CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set G

LOGICAL REASONING

Question. In the given letter series, some of the letters are missing which are given in that order as one of the options below it. Choose the correct option. mnonopqopqrs
A. knopq
B. oqrst
C. pqrst
D. qrstu
Answer : C

Question. How many such pair of letters are there in the word PRODUCTION each of which has as many letters between them as in the English alphabet?
A. None
B. One
C. Two
D. Three
Answer : D

Question. In a certain code 'a friend of mine' is written as '4 9 1 6', 'mine lot of metal' is written as '3 1 0 9' and 'a piece of metal' is written as '7 1 6 3'. How can '8 7 3' be written in same code?
A. a metal piece
B. metal for friend
C. piece of advice
D. large metal piece
Answer : D

Question. Boys and girls are made to sit in 7 rows and 7 columns all of them facing West. P is at the centre of the arrangement, Q is just behind P, R is on the immediate left of Q, and S is to the West of P. Then R is in which direction with respect to S?
A. South-West
B. North-West
C. South-East
D. Data inadequate
Answer : C

Question. There is a certain relationship between figures (i) and (ii). Establish a similar relationship between figures (iii) and (iv) by selecting a suitable figure from the options that would replace the question mark in figure (iii).

Answer : D

Question. Read the following information and answer the question given below :
(i) Four persons A, B, C and D eat Mango, Banana, Orange or Peach from Monday to Thursday. No two persons eat the same fruit on a day. Each of them eats only one fruit on a day and does not repeat it on any other day.
(ii) Neither C nor D eats orange or peach on Tuesday.
(iii) B eats banana on Wednesday.
(iv) A eats peach on Monday.
(v) C does not eat mango on Thursday.
(vi) D eats banana on Monday.
Which fruit does D eat on Thursday?
A. Banana
B. Peach
C. Either banana or mango
D. Either peach or orange
Answer : D

Question. A, B, C, D, E and F are seated in a circle facing the centre. A and C are seated adjacent to each other and E and B are also seated adjacent to each other. B is to the immediate left of F. There are two persons between D and E. A is not seated adjacent to E. Who is to the immediate left of E?
A. C
B. B
C. F
D. Cannot be determined
Answer : A

Question. The six faces of a cube are coloured, each with a different colour.
I. The white face is between yellow and green.
II. The red face is adjacent to brown.
III. The green face is opposite the yellow side.
IV. The blue face is adjacent to red.
V. The yellow face is the top face of the cube.
The faces adjacent to white bear the colours
A. Yellow, green, brown and red
B. Yellow, brown, blue and green
C. Yellow, green, blue and red
D. None of these
Answer : B

Question. Find the correct mirror-image of the given word, if the mirror is placed vertically to the right.

Answer : A

Question. The given Venn diagram represents the students who study Physics, Chemistry and Mathematics. 
Identify the region which represents the students who study Physics and Mathematics but not Chemistry.

A. T
B. P + T + S
C. V
D. P + T + S + R + U + V
Answer : A

Question. The given set of figures carry certain characters. Assuming that the characters in each set follow a similar pattern, identify the pattern and find the missing character.

A. 49
B. 50
C. 60
D. 69
Answer : D

Question. Three figures X, Y and Z shows a sequence of folding a sheet of paper. Figure Z shows the manner in which the folded sheet has been cut out. Select a figure from the options which most closely resembles the unfolded form of sheet.

Answer : B

Question. Select a figure from the options which will continue the series established by the problem figures.

Answer : A

Question. There is a certain relationship between figures (i) and (ii). Establish a similar relationship between figures (iii) and (iv) by selecting a figure from the options that would replace the question mark in figure (iv).

Answer : C

Question. How many such 1's are there in the given arrangement, each of which is immediately preceded by a perfect square?
1 8 5 9 4 7 1 2 5 8 3 6 5 9 2 7 6 4 5 2 9 2 6 4 1 2 3 5 1 4 2 8 3
A. None
B. One
C. Two
D. Three
Answer : B

MATHEMATICAL REASONING

Question. If one of the zeroes of the cubic polynomial x³ + ax² + bx + c is –1, then the product of the other two zeroes is
A. b – a + 1
B. b – a – 1
C. a – b + 1
D. a – b – 1
Answer : A

Question. Solve for x and y : bx + ay = a + b,
ax(1/a-b - 1/a+b) + by (1/b-a - 1/b+a) = 2a/a+b
A. x = (b²+a²/ab+b², y = 2b/a+b
B. x = 2a/a-b, y = 2b/b-a
C. x = a²+b²/a²+ab, y = -2b/a-b
D. x = a² + b², y = 2ab + a²
Answer : A

Question. If tanq + secq = l, then secq = l² + m / nl, find m : n.
A. 1 : 2
B. 2 : 1
C. 4 : 1
D. 1 : 4
Answer : A

Question. In DPQR, point M is on side PQ and point S is on the side PR such that QRSM is a trapezium.
If MS : QR = 3 : 5, then find area (DPMS) : area (QRSM).
A. 9 : 16
B. 10 : 17
C. 3 : 5
D. 9 : 25
Answer : A

Question. The median of the given data is 50. Find the values of p and q respectively, if the sum of all the frequencies is 90.

