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Number System CAT 2025 Slot 1 Actual Questions

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Visual Lens | Odd even numbers | MEDIUM | CAT 2025 Slot 1

In the set of consecutive odd numbers {1, 3, 5, …, 57}, there is a number k such that the sum of all the elements less than k is equal to the sum of all the elements greater than k. Then, k equals

  1. 37
  2. 43
  3. 39
  4. 41
Answer

Let the diagonals be d1 and d2.

Area of a rhombus = 1/2 × d1 × d2 = 396 → d1 × d2 = 792.

In a rhombus, diagonals bisect each other at right angles.

So by Pythagoras: (d1/2)^2 + (d2/2)^2 = 36^2.

Multiply by 4: d1^2 + d2^2 = 4 × 1296 = 5184.

Use identity: (d1 − d2)^2 = (d1^2 + d2^2) − 2(d1 × d2).

Substitute: (d1 − d2)^2 = 5184 − 2×792 = 5184 − 1584 = 3600.

So |d1 − d2| = √3600 = 60.

Final Answer: 60

Breakup G Strategy | Rhombus + Algebra | MEDIUM | CAT 2025 Slot 1

If the length of a side of a rhombus is 36 cm and the area of the rhombus is 396 sq. cm, then the absolute value of the difference between the lengths, in cm, of the diagonals of the rhombus is ____

Answer

Let the diagonals be d1 and d2.

Area of a rhombus = 1/2 × d1 × d2 = 396 → d1 × d2 = 792.

In a rhombus, diagonals bisect each other at right angles.

So by Pythagoras: (d1/2)^2 + (d2/2)^2 = 36^2.

Multiply by 4: d1^2 + d2^2 = 4 × 1296 = 5184.

Use identity: (d1 − d2)^2 = (d1^2 + d2^2) − 2(d1 × d2).

Substitute: (d1 − d2)^2 = 5184 − 2×792 = 5184 − 1584 = 3600.

So |d1 − d2| = √3600 = 60.

Final Answer: 60

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