SSC CGL 2025 Tier-II · Paper-I, 19 January 2026 · Mathematical Abilities · Question 1 of 150

We transcribed this text from the pictures of a candidate's response sheet for this paper; the answer is the key SSC marked with a green tick on that sheet. Solution last reviewed 11 September 2026.

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SSC does not publish CGL question papers or a public answer key. This paper reaches us as a candidate's response sheet saved as pictures, so we transcribed the text from those pictures, and a second reader checked every question against them; where the layout carries meaning, we show the picture too. Each answer is the key SSC marked with a green tick on that sheet. SSC released a tentative key for challenge and later its final one, both only to candidates behind a login, and the sheet does not say which of the two it carries. UnlockIAS wrote the solutions.

The diagonals of three faces of a rectangular parallelepiped that meet at a single corner measure √5 cm, √10 cm, and √13 cm. What is the volume of this parallelepiped?

(a)12 cm³
(b)24 cm³
(c)6 cm³
(d)18 cm³

Answer

Why

Correct — C. Call the three edges at the corner a, b and c. Each face diagonal is the hypotenuse of a right triangle whose legs are two edges.

Write the three faces: a² + b² = 5, b² + c² = 10, c² + a² = 13
Add all three: 2(a² + b² + c²) = 28
Halve: a² + b² + c² = 14

Subtract the first face: c² = 14 − 5 = 9, so c = 3
Subtract the second: a² = 14 − 10 = 4, so a = 2
Subtract the third: b² = 14 − 13 = 1, so b = 1

Multiply the edges: V = 2 × 1 × 3 = 6 cm³ → option (c)

Why the others are wrong

(a)
12 cm³ — 12 = 4 × 1 × 3 — the square a² = 4 used in place of the edge a = 2. Take the root of every edge-square first: the edges are 2, 1 and 3.
(b)
24 cm³ — 24 is the total edge length, 4 × (1 + 2 + 3), not the volume. A cuboid has twelve edges, four of each length, and their sum is a length in cm.
(d)
18 cm³ — 18 = 2 × 1 × 9 — c² = 9 left unrooted while a and b were rooted. With c = 3 the product is 2 × 1 × 3 = 6.

Concept

A rectangular parallelepiped is a cuboid: six rectangular faces, with three edges a, b and c meeting at every corner.

Each face is a rectangle, so its diagonal is Pythagoras on two edges: √(a² + b²), √(b² + c²) and √(c² + a²).

Adding the three squared diagonals counts every edge-square twice. Halve the sum to get a² + b² + c², then subtract each squared diagonal in turn to free one edge-square at a time.

The stem says the three faces meet at a single corner. That makes them three different faces, ab, bc and ca, so each pair of edges appears in exactly one equation. Two opposite faces would repeat the same diagonal.

Key facts

The face diagonals of a cuboid with edges a, b, c are √(a² + b²), √(b² + c²) and √(c² + a²).
The three squared face diagonals add up to 2(a² + b² + c²).
The space diagonal of a cuboid is √(a² + b² + c²), which is √14 cm for this block.
A cuboid has 12 edges, four of each length, so its total edge length is 4(a + b + c).

Study next

Space diagonal of a cuboid, √(l² + b² + h²)Volume of a cuboid from the areas of three adjacent faces, √(ab × bc × ca)Total surface area 2(ab + bc + ca) from the same three edges

Common traps

Forgetting to halve the sum: a² + b² + c² is 14, not 28.
Multiplying the edge-squares 4, 1 and 9, or a mix of squares and roots, instead of the edges 2, 1 and 3.

Here the diagonals are given and the edges must be recovered. 17 Sep 2024, 16:00, Quant Q.6 runs the other way: it gives a cuboidal box of 1.2 cm × 1.3 cm × 1.5 cm and asks for the longest diagonal, √(l² + b² + h²).

Related PYQs

No directly related past PYQ was found.

Topic: Right Prism, Right Circular Cone, Right Circular Cylinder, Sphere, Hemispheres, Rectangular Parallelepiped, Regular Right Pyramid with triangular or square base — all 56 SSC CGL questions on it, every solved shift.