CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set M

LOGICAL REASONING

Question. How many such pair of letters are there in the word ORGANISED each of which has as many letters between them in the word as in the English alphabet?
A. One
B. Three
C. Two
D. More than three
Answer : C

Question. Consider the following letter/number/symbol arrangement and answer the question that follows :
@ C F S 9 W A ★ X 6 9 Q J % H 8 U N 1 2 T # 4 5 7 K $
What should come in the place of the question mark (?) in the following series based on the above arrangement?
★ 6 Q 9 J H % 8 N U 1 T ?
A. N
B. 4
C. 2
D. None of these
Answer : C

Question. Rishu started walking towards East from a point ‘X’, walked 250 metres and reached the point ‘P’. Again, he turned towards South, walked 50 metres and reached point ‘Q’. Again, he turned towards East, walked 250 metres and reached point ‘R’. He again turned South, walked 50 metres and reached point ‘S’. What is the shortest distance between ‘X’ and ‘S’?
A. 509.9 metres
B. 1100 metres
C. 561.2 metres
D. 590.9 metres
Answer : A

Question. Select a figure from the options which do not satisfy the same conditions of placement of the dots as in Fig. (X).

Answer : D

Question. Study the following information carefully and answer the given 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 : would 32 73 verb 26 new desk 19
Step I : 73 would 32 verb 26 new desk 19
Step II : 73 desk would 32 verb 26 new 19
Step III : 73 desk 32 would verb 26 new 19
Step IV : 73 desk 32 new would verb 26 19
Step V : 73 desk 32 new 26 would verb 19
Step VI : 73 desk 32 new 26 verb would 19
Step VII : 73 desk 32 new 26 verb 19 would
Step VII is the last step of the above input.
As per the rules followed in the above steps, answer the question.
Step IV of an input is :
63 car 51 eyes 25 36 store lane
Which of the following is definitely the input?
A. eyes car 25 63 36 51 store lane
B. eyes 25 car 63 51 36 store lane
C. eyes car 51 63 36 store lane
D. Can't be determined
Answer : D

Question. Two positions of a dice are shown here. Find the number on face opposite to the face having number 3.

A. 2 
B. 6
C. 5
D. 1
Answer : B

Question. Select the correct water image of the given combination of numbers and alphabets.

Answer : A

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

Answer : D

Question. Group the given figures into three classes on the basis of their identical properties by using each figure only once.

A. 1, 2, 3; 4, 5, 7; 6, 8, 9
B. 1, 3, 5; 2, 6, 7; 4, 8, 9
C. 1, 3, 5; 2, 6, 8; 4, 7, 9
D. 1, 4, 5; 2, 6, 7; 3, 8, 9
Answer : B

Question. There is a definite relationship between figures (1) and (2). Establish a similar relationship between figures (3) and (4) by selecting a suitable figure from the options which will replace the (?) in figure (4).

Answer : D

Question. If 'P – Q' means 'P is the daughter of Q', 'P × Q' means 'P is the father of Q', 'P ÷ Q' means 'P is the wife of Q' and 'P + Q' means 'P is the brother of Q', then how is M related to N in M – O × R × N?
A. Daughter
B. Aunt
C. Mother
D. Sister
Answer : B

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. The operations an numbers progress from left to right.
Rules:
(i) If a two digit odd number is followed by a prime number, the first number is to be multiplied by the prime number.
(ii) If an even number is followed by another even number, the first number is to be divided by the second number.
(iii) If an odd number is followed by a composite odd number, the first number is to be added to the second number.
(iv) If an even number which is multiple of 5 is followed by another number which is a multiple of 5, the second number is to be subtracted from the first number.
(v) If a number which is perfect square is followed by another number which is a perfect square, the resultant number is the product of the square roots of the two numbers.
21     9    15
81    25   5
What is the sum of the resultants of the two rows?
A. 240
B. 270
C. 280
D. None of these
Answer : A

Question. A set of three figures X, Y and Z showing a sequence of folding of a piece of paper. Fig. (Z) shows the manner in which the folded paper has been cut. Select a figure from the options which would most closely resembles the unfolded form of Fig. (Z).

