JEE Main Paper 2 (B.Arch) Aptitude - Visualisation and Reasoning Flashcards

24 sample cards from the Aptitude - Visualisation and Reasoning deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. If a die shows $2$ on top and $3$ facing you, what number is on the bottom and the back?

   Bottom $= 7-2 = 5$, back $= 7-3 = 4$.

2. What is a 'net' (surface development) of a solid?

   A net is the unfolded, flattened two-dimensional pattern that, when folded along its edges, forms the surface of the 3D solid.

3. How many distinct nets (hexomino folding patterns) can form a cube?

   There are $11$ distinct nets that fold into a cube.

4. In a cube net, how can you identify which two squares become opposite faces?

   Two squares are opposite if they are separated by exactly one square in a straight line, or if they sit at the two ends of an 'L'/zig-zag such that they never share an edge after folding; opposite faces never touch along an edge in the assembled cube.

5. What does the lateral surface development of a cylinder of radius $r$ and height $h$ look like, and what are its dimensions?

   It is a rectangle of width equal to the circumference $2\pi r$ and height $h$: $$\text{width} = 2\pi r,\quad \text{height} = h.$$

6. What is the surface development of a right circular cone of base radius $r$ and slant height $l$?

   A sector (part of a circle) of radius $l$ whose arc length equals the base circumference $2\pi r$. The sector angle is $$\theta = \frac{r}{l}\times 360^{\circ} = \frac{2\pi r}{l}\ \text{radians}.$$

7. State the total surface area of a cube of edge $a$.

   $$\text{TSA} = 6a^{2}.$$

8. State the total surface area of a closed right circular cylinder of radius $r$ and height $h$.

   $$\text{TSA} = 2\pi r h + 2\pi r^{2} = 2\pi r(h+r).$$

9. State the total surface area of a right circular cone of radius $r$ and slant height $l$.

   $$\text{TSA} = \pi r l + \pi r^{2} = \pi r(l+r).$$

10. State the surface area of a sphere of radius $r$.

    $$\text{SA} = 4\pi r^{2}.$$

11. In paper folding problems, after a sheet is folded and a hole is punched, how do you find the number of holes when unfolded?

    Each fold doubles a punched hole by reflection across the fold line. With $n$ thicknesses of paper at the punch location, unfolding produces holes mirror-symmetric about each fold crease; count by reflecting the punched position back across every fold in reverse order.

12. A square paper folded once in half then punched once has how many holes when fully unfolded?

    $2$ holes — the single fold creates two layers, so one punch passes through both, giving two symmetric holes.

13. A square paper folded in half twice (into quarters) and punched once gives how many holes?

    $4$ holes (two folds create four layers, so $2^{2}=4$ punched holes after unfolding).

14. What is the rule relating an object and its mirror image regarding left-right and up-down?

    A plane mirror produces lateral inversion: left and right appear swapped, while top and bottom remain unchanged. The image is virtual, erect, and the same size as the object.

15. How does a letter like 'b' appear in a vertical plane mirror placed to its side?

    It appears laterally inverted as 'd'; letters with vertical-axis symmetry (A, H, I, M, O, T, U, V, W, X, Y) look unchanged.

16. Which capital letters remain unchanged in a vertical (left-right) mirror reflection?

    Letters symmetric about a vertical axis: A, H, I, M, O, T, U, V, W, X, Y.

17. What distinguishes a mirror image from a water (water-surface) image?

    A mirror (vertical mirror) inverts left-right, while a water image inverts top-bottom (vertical flip), as the reflection appears beneath the object as if mirrored across a horizontal water surface.

18. In a water reflection, how does the object appear?

    It appears inverted top-to-bottom (upside down) directly below the original, with vertical features flipped while left-right order is preserved.

19. Which capital letters look unchanged in a water (horizontal-axis) reflection?

    Letters symmetric about a horizontal axis: B, C, D, E, H, I, K, O, X.

20. A clock shows time $T$ in a mirror. How do you find the actual time?

    Subtract the shown time from $11{:}60$ (i.e. $12{:}00$): $$\text{actual} = 11{:}60 - T.$$ For example a mirror reading of $4{:}20$ corresponds to $11{:}60 - 4{:}20 = 7{:}40$.

21. When matching rotated views of an asymmetric solid, what feature is most useful to track?

    Track a unique 'handedness' or distinctive marker (an asymmetric notch, arrow, or differently coloured face) and follow its position relative to neighbouring faces; rotation preserves the cyclic order of faces around an axis, while reflection reverses it.

22. How can you tell if a transformation between two figures is a pure rotation rather than a reflection, using their corner orderings?

    List the vertices in clockwise order for both figures: a pure rotation preserves the clockwise (cyclic) order, whereas a reflection reverses it to counter-clockwise.

23. For a regular tetrahedron, how many distinct rotational orientations map it onto itself?

    $12$ rotational symmetries (the rotation group has order $12$).

24. How many rotational symmetries does a cube have?

    $24$ distinct rotations map a cube onto itself.

What this deck covers

Aptitude - Visualisation and Reasoning flashcards FAQ