SOF IMO Sample Paper 2026 Class 12

CLASS

12
SAMPLE PAPER
2025-26 SOF INTERNATIONAL
MATHEMATICS OLYMPIAD

Total Questions : 50 Time : 1 hr.
PATTERN & MARKING SCHEME
(2) Mathematical
(1) Logical Reasoning (3) Everyday (4) Achievers
Section
Reasoning or Applied Mathematics Section
Mathematics
No. of Questions 15 20 10 5
Marks per Ques. 1 1 1 3

SYLLABUS
Section – 1 : Verbal and Non-Verbal Reasoning.
Section – 2 : Relations and Functions, Inverse Trigonometric Functions, Matrices and
Determinants, Continuity and Differentiability, Application of Derivatives, Integrals, Application of
Integrals, Differential Equations, Vector Algebra, Three Dimensional Geometry, Probability, Linear
Programming.
OR
Section – 2 : Numbers, Quantification, Numerical Applications, Solutions of Simultaneous
Linear Equations, Matrices, Determinants, Application of Derivatives, Integration, Application of
Integrations, Differential Equations, Probability, Inferential Statistics, Index numbers, Time-based
data, Financial Mathematics, Linear Programming.
Section – 3 : The syllabus of this section will be based on the syllabus of Quantitative Aptitude.
Section – 4 : Matrices, Determinants, Application of Derivatives, Integration, Application of
Integrations, Differential Equations, Linear Programming, Probability.

LOGICAL REASONING
1. In the following equation, two signs and two numbers need to be interchanged to
make it correct. Select the correct interchange of signs and numbers from the given
options.
1050 × 60 ÷ 7 + 13 – 50 = 100
(A) ÷, × and 7, 50 (B) ×, – and 60, 7
(C) ×, ÷ and 60, 50 (D) +, ÷ and 50, 13
2. Select the odd one out.

(A) (B)

(C) (D)

Sample Paper | Class-12 | 1

3. A number arrangement machine when given an input of numbers, rearranges
them following a particular rule in each step. The following is an illustration of
input and steps of rearrangement.
Input : 25 280 345 36 93 147 550
Step-I : 550 280 345 36 93 147 25
Step-II : 550 345 280 93 147 36 25
Step-III : 550 345 280 147 93 36 25
What will be the third step for the following input?
Input : 113 18 48 225 462 175 288
(A) 462 288 48 225 113 175 18
(B) 462 288 225 175 113 48 18
(C) 462 225 288 48 113 175 18
(D) 462 288 225 48 113 175 18

MATHEMATICAL REASONING

1 4  9 −4 
4. If A is skew-symmetric matrix of order 2 and B, C are matrices 
2 9  and  −2 1 
   
respectively, then A3BC + A5(B2C2) + A7(B3C3) + ...... + A2n+1 BnC n is
(A) A skew-symmetric matrix
(B) A symmetric matrix
(C) An identity matrix

5. Solve the differential equation (x + y)dy + (x – y)dx = 0, given that y = 1 when
x = 1.
π
(A) log ( x + y ) + = log 2
2
2 −1  y 
(B) log y + tan   = log 2
x
y π
(C) log ( x 2 + y 2 ) + 2 tan−1   = + log 2
x 2

y π
(D) log x 2 + 2 tan−1   =
x 2

6. It has been found that if A and B play a game 12 times, A wins 6 times, B wins 4
times and they draw twice. A and B take part in a series of 3 games. The probability
that they will win alternately is
5 5
(A) (B)
72 36
19
(C) (D) None of these
27

2 | Sample Paper | Class-12

APPLIED MATHEMATICS
4. The following table gives information regarding weekly income of labourers
working at a dam site:
Income (in `) 600-700 700-800 800-900 900-1000 1000-1100 1100-1200 1200-1300
Number of
40 68 86 120 90 40 26
labourers

Estimate the quartile deviation.
(A) 115.87 (B) 231.77
(C) 7.99 (D) 8.88

5. For a Poisson's distribution, 3P(X = 2) = P(X = 4). Find P(X = 3). (Use : e–6 = 0.00248)
(A) 0.07297 (B) 0.08928
(C) 0.06261 (D) 0.0216

6. The amount of money today which is equal to series of payments in future is
(A) Nominal value of annuity (B) Present value of annuity
(C) Future value of annuity (D) None of these

EVERYDAY MATHEMATICS
7. In a class, 25% of the students get a scholarship. There are 40 girls and 50% of
them get a scholarship. If the girls constitute 25% of the strength of the class, then
find the percentage of boys who get a scholarship.
(A) 15% (B) 25%
(C) 20% (D) None of these

8. A man 2 metres tall walks away from a lamp post 5 metres height at the rate of
4.8 km/hr. The rate of increase of the length of his shadow, is
(A) 1.6 km/hr (B) 6.3 km/hr
(C) 5 km/hr (D) 3.2 km/hr

ACHIEVERS SECTION
9. Read the given statements carefully and select the correct option.
7
 dy  3
3
d2 y
Statement-I : The degree and order of the differential equation 1+    = 7
  dx   dx 2
respectively are 3 and 2.
4
Statement-II : If x3dy + xydx = x2dy + 2ydx, y(2) = e, then y(–1) = .
e
(A) Both Statement-I and Statement-II are true.
(B) Both Statement-I and Statement-II are false.
(C) Statement-I is true but Statement-II is false.
(D) Statement-I is false but Statement-II is true.

Sample Paper | Class-12 | 3

10. Solve the following and select the correct option.
(i) If the function f(x) = 2x2 – kx + 5 is increasing on [1, 2], then k lies in the
interval
(ii) The interval in which the function y = x3 + 5x2 – 1 is decreasing, is
(i) (ii)
(A) (– ∞, 4) (0, 10)
 −10 
(B) (4, ∞)  , 0
3 
 −10 
(C) (– ∞, 4)  , 0
3 

ANSWER KEY
1. (C) 2. (C) 3. (B)
MATHEMATICAL REASONING 4. (A) 5. (C) 6. (B)
APPLIED MATHEMATICS 4. (A) 5. (B) 6. (B)
7. (D) 8. (D) 9. (C) 10. (C)

4 | Sample Paper | Class-12