EVERY GATE PAPER

Spatial aptitude

Spatial aptitude tests whether you can predict how a figure looks after it is moved, turned, mirrored, resized, folded, cut, assembled or viewed from another side, in two and three dimensions. A small set of exact rules settles every such question: coordinate rules for rotation and reflection, reflection in each fold line when a sheet is unfolded, opposite-face rules for cube nets and dice, and counting formulas for grids. Working through them step by step is more reliable than guessing from the picture.

23 previous-year questions · 22 solved · 5 practice questions

Translation, rotation, reflection and scaling

A spatial question gives a figure and asks what it looks like after one or more operations. Four operations make up the transformation of shapes. Translation slides every point the same distance in the same direction, so nothing turns. Rotation turns the figure about a fixed centre through a stated angle. The direction matters: clockwise is the way a clock's hands move, and anticlockwise is the opposite way. Reflection (mirroring) flips the figure across a mirror line, and each point lands the same distance away on the other side. Scaling enlarges or shrinks the figure about a centre by a factor $k$ . The shape and every angle stay the same, and every length is multiplied by $k$ .

Translation, rotation and reflection keep lengths, angles and area unchanged. Translation and rotation also keep handedness: a letter F stays a readable F however it is turned. A reflection reverses handedness, and no rotation can undo that. This settles whether an option is a rotation of the given figure or its mirror image. The one exception is a figure that has a line of symmetry: its mirror image is then also a rotation of it, and the two cannot be told apart.

Conventions used below. Coordinates have $x$ to the right and $y$ upward. Angles are in degrees, and a positive angle means anticlockwise. A vertical mirror is a mirror line running up and down (the line $x = 0$ ), and it swaps left and right. A horizontal mirror runs left to right (the line $y = 0$ ). It swaps top and bottom, and its result is often called the water image.

TRANSLATION BY (A, B)

(x, y) \to (x + a, y + b)

Every point moves by the same amount. Size, shape and orientation are unchanged.

ROTATION ABOUT THE ORIGIN THROUGH AN ANGLE Θ ANTICLOCKWISE

x^{'} y^{'} = x cos θ - y sin θ = x sin θ + y cos θ

θ is in degrees and positive for anticlockwise; use −θ for a clockwise turn. For a centre other than the origin, use the next formula.

ROTATION ABOUT A POINT (A, B) THROUGH AN ANGLE Θ ANTICLOCKWISE

x^{'} y^{'} = a + (x - a) cos θ - (y - b) sin θ = b + (x - a) sin θ + (y - b) cos θ

This is the origin rule applied to each point's position relative to $(a, b)$ : shift $(a, b)$ to the origin, rotate, then shift back. For a quarter turn anticlockwise it gives $(x, y) \to (a - (y - b), b + (x - a))$ .

QUARTER AND HALF TURNS ABOUT THE ORIGIN

9 0^{\circ} anticlockwise: 9 0^{\circ} clockwise: 18 0^{\circ} either way: (x, y) \to (- y, x) (x, y) \to (y, - x) (x, y) \to (- x, - y)

These are exact special cases of the rotation rule. For any other centre, apply them to each point's position relative to that centre.

CLOCKWISE AND ANTICLOCKWISE

θ clockwise = (36 0^{\circ} - θ) anticlockwise

So $27 0^{\circ}$ clockwise is $9 0^{\circ}$ anticlockwise. Reduce any angle modulo $36 0^{\circ}$ first: $45 0^{\circ}$ clockwise is $9 0^{\circ}$ clockwise.

REFLECTION IN A MIRROR LINE THROUGH THE ORIGIN

x = 0 (vertical): y = 0 (horizontal): y = x : y = - x : (x, y) \to (- x, y) (x, y) \to (x, - y) (x, y) \to (y, x) (x, y) \to (- y, - x)

A vertical mirror (the y-axis) swaps left and right. A horizontal mirror (the x-axis) swaps top and bottom. Reflecting twice in the same line gives back the original figure.

REFLECTION IN A LINE NOT THROUGH THE ORIGIN

x = a : y = b : (x, y) \to (2 a - x, y) (x, y) \to (x, 2 b - y)

The mirror line is the perpendicular bisector of the segment joining each point to its image. Points on the mirror line stay where they are.

SCALING BY A FACTOR K

length \times k, area \times k^{2}, volume \times k^{3}

$k > 1$ enlarges and $0 < k < 1$ shrinks. Angles are unchanged. The same factors apply to similar solids: surface area $\times k^{2}$ , volume $\times k^{3}$ .

SCALING ABOUT A CENTRE (A, B) BY A FACTOR K

(x, y) \to (a + k (x - a), b + k (y - b))

Each point moves along the line from the centre through it, to $k$ times its original distance from the centre. The centre itself does not move. With the origin as centre this is $(x, y) \to (k x, k y)$ .

TWO REFLECTIONS MAKE A ROTATION

M_{2} \circ M_{1} = R_{2 α}

Reflect in line $m_{1}$ first, then in line $m_{2}$ . Let the lines meet at a point O, and let $α$ be the angle through which $m_{1}$ must turn about O to lie along $m_{2}$ . The result is a rotation about O through $2 α$ , in that same direction. Two parallel lines a distance $d$ apart give a translation through $2 d$ , at right angles to the lines and in the direction from $m_{1}$ towards $m_{2}$ . A vertical mirror followed by a horizontal one (or the other way round) gives a $18 0^{\circ}$ turn about the point where they cross.

CLOCK SEEN IN A VERTICAL MIRROR

t_{mirror} = 11 : 60 - t

The mirror stands beside the clock, so it reflects in the line through 12 and 6. Subtract the actual time from 11:60 (that is, from 12:00): 4:25 gives 7:35, and 9:10 gives 2:50. For a time from 12:00 to 12:59, subtract from 23:60 instead (12:10 gives 11:50). Read a result of 0:mm as 12:mm (11:25 gives 0:35, that is 12:35). The rule is exact for both hands.

To follow a figure through several moves, follow the directions its features point in. The table shows where each direction goes under each operation.

Where a direction goes

Before	$9 0^{\circ}$ clockwise	$9 0^{\circ}$ anticlockwise	$18 0^{\circ}$	Vertical mirror	Horizontal mirror
Up	Right	Left	Down	Up	Down
Right	Down	Up	Left	Left	Right
Down	Left	Right	Up	Down	Up
Left	Up	Down	Right	Right	Left

WORKED EXAMPLE · ROTATION FOLLOWED BY REFLECTION

A point $P (3, 1)$ belongs to a figure. The figure is rotated $9 0^{\circ}$ anticlockwise about the origin, and the result is then reflected in the $x$ -axis. Find the final position of $P$ and the single transformation that has the same effect on every point.

Solution

Rotation $9 0^{\circ}$ anticlockwise: $(x, y) \to (- y, x)$ , so $P (3, 1) \to (- 1, 3)$ .
Reflection in the $x$ -axis: $(x, y) \to (x, - y)$ , so $(- 1, 3) \to (- 1, - 3)$ .
For a general point: $(x, y) \to (- y, x) \to (- y, - x)$ .
The rule $(x, y) \to (- y, - x)$ is reflection in the line $y = - x$ . This is consistent with counting reflections: one rotation and one reflection make one reflection overall, so handedness is reversed.

Answer: $P$ ends at $(- 1, - 3)$ . The combined effect is a reflection in the line $y = - x$ .

EXAM TIP

Count the reflections. With an even number (including zero), handedness is unchanged and the net effect is a rotation or a translation. With an odd number, handedness is reversed and the net effect is a reflection, possibly combined with a slide along the mirror line (a glide reflection). So if an option is the mirror image of a figure that has no line of symmetry, no sequence of rotations and translations can produce it.

EXAM TIP

A turn that looks clockwise from the front looks anticlockwise from behind. If a figure is drawn on a transparent sheet and the sheet is turned over, the figure you now see is its mirror image.

EXAM TIP

In simple block capitals, the letters A, H, I, M, O, T, U, V, W, X and Y look the same in a vertical mirror, and B, C, D, E, H, I, K, O and X look the same in a horizontal mirror. A word seen in a vertical mirror also has its letter order reversed. In a horizontal mirror the order stays the same and each letter turns upside down.

EXAM TIP

Do not use a subtraction rule for a clock seen in a horizontal mirror (its water image): the two reflected hands usually do not show a time a real clock could show. The minute hand at $m$ minutes appears at $30 - m$ minutes (add 60 if this is negative), and the hour hand must be reflected separately in the line through 3 and 9. Read each hand on its own.

EXAM TIP

Follow two features rather than the whole figure, for example an arrowhead and a dot. Where they point after the move fixes the answer.

Symmetry, assembling and grouping

A figure has line symmetry about a line if reflecting it in that line leaves it unchanged. It has rotational symmetry of order $N$ if it looks the same in $N$ positions during one full turn about its centre. Order $N = 1$ means no rotational symmetry. A figure that looks the same after a $18 0^{\circ}$ turn is said to have point symmetry; its order of rotational symmetry is even.

SMALLEST ANGLE OF ROTATIONAL SYMMETRY

θ_{m i n} = \frac{36 0 ^{\circ}}{N}

N is the order of rotational symmetry. A regular polygon with n sides has N = n and n lines of symmetry.

Symmetry of common figures

Figure	Lines of symmetry	Order of rotational symmetry
Equilateral triangle	3	3
Isosceles triangle (not equilateral)	1	1
Square	4	4
Rectangle (not a square)	2	2
Rhombus (not a square)	2 (the diagonals)	2
Parallelogram (not a rectangle or rhombus)	0	2
Kite (not a rhombus)	1	1
Isosceles trapezium	1	1
Regular hexagon	6	6
Circle	Infinitely many	Infinite (any angle)
Capital letters N, S, Z (block style)	0	2

Assembling questions give a set of pieces and ask which target shape they form, or which piece completes a shape. First read whether the pieces may be turned over. If they may not (for example, because they are printed on one side only), allow only translations and rotations: a piece shaped like an L or an F and its mirror image are then different pieces. If the question does not say, check whether an option needs a piece to be turned over before you accept it.

Procedure for assembling pieces

Check area first. Add the areas of the pieces (count unit squares or unit triangles) and compare with the target. Unequal areas rule out an option at once.
Check the longest straight edges. A straight edge of the target of length L must be made from piece edges that add up to exactly L.
Check the corners. Every corner of the target must be filled by one piece corner of that angle, or by several piece corners whose angles add up to it.
Place the largest or most awkward piece first, then fill the remaining gap. The shape of the gap usually forces the next piece.
If the pieces are drawn on a grid, colour the grid like a chessboard and compare the colours each piece covers (see the example).

SIDE OF A SQUARE ASSEMBLED FROM PIECES

s = A_{1} + A_{2} + \dots + A_{m}

The pieces fit together with no overlap and no gap, so their areas add. For example, pieces with a total area of 20 cm² can only form a square of side $20 = 25$ cm. This is a necessary condition, not a guarantee that the pieces fit.

Grouping questions ask for the odd one out, or for the figures that belong together. They usually hide a transformation: three options are rotations of one figure and the fourth is its mirror image, or three are the same shape at different scales. The quickest check is the clockwise-order test in the tip below.

WORKED EXAMPLE · AREA PASSES, COLOURING FAILS

A T-piece is made of four unit squares: three in a straight row, with a fourth attached to the middle square of the row. Can nine T-pieces be assembled into a $6 \times 6$ square, if pieces may be turned over?

Solution

Area check: $9 \times 4 = 36 = 6 \times 6$ . The area test is passed, so it cannot settle the question.
Colour the $6 \times 6$ square like a chessboard. It has 18 dark and 18 light unit squares.
Wherever a T-piece is placed, its middle square and the square attached to it have opposite colours, and the two end squares of the row have the same colour as the attached one. So each piece covers 3 squares of one colour and 1 of the other, and adds $+ 2$ or $- 2$ to the difference (dark minus light).
Suppose $a$ pieces give $+ 2$ and $9 - a$ give $- 2$ . The total difference is $2 a - 2 (9 - a) = 4 a - 18$ .
The square needs a difference of 0, which would require $a = 4.5$ . That is impossible because $a$ is a whole number.

Answer: No. The area is right, but the colour count rules out every arrangement. In general, a region with equal numbers of dark and light squares can only be covered by an even number of T-pieces.

EXAM TIP

Clockwise-order test for rotation versus reflection: pick three distinct features of the figure, such as a dot, a shaded corner and an arrowhead, and read them going clockwise. In a rotated copy they come in the same cyclic order. In a mirror image the order is reversed.

EXAM TIP

Counting distinct shapes made of unit squares joined full edge to full edge: two such shapes are the same if one can be moved exactly onto the other. When both rotations and reflections are allowed, there are 2 shapes made of 3 squares, 5 made of 4 squares and 12 made of 5 squares. If turning over is not allowed, the counts are 2, 7 and 18.

Paper folding, cutting and punching

Folding a sheet along a line is a reflection: the part that moves lands on its own mirror image in the fold line. So anything punched or cut through the folded packet appears again, after unfolding, at its mirror image in each fold line. The folds are undone in reverse order. Every layer of paper under the punch gets its own hole.

HOLES FROM ONE PUNCH THROUGH N COMPLETE HALF-FOLDS

holes = layers under the punch = 2^{n}

Valid when every fold doubles the whole packet and the punch does not touch any fold line. A punch that touches a fold line gives fewer holes, because holes merge across the fold.

STRIP FOLDED IN HALF N TIMES, THEN CUT ONCE ACROSS

pieces = 2^{n} + 1

All the folds are parallel and each one folds the whole strip in half, so the packet is $2^{n}$ layers thick. One straight cut runs right across the packet without touching a fold. It cuts the strip at $2^{n}$ places, which leaves $2^{n} + 1$ pieces. For example, 3 folds and one cut give 9 pieces.

Unfolding procedure

Write down each fold: its fold line, and which part moved onto which. Coordinates, with the sheet running from (0, 0) to (s, s), make diagonal folds painless.
Mark the punch or cut on the final folded shape.
Undo the last fold first. Reflect every hole in that fold line and keep the original as well, because both layers were punched.
Undo each earlier fold the same way, working back to the first fold. The number of holes doubles at each step, except for a hole that touches the fold line.
A hole or notch touching a fold line opens into a single hole that is symmetric about that line. A semicircular notch whose diameter lies on the fold becomes a full circle. A triangular notch whose base lies on the fold becomes a kite with the fold as its axis. The kite is a rhombus if the two sides meeting at the notch's apex are equal (a square if they also meet at a right angle), and an arrowhead shape if one of the angles at the base is obtuse.
Check the answer: when every fold is a complete half-fold, the final pattern is mirror-symmetric about the first fold line.