Answer : B

Question. Rita stands third in a row of twenty students, arranged in ascending order of height. Five new students join the group, all taller than Rita. What will be Rita's position if the students are now arranged in descending order of their heights?
A. 22nd
B. 23rd
C. 18th
D. None of these
Answer : B

Question. In a certain code language, if COIMBATORE is written as DOKQEFSPUB, then how is POPULARISE written in that code language?
A. QPQVMBSJTF
B. MWRQRFTJSB
C. NWRQRFTJSB
D. MVQPQFTJSB
Answer : C

MATHEMATICAL REASONING

Question. The graph of y = ax2 + bx + c is given. Identify the signs of a, b and c.

A. a > 0, b < 0 and c < 0
B. a > 0, b > 0 and c < 0
C. a < 0, b < 0 and c > 0
D. a > 0, b < 0 and c > 0
Answer : D

Question. Solve for x and y :

Answer : D

Question. In the given figure, if AD is the bisector of ∠BAC.
If AB = 10 cm, AC = 14 cm and BC = 6 cm, then find BD and DC.

A. 2.5 cm, 3.5 cm
B. 3 cm, 3 cm
C. 3.5 cm, 4.5 cm
D. 4 cm, 2 cm
Answer : A

Question. In the given figure, AB is a diameter of a circle with centre O, AT is a tangent and ∠AOQ = 58°, then find ∠ATQ.

A. 61°
B. 60°
C. 55°
D. 51°
Answer : A

Question. If the 10th term of an A.P. is 52 and 17th term is 20 more than the 13th term, then find the A.P.
A. 40, 45, 50,.....
B. 45, 50, 55,.....
C. 17, 22, 27,.....
D. 7, 12, 17,.....
Answer : D

Question. If the coordinates of the mid-points of the sides of a triangle are (1, 1), (2, –3) and (3, 4). Find its centroid.
A. (2, 2)
B. (2, –2)
C. (2, 2/3)
D. (–2, 2/3)
Answer : C

Question. The given bar graph shows the sales of books (in thousand number) from six branches (B1 to B6) of a publishing company during two consecutive years 2016 and 2017.

(a) What percent of the average sales of branches B1, B2 and B3 in 2017 is the average sales of branches B1, B3 and B6 in 2016?
(b) What is the total sales branches B1, B3 and B5 together for both the years (in thousand numbers) ?
      (a)             (b)
A. 45%           250
B. 82.5           315
C. 90.6           560
D. 85.18%      520
Answer : D

Question. If a and b are the zeros of the quadratic polynomial f(t) = t2 – 4t + 3, then find the value of a4b3 + a3b4.
A. 128
B. 116
C. 108
D. 98
Answer : C

Question. Which of the following rational numbers have terminating decimal expansion?
(i) 17/8
(ii) 64/455
(iii) 15/1600
(iv) 13/3125
(v) 129/2² X 5⁷ × 7¹⁷
(vi) 987/10500
A. Only (iv) and (vi)
B. Only (ii) and (v)
C. Only (i), (iii), (iv) and (vi)
D. Only (i), (iii) and (iv)
Answer : C

Question. The following table gives weekly wages in rupees of workers in a certain commercial organization. The frequency of class 49-52 is missing. It is known that the mean of the frequency distribution is 47.2. Find the missing frequency.

