CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set H

LOGICAL REASONING

Question. How many such symbols are there each of which is immediately preceded by a digit and how many such digits are there each of which is immediately preceded by a symbol. What is total number of both those?
A F 9 V d 9 V @ S 2 G R A 3 Q £ B Z 7 C Y D
T 4 8 P F 6 @ $ 5 E H B J d
A. 3
B. 1
C. 2
D. None of these
Answer : A

Question. If in a certain code language, MANGO is coded as 31–1– 41–7–51, then how will LEAF be coded in that language?
A. 21–10–8–9
B. 21–10–9–8
C. 21–5–6–1
D. 21–5–1–6
Answer : D

Question. There is a certain relationship between figures (i) and
(ii). Establish the similar relationship between figures
(iii) and (iv) by selecting a suitable figure from the options which will replace the (?) in fig. (iv).

Answer : A

Question. In the given question, two rows of numbers are given. The resultant number in each row is to be worked out separately based on the following rules and the question below the rows of numbers is to be answered.
Rules:
(i) If an odd number is followed by another composite odd number, they are to be multiplied.
(ii) If an even number is followed by an odd number, they are to be added.
(iii) If an even number is followed by a number which is a perfect square, the even number is to be subtracted from the perfect square.
(iv) If an odd number is followed by an even number, the even number is to be subtracted from the odd number.
(v) If an odd number is followed by a prime odd number, the first number is to be divided by the second number.
35 7 2
12 16 x
If x is the resultant of the first row, then what will be the product of the resultants of the two rows?
A. 7
B. 10
C. 24
D. 21
Answer : D

Question. Find the missing number, if same rule is followed in all the three figures.

A. 47
B. 45
C. 37
D. 35
Answer : C

Question. Study the statements carefully.
(1) 'A Δ B' means 'A is the mother of B'.
(2) 'A ⬜ B' means 'A is the sister of B'.
(3) 'A * B' means 'A is the father of B'.
(4) 'A b B' means 'A is the brother of B'.
Which of the following means 'N is the maternal uncle of M'?
A. N b P ⬜ L D E ⬜ M
B. N ⬜ Y D A b M
C. M ⬜ Y * P ⬜ N
D. None of these
Answer : A

Question. Select a figure from the options which will continue the same series as established by the Problem Figures.

Answer : D

Question. Four friends Mohit, Raj, Varun and Anuj live in a same locality. The house of Raj is in the East of Mohit's house and in the North of Varun's house. The house of Varun is in the West of Anuj's house. Anuj's house is in which direction w.r.t. Mohit's house?
A. South-East
B. South
C. North-West
D. West
Answer : A

Question. Six lectures – A, B, C, D, E and F – are to be delivered from Monday to Sunday, one lecture everyday.
(i) Lecture C cannot be delivered on Friday.
(ii) Lecture A is delivered immediately after lecture D.
(iii) There should be a gap of two days between the lectures B and F.
(iv) There is one holiday except Saturday. Lecture F is delivered on the next day of holiday.
(v) Lecture E is delivered on Wednesday and it is not immediately followed by lecture F.
How many lectures were delivered between lectures F and D?
A. None
B. Two
C. Three
D. Four
Answer : A

Question. Which of the following Venn diagrams best represents the relationship amongst, 'Gas, Water and Helium'?

Answer : C

Question. If 'P' means '×', 'Q' means '÷', 'R' means '+' and 'S' means '–', then what is the value of 154 Q 14 S 7 P 3 R 25 ?
A. 35
B. 15
C. 42
D. 25
Answer : B

Question. A word and number arrangement machine when given an input line of words and numbers rearranges them following a particular rule in each step. The following is an illustration of input and rearrangement.
Input : 51 pour 32 start now 23 46 house
Step I : 23 51 pour 32 start now 46 house
Step II : 23 start 51 pour 32 now 46 house
Step III : 23 start 32 51 pour now 46 house
Step IV : 23 start 32 pour 51 now 46 house
Step V : 23 start 32 pour 46 51 now house
Step VI : 23 start 32 pour 46 now 51 house
If Step VI is the last step of the rearrangement, and the same rule is followed for below input, then which of the following will be step III for the given Input?
Input : nice flower 34 12 costly height 41 56
A. 12 nice 34 height flower costly 41 56
B. 12 nice 34 height 41 flower costly 56
C. 12 nice 34 flower costly height 41 56
D. 12 nice flower 34 costly height 41 56
Answer : A