WORKED EXAMPLE · TWO FOLDS, ONE PUNCH AND ONE NOTCH

A square sheet of side 8 cm has corners at (0, 0), (8, 0), (8, 8) and (0, 8), in cm. It is folded so that its right half lies on its left half (fold line $x = 4$ ), and then so that its top half lies on its bottom half (fold line $y = 4$ ). The folded packet now occupies $0 \leq x \leq 4$ , $0 \leq y \leq 4$ . A small round hole is punched at (1, 1). A semicircular notch of radius 0.5 cm is also cut, with its diameter on the folded edge $y = 4$ and its centre at (2, 4). How many holes does the unfolded sheet have, and where are they?

Solution

The punch at (1, 1) does not touch either fold line and passes through all 4 layers.
Undo the second fold by reflecting in $y = 4$ , that is $(x, y) \to (x, 8 - y)$ . The punch gives holes at (1, 1) and (1, 7). The notch lay on this fold line, so it opens into one full circle of radius 0.5 cm centred at (2, 4).
Undo the first fold by reflecting in $x = 4$ , that is $(x, y) \to (8 - x, y)$ . The holes at (1, 1) and (1, 7) gain partners at (7, 1) and (7, 7). The circle at (2, 4) gains a partner at (6, 4).
The round hole was doubled twice (4 holes). The notch was doubled only once (2 circles) because it lay on a fold line.

Answer: 6 holes: four small holes near the corners at (1, 1), (1, 7), (7, 1) and (7, 7), and two circles of radius 0.5 cm on the horizontal centre line, centred at (2, 4) and (6, 4).

EXAM TIP

Fold a square sheet in half along one midline, then in half again along the other midline. The corner of the packet where the two fold lines meet is the centre of the original sheet. Cut off a small right-angled triangle at that corner, with its two shorter sides along the folds. Unfolding leaves one hole at the centre of the sheet: a square standing on a corner if the two cut sides are equal, and a rhombus otherwise.

EXAM TIP

Holes appear only where there was paper under the punch. If a fold covers only part of the sheet, count the layers at the exact punch point instead of using $2^{n}$ .

EXAM TIP

Take a square with corners (0, 0) and (s, s) on one diagonal. Reflection in that diagonal is $(x, y) \to (y, x)$ . Reflection in the other diagonal, the line $x + y = s$ , is $(x, y) \to (s - y, s - x)$ .

Cubes and dice: nets, opposite faces and painted cubes

A cube has 6 faces, 12 edges and 8 corners. Each face shares an edge with 4 others and is opposite exactly one. The usual drawing of a cube or die shows the 3 faces that meet at one corner. So the three faces seen in such a view are always mutually adjacent, and none of them is opposite another.

Rules for a net (the flat six-square pattern that folds into a cube)

Squares that share an edge in the net are adjacent on the cube.
In any straight row or column of the net, alternate squares are opposite. In a row of three, the 1st and 3rd are opposite. In a row of four, the 1st and 3rd are opposite and so are the 2nd and 4th. A net never has five squares in a line.
Z rule: suppose four squares sit in grid cells (column, row) = (1, 1), (2, 1), (2, 2) and (3, 2), or in any rotation or mirror image of this shape. That is two side-by-side pairs, with the second pair one row up and one column across. Then the two end squares, (1, 1) and (3, 2), are opposite.
A 2 × 2 block of squares can never appear in a net, because only three faces meet at a corner of a cube. There are exactly 11 different nets of a cube.

Rules for several views of one die

Two views with two faces in common: if their third faces differ, those third faces are opposite. Only two faces are adjacent to both common faces, and they are opposite each other.
Two views with exactly one face in common: in each view, list the three faces going clockwise round the visible corner, starting with the common face. The second faces of the two lists are opposite, and so are the third faces.
Elimination: a face seen in any view together with X is not opposite X. Once four different faces have been seen with X, the remaining face is opposite X.

OPPOSITE FACES OF A STANDARD DIE

1 + 6 = 2 + 5 = 3 + 4 = 7

Use this only when the question says the die is standard or ordinary. For any other die, work out the opposite faces from the views.

PAINTED N × N × N CUBE CUT INTO UNIT CUBES

3 painted faces 2 painted faces 1 painted face no paint = 8 = 12 (n - 2) = 6 (n - 2)^{2} = (n - 2)^{3}

Valid for all six faces painted and $n \geq 2$ . The four counts add up to $n^{3}$ . Cubes with at least one painted face: $n^{3} - (n - 2)^{3} = 6 n^{2} - 12 n + 8$ .

PAINTED A × B × C CUBOID CUT INTO UNIT CUBES

3 painted faces 2 painted faces 1 painted face no paint = 8 = 4 (p + q + r) = 2 (pq + q r + r p) = pq r

Here p = a − 2, q = b − 2 and r = c − 2. Valid for all six faces painted and a, b, c ≥ 2. If some faces are unpainted, count directly by layer, row and column.

PIECES FROM PLANE CUTS PARALLEL TO THE FACES

pieces = (x + 1) (y + 1) (z + 1)

x, y and z are the numbers of cuts in the three directions, with no pieces moved between cuts. For a fixed total number of cuts, the number of pieces is greatest when x, y and z are as nearly equal as possible. To cut a cube into $n^{3}$ equal cubes without moving pieces between cuts, the minimum number of plane cuts is $3 (n - 1)$ .

WORKED EXAMPLE · OPPOSITE FACES FROM THREE VIEWS

The faces of a cube carry the letters A to F. Three views each show the three faces that meet at one corner. View 1 shows A, B and C. View 2 shows A, C and E. View 3 shows A, B and F. Which letter is opposite each of A, B and C?

Solution

Views 1 and 2 share A and C, so their third faces, B and E, are opposite.
Views 1 and 3 share A and B, so their third faces, C and F, are opposite.
A is seen with B, C, E and F, so none of these is opposite A. The only letter left is D, so A and D are opposite.
Check: the pairs (A, D), (B, E) and (C, F) use each letter exactly once.

Answer: A is opposite D, B is opposite E, and C is opposite F.

WORKED EXAMPLE · PAINTED CUBE AND CUTS

A cube of edge 4 cm is painted on all six faces and then cut into cubes of edge 1 cm, with no pieces moved between cuts. Find (a) the minimum number of cuts, (b) the number of small cubes with exactly two painted faces, and (c) the number with at least one painted face.

Solution

(a) Each direction needs $4 - 1 = 3$ cuts, so the total is $3 (4 - 1) = 9$ .
(b) $12 (n - 2) = 12 \times 2 = 24$ . These are the cubes along each edge, excluding the corners.
(c) The unpainted cubes form the inner $2 \times 2 \times 2$ block of $(4 - 2)^{3} = 8$ cubes, so $64 - 8 = 56$ cubes have paint.
Check: $8 + 24 + 6 (2)^{2} + 8 = 8 + 24 + 24 + 8 = 64$ .

Answer: (a) 9 cuts; (b) 24; (c) 56.

EXAM TIP

The painted-cube formulas assume paint on all six faces. If the bottom face (or any other face) is left unpainted, the layer next to it can contain unpainted cubes. Count by position instead of using the formula.

EXAM TIP

A face can never be opposite a face it appears beside in any view. Use this to rule out wrong pairs quickly before applying the two-view rules.

EXAM TIP

Turning a cube: a quarter turn about a vertical axis leaves the top and bottom faces where they are and moves each side face one place round. Tipping the cube forward over its front bottom edge moves the top face to the front, the front to the bottom, the bottom to the back and the back to the top, and leaves the left and right faces where they are.

EXAM TIP

Knowing the three opposite pairs does not fix a die completely. On a standard die, 1, 2 and 3 meet at one corner, and they can run either clockwise or anticlockwise round it, so there are two mirror-image standard dice. When an answer depends on which way round the faces run, decide it from the views given.

Counting squares, rectangles, triangles and cubes

Counting by eye misses figures. Count by a rule instead: either by size (every $1 \times 1$ , then every $2 \times 2$ , and so on) or by choosing the lines that form each figure. For a complete grid, choosing lines gives closed formulas. For an irregular figure, list the figures size by size and write down each count before adding. An $m \times n$ grid here means $m$ rows and $n$ columns of unit squares, drawn with $m + 1$ horizontal and $n + 1$ vertical lines.

SQUARES IN AN N × N GRID

k = 1 \sum n k^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}

All sizes counted. For example, a 3 × 3 grid has 9 + 4 + 1 = 14 squares, and an 8 × 8 chessboard has 204.

SQUARES IN AN M × N GRID (M ≥ N)

k = 1 \sum n (m - k + 1) (n - k + 1) = \frac{n ( n + 1 ) ( 3 m - n + 1 )}{6}

A k × k square fits in (m − k + 1)(n − k + 1) positions, for k = 1 to n. For example, a 4 × 3 grid has 12 + 6 + 2 = 20 squares, and the closed form gives $\frac{3 \times 4 \times 10}{6} = 20$ .

RECTANGLES (SQUARES INCLUDED) IN AN M × N GRID

^{m + 1} C_{2} \times^{n + 1} C_{2} = \frac{m ( m + 1 ) n ( n + 1 )}{4}

Choose 2 of the m + 1 horizontal lines and 2 of the n + 1 vertical lines. For example, a 4 × 3 grid has 10 × 6 = 60 rectangles, and an 8 × 8 chessboard has 36 × 36 = 1296.

CUBES IN AN N × N × N BLOCK

k = 1 \sum n k^{3} = (\frac{n ( n + 1 )}{2})^{2}

Cubes of every size are counted. A k × k × k cube fits in $(n - k + 1)^{3}$ positions.

CUBOIDS IN AN A × B × C BLOCK

^{a + 1} C_{2} \times^{b + 1} C_{2} \times^{c + 1} C_{2}

Choose 2 of the dividing planes in each of the three directions (outer faces included).

TRIANGLES: K LINES FROM A VERTEX TO THE OPPOSITE SIDE, H LINES PARALLEL TO THAT SIDE

^{k + 2} C_{2} \times (h + 1)

The k lines run from the vertex to different interior points of the opposite side, and each parallel line crosses the whole triangle. Every triangle uses two of the k + 2 lines through the vertex and one of the h + 1 lines parallel to the base (the base included).

TRIANGLES IN A TRIANGULAR GRID OF SIDE N

⌊ \frac{n ( n + 2 ) ( 2 n + 1 )}{8} ⌋

An equilateral triangle is divided into $n^{2}$ small equilateral triangles by lines parallel to its sides. The formula counts upright and inverted triangles of all sizes: n = 2 gives 5, n = 3 gives 13 and n = 4 gives 27.

LINES AND TRIANGLES THROUGH GIVEN POINTS

lines triangles =^{n} C_{2} -^{p} C_{2} + 1 =^{n} C_{3} -^{p} C_{3}

There are n points, exactly p of them (p ≥ 2) on one straight line, and no other three collinear. With no three collinear, lines = $^{n} C_{2}$ and triangles = $^{n} C_{3}$ .

DIAGONALS AND CORNER TRIANGLES OF A CONVEX POLYGON WITH N CORNERS

diagonals triangles with all three corners at the polygon’s corners = \frac{n ( n - 3 )}{2} =^{n} C_{3}

For example, a hexagon has 9 diagonals and 20 such triangles. The second count is not the number of triangles visible when all the diagonals are drawn, because that also includes triangles with corners where diagonals cross.

Counting triangles by line directions. A triangle is formed by three drawn lines that are pairwise non-parallel and do not all pass through one point, with all three sides lying along drawn segments. Group the drawn lines by direction. Each triangle then uses three different directions, and the choices can be counted one group at a time.

WORKED EXAMPLE · SQUARE WITH DIAGONALS AND MIDLINES

A square has both diagonals drawn, and also the two lines joining the midpoints of opposite sides. How many triangles are there in the figure?

Solution

Take the square from (0, 0) to (2, 2). The drawn lines fall into four directions: horizontal $y = 0, 1, 2$ ; vertical $x = 0, 1, 2$ ; diagonal $y = x$ ; diagonal $y = 2 - x$ . All of them run fully across the square.
Both diagonals with one horizontal line: the diagonals meet at the centre (1, 1). With $y = 0$ or $y = 2$ they form a triangle whose base is a side of the square and whose apex is the centre. $y = 1$ passes through the centre and forms nothing. That gives 2 triangles, and using a vertical line instead gives 2 more, making 4.
One horizontal line $y = a$ , one vertical line $x = b$ and the diagonal $y = x$ : the vertices are $(b, a)$ , $(a, a)$ and $(b, b)$ , which form a triangle whenever $a \neq = b$ . There are $3 \times 3 - 3 = 6$ such choices of $(a, b)$ .
The same with the diagonal $y = 2 - x$ gives 6 more.
Two lines in the same direction are parallel, so no other choice of three lines works. Total: $4 + 6 + 6 = 16$ .
Cross-check by size: 8 smallest triangles; 4 made of two smallest ones (a side of the square with the centre as apex); 4 half-squares cut off by a diagonal. $8 + 4 + 4 = 16$ .

Answer: 16 triangles.

EXAM TIP

The grid formulas apply only to complete grids, where every line runs the full width or height. If a line stops short, count the affected sizes by hand.

EXAM TIP

Read the question carefully: in the same grid, 'how many squares' and 'how many rectangles' have different answers. A 4 × 3 grid has 20 squares but 60 rectangles.

Views of solids, cube stacks and figure series

Views. The top view (plan) is what you see looking straight down. The front view (elevation) is what you see looking horizontally at the front. In these notes the side view is the view from the right-hand side; check which side a question asks for. Each view is drawn without perspective, like a shadow with the visible edges drawn in.

Views of simple solids

Solid (resting on its base)	Top view	Front view	Side view
Cube	Square	Square	Square
Cuboid, length l left to right, breadth b front to back, height h	Rectangle l × b	Rectangle l × h	Rectangle b × h
Cylinder standing on a circular face	Circle	Rectangle	Rectangle
Cone standing on its base, apex up	Circle with a dot at the centre (the apex)	Isosceles triangle	Isosceles triangle
Sphere	Circle	Circle	Circle
Hemisphere, flat face down	Circle	Semicircle	Semicircle
Square pyramid, base edges parallel to the viewing directions	Square with both diagonals	Isosceles triangle	Isosceles triangle

EULER'S FORMULA FOR A CONVEX POLYHEDRON

V - E + F = 2

V vertices, E edges, F faces. For a cube, 8 − 12 + 6 = 2. A prism on an n-sided base has V = 2n, E = 3n, F = n + 2. A pyramid on an n-sided base has V = n + 1, E = 2n, F = n + 1.