A. 4, 6
B. 7, 5
C. 5, 7
D. 9, 3
Answer : C

Question. The given figure is made up of a large circle ABCDE with centre O, a small circle BQOP, two semi-circles and a sector OED. AC is 28 cm. Find the total shaded area of the figure.

A. (198π – 98) cm²
B. (49π + 198) cm²
C. (150π + 100) cm²
D. (147π – 196) cm²
Answer : D

Question. The sum of the remainders obtained when 2x³ + (p + 2)x + p – 2 is divided by x – 2 and when it is divided by x + 1 is 0. Find the value of p.
A. 3
B. –2
C. – 4
D. 8
Answer : C

Question. Find the values of p and q respectively for which the equation 2x² + px = q has root –3 and factor (x – 5).
A. –5, –25
B. – 4, 30
C. 5, –25
D. – 4, –30
Answer : B

Question. Find the sum from the sixth term to the twelfth term of the arithmetic progression 6, 10, 14, …
A. 266
B. 240
C. 256
D. 276
Answer : A

Question. Two dice are thrown simultaneously. Find the probability of getting a multiple of 2 on one dice and a multiple of 3 on the other dice.
A. 1/3
B. 11/36
C. 1/4
D. 13/36
Answer : B

Question. In the given diagram, ST is a tangent to the circle with centre O. Find the value of x.

A. 25°
B. 35°
C. 40°
D. 45°
Answer : D

Question. In the arithmetic progression 7, 10, 13, … how many terms will add up to a sum of 920?
A. 25
B. 16
C. 27
D. 23
Answer : D

Question. A solid rectangular block has a square base with side (√3 − √2)m. The volume of the block is (3√18 − 7√3)m³.
Find the height of the block in the form (a√2 + b√3)m, where a and b are integers.
A. 3√2 + √3
B. 5√3 + 4√2
C. 15√2 + √3
D. 5√5 + 3√3
Answer : A

Question. Solve : 3(81)x + 3 = 9x + 1 + 9x.
A. 3, 2
B. ±1/2
C. 4, 5
D. ±1
Answer : B

Question. For a frequency distribution, mean, median and mode are connected by which of the following relations?
A. Mode = 3 Mean – 2 Median
B. Mode = 2 Median – 3 Mean
C. Mode = 3 Median – 2 Mean
D. Mode = 3 Median + 2 Mean
Answer : C

Question. Two circles of radii 10 cm and 8 cm intersect each other and the length of common chord is 12 cm. The distance between their centres is

A. √7 cm
B. 3√7 cm
C. 4√7 cm
D. (8 + 2√7) cm
Answer : D

Question. O is the centre of a circle of diameter 4 cm and OABC is a square. If the shaded area is 1/3 area of the square, then the side of the square is

A. π√3 cm
B. √3π cm
C. 3 √π cm
D. 3 π cm
Answer : B

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?
Step-I : 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.
Step-II : 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.
Step-III : Join BP and BQ to get the desired tangents.

A. Only Step-I
B. Only Step-II
C. Both Step-I and Step-II
D. Both Step-II and Step-III
Answer : B

Question. The value of a+√a² - b² / a - √a² - b² + a - √a² - b² / a + √a² - b² is
A. a²/b²
B. b²/a²
C. a/b
D. 2(2a² - b²)/b²
Answer : D

Question. If the sums of n, 2n and 3n terms of an A.P. are S₁, S₂ and S₃ respectively, then S₃(S₂ - S₁) is
A. 0
B. 1
C. 2
D. 3
Answer : D

EVERYDAY MATHeMATICS

Question. There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field, while Ravish takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?
A. 54
B. 24
C. 36
D. 72
Answer : C

