2-D Transformations and Algorithms MCQs: 12 Solved Questions with Explanations

2-D transformation questions turn on two things: which quantity a transformation leaves unchanged, and whether the paper is using column vectors, row vectors or a reference line. Get those two right and most of the marks follow. The twelve questions below run from scaling factors to the viewing pipeline, and ten of them carry an exam attribution across UGC NET, ISRO, UPPSC, UPLT and DSSSB. Attempt each one before reading its explanation, and when you miss one, go back to the formula table in the next section and find the rule you skipped.

1. 2-D transformation formulas and the composite-transformation algorithm

Use column vectors. Write a point as \((x,y,1)^T\) when homogeneous coordinates are required.

Translation needs homogeneous coordinates because a \(2\times2\) linear matrix cannot add \(t_x\) and \(t_y\).

For \(P(3,2)\), scaling by \(s_x=2,s_y=4\) gives \((6,8)\), then a 90-degree counter-clockwise rotation gives \((-8,6)\). Reversing the order gives \((3,2)\mapsto(-2,3)\mapsto(-4,12)\). The rightmost matrix acts first.

A transformation about any point other than the origin follows one fixed algorithm: translate the fixed point to the origin, apply the transformation there, then translate back. For a rotation by \(\theta\) about a pivot \((x_p,y_p)\) the composite matrix is \(M=T(x_p,y_p)\,R(\theta)\,T(-x_p,-y_p)\), and scaling about a fixed point is the same three steps with \(S(s_x,s_y)\) in the middle.

Worked: rotate \(P(4,3)\) by 90 degrees counter-clockwise about \((1,1)\). Translating gives \((3,2)\), rotating gives \((-2,3)\), and translating back gives \((-1,4)\). The pivot itself must come back to \((1,1)\), and that is the check to run whenever a composite answer looks wrong.

Rotation, translation and reflection preserve lengths and angles; reflection reverses orientation. Uniform scaling preserves shape but changes size; non-uniform scaling can change proportions. Shear changes angles and usually shape.

2. 2-D scaling MCQs: factors, size and invariants

Question 1, UGC NET December 2022

Open this PYQ in the learn module.

Given a vector with cartesian components \((x, y)\), if scaling is done with matrix \(\left[\begin{array}{cc}0.5 & 0 \\ 0 & 1.5\end{array}\right]\), which of the following are true.

A. Decreases the vertical by three halves

B. Increases the vertical by three halves

C. Doubles the horizontal

D. Halves the horizontal

Choose the correct answer from the options given below:

• A. A and C only

• B. A and D only

• C. B and C only

• D. B and D only

Correct answer: D. B and D only. Multiplication gives \((0.5x,1.5y)^T\). The horizontal component is halved and the vertical component becomes three halves of its original value, so statements B and D are true.

Question 2, scaling and size

Open this question in the practice module.

Which of the following transformations can change the size of a shape?

• A. Rotation

• B. Reflection

• C. Scaling

• D. Translation

Correct answer: C. Scaling. Rotation, reflection and translation are rigid transformations, so they preserve size. Scaling multiplies coordinates by scale factors and can enlarge or shrink the object.

Question 3, UPLT 2018

Open this PYQ in the learn module.

A scaling transformation changes the

• A. size of an object

• B. location of an object

• C. shape of an object

• D. Both (a) and (b)

Correct answer: A. size of an object. Origin-based scaling multiplies each coordinate by its own factor, so the object grows or shrinks while the origin stays fixed. Points other than the origin do shift, but that displacement is a consequence of the size change rather than an independent translation, which rules out B and D. Shape changes only when \(s_x\) and \(s_y\) differ, so size is the change that always happens.

3. 2-D reflection MCQs: congruence, mirror images and determinants

Question 4, congruence and rigid motions

Open this question in the practice module.

Which of the following transformations results in a shape that is congruent to the original shape?

• A. Scaling by a factor of 2

• B. Reflection

• C. Shearing

• D. Dilation

Correct answer: B. Reflection. Reflection preserves every distance and angle, so its image is congruent to the original even though orientation is reversed. Scaling by 2 and dilation change size, while shear generally changes angles.

Question 5, UPPSC Polytechnic Lecturer 2022

Open this PYQ in the learn module.

For getting the mirror image of a triangle, which of the following transformation is needed?

• A. Rotation

• B. Scaling

• C. Rotation and Scaling both

• D. Reflection

Correct answer: D. Reflection. Reflection across a chosen line produces a mirror image. Rotation does not reverse handedness, and scaling changes dimensions, so neither alone does this.

Question 6, UGC NET June 2019

Open this PYQ in the learn module.

Consider the following statements regarding \(2𝐷\) transforms in computer graphics:

\(S1: \: \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) is a \(2×2\) matrix that reflects (mirrors) only \(2𝐷\) point about the X-axis.

\(𝑆_2\) : A \(2×2\) matrix which mirrors any \(2𝐷\) point about the \(𝑋\)-axis, is a rotation matrix.

What can you say about the statements \(𝑆_1\) and \(𝑆_2\)?

• A. Both \(𝑆_1\) and \(𝑆_2\) are true

• B. Only \(𝑆_1\) is true

• C. Only \(𝑆_2\) is true

• D. Both \(𝑆_1\) and \(𝑆_2\) are false

Correct answer: B. Only \(𝑆_1\) is true. The matrix \(\operatorname{diag}(1,-1)\) maps \((x,y)\) to \((x,-y)\), so it reflects about the x-axis and S1 is true. Its determinant is \(-1\), while a proper 2-D rotation matrix has determinant \(+1\), so S2 is false.

