SOLVED PAPER 2022 (CDS) (II) (Elementary Mathematics)

Total Questions: 100

1. A closed box is built of wood of uniform thickness. Its external dimensions are 12 cm, 10 cm and 8 cm. If the inner surface area is 376 square cm, then what is the thickness of the wood?

Correct Answer: (b) 1 cm
Solution:Given, external dimensions of the
box are length 12 cm, breadth 10
cm and height 8 cm and inner surface
area is 376 cm².

Let the thickness of the wood be x cm.
Then internal dimensions of the box are
Length (l) = (12 - 2x) cm
Breadth (b) = (10 - 2x) cm
Height (h) = (8 - 2x) cm

We know that,
Inner surface area = 2(lb + bh + hI)

376 = 2[ (12 - 2x) (10 - 2x) + (10 - 2x) (8 - 2x)]
+ (8 - 2x) (12 - 2x)]
⇒ 188 = [120 - 44x + 4x² + 80
- 36x + 4x² + 96 - 40x + 4x²]
⇒ 47 = 74 - 30x + 3x²
⇒ x² - 10x + 9 = 0
⇒ (x - 1)(x - 9) = 0
⇒ x = 1 and x = 9 (not possible)
Hence, the thickness of wood is 1 cm.

2. X, Y and Z are three equilateral triangles. The sum of the areas of X and Y is equal to the area of Z. If the side lengths of X and Y are 6 cm and 8 cm respectively, then what is the side length of Z?

Correct Answer: (c) 10 cm
Solution:

3. What is the time taken by a person to cover one round of a circular park of diameter 210 m if he walks at a speed of 6 km/h? (Take π = ²²⁄₇)

Correct Answer: (a) 6.6 min
Solution:

4. In an equilateral triangle of side 2√3 cm, a circle is inscribed touching the sides. What is the area of the remaining portion of the triangle?

Correct Answer: (b) (3√3 - π) square cm
Solution:

5. A triangle has sides of length x cm, (x + 13) cm and (x + 26) cm. If its area is 126 square cm, then what is the value of x?

Correct Answer: (d) 15
Solution:

6. ABCD is a parallelogram. A circle through A, B and C intersects CD (produced) at E. Which of the following is/are correct?

1. AE = AD
2. CD = DE

Select the correct answer using the code given below

Correct Answer: (a) Only 1
Solution:

7. Consider the following statements

1. In an equilateral triangle, the centroid and centre of circumcircle coincide.
2. Angle bisectors of a cyclic quadrilateral form another cyclic quadrilateral.
3. Every cyclic parallelogram is a rectangle.

Which of the statements given above are correct?

Correct Answer: (d) 1, 2 and 3
Solution:Statement 1. In an equilateral triangle the centroid and centre of circumcircle coincide.
Statement 2. We know that angle bisectors of a cyclic quadrilateral form another cyclic quadrilateral.
Statement 3. We know that, if one of the interior angles of a parallelogram is 90°, all the other angles will also be equal to 90°. Since, all the angles in the parallelogram ABCD is a cyclic rectangle. Hence, a parallelogram is, a rectangle.

Hence, Statements 1, 2 and 3 are correct.

8. ABCD is a trapezium in which AB is parallel to DC. The vertices A, B, C and D pass through a circle. Which of the following are correct?

1. AD = BC
2. ∠A + ∠C=180°
3. ∠A + ∠D = 180°

Select the correct answer using the code given below.

Correct Answer: (b) Only 2 and 3
Solution:
Here, ABCD is a trapezium, AB || DC and
vertices A, B, C and D pass through a circle.
We can say that, ABCD is a cyclic
quadrilateral.

So, ∠A + ∠C = 180°  ...(i)
[∴ ABCD is a cyclic trapezium]
and ∠b + ∠d = 180°  ...(ii)
Here, AB || CD
So, ∠A + ∠D = 180°  ...(ii)
[∴ sum of interior angles is equal to 180°]
From Equation (i) and (iii), we get
∠C = ∠D
Similarly, from Equation (ii) and (iii), we get
∠A = ∠B
Therefore, Statements 2 and 3 are the
correct.

9. ABCD is a cyclic quadrilateral. AB and DC when produced, meet in E. Which of the following statements is/are correct?

1. Δ EBC is similar to ΔEAD
2. ∠CDE+ ∠DAE = 180°

Select the correct answer using the code given below.

Correct Answer: (a) Only 1
Solution:

10. In a triangle ABC, DE is a line segment which intersects AB at D and AC at E such that DE is parallel to BC. The line segment divides the triangle in two parts of equal area. What is BD/AB equal to?

Correct Answer: (b)
Solution: