CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set N

LOGICAL REASONING

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

A. 12
B. 9
C. 7
D. 8
Answer : B

Question. Find the number of rectangles formed in the given figure.

A. 24
B. 28
C. 36
D. None of these
Answer : D

Question. If ‘*’ stands for ‘+’, ‘%’ stands for ‘–’, ‘#’ stands for ‘×’ and ‘e’ stands for ‘÷’, then which of the following options is correct?
A. 5 * 8 e 2 % 7 # 2 = –7
B. 18 e 9 * 4 # 2 % 8 # 2 * 42 e 21 = – 4
C. 9 * 5 # 4 % 12 e 3 * 6 = 25
D. 8 * 7 % 6 * 4 % 3 * 9 % 8 = 10
Answer : B

Question. In a certain code language,
‘only in serial order’ is coded as ‘ve pu na to’;
‘order in the state’ is coded as ‘li ve su pu’;
‘the logical idea only’ is coded as ‘su na ri jo’ and
‘in idea or theory’ is coded as ‘zt jo bk pu’.
Which of the following is the code for ‘theory’?
A. zt
B. bk
C. jo
D. Either zt or bk
Answer : D

Question. A square transparent Sheet (X) with a pattern and a dotted line on it is shown here. If the given sheet is folded along the dotted line, then which pattern would appear from the given options?

Answer : A

Question. Which of the following options will continue the same series as established by the Problem Figures?

Answer : A

Question. If ‘M × N’ means ‘M is the daughter of N’, ‘M + N’ means ‘M is the father of N’, ‘M ÷ N’ means ‘M is the mother of N’ and ‘M – N’ means ‘M is the brother of N’, then in the expression ‘P ÷ Q + R – T × K’, how is P related to K?
A. Daughter-in-law
B. Sister-in-law
C. Daughter
D. Mother-in-law
Answer : D

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

Answer : B

Question. Ashok started walking towards South. After walking 50 metres, he took a right turn and walked 30 metres.
He then again took a right turn and walked 100 metres.
He again took a right turn and walked 30 metres and stopped. How far and in which direction is he now from the starting point?
A. 50 metres, South
B. 150 metres, North
C. 180 metres, East
D. 50 metres, North
Answer : D

Question. A sheet of paper shown in Fig. (X) is folded to form a box. Choose the box(es) from the options that is/are similar to the box so formed.

A. (P) and (Q) only
B. (Q) and (S) only
C. (R) only
D. (P) and (S) only
Answer : C

Question. If it is possible to make only one meaningful word with 3rd, 4th, 8th and 9th letters (using each letter only once) of the word ‘CENTURIES’, then which would be the 2nd letter of the word formed? If more than one such word can be formed, then give ‘Y’ as your answer. If no such word can be formed, then give ‘Z’ as your answer.
A. Y
B. N
C. Z
D. E
Answer : A

Question. Study the given Venn diagram carefully and answer the following question.

Which of the following numbers represents the tall persons who are swimmers and tennis players but not runners?
A. 12
B. 3
C. 6
D. 7
Answer : B

Question. In the 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 on numbers progress from left to right.
Rules:
(i) If an odd number is followed by another composite odd number, then they are to be multiplied.
(ii) If an even number is followed by an odd number, then they are to be added.
(iii) If an even number is followed by a number which is a perfect square, then the even number is to be subtracted from the perfect square.
(iv) If an odd number is followed by a prime odd number, then the first number is to be divided by the second number.
(v) If an odd number is followed by an even number, then the second one is to be subtracted from the first one.
24      49      5
x        33      94
If x is the resultant of first row, then what is the resultant of second row?
A. 54
B. 71
C. 45
D. 69
Answer : B

Question. Which of the following options satisfies the same conditions of placement of the dots as in the given figure?

Answer : C

Question. Which of the following figures is exactly embedded in the given figure as one of its parts?

Answer : B

MATHEMATICAL REASONING

Question. Simplify: sin²θ + cos²θ / sinθ + cosθ + sinθ cosθ
A. 0
B. 1
C. sin²θ – cos²θ
D. sinθ – cosθ
Answer : B

Question. Study the given co-ordinate system carefully and find the area of the figure ABCD.

A. 24 sq. units
B. 12√6 sq. units
C. 18 sq. units
D. None of these
Answer : A

Question. A square lawn has a circular pond in the centre. Find the area of the shaded region, given that the diameter of the circle is 1/3 times the diagonal of square with side 3 cm.

A. 9 - π / 2 cm²
B. (18 - π)/4 cm²
C. (18 - π)/18 cm²
D. (2 + π)/8 cm²
Answer : B

Question. If the roots of the equation (a² + b²) x² – 2(ac + bd) x + (c² + d²) = 0 are equal, then
A. ab = cd
B. a/d = b/c
C. ad = bc
D. None of these
Answer : C

Question. The ratio of the outer and inner perimeters of a circular path is 23 : 22. If the path is 5 metres wide, then the diameter of the inner circle is
A. 55 m
B. 110 m
C. 220 m
D. 230 m
Answer : C

Question. Find the largest possible positive integer that will divide 125, 162 and 259 leaving remainder 5, 6 and 7 respectively.
A. 6
B. 8
C. 12
D. 13
Answer : C

Question. If the polynomial 8x4 + 14x³ – 2x² + px + q is exactly divisible by 4x² + 3x – 2, then the values of p and q respectively are _______ .
A. 2 and 0
B. –7 and 2
C. 5 and –3
D. 4 and –1
Answer : B

Question. Following table shows the marks scored by a group of 90 students in a Mathematics test of 100 marks.
Marks       Number of students
0-20                     7
20-30                10
30-40                10
40-50                20
50-60                20
60-70                15
70-100                8
Find the probability that a student obtained :
(i) Less than 60% marks.
(ii) 70 or more marks.
        (i)           (ii)
A. 4/5           4/45
B. 19/45       4/45
C. 37/90       23/90
D. 67/90       4/45
Answer : D

Question. A real number 2² x 3² x 7³ / 2² x 5³ x 3⁵ x 7⁴ will have ________.
A. Terminating decimal expansion
B. Non-terminating decimal