Question. If it is possible to make only one meaningful word from the second, the sixth, the seventh, the eighth and the tenth letters of the word PERFORMANCE using each letter only once, then first letter of the word is your answer. If no such word can be formed your answer is X and if more than one such word can be formed your answer is Y.
A. C
B. R
C. Y
D. X
Answer : C

Question. In which of the following options, Fig. (X) is exactly embedded as one of its part?

Answer : C

Question. Select the correct water image of Fig. (X) from the given options.

Answer : B

MATHEMATICAL REASONING

Question. In the given diagram, ∠ABC = ∠AED, AD = 3 cm,
AE = 5 cm and EC = 2 cm. Find :

(i) BD
(ii) Area of ΔAED / Area of ΔABC
          (i)              (ii)
A. 8(2/3)cm       9/49
B. 8(2/3)cm       9/23
C. (2/5)cm         9/49
D. (2/5)cm         9/23
Answer : A

Question. The number 0.211211121111211111..... is a
A. Terminating decimal
B. Non-terminating repeating decimal
C. Non-terminating and non-repeating decimal
D. None of these
Answer : C

Question. Solve the following system of equations :
(a – b)x + (a + b)y = a² – 2ab – b²
(a + b)(x + y) = a² + b²
A. a – b, –2ab
B. a + b, –2ab
C. a² + b², -2ab/a+b
D. a+b. -2ab/a+b
Answer : D

Question. If sin θ = 3/5, then evaluate cos θ - 1/tanθ/2cotθ 
A. –1/5
B. 1/5
C. 2/5
D. –2/5
Answer : A

Question. If a and b are the zeros of the quadratic polynomial x² – 3x + 2, then a quadratic polynomial whose zeros are 1/2α+β and 1/2β+α and is
A. 20x² + 9x + 1
B. 20x² – 9x – 1
C. 20x² – 9x + 1
D. 20x² + 9x – 1
Answer : C

Question. Find the median from the following data :

A. 10
B. 20
C. 30
D. 40
Answer : D

Question. Given below are the steps of construction of a pair of tangents to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm. Find which of the following step is wrong?
(P) Take a point O on the plane paper and draw a circle of radius OA = 4 cm. Also, draw a concentric circle of radius OB = 6 cm.
(Q) Find the mid-point A of OB and draw a circle of radius BA = AO. Suppose this circle intersects the circle of radius 4 cm at P and Q.
(R) Join BP and BQ to get the desired tangents from a point B on the circle of radius 6 cm.
A. Only (P)
B. Only (Q)
C. Both (P) & (Q)
D. Both (Q) & (R)
Answer : B

Question. In the Maths test two representatives, while solving a quadratic equation, committed the following mistakes:
(i) One of them made a mistake in the constant term and got the roots as 5 and 9.
(ii) Another one committed an error in the coefficient of x and got the roots as 12 and 4.
But in the meantime, they realised that they are wrong and they managed to get it right jointly. Find the correct quadratic equation.
A. x² + 4x + 14 = 0
B. 2x² + 7x – 24 = 0
C. x² – 14x + 48 = 0
D. 3x² – 17x + 52 = 0
Answer : C

Question. If the p^th, q^th and r^th terms of an A.P. are P, Q, R respectively, then P(q – r) + Q(r – p) + R(p – q) is equal to ______.
A. 0
B. 1
C. pqr
D. p + qr
Answer : A

Question. In the diagram, O is the centre of the circle.
Find the value of x.

A. 111°
B. 123°
C. 69°
D. 49°
Answer : A

Question. The area of a rectangular garden, ABCD, is 100 m².
Inside the garden there is a rectangular lawn, EFGH, whose sides are parallel to those of the garden.

