 0, 2)$… 2025 · 4 marks · Short answer
- In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. Consider… 2025 · 5 marks · Short answer
- 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… 2025 · 1 mark · MCQ
- The plane $px + y + pz = r$ intersects the co-ordinate axes at A, B and C. Find the co-ordinates of the points A, B and C. Find… 2025 · 4 marks · Long answer
- Find the equation of the straight line passing through $(1, 2, 1)$ and parallel to the line joining the points $(1, 4, 6)$ and… 2025 · 2 marks · Short answer
- Two friends are planning a road trip. One friend stays in City A represented by the position vector… 2025 · 4 marks · Long answer
- Show that the line whose vector equation is $\vec{r} = (2\hat{i} - 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - \hat{j} + 4\hat{k})$… 2025 · 4 marks · Derivation
- If $L$ is the foot of the perpendicular drawn from a point $P(a, b, c)$ on the $XY$ plane, what is the co-ordinate of $L$? 2025 · 1 mark · MCQ
- A line makes angles $\alpha, \beta$ and $\gamma$ with the x, y and z axes respectively. Given that… 2025 · 1 mark · Short answer
- Find the shortest distance between the lines… 2025 · 2 marks · Short answer
- Find the equation of the plane passing through the points $(2, 3, 1)$, $(4, -5, 3)$ and parallel to $x$-axis. 2025 · 2 marks · Short answer
- Find the equation of the plane passing through the points $(2, 2, -1)$, $(3, 4, 2)$ and $(7, 0, 6)$. 2025 · 2 marks · Short answer
- In the beautiful town of Darjeeling in the Himalayan foothills, the city planning committee wants to construct two major roads to… 2025 · 4 marks · Long answer
- A mobile tower is situated at the top of a hill. Consider the surface on which the tower stands as a plane having points… 2024 · 4 marks · Short answer
- 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… 2023 · 1 mark · MCQ
- Find the equation of the plane passing through the points $(0, 4, -3)$ and $(6, -4, 3)$ if the sum of their intercepts on three… 2023 · 4 marks · Short answer
- Show that $\frac{4-x}{-1} = \frac{y+3}{-4} = \frac{z+1}{7}$ and $\frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+10}{8}$ intersect each… 2023 · 4 marks · Derivation
- Find the equation of the plane passing through the point $(1, 1, -1)$ and perpendicular to the planes $x + 2y + 3z = 7$ and… 2023 · 4 marks · Short answer
- A line passes through the point $(2, -1, 3)$ and is perpendicular to the lines… 2023 · 4 marks · Short answer
- In subparts (i) and (ii) choose the correct options and in subparts (iii) to (v), answer the questions as instructed. If… 2023 · 5 marks · Short answer
- Write the equation of the plane passing through the point $(2, 4, 6)$ and making equal intercepts on the coordinate axes. 2023 · 1 mark · Short answer
- Choose the correct option for the following questions. The equation of the plane which is parallel to $2x - 3y + z = 0$ and which… 2022 · 2 marks · Short answer
- The equation of the line passing (1, -1, 0) and parallel to the line $\frac{x-1}{1} = \frac{y+2}{-2} = \frac{z+1}{-1}$ is: 2022 · 2 marks · MCQ
- Find the equation of the plane passing through the point $(1, 1, 1)$ and is perpendicular to the line… 2022 · 2 marks · Short answer
- What are the direction ratios of the line passing through two points (-2, 4, 5) and (1, 2, 3)? 2022 · 2 marks · MCQ
- What will be the angle between the two lines $\frac{-x+2}{-2} = \frac{y-1}{7} = \frac{z+3}{-3}$ and… 2022 · 2 marks · MCQ
- Find the equation of the lines passing through the point $(2, 1, 3)$ and perpendicular to the lines… 2021 · 4 marks · Short answer
- Find the distance of the point $A(3, 4, 4)$ from the point, where the line joining points $P(3, -4, -5)$ and $Q(2, -3, 1)$… 2021 · 4 marks · Short answer
- Determine the equation of the line passing through the point $(-1, 3, -2)$ and perpendicular to the lines… 2020 · 4 marks · Short answer
- Find the image of the point $(3, -2, 1)$ in the plane $3x - y + 4z = 2$. 2020 · 4 marks · Short answer
- Find the equation of the lines passing through the point $(2, 1, 3)$ and perpendicular to the lines… 2019 · 4 marks · Short answer
- Find the equation of the plane passing through the intersection of the planes $2x + 2y - 3z - 7 = 0$ and $2x + 5y + 3z - 9 = 0$… 2019 · 4 marks · Short answer
- [3×2] If $\vec{a}$ and $\vec{b}$ are perpendicular vectors, $|\vec{a} + \vec{b}| = 13$ and $|\vec{a}| = 5$, find the value of… 2019 · 6 marks · Short answer
- Find the shortest distance between the lines $\frac{x-8}{3} = \frac{y+9}{-16} = \frac{z-10}{7}$ and… 2018 · 4 marks · Short answer
- Find the image of a point having position vector $3\hat{\imath} - 2\hat{\jmath} + \hat{k}$ in the plane… 2018 · 6 marks · Short answer
- [3×2] Find the area of the parallelogram whose adjacent sides are given by the vectors… 2018 · 6 marks · Short answer
- Find the cartesian equation of the plane passing through the intersection of the planes… 2018 · 4 marks · Short answer
- Show that the lines $\frac{x-4}{1} = \frac{y+3}{-4} = \frac{z+1}{7}$ and $\frac{x-1}{2} = \frac{y+1}{-3} = \frac{z+10}{8}$… 2017 · 5 marks · Short answer
- Find the equation of the plane passing through the point $(1, -2, 1)$ and perpendicular to the line joining the points… 2017 · 5 marks · Short answer
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