CDS Practice Test 2026

About the CDS exam

The Combined Defence Services Examination (CDS) is a defence examination administered in India. It is designed to assess candidates' knowledge and proficiency across 3 key subject areas including English, General Knowledge, Elementary Mathematics. This exam is widely recognized as an essential benchmark for students and professionals.

Use short CDS practice sets to identify topics to revisit. Choose a subject and difficulty, answer questions, then review the explanations. New accounts get 30 question credits; further new questions use paid credits. Retaking previously seen questions is free.

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8 free CDS practice questions (2026)

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1. What are the last two digits of the sum 1! + 2! + 3! + ... + 100!?

   • 13
   • 33
   • 23
   • 43
   Show answer & explanation

   Answer: A. 13

   The terms from 10! onwards end in 00 because they contain at least two 5s and multiple 2s. Thus, the last two digits of the sum depend only on the sum 1! + 2! + ... + 9! modulo 100. Summing the last two digits of each term: 01 + 02 + 06 + 24 + 20 + 20 + 40 + 20 + 80 = 213. 213 mod 100 is 13.

   Topic: Elementary Mathematics · hard

2. If a_1, a_2, a_3, ..., a_16 are in Arithmetic Progression (AP) and a_1 + a_4 + a_7 + a_10 + a_13 + a_16 = 114, then what is the value of a_1 + a_6 + a_11 + a_16?

   • 64
   • 76
   • 82
   • 98
   Show answer & explanation

   Answer: B. 76

   In an AP, terms equidistant from the beginning and the end have a constant sum. Thus, a_1 + a_16 = a_4 + a_13 = a_7 + a_10. The first sum consists of three such pairs, so 3(a_1 + a_16) = 114, which means a_1 + a_16 = 38. The required sum is a_1 + a_6 + a_11 + a_16. Since a_6 + a_11 = a_1 + a_16, this sum is 2(a_1 + a_16) = 2 * 38 = 76.

   Topic: Elementary Mathematics · hard

3. If cosθ + sinθ = √2 cosθ, then what is the value of cosθ - sinθ?

   • √2 sinθ
   • -√2 sinθ
   • (1/√2) sinθ
   • √2 cosθ
   Show answer & explanation

   Answer: A. √2 sinθ

   We know that (cosθ + sinθ)^2 + (cosθ - sinθ)^2 = 2(sin^2θ + cos^2θ) = 2. Substituting the given equation: (√2 cosθ)^2 + (cosθ - sinθ)^2 = 2 => 2cos^2θ + (cosθ - sinθ)^2 = 2 => (cosθ - sinθ)^2 = 2(1 - cos^2θ) = 2sin^2θ. Taking the square root, cosθ - sinθ = √2 sinθ.

   Topic: Elementary Mathematics · hard

4. Three circles of radii r_1, r_2, and r_3 touch each other externally and also touch a common straight line. If r_1 = 9 cm and r_2 = 4 cm are the radii of the two larger circles, and r_3 is the radius of the smallest circle touching both of them and the line, what is the value of r_3?

   • 1.22 cm
   • 1.44 cm
   • 1.69 cm
   • 2.25 cm
   Show answer & explanation

   Answer: B. 1.44 cm

   For three externally touching circles that also touch a common line, the relationship between their radii is given by 1/√r_3 = 1/√r_1 + 1/√r_2. Substituting the values: 1/√r_3 = 1/√9 + 1/√4 = 1/3 + 1/2 = 5/6. Therefore, √r_3 = 6/5 => r_3 = 36/25 = 1.44 cm.

   Topic: Elementary Mathematics · hard

5. A solid right circular cone of height 24 cm and base radius 12 cm is cut into two parts by a plane parallel to its base at a height of 12 cm from the base. What is the ratio of the volume of the upper cone to the volume of the lower frustum?

   • 1:7
   • 1:8
   • 1:3
   • 1:4
   Show answer & explanation

   Answer: A. 1:7

   The height of the original cone is H = 24 cm. The cut is made at a height of 12 cm from the base, so the upper cone has height h = 24 - 12 = 12 cm. Since the cut is parallel to the base, the upper cone is similar to the original cone. The ratio of heights is h/H = 12/24 = 1/2. The ratio of their volumes is (h/H)^3 = (1/2)^3 = 1/8. Thus, if the original cone volume is 8V, the upper cone is V, and the frustum is 8V - V = 7V. The ratio is 1:7.

   Topic: Elementary Mathematics · hard

6. A train leaves station A at 5 AM and reaches station B at 9 AM. Another train leaves station B at 7 AM and reaches station A at 10:30 AM. At what time do the two trains cross each other?

   • 7:36 AM
   • 7:56 AM
   • 8:12 AM
   • 8:24 AM
   Show answer & explanation

   Answer: B. 7:56 AM

   Let the distance between A and B be D. Train 1 takes 4 hours (Speed S_1 = D/4). Train 2 takes 3.5 hours = 7/2 hours (Speed S_2 = 2D/7). By 7 AM, Train 1 has already traveled for 2 hours, covering 2 * (D/4) = D/2 distance. The remaining distance between them at 7 AM is D/2. Their relative speed is S_1 + S_2 = D/4 + 2D/7 = 15D/28. Time taken to meet after 7 AM = (D/2) / (15D/28) = 14/15 hours = 56 minutes. Therefore, they cross at 7:56 AM.

   Topic: Elementary Mathematics · hard

7. If α and β are the roots of the quadratic equation x^2 - 5x + 6 = 0, what is the equation whose roots are α^2 + β and β^2 + α?

   • x^2 - 18x + 77 = 0
   • x^2 - 15x + 56 = 0
   • x^2 - 12x + 35 = 0
   • x^2 - 20x + 99 = 0
   Show answer & explanation

   Answer: A. x^2 - 18x + 77 = 0

   The roots of x^2 - 5x + 6 = 0 are α = 2 and β = 3. Let the new roots be R_1 = α^2 + β = 2^2 + 3 = 7 and R_2 = β^2 + α = 3^2 + 2 = 11. The sum of the new roots S = 7 + 11 = 18 and the product P = 7 * 11 = 77. The required quadratic equation is x^2 - Sx + P = 0 => x^2 - 18x + 77 = 0.

   Topic: Elementary Mathematics · hard

8. A dishonest dealer sells articles at a profit of 10% on the cost price and also uses a false weight of 800 g instead of 1 kg. What is his actual total profit percentage?

   • 32.5%
   • 35%
   • 37.5%
   • 40%
   Show answer & explanation

   Answer: C. 37.5%

   Let the cost price of 1 g of article be Re 1. The CP of 1000 g is Rs 1000. He sells at 10% profit on CP, so SP = Rs 1100. However, he only gives 800 g, which actually costs him Rs 800. Profit = 1100 - 800 = Rs 300. Profit percentage = (300 / 800) * 100% = 37.5%.

   Topic: Elementary Mathematics · hard

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CDS Exam Subjects

Available practice subjects: English, General Knowledge, Elementary Mathematics. Select a subject for focused preparation. Coverage is not a guarantee of the complete official syllabus.

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