Find the area of the lawn, EFGH (in sq. metres).
A. 110 − 5x − 200/x
B. 110 + 5x − 200/x
C. 110 + 5x + 200/x
D. 110 − 5x + 200/x
Answer : A

Question. Which of the following is not the graph of a quadratic polynomial ?

Answer : D

Question. Find the values of a and b respectively for which the following system of linear equations has infinite number of solutions.
2x – 3y = 7 (a + b)x – (a + b – 3)y = 4a + b
A. –1, –5
B. –1, 5
C. 1, 5
D. –5, –1
Answer : D

Question. If tan²q = 1 – a², then secq + tan³q cosecq = (2 – a²)n, where n =
A. 2
B. 3/2
C. 1
D. 2/9
Answer : B

Question. ABCD is a parallelogram and L is a point on DB. The produced line AL meets BC at M and DC produced at N.
Given that DL = 3LB, find AB/CN

A. 3/2
B. 1/2
C. 4/5
D. 1/4
Answer : B

Question. n² – 1 is divisible by 8, if n is
A. An even integer
B. A natural number
C. An odd integer
D. None of these
Answer : C

Question. If α and β are the zeros of the polynomial 3t2 – 6t + 4, find the value of α/β + β/α + 2 (1/α + 1/β) + 3αβ
A. 5
B. 8
C. 10/3
D. 1/2
Answer : B

Question. Evaluate :
- tan θ cot(90° - θ) + sec θ cosec (90° - θ)
+ sin2 35 ° + sin2 55 ° / tan 10° tan 20° tan 30° tan 70° tan 80°
A. √3
B. 2√3
C. √3/2
D. 1
Answer : B

Question. Which of the following equations are not quadratic?
A. x(2x + 3) = x + 2
B. (x – 2)2 + 1 = 2x – 3
C. y(8y + 5) = y2 + 3
D. y(2y + 15) = 2(y2 + y + 8).
Answer : D

Question. Which of the following statements is correct ?
A. Tossing a fair coin is a fair way to decide that which one team out of the two cricket teams should bat first.
B. If a coin is tossed once, there are two possible outcomes : a head or a tail. Therefore, for each of these outcomes, the probability is 1/3.
C. If two coins of different denominations are tossed simultaneously, there are three possible outcomes: two heads, two tails or one of each. Here, the probability of getting two heads is 1/3 whereas the probability of getting one head and one tail is 1/2.
D. If a dice is tossed once, there are only two possible outcomes : getting a number greater than 4 or less than equal to 4. Therefore, the probability of getting a number greater than 4 is 1/2
Answer : A
.

EVERYDAY MATHeMATICS

Question. Mohit has two coins, one of ₹ 1 denomination and the other of ₹ 2 denomination. He tosses the two coins simultaneously. What is the probability that he gets atleast one head?
A. 1/4
B. 1/2
C. 3/4
D. 0
Answer : C

Question. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ?
A. 2520 cm
B. 5230 cm
C. 4296 cm
D. 3893 cm
Answer : A

Question. A man travels 370 km partly by train and partly by car. If he covers 250 km by train and the rest by car, it takes him 4 hours. But, if he travels 130 km by train and the rest by car, he takes 18 minutes longer.
Find the speed of the train and car respectively.
A. 80 km/hr, 100 km/hr
B. 98 km/hr, 100 km/hr
C. 100 km/hr, 98 km/hr
D. 100 km/hr, 80 km/hr
Answer : D

Question. The foot of a ladder is 6 m away from a wall and its top reaches a window 8 m above the ground. If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reaches?
A. 6 m
B. 4 m
C. 5 m
D. 3 m
Answer : A

Question. There is a square field whose side is 44 m. A square flower bed is prepared in its centre leaving a gravel path all around the flower bed. The total cost of laying the flower bed and gravelling the path at ₹ 2.75 and ₹ 1.50 per square metre respectively, is ₹ 4904. Find the width of the gravel path.
A. 1 m
B. 2 m
C. 4 m
D. 6 m
Answer : B