Stacks of unit cubes. Describe a stack by its top-view grid, writing the number of cubes in each column in its cell. The total number of cubes is the sum of all the entries. The front view shows, for each column of the grid from left to right, the tallest stack in that column. The right side view shows, for each row, the tallest stack in that row, with the front row on the viewer's left. Unless the question says otherwise, assume no cube floats: a column of height $h$ contains $h$ cubes.

COMPLETING A STACK TO AN L × B × H CUBOID

cubes to add = l \times b \times h - N

N is the number of cubes already in the stack. The cuboid must enclose the whole stack.

WORKED EXAMPLE · READING A STACK

A stack of unit cubes stands on a $3 \times 3$ square base. Its top view gives the number of cubes in each column as follows. Back row, left to right: 3, 1, 2. Middle row: 2, 0, 1. Front row: 1, 1, 1. No cube floats. Find (a) the number of cubes, (b) the heights shown in the front view and in the right side view, and (c) the number of cubes needed to turn the stack into a solid $3 \times 3 \times 3$ cube.

Solution

(a) Add the entries: $(3 + 1 + 2) + (2 + 0 + 1) + (1 + 1 + 1) = 6 + 3 + 3 = 12$ .
(b) Front view, grid columns from left to right: $max (3, 2, 1) = 3$ , $max (1, 0, 1) = 1$ , $max (2, 1, 1) = 2$ . So the heights are 3, 1, 2.
Right side view: seen from the right, the front row is on the viewer's left. Row maxima: front row $max (1, 1, 1) = 1$ , middle row $max (2, 0, 1) = 2$ , back row $max (3, 1, 2) = 3$ . So the heights are 1, 2, 3 from left to right.
(c) $27 - 12 = 15$ .

Answer: (a) 12 cubes; (b) front view heights 3, 1, 2 and right side view heights 1, 2, 3, both read left to right; (c) 15 cubes.

Figure series and patterns. A figure series often runs several independent rules at once. One element turns by a fixed angle, another moves round the corners, a count goes up by one, a shading alternates. Split the figure into its elements and find each element's rule separately. The answer must satisfy all the rules at once.

ELEMENT MOVING ROUND N EQUALLY SPACED POSITIONS

p_{t} = (p_{0} + s t) mod n

The element starts at position $p_{0}$ and moves s positions per step, counted in the stated direction. For an angle, the turn after t steps is $tθ$ reduced modulo $36 0^{\circ}$ .

TOTAL MOVEMENT WHEN THE STEP GROWS BY ONE EACH TIME

1 + 2 + \dots + t = \frac{t ( t + 1 )}{2}

Use this when an element moves 1 position, then 2, then 3, and so on. Reduce the total modulo the number of positions.

WORKED EXAMPLE · TWO RULES AT ONCE

Each figure in a series is a square with 8 marked positions: its 4 corners and the midpoints of its 4 sides. In figure 1 a dot is at the top-left corner. To reach figure 2 it moves clockwise by 1 position, then by 2 positions to reach figure 3, then by 3, and so on. An arrow at the centre points up (north) in figure 1 and turns $4 5^{\circ}$ anticlockwise from each figure to the next. Describe figure 6.

Solution

From figure 1 to figure 6 there are 5 moves, so the dot travels $1 + 2 + 3 + 4 + 5 = 15$ positions clockwise.
Positions repeat every 8, and $15 mod 8 = 7$ . Number the positions clockwise from the top-left corner: 0 top-left, 1 top midpoint, 2 top-right, 3 right midpoint, 4 bottom-right, 5 bottom midpoint, 6 bottom-left, 7 left midpoint. The dot is at position 7, the midpoint of the left side.
The arrow turns $5 \times 4 5^{\circ} = 22 5^{\circ}$ anticlockwise from north. $9 0^{\circ}$ takes it to west, $18 0^{\circ}$ to south, and $22 5^{\circ}$ to south-east.

Answer: In figure 6 the dot is at the midpoint of the left side and the arrow points south-east.

EXAM TIP

Different solids can share a view. A cylinder, a cone and a sphere all have a circular top-view outline, and a cube and a square pyramid both have a square one. Use at least two views to identify a solid, and the third as well when two still leave a choice.

EXAM TIP

A turn of $4 5^{\circ}$ per step returns to the start after 8 steps, and a turn of $9 0^{\circ}$ per step after 4. When a rule repeats every p steps, figure $t + p$ looks the same as figure $t$ for that element.

EXAM TIP

Pattern completion: when part of a symmetric pattern is missing, find the symmetry the visible parts share, such as a mirror line or a quarter turn about the centre. Build the missing part from the part that corresponds to it under that symmetry, then check the chosen option against every neighbouring part, not just one.

PREVIOUS-YEAR QUESTIONS

23 questions from the EC, EE and IN papers

Try each question first, then open the answer. One shows the official answer only. We placed each question under a topic ourselves; the exam paper does not label topics.

See these papers in full: GATE ECE 2026 · GATE ECE 2024 · GATE ECE 2023 · GATE ECE 2022 · GATE EE 2026 · GATE EE 2024 · GATE EE 2023 · GATE EE 2022 · GATE IN 2026 · GATE IN 2025 · GATE IN 2024 · GATE IN 2023 · GATE IN 2022

GATE 2026 EC GA Q.3

GATE 2026 · ECGA Q.3 · 1 mark · MCQ

The figure below has exactly three intersecting line segments with a rectangular portion WXYZ missing. Which one of the following options P, Q, R, and S is the missing portion WXYZ?

GATE 2026 EC GA question 3: MAIN FIGURE: a blank rectangle, wider than it is tall, with a "?" in its centre. — Figure from the GATE 2026 EC paper. It is part of the question.

(A)
P
(B)
Q
(C)
R
(D)
S

Answer and solution

GATE 2026 EC GA Q.3: answer and solution

GATE 2026 EC GA Q.10

GATE 2026 · ECGA Q.10 · 2 marks · MCQ

[Figure: two-line text "Ref Fig" over "12 34" $\to$ its transformed image]

Then

[Figure: two-line text "GATE - 2026" over "Aptitude Test" $\to$ ?]

[Figures P, Q, R, S: four candidate images, described below]

GATE 2026 EC GA question 10: Reference pair: on the left a two-line block of bold text, top line "Ref Fig", bottom line "12 34". — Figure from the GATE 2026 EC paper. It is part of the question.

(A)
P
(B)
Q
(C)
R
(D)
S

Answer and solution

GATE 2026 EC GA Q.10: answer and solution

GATE 2026 EE GA Q.2

GATE 2026 · EEGA Q.2 · 1 mark · MCQ

The figure below is supposed to show three non-overlapping shapes – one oval and two triangles. Which one of the following figures P, Q, R, or S fits the missing portion indicated by ‘?’ and completes the oval and the two triangles?

GATE 2026 EE GA question 2: Main figure: a white rectangular box marked '?' in the centre, about 1.5 times as wide as it is tall, hides parts of… — Figure from the GATE 2026 EE paper. It is part of the question.

