Non-Verbal Reasoning - MCQ Practice Questions

Understanding:

We need to determine how many holes appear on a flat sheet of paper after it is folded twice and a single hole is punched through all layers.

Step 1: Count layers after each fold.

Each fold doubles the number of layers. After the first fold (vertical), there are 2 layers. After the second fold (horizontal), there are 4 layers.

Step 2: Effect of punching.

Punching a hole through all 4 layers means when unfolded, 4 holes will appear — one in each quadrant formed by the two fold lines.

Step 3: Visualize positions.

The vertical fold mirrors left and right. The horizontal fold mirrors top and bottom. So the single punch creates one hole per quadrant: top-left, top-right, bottom-left, and bottom-right.

Answer:

When the paper is unfolded, 4 holes will appear.

Quick Tip:

Each fold doubles the number of holes produced by one punch. Two folds → $2^2=4$ holes. Three folds would give $2^3=8$ holes.

Understanding:

We are asked what happens when a transparent sheet is folded along a vertical central dotted line.

Step 1: Nature of folding along a vertical axis.

Folding along a vertical line brings the right half onto the left half. Since the sheet is transparent and the fold is a reflection, the right side appears as the mirror image of the left side when viewed from above.

Step 2: Result of the fold.

The right half lands exactly on top of the left half with each point on the right being the mirror image of the corresponding point on the left — this is the definition of a vertical reflection.

Step 3: Eliminate wrong options.

A slide (translation) would not bring the halves together. A 90° rotation would not align a vertical fold. A 180° rotational match describes point symmetry, which is different from reflection symmetry produced by a fold.

Answer:

Folding along a vertical line makes the right half a mirror image of the left half, and the two halves overlap exactly.

Understanding:

A solid cube is divided into $3\times 3\times 3=27$ smaller cubes by making 2 cuts along each axis. We need to find how many smaller cubes have no exposed face.

Step 1: Identify interior cubes.

A smaller cube is completely interior only if it is not on any face of the original cube. Along each axis, only the middle slice position (position 2 out of 3) is not on an outer face.

Step 2: Count interior positions.

There is exactly one cube at the very center of the $3\times 3\times 3$ arrangement.

Step 3: Verify.

In a $3\times 3\times 3$ grid, along each dimension the two end positions are on outer faces. Only the middle position (1 per axis) is interior. So $1\times 1\times 1=1$ cube is fully hidden.

Answer:

Only 1 smaller cube is completely interior with no face exposed.

Quick Tip:

For an $n\times n\times n$ cube cut into unit cubes, the number of fully interior cubes is $(n-2{)}^3$. For $n=3$: $(3-2{)}^3=1$.

Understanding:

We cut squares of side 2 cm from each corner of a $10\text{ cm}\times 8\text{ cm}$ sheet and fold up the flaps to form an open box.

Formula:

Step 1: Find new dimensions after cutting.

Step 2: Calculate volume.

Answer:

The volume of the open box is 96 cubic centimetres.

Quick Tip:

Always subtract twice the cut size from each dimension (both ends are cut). The height of the box equals the side of the square cut.

Understanding:

We start with a cube where P is on top and Q faces the observer. We rotate the cube 90° to the right (clockwise when seen from above) and observe that R now faces us.

Step 1: Identify the rotation axis.

Rotating 90° to the right about the vertical (top-bottom) axis means the top and bottom faces do NOT change. Only the four side faces cycle.

Step 2: Determine what changes.

Before rotation: Q faces front. After rotating 90° clockwise (viewed from top), the face that was on the right now faces front. The top and bottom remain the same.

Step 3: Identify the top face.

Since the rotation is about the vertical axis (top to bottom), the face on top before the rotation remains on top after the rotation. P was on top before the rotation, so P is still on top.

Step 4: Confirm consistency.

Q was facing front; after a 90° clockwise rotation (from above), Q moves to the left side. R, which was on the right, now faces front — consistent with the information given.

Answer:

The face on top remains P, since a rotation about the vertical axis does not change the top face.

Quick Tip:

A rotation about the vertical axis only cycles the four lateral faces (front, right, back, left). The top and bottom faces always stay fixed during such a rotation.

Understanding:

The sheet is rolled along the longer side (22 cm), so the circumference of the base circle equals 22 cm and the height of the cylinder equals 10 cm.

Formula:

Step 1: Find the radius.

Step 2: Calculate the volume.

Answer:

The volume of the hollow cylinder is 385 cubic centimetres.

Quick Tip:

When a rectangle is rolled into a cylinder, the longer side becomes the circumference and the shorter side becomes the height (or vice versa depending on which edge meets). Always identify which edge forms the circumference before computing the radius.

Understanding:

A paper is folded three times along a vertical axis. A single triangular notch is cut from the folded right edge (which is a multi-layer edge). We need to find the number of cut-outs after unfolding.

Step 1: Count the layers after three folds.

Each vertical fold doubles the number of layers.

Step 2: Effect of cutting through all layers.

Cutting a notch through all 8 layers produces 8 separate triangular cut-outs when the paper is unfolded. The cut-out appears once for each layer.

Step 3: Why they appear on the edge.

Each fold brings more paper to the folded edge. Since the notch is cut from the folded right edge, upon unfolding, each layer reveals one triangular hole positioned along that edge.

Answer:

When the paper is unfolded, 8 triangular cut-outs will appear.

Quick Tip:

For edge cuts (not interior punches), the number of cut-outs equals the number of layers, which is $2^n$ for $n$ folds. For interior punches that do not touch any fold line, each layer also gives one hole, totalling $2^n$.

Understanding:

We need to find how many cubes of side 3 cm fit into a box of dimensions $12\text{ cm}\times 9\text{ cm}\times 6\text{ cm}$.

Formula:

where $a$ is the side of the small cube.

Step 1: Find how many cubes fit along each dimension.

Step 2: Multiply to get total.

Verification:

Answer:

24 cubes of side 3 cm can be packed into the box.

Understanding:

A square sheet is folded twice — first diagonally and then in half — and a hole is punched through all layers. We need to determine how many holes appear on unfolding.

Step 1: Count layers after each fold.

After the first diagonal fold: 2 layers.

After the second fold (in half): 4 layers.

Step 2: Count holes produced.

Punching once through 4 layers creates 4 holes when fully unfolded, one per layer.

Step 3: Position of holes.

The first diagonal fold creates reflection symmetry along one diagonal of the square. The second fold creates another line of symmetry. The punch point is replicated by each unfolding across these two axes, producing 4 symmetrically placed holes.

Step 4: Confirm no holes coincide on a fold line.

Since the punch is through the center of the smallest triangle (not on a fold edge), all 4 holes are distinct and separate on the fully opened sheet.

Answer:

When the paper is fully unfolded, 4 circular holes will appear.

Quick Tip:

Each fold that does not pass through the punch location doubles the number of holes. Two such folds → $2^2=4$ holes. If the punch lands exactly on a fold line, it counts as a half-hole on that edge, giving fewer distinct holes.