Question. A person invested some amount at the rate of 10% simple interest and some other amount at the rate of 12% simple interest. He received yearly interest of ₹ 130. But if he had interchanged the amounts invested, he would have received ₹ 4 more as interest. How much amount did he invest at the rate of 10% and 12% respectively?
A. ₹ 1200, ₹ 100
B. ₹ 700, ₹ 500
C. ₹ 1000, ₹ 1200
D. ₹ 500, ₹ 700
Answer : B

Question. Father is aged three times more than his son Ronit. After 8 years, he would be two and a half times of Ronit's age. After further 8 years, how many times would he be of Ronit's age?
A. 2 times
B. 2(1/2)times
C. 2(3/4)times
D. 3 times
Answer : A

Question. By selling an article, Mohit earned a profit equal to one-fourth of the price he bought it. If he sold it for ₹ 375, what was the cost price?
A. ₹ 281.75
B. ₹ 300
C. ₹ 312.50
D. ₹ 350
Answer : B

Question. Simran started a software business by investing ₹ 50,000. After six months, Nancy joined her with a capital of ₹ 80,000. After 3 years, they earned a profit of ₹ 24,500. What was Simran's share in the profit?
A. ₹ 9,423
B. ₹ 10,250
C. ₹ 12,500
D. None of these
Answer : D

Question. 10 women can complete a work in 7 days and 10 children take 14 days to complete the work. How many days will 5 women and 10 children take to complete the work?
A. 3
B. 5
C. 7
D. None of these
Answer : C

Question. In covering a certain distance, the speeds of A and B are in the ratio of 3 : 4. A takes 30 minutes more than B to reach the destination. The time taken by A to reach the destination is
A. 1 hour
B. 1(1/2)hours
C. 2 hours
D. 2(1/2)hours
Answer : C

Question. Three containers have their volumes in the ratio 3 : 4 : 5. They are full of mixtures of milk and water. The mixtures contain milk and water in the ratio of (4 : 1), (3 : 1) and (5 : 2) respectively. The contents of all these three containers are poured into a fourth container. The ratio of milk and water in the fourth container is
A. 4 : 1
B. 151 : 48
C. 157 : 53
D. 5 : 2
Answer : C

Question. Pure ghee costs ₹ 100 per kg. After adulterating it with vegetable oil costing ₹ 50 per kg, a shopkeeper sells the mixture at the rate of ₹ 96 per kg, thereby making a profit of 20%. In what ratio does he mix the two?
A. 3 : 2
B. 4 : 3
C. 5 : 2
D. 7 : 6
Answer : A

Question. A manufacturer offers a 20% rebate on the marked price of a product. The retailer offers another 30% rebate on the reduced price. The two reductions are equivalent to a single reduction of
A. 40%
B. 44%
C. 46%
D. 50%
Answer : B

ACHIEVERS SECTION

Question. The graph of ax² + bx + c is shown here and A(-b/2a, - D/4a) Identify the signs of a, b and c.

        a     b    c
A. +ve +ve –ve
B. +ve –ve –ve
C. +ve –ve +ve
D. –ve +ve –ve
Answer : B

Question. Match the columns.

A. (a) → (i), (b) → (iii), (c) → (i), (d) → (ii)
B. (a) → (ii), (b) → (iii), (c) → (ii), (d) → (iv)
C. (a) → (iii), (b) → (iii), (c) → (i), (d) → (ii)
D. (a) → (ii), (b) → (iii), (c) → (i), (d) → (ii)
Answer : C

Question. Study the statements carefully.
Statement-1 : Three times the square of one side of an equilateral triangle is equal to four times the square of one of its altitudes.
Statement-2 : Three times the sum of the squares of the sides of a triangle is equal to the three times the sum of the squares of the medians of the triangle.
A. Statement-1 is true and statement-2 is false.
B. Statement-1 is false and statement-2 is true.
C. Both statements are true.
D. Both statements are false.
Answer : A

Question. Which of the following is incorrect?
A. For K = 9/8, the equation 2x² + 3x + K = 0 will have real and equal roots.
B. For K = –1, the equation x² + K(4x + K – 1) + 2 = 0 will have equal roots.
C. For K = 2, the equation x² – 2x(1 + 3K) + 7(3 + 2K) = 0 will have equal roots.
D. For K = –3, the equation (K + 1)x² – 2(K – 1)x + 1 = 0 will have equal roots.
Answer : D

Question. Fill in the blanks:
Every ___(_a) __ number can be expressed (factorised) as the product of ___(b) __ factors and this factorisation is ___(_c) __ except for the order in which the prime factor occur.
        (a)                     (b)                 (c)
A. Prime                Composite     Unique
B. Composite        Prime             Unique
C. Odd                  Even              Universal
D. Even                Odd                Universal
Answer : B