(A)
P
(B)
Q
(C)
R
(D)
S

Answer and solution

GATE 2026 EE GA Q.2: answer and solution

GATE 2026 EE GA Q.5

GATE 2026 · EEGA Q.5 · 1 mark · MCQ

Which one of the following figures P, Q, R, or S, correctly shows the 45° clockwise-rotated version of figure (I)?

GATE 2026 EE GA question 5: Figure (I), at the top centre, is an upright square pattern of solid black unit squares on a 12 x 12 grid. — Figure from the GATE 2026 EE paper. It is part of the question.

(A)
P
(B)
Q
(C)
R
(D)
S

Answer and solution

GATE 2026 EE GA Q.5: answer and solution

GATE 2026 IN GA Q.7

GATE 2026 · INGA Q.7 · 2 marks · MCQ

A paper shown in Panel I is folded along the dashed lines (- - -) to construct a cube. The shaded regions shown in Panel I appear on the outer surface of the cube. Referring to cubes shown in Panel II, which one of the options is correct?

GATE 2026 IN GA question 7: A vertical line separates Panel I (left) from Panel II (right). — Figure from the GATE 2026 IN paper. It is part of the question.

(A)
Only (i) can correspond to the unfolded cube in Panel I.
(B)
Only (ii) can correspond to the unfolded cube in Panel I.
(C)
Both (i) and (ii) can correspond to the unfolded cube in Panel I.
(D)
Neither (i) nor (ii) can correspond to the unfolded cube in Panel I.

Answer and solution

GATE 2026 IN GA Q.7: answer and solution

GATE 2025 IN GA Q.4

GATE 2025 · INGA Q.4 · 1 mark · MCQ

The paper as shown in the figure is folded to make a cube where each square corresponds to a particular face of the cube. Which one of the following options correctly represents the cube?

Note: The figures shown are representative.

GATE 2025 IN GA question 4: A cube net of six equal squares in a cross shape. — Figure from the GATE 2025 IN paper. It is part of the question.

(A)
Figure — cube: front face an open triangle pointing up, top face a filled black circle, right face blank
(B)
Figure — cube: front face an open triangle, top face blank, right face a filled black triangle
(C)
Figure — cube: front face an open triangle, top face blank, right face blank
(D)
Figure — cube: front face an open triangle, top face a filled black triangle, right face a filled black circle

Answer and solution

GATE 2025 IN GA Q.4: answer and solution

GATE 2024 EC GA Q.9

GATE 2024 · ECGA Q.9 · 2 marks · MCQ

Examples of mirror and water reflections are shown in the figures below:

[Figure: MIRROR REFLECTION and WATER REFLECTION examples]

An object appears as the following image after first reflecting in a mirror and then reflecting on water.

[Figure: seven-segment characters reading 5·24]

The original object is

GATE 2024 EC GA question 9: Every character is a seven-segment-display glyph drawn with double (outlined) strokes. — Figure from the GATE 2024 EC paper. It is part of the question.

(A)
Figure: seven-segment glyphs reading h2·5. They are h (full left vertical, middle bar, lower-right vertical), 2, a dot slightly above mid-height, and 5.
(B)
Figure: seven-segment glyphs reading 2·5 (dot slightly above mid-height), then a mirror-image h (full right vertical, middle bar, lower-left vertical).
(C)
Figure: seven-segment glyphs. They are a left-right mirror image of 4 (full left vertical, middle bar, upper-right vertical), 5, a dot slightly below mid-height, and 2.
(D)
Figure: seven-segment glyphs reading h·25. They are h, dot, 2, 5.

Answer and solution

GATE 2024 EC GA Q.9: answer and solution

GATE 2024 EC GA Q.10

GATE 2024 · ECGA Q.10 · 2 marks · MCQ

Two identical sheets A and B, of dimensions $24 cm \times 16 cm$ , can be folded into half using two distinct operations, FO1 or FO2.

In FO1, the axis of folding remains parallel to the initial long edge, and in FO2, the axis of folding remains parallel to the initial short edge.

If sheet A is folded twice using FO1, and sheet B is folded twice using FO2, the ratio of the perimeters of the final shapes of A and B is

(A)
14:11
(B)
11:14
(C)
18:11
(D)
11:18

Answer and solution

GATE 2024 EC GA Q.10: answer and solution

See the question in the paper

Tap to open full size

GATE 2024 EC GA Q.10: answer and solution

GATE 2024 EE GA Q.9

GATE 2024 · EEGA Q.9 · 2 marks · MCQ

The assembly shown below has three teethed circular objects (Pinions) and two teethed flat objects (Racks), which are perfectly mating with each other. Pinions can only rotate clockwise or anti-clockwise staying at its own center. Racks can translate towards the left (←) or the right (→) direction.

[Figure]

If the object A (Rack) is translating towards the right (→) direction, the correct statement among the following is

GATE 2024 EE GA question 9: Black silhouette drawing of a gear train. — Figure from the GATE 2024 EE paper. It is part of the question.

(A)
Object B translates towards the right direction.
(B)
Object B translates towards the left direction.
(C)
Object R rotates in the anticlockwise direction.
(D)
Object Q rotates in the clockwise direction.

Answer and solution

GATE 2024 EE GA Q.9: answer and solution

GATE 2024 IN GA Q.9

GATE 2024 · INGA Q.9 · 2 marks · MCQ

Five cubes of identical size and another smaller cube are assembled as shown in Figure A. If viewed from direction X, the planar image of the assembly appears as Figure B.

[Figure A] [Figure B]

If viewed from direction Y, the planar image of the assembly (Figure A) will appear as

GATE 2024 IN GA question 9: Figure A (isometric line drawing): four identical cubes form a 2 x 2 bottom layer. — Figure from the GATE 2024 IN paper. It is part of the question.

(A)
Figure — two equal squares side by side on the bottom and one equal square above the right-hand one (an L-shape); inside the upper square, a small square of half the edge sits in its lower-right corner.
(B)
Figure — the same L-shape (two squares on the bottom, one above the right-hand one); the upper square is split into two equal halves by a full-height vertical line.
(C)
Figure — the same L-shape; inside the upper square, a small square of half the edge sits in its lower-left corner.
(D)
Figure — two equal squares on the bottom and one equal square above the right-hand one; a small square of half the edge sits on top of the left bottom square, outside the upper square and touching its left side.

Answer and solution

GATE 2024 IN GA Q.9: answer and solution

GATE 2024 IN GA Q.10

GATE 2024 · INGA Q.10 · 2 marks · MCQ

Visualize a cube that is held with one of the four body diagonals aligned to the vertical axis. Rotate the cube about this axis such that its view remains unchanged. The magnitude of the minimum angle of rotation is

(A)
$12 0^{\circ}$
(B)
$6 0^{\circ}$
(C)
$9 0^{\circ}$
(D)
$18 0^{\circ}$

Answer and solution

GATE 2024 IN GA Q.10: answer and solution

See the question in the paper

Tap to open full size

GATE 2024 IN GA Q.10: answer and solution

GATE 2023 EC GA Q.5

GATE 2023 · ECGA Q.5 · 1 mark · MCQ

How many rectangles are present in the given figure?

GATE 2023 EC GA question 5: A square outline (drawn in blue lines, no shading). — Figure from the GATE 2023 EC paper. It is part of the question.