4. 2-D rotation, shearing and coordinate MCQs

Question 7, ISRO 2011

Open this question in the practice module.

What is the matrix that represents rotation of an object by θ degree about the origin in 2D?

• A.

  cos θ    −sin θ
  sin θ     cos θ

• B.

  sin θ   −cos θ
  cos θ    sin θ

• C.

  cos θ   −sin θ
  cos θ    sin θ

• D.

  cos θ     sin θ
  −sin θ    cos θ

Correct answer: A. Column-vector rotation gives \(x'=x\cos\theta-y\sin\theta\) and \(y'=x\sin\theta+y\cos\theta\). At \(90^\circ\), option A maps \((1,0)\) to \((0,1)\), confirming the signs.

Question 8, UGC NET June 2016

Open this PYQ in the learn module.

Let us consider that the original point is \((x,y)\) and new transformed point is \((x’,y’)\). Further \(Sh_𝑥\) and \(Sh_𝑦\) are shearing factors in \(x\) and \(y\) directions. If we perform the \(y\) direction shear relative to \(x=x_{ref}\) then the transformed point is given by

• A. \(x’=x+Sh_x.(y-y_{ref}); \\ y’=y\)

• B. \(x’=x; \\ y’=y.Sh_x\)

• C. \(x’=x; \\ y’=Sh_y(x-x_{ref})+y\)

• D. \(x’=Sh_y.y; \\ y’=y.(x-x_{ref})\)

Correct answer: C. \(x’=x; \\ y’=Sh_y(x-x_{ref})+y\). A y-shear leaves x unchanged and adds an x-dependent displacement to y. At \(x=x_{ref}\), the displacement is \(Sh_y(x_{ref}-x_{ref})=0\), so the reference line stays fixed.

Question 9, DSSSB TGT Shift 2 2021

Open this PYQ in the learn module.

Consider a triangle with co-ordinates points: A(0, 0), B(3, 3) and C(2, 5). When scaling parameter is 2 towards x-axis and 4 towards y-axis, then what are the new co-ordinates of triangle?

• A. A(0, 0), B(6, 12), C(4, 20)

• B. A(0, 0), B(5, 7), C(4, 9)

• C. A(2, 4), B(5, 7), C(4, 9)

• D. A(0, 0), B(3/2, 3/4), C(1, 5/4)

Correct answer: A. A(0, 0), B(6, 12), C(4, 20). Apply \((x',y')=(2x,4y)\): \(A\mapsto(0,0)\), \(B\mapsto(2\times3,4\times3)=(6,12)\), and \(C\mapsto(2\times2,4\times5)=(4,20)\). Origin-based scaling keeps the origin fixed.

5. 2-D viewing and aspect-ratio MCQs

Question 10, DSSSB 2021

Open this PYQ in the learn module.

If an image has a height of 4 inches and an aspect ratio of 3 : 2, then what will be the width of the image?

• A. 3 inches

• B. 6 inches

• C. 4 inches

• D. 5.5 inches

Correct answer: B. 6 inches. Read \(3:2\) as width:height. Two parts equal 4 inches, so three parts equal 6 inches; directly, \(w/4=3/2\), hence \(w=6\).

Question 11, DSSSB TGT Shift 3 2021

Open this PYQ in the learn module.

If an image has a width of 4 inch and an aspect ratio of 3 : 2, then what is its height?

• A. 4/3 inch

• B. 8/3 inch

• C. 2/3 inch

• D. 16/3 inch

Correct answer: B. 8/3 inch. Use width:height \(=3:2\), so \(4/h=3/2\). Cross-multiplication gives \(3h=8\), hence \(h=8/3\) inch, so do not invert the ratio from Question 10.

Question 12, DSSSB TGT Shift 3 2021

Open this PYQ in the learn module.

Rearrange steps involved in two dimensional viewing transformation.

I. Map normalized viewport to device co-ordinates

II. Convert world co-ordinates to viewing co-ordinates

III. Construct world co-ordinate scene using modeling coordinate transformation

IV. Map viewing co-ordinates to normalized viewing co-ordinates using window-viewport specification

• A. III → II → IV → I

• B. II → III → IV → I

• C. I → II → IV → III

• D. III → II → I → IV

Correct answer: A. III → II → IV → I. Build the world scene, convert it to viewing coordinates, normalize the view, then map it to device coordinates. The chain is model/world → view → normalized view → device.

6. How exams test 2-D transformations and where students lose marks

These are single-correct MCQs. Use the GATE question-types guide to distinguish MCQ, MSQ and NAT. Check the invariant before you multiply any coordinates: on Questions 2, 4 and 5 it settles the answer with no arithmetic at all.

After each error, use Why PYQs Beat Buying Another Question Bank: record the transformation, invariant and a check such as \((1,0)\mapsto(0,1)\) for a 90-degree counter-clockwise rotation.

7. 2-D transformations MCQs: the short version and next step

• Scaling changes size.

• Reflection creates a congruent mirror image.

• Counter-clockwise rotation uses \([\cos\theta,-\sin\theta;\sin\theta,\cos\theta]\).

• A y-shear changes y from x while preserving x.

• A transformation about a pivot is translate, transform, translate back.

• The viewing chain is world, view, normalized view, device.

Reattempt Questions 1, 6, 8, 9 and 12 without looking. They cover matrices, coordinates and viewing. GATE CS Exam Preparation is the broader subject route.

GATE Guidance by Sanchit Sir offers an exam plan. Redo the 2-D transformations module until you can justify every matrix from coordinates.