Question. A man repays a loan of ₹ 3250 by paying ₹ 20 in the first month and then increases the payment by ₹ 15 every month. How long will it take him to clear the loan?
A. 1 year 3 months
B. 1 year 8 months
C. 1 year
D. 1 year 5 months
Answer : B

Question. A merchant purchases a wrist watch for ₹ 450 and fixes the list price in such a way that after allowing a discount of 10% he earns a profit of 20%. The list price of the watch is
A. ₹ 500
B. ₹ 600
C. ₹ 700
D. ₹ 750
Answer : B

Question. A man bought 5 shirts at ₹ 450 each, 4 trousers at ₹ 750 each and 12 pairs of shoes at ₹ 750 each. What is the average expenditure per article?
A. ₹ 678.57
B. ₹ 800
C. ₹ 900
D. ₹ 1000
Answer : A

Question. When the capacity of the bucket is 13.5 litres, 12 buckets of water will fill a tank. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 9 litres ?
A. 8
B. 15
C. 16
D. 18
Answer : D

Question. From the salary of a worker, 10% is deducted as house rent, 15% of the rest he spends on children's education and 10% of the balance he spends on clothes. Now, he is left with ₹ 1377. His salary is _______.
A. ₹ 2000
B. ₹ 2040
C. ₹ 2100
D. ₹ 2200
Answer : A

ACHIEVERS SECTION

Question. ABCD is a rectangle and M is a point on CD. AC and BM meet at X.

It is given that CM = 3MD. Find :
(i) area of DCXM : area of DAXB
(ii) area of DBXC : area of rectangle ABCD  
      (i)             (ii)
A. 9:16         3:11
B. 9:16         3:14
C. 16:9         14:3
D. 16:9         11:3
Answer : B

Question. Fill in the blanks.
In 'Less Than' Ogive, the cumulative frequencies  are written corresponding to the ___P___ limits of the classes of the given data.
In 'More Than' Ogive, the cumulative frequencies are written corresponding to the ___Q___ limit of the classes.
On an ogive, point P whose y-coordinate = n/2 (i.e., half of the total number of the entries), has its x-coordinate equal to the ___R___ of the data.
Two ogives, one 'less than' type and the other 'more than' type for the same data when drawn simultaneously on the same graph, intersect each other at the point P whose ___S___ = n/2 and ___T___ = median of the data where n is the total number of the entries of the data.
        P         Q            R                S                   T
A. Upper    Lower    Median    y-coordinate    x-coordinate
B. Lower    Upper    Median    y-coordinate    x-coordinate
C. Upper    Lower    Median    x-coordinate    y-coordinate
D. Upper    Lower    Upper      y-coordinate    x-coordinate limit
Answer : A

Question. Study the statements carefully.
Statement I : Both the roots of the equation
x² – x + 1 = 0 are real.
Statement II : The roots of the equation
ax² + bx + c = 0 are real if and only if b² – 4ac ≥ 0.
Which of the following options hold?
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 and Statement II is true.
Answer : D

Question. Match the columns.
                      Column-I                            Column-II
(i) When an odd integer 'a' is divided           (p) 5
by 2, then remainder r can be
(ii) When an integer 'a' whose unit               (q) 1
digit is 4 is divided by 5, then
remainder r can be
(iii) A whole number lying between               (r) 2
√2 and 2√2 is
(iv) If the digit at the units place of a            (s) 4
number is 5, then it is surely divisible by
A. (i) → (p), (ii) → (r), (iii) → (s), (iv) → (q)
B. (i) → (q), (ii) → (s), (iii) → (p), (iv) → (r)
C. (i) → (p), (ii) → (r), (iii) → (q), (iv) → (s)
D. (i) → (q), (ii) → (s), (iii) → (r), (iv) → (p)
Answer : D

Question. The two circles intersects at C and D, where O is the centre of the second circle. AD produced and cuts the second circle at F. BD produced and cuts the second circle at E. ∠DEF = 110°, ∠ACB = 32° and ∠DAB = 118°.

Find :
(i) ∠ACE (ii) ∠COD
(i) (ii)
A. 96° 68°
B. 72° 59°
C. 59° 72°
D. 68° 96°
Answer : D