(A)
8
(B)
9
(C)
10
(D)
12

Answer and solution

GATE 2023 EC GA Q.5: answer and solution

GATE 2023 EC GA Q.10

GATE 2023 · ECGA Q.10 · 2 marks · MCQ

A 100 cm $\times$ 32 cm rectangular sheet is folded 5 times. Each time the sheet is folded, the long edge aligns with its opposite side. Eventually, the folded sheet is a rectangle of dimensions 100 cm $\times$ 1 cm.

The total number of creases visible when the sheet is unfolded is ______.

(A)
32
(B)
5
(C)
31
(D)
63

Answer and solution

GATE 2023 EC GA Q.10: answer and solution

See the question in the paper

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GATE 2023 EC GA Q.10: answer and solution

GATE 2023 EE GA Q.5

GATE 2023 · EEGA Q.5 · 1 mark · MCQ

How many triangles are present in the given figure?

GATE 2023 EE GA question 5: A parallelogram drawn in blue lines. — Figure from the GATE 2023 EE paper. It is part of the question.

(A)
12
(B)
16
(C)
20
(D)
24

Answer and solution

GATE 2023 EE GA Q.5: answer and solution

GATE 2023 IN GA Q.5

GATE 2023 · INGA Q.5 · 1 mark · MCQ

In the given diagram, ovals are marked at different heights ( $h$ ) of a hill. Which one of the following options P, Q, R, and S depicts the top view of the hill?

GATE 2023 IN GA question 5: Main figure: a graph. — Figure from the GATE 2023 IN paper. It is part of the question.

(A)
P
(B)
Q
(C)
R
(D)
S

Answer and solution

GATE 2023 IN GA Q.5: answer and solution

GATE 2023 IN GA Q.10

GATE 2023 · INGA Q.10 · 2 marks · MCQ

An opaque cylinder (shown below) is suspended in the path of a parallel beam of light, such that its shadow is cast on a screen oriented perpendicular to the direction of the light beam. The cylinder can be reoriented in any direction within the light beam. Under these conditions, which one of the shadows P, Q, R, and S is NOT possible?

GATE 2023 IN GA question 10: Top: a 3D sketch of a solid circular cylinder, shaded light grey. — Figure from the GATE 2023 IN paper. It is part of the question.

(A)
P
(B)
Q
(C)
R
(D)
S

Answer and solution

GATE 2023 IN GA Q.10: answer and solution

GATE 2022 EC GA Q.4

GATE 2022 · ECGA Q.4 · 1 mark · MCQ

The figure shows a grid formed by a collection of unit squares. The unshaded unit square in the grid represents a hole.

[Figure]

What is the maximum number of squares without a “hole in the interior” that can be formed within the $4 \times 4$ grid using the unit squares as building blocks?

GATE 2022 EC GA question 4: A 4 by 4 square grid made of 16 unit squares, with a black dot at each of its 25 grid-line intersections (5 by 5 lattice… — Figure from the GATE 2022 EC paper. It is part of the question.

(A)
15
(B)
20
(C)
21
(D)
26

Answer and solution

GATE 2022 EC GA Q.4: answer and solution

GATE 2022 EC GA Q.5

GATE 2022 · ECGA Q.5 · 1 mark · MCQ

An art gallery engages a security guard to ensure that the items displayed are protected. The diagram below represents the plan of the gallery where the boundary walls are opaque. The location the security guard posted is identified such that all the inner space (shaded region in the plan) of the gallery is within the line of sight of the security guard.

If the security guard does not move around the posted location and has a 360 $^{\circ}$ view, which one of the following correctly represents the set of ALL possible locations among the locations P, Q, R and S, where the security guard can be posted to watch over the entire inner space of the gallery.

GATE 2022 EC GA question 5: A grey-shaded V-shaped (chevron) floor plan drawn as a concave quadrilateral with four vertices. — Figure from the GATE 2022 EC paper. It is part of the question.

(A)
P and Q
(B)
Q
(C)
Q and S
(D)
R and S

Answer and solution

GATE 2022 EC GA Q.5: answer and solution

GATE 2022 EC GA Q.10

GATE 2022 · ECGA Q.10 · 2 marks · MCQ

Consider the following square with the four corners and the center marked as P, Q, R, S and T respectively.

Let X, Y and Z represent the following operations:

X: rotation of the square by 180 degree with respect to the S-Q axis.

Y: rotation of the square by 180 degree with respect to the P-R axis.

Z: rotation of the square by 90 degree clockwise with respect to the axis perpendicular, going into the screen and passing through the point T.

Consider the following three distinct sequences of operation (which are applied in the left to right order).

(1) XYZZ
(2) XY
(3) ZZZZ

Which one of the following statements is correct as per the information provided above?

GATE 2022 EC GA question 10: A square drawn upright (sides horizontal and vertical) with both diagonals drawn as thin lines. — Figure from the GATE 2022 EC paper. It is part of the question.

(A)
The sequence of operations (1) and (2) are equivalent
(B)
The sequence of operations (1) and (3) are equivalent
(C)
The sequence of operations (2) and (3) are equivalent
(D)
The sequence of operations (1), (2) and (3) are equivalent

Answer and solution

GATE 2022 EC GA Q.10: answer and solution

GATE 2022 EE GA Q.5

GATE 2022 · EEGA Q.5 · 1 mark · MCQ

The figure below shows the front and rear view of a disc, which is shaded with identical patterns. The disc is flipped once with respect to any one of the fixed axes 1-1, 2-2 or 3-3 chosen uniformly at random.

What is the probability that the disc DOES NOT retain the same front and rear views after the flipping operation?

GATE 2022 EE GA question 5: Two equal circles side by side, captioned "Front View" (left) and "Rear View" (right). — Figure from the GATE 2022 EE paper. It is part of the question.

(A)
0
(B)
$\frac{1}{3}$
(C)
$\frac{2}{3}$
(D)
1

Answer and solution

GATE 2022 EE GA Q.5: answer and solution

GATE 2022 EE GA Q.10

GATE 2022 · EEGA Q.10 · 2 marks · MCQ

[Figure: a 4 x 4 square grid with P at its bottom-left corner]

An ant is at the bottom-left corner of a grid (point P) as shown above. It aims to move to the top-right corner of the grid. The ant moves only along the lines marked in the grid such that the current distance to the top-right corner strictly decreases.

Which one of the following is a part of a possible trajectory of the ant during the movement?

GATE 2022 EE GA question 10: Question figure: a 4 x 4 grid of equal squares (5 horizontal lines and 5 vertical lines, so 5 x 5 lattice points). — Figure from the GATE 2022 EE paper. It is part of the question.

(A)
Figure: from P, one unit segment goes right along the bottom (and ends) and one goes up. From the top of the upward segment the path goes right 1, up 1, right 1, then down 1, where it ends.
(B)
Figure: a stroke that starts one unit above P and comes down to P, then goes right 1, up 2, right 1, then down 1, where it ends.
(C)
Figure: a staircase from P going up 1, right 1, up 1, right 1.
(D)
Figure: a closed unit square with P at its bottom-left corner (right 1, up 1, left 1, down 1 back to P).

Answer and solution

GATE 2022 EE GA Q.10: answer and solution

GATE 2022 IN GA Q.5

GATE 2022 · INGA Q.5 · 1 mark · MCQ

A building has several rooms and doors as shown in the top view of the building given below. The doors are closed initially.

What is the minimum number of doors that need to be opened in order to go from the point P to the point Q?

GATE 2022 IN GA question 5: Top view (floor plan) of a building drawn as a large, nearly square outer rectangle. — Figure from the GATE 2022 IN paper. It is part of the question.