A. 44
B. 49
C. 58
D. 60
Answer : A

Question. Find the roots of the given quadratic equation.
4x² + 4bx – (a² – b²) = 0
A. a+b/2, a+b/2
B. a−b/2, a+b/2
C. a−b/2, a−b/2
D. a−b/2, - (a+b)/2
Answer : D

Question. If three times the larger of the two numbers is divided by the smaller one, we get 4 as quotient and 3 as the remainder. Also, if seven times the smaller number is divided by the larger one, we get 5 as quotient and 1 as remainder. Find the numbers.
A. 35 and 28
B. 45 and 18
C. 25 and 18
D. 25 and 28
Answer : C

Question. If secθ = x+1/4x, then secθ + tanθ =
A. 2x or 1/2x
B. Only 2x
C. Only 1/2x
D. 2x or 1/x
Answer : A

Question. In the given diagram, a fair spinner is placed at the centre O of the circle. Diameter AOB and radius OC divides the circle into three regions labelled X, Y and Z. If ∠BOC = 45°. What is the probability that the spinner will land in the region X?

A. 2/7
B. 3/8
C. 3/5
D. 2/5
Answer : B

Question. In the given figure, we have AB || CD || EF.
If AB = 6 cm, CD = x cm, EF = 10 cm, BD = 4 cm and DE = y cm, then find the values of x and y respectively.

A. 3.75 cm, 6.67 cm 
B. 2.75 cm, 3.67 cm
C. 6.75 cm, 3.67 cm
D. 3.75 cm, 2.67 cm
Answer : A

Question. Evaluate:
2sin68°/cos22° - 2cot15°/5tan75° - 3tan45° tan20° tan40° tan50° tan70°/5
A. 1
B. 4
C. 2
D. 3
Answer : A

Question. Sum of the first 14 terms of an A.P. is 1505 and its first term is 10. Find its 25th term.
A. 370
B. 320
C. 380
D. 390
Answer : A

Question. With vertices A, B and C of a triangle ABC as centres, arcs are drawn with radii 5 cm each as shown in figure.

If AB = 14 cm, BC = 48 cm and CA = 50 cm, then find the area of the shaded region. (Use π = 3.14)
A. 257.5 cm²
B. 277.5 cm²
C. 255.7 cm²
D. 296.75 cm²
Answer : D

Question. The following is the distribution of height of students of a certain class in a certain city :

Find the average height (in approx) of maximum number of students.
A. 168.35
B. 157.35
C. 170.38
D. None of these.
Answer : B

Question. In the given figure, if O is the centre of two concentric circles with OA = 6 cm, OB = 3 cm and AP = 10 cm, then BP = __________.

A. √91 cm
B. √127 cm
C. √119 cm
D. √109 cm
Answer : B

EVERYDAY MATHEMATICS

Question. The car hire charges in a city comprise of a fixed charges together with the charge for the distance covered. For a journey of 12 km, the charge paid is ₹ 89 and for a journey of 20 km, the charge paid is ₹ 145. What will a person have to pay for travelling a distance of 30 km?
A. ₹ 315
B. ₹ 215
C. ₹ 415
D. ₹ 210
Answer : B

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 tip reach?
A. 6 m
B. 8 m
C. 10 m
D. 4 m
Answer : A

Question. A peacock is sitting on the top of a pillar, which is 9 m high. From a point 27 m away from the bottom of the pillar, a snake is coming to its hole at the pillar.
Seeing the snake the peacock pounces on it. If their speeds are equal, at what distance from the hole is the snake caught?
A. 12 m
B. 11 m
C. 10 m
D. 13 m
Answer : A

Question. Kavleen wanted to save at least ₹ 6500 for sending her daughter to school next year (after 12 months).
She saved ₹ 450 in the first month and raised her savings by ₹ 20 every next month. How much will she be able to save in next 12 months?
A. ₹ 5600
B. ₹ 6500
C. ₹ 6720
D. ₹ 6550
Answer : C

Question. As observed from the top of a light house, 100 m above sea level, the angle of depression of a ship, sailing directly towards it, changes from 30° to 45°. Determine the distance travelled by the ship during the period of observation.
A. 75.3 m
B. 68.59 m
C. 73.2 m
D. 74.56 m
Answer : C

Question. 1000 tickets of a lottery were sold and there are 5 prizes on these tickets. If Saket has purchased one lottery ticket, what is the probability of winning a prize?
A. 0.005
B. 0.006
C. 0.004
D. 0.003
Answer : A