(A)
4
(B)
3
(C)
2
(D)
1

Answer and solution

GATE 2022 IN GA Q.5: answer and solution

GATE 2022 IN GA Q.10

GATE 2022 · INGA Q.10 · 2 marks · MCQ

Which one of the following sets of pieces can be assembled to form a square with a single round hole near the center? Pieces cannot overlap.

GATE 2022 IN GA question 10: Four options. — Figure from the GATE 2022 IN paper. It is part of the question.

(A)
Figure — P with no kink and a large semicircular notch on its right side, with a short step below the notch; Q with all straight edges; R with a semicircular notch on the steep edge that ends at its peak
(B)
Figure — P with kink and a small arc notch; R (middle) with a large near-semicircular notch between its V and its steep edge; right-hand piece a four-sided shape (Q without its V-cut) with a small arc notch at its lower left
(C)
Figure — P with kink and a small arc notch on its lower-right edge; Q with a small arc notch on its short edge next to the tip; R with a large near-semicircular notch between its V and its steep edge
(D)
Figure — P, Q and R with all edges straight, no notches

Official answer

Official answer: CSource: Official answer key, GATE 2022 IN (IIT Kharagpur)

GATE 2022 IN GA Q.10: official answer

PRACTICE

5 questions to test yourself

PRACTICE 11 mark · MCQ

An L-shaped figure on a card is made of two straight arms meeting at a right-angled corner. A long arm runs from the corner towards the top of the card (north), and a short arm runs from the corner towards the right-hand edge (east). The card is (i) rotated $9 0^{\circ}$ clockwise in its own plane, then (ii) reflected in a vertical mirror line, which swaps east and west, and then (iii) rotated $18 0^{\circ}$ . In which directions do the two arms now point from the corner?

(A)
Long arm east, short arm north
(B)
Long arm east, short arm south
(C)
Long arm west, short arm north
(D)
Long arm south, short arm west

Answer and solution

Answer: (A)

Follow the direction of each arm. Start: long arm north, short arm east.
(i) $9 0^{\circ}$ clockwise sends north to east and east to south: long arm east, short arm south.
(ii) The vertical mirror swaps east and west and leaves north and south unchanged: long arm west, short arm south.
(iii) $18 0^{\circ}$ reverses every direction: long arm east, short arm north.
Check: the sequence contains one reflection, so the result must be a mirror image of the start. It is: north and east have swapped, which is a reflection in the line running from north-east to south-west.

PRACTICE 22 marks · NAT

A solid wooden block measuring 5 cm (length) × 4 cm (width) × 3 cm (height) stands on its 5 cm × 4 cm face. Its top face and its four vertical faces are painted; the bottom face is not painted. The block is then cut into cubes of edge 1 cm. The number of these small cubes that have no paint on any face is ______.

Answer and solution

Answer: $12$

Label each small cube by its position: length 1 to 5, width 1 to 4, height 1 (bottom) to 3 (top).
To have no paint, a cube must not touch a painted vertical face. So its length position is 2, 3 or 4 (3 choices) and its width position is 2 or 3 (2 choices).
It must not be in the painted top layer. The bottom layer is allowed because the bottom face is unpainted. So its height is 1 or 2 (2 choices).
Unpainted cubes $= 3 \times 2 \times 2 = 12$ .
For comparison, with all six faces painted the answer would be $(5 - 2) (4 - 2) (3 - 2) = 6$ . The unpainted bottom adds a second layer of 6.

PRACTICE 32 marks · MSQ

A net of a cube is made of six squares. Four of them, labelled P, Q, R and S from left to right, lie in one horizontal row, each sharing an edge with the next. Square T is attached to the top edge of Q, and square U is attached to the bottom edge of R. The net is folded into a cube with the labels on the outside. Which of the following statements is/are correct?

(A)
P and R are on opposite faces.
(B)
T and U are on opposite faces.
(C)
Q and T are on opposite faces.
(D)
S and U share an edge of the cube.

Answer and solution

Answer: (A), (B), (D)

In a row of four squares, alternate squares are opposite: P is opposite R, and Q is opposite S. So A is correct.
The row P, Q, R, S folds into a ring of four side faces, open at the top and at the bottom. T, hinged on the top edge of Q, closes the top opening. U, hinged on the bottom edge of R, closes the bottom opening. So T and U are opposite, and B is correct.
Q and T share an edge in the net, so they are adjacent on the cube, not opposite. C is wrong.
Every face of the ring shares an edge with both end faces, so S and U share an edge. D is correct.

PRACTICE 42 marks · MCQ

A square sheet of paper ABCD has corner A at the top left, B at the top right, C at the bottom right and D at the bottom left. It is folded along the diagonal AC so that corner B lies exactly on corner D. The resulting triangle is then folded in half so that corner A lies exactly on corner C. A single small hole is punched through all the layers, close to the corner of the folded triangle where A and C now lie together, and away from both fold lines. When the sheet is completely unfolded, where are the holes?

(A)
One hole near each of the four corners A, B, C and D
(B)
Two holes near corner A and two holes near corner C
(C)
Two holes near corner B and two holes near corner D
(D)
Four holes, all near corner C

Answer and solution

Answer: (B)

Place D at (0, 0), C at (1, 0), B at (1, 1) and A at (0, 1).
Fold 1 is along AC, the line $x + y = 1$ . The sheet now covers triangle ACD in 2 layers. Fold 2 brings A onto C, so its fold line is the perpendicular bisector of AC, the line $y = x$ , which passes through D. The packet is now triangle D, C, O with O = (0.5, 0.5), 4 layers thick, and A lies on C.
Take the punch at (0.8, 0.1), near C. Undo fold 2 by reflecting in $y = x$ : holes at (0.8, 0.1) and (0.1, 0.8). The second one is near A.
Undo fold 1 by reflecting in $x + y = 1$ , that is $(x, y) \to (1 - y, 1 - x)$ . (0.8, 0.1) gives (0.9, 0.2), near C, and (0.1, 0.8) gives (0.2, 0.9), near A.
Result: four holes, two near C and two near A, and none near B or D. B and D ended up at the other corner of the folded triangle, which was not punched.

PRACTICE 51 mark · NAT

Triangle PQR is drawn with P at the top and QR as its horizontal base. Three straight lines are drawn from P to three different points lying strictly between Q and R. Two straight lines parallel to QR are also drawn, at two different heights, each running from side PQ to side PR. The total number of triangles of all sizes in the figure is ______.

Answer and solution

Answer: $30$

Lines through P: PQ, PR and the three new lines, 5 in all. Lines parallel to the base: QR and the two new lines, 3 in all.
Two parallel lines cannot both be sides of one triangle, and three lines through P all meet at P. So every triangle uses exactly two lines through P and one line parallel to QR.
Every such choice does give a triangle, because each parallel line crosses all five lines through P.
Number of triangles $=^{5} C_{2} \times 3 = 10 \times 3 = 30$ . This is the formula $^{k + 2} C_{2} \times (h + 1)$ with $k = 3$ and $h = 2$ .

OFFICIAL GATE 2027 SYLLABUS WORDING

Copied from the GATE 2027 syllabus; see all 2027 syllabi.

EVERY GATE PAPER

Transformation of shapes: translation, rotation, scaling, mirroring, assembling, and grouping paper folding, cutting, and patterns in 2 and 3 dimensions

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