Question. A cylindrical container of radius 6 cm and height 15 cm is filled with ice-cream. The whole ice-cream has to be distributed to 10 children in equal cones with hemispherical tops. If the height of the conical portion is four times the radius of its base, find the radius of the ice-cream cone.
A. 3 cm
B. 4 cm
C. 6 cm
D. 5 cm
Answer : A

Question. A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48 km, 60 km and 72 km a day, round the field. When will they meet again?
A. 30 days
B. 20 days
C. 22 days
D. 15 days
Answer : A

Question. Kunal bought a suitcase with 15% discount on the labelled price. He sold the suitcase for ₹ 2880 with 20% profit on the labelled price. At what price did he buy the suitcase?
A. ₹ 2040
B. ₹ 2400
C. ₹ 2604
D. ₹ 2640
Answer : A

Question. The cost of a table and a chair are in the ratio of 5 : 7. If the cost of chair and table is increased by 20% and 10% respectively, then what will be the new ratio?
A. 16 : 17
B. 55 : 84
C. 60 : 77
D. None of these
Answer : B

Question. In figure, a crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF it is 5 cm. Find:
(i) the radii of two circles
(ii) the area of the shaded region. [Take π = 3.14]

                 (i)                                 (ii)
A. 20.5 cm and 25 cm         642.915 cm²
B. 20.5 cm and 26 cm         642.915 cm²
C. 25 cm and 27.5 cm         642.915 cm²
D. 25 cm and 26.5 cm         642.915 cm²
Answer : A

Question. Cards numbered from 11 to 60 are kept in a box.
If a card is drawn at random from the box, find the probability that the number on the drawn card is (i) an odd number
(ii) a perfect square number
(iii) divisible by 5
(iv) a prime number less than 20
      (i)      (ii)        (iii)       (iv)
A. 1/2     1/5       2/25     4/25
B. 2/25   1/2       1/5       2/25
C. 1/2     1/5       2/25     2/25
D. 1/2     2/25     1/5       2/25
Answer : D

Question. Fill in the blanks and select the CORRECT option.
(i) The coordinates of the circumcentre of the triangle whose vertices are (8, 6) (8, –2) and (2, –2) is P . Its circumradius is Q .
(ii) If two vertices of an equilateral triangle be (0,0) and (3, 3 ), then the third vertex is R .
(iii) Three consecutive vertices of a parallelogram are (–2, –1), (1, 0) and (4, 3). The fourth vertex is S .
         P         Q           R             S
A. (5, 2)       5     (−3, − 3)     (1, –2)
B. (5, 2)       5     (3, − 3)        (1, 2)
C. (5, 2)       5     (3, 3)           (–1, 2)
D. (–5, 2)     5     (− 3, 3)        (1, 2)
Answer : B

Question. Read the statements carefully and select the correct option.
Statement-I : If a hemisphere of lead of radius 7 cm is melted and recast into a right circular cone of height 49 cm, then the radius of the base is 7 cm.
Statement-II : Lead spheres of diameter 6 cm are dropped into a cylindrical beaker containing some water and are fully submerged. If the diameter of the beaker is 18 cm and water rises by 40 cm, then the number of lead spheres dropped in the water is 40.
A. Both Statement-I and Statement-II are false.
B. Both Statement-I and Statement-II are true.
C. Statement-I is true but Statement-II is false.
D. Statement-I is false but Statement-II is true.
Answer : A

Question. Read the statements carefully and state 'T' for true and 'F' for false.
1. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side of the triangle.
2. The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle .
3. If a line through one vertex of a triangle divides the opposite side in the ratio of other two sides, then the line bisects the angle at the vertex.
4. Any line parallel to the parallel sides of a trapezium divides the non-parallel sides proportionally.
5. Two times the sum of the squares of the sides of a triangle is equal to four times the sum of the square of the medians of the triangle.
    1     2    3     4     5
A. T    T    T     T     T
B. T    T    T     T     F
C. F    T    F     T     F
D. T    T    F     T     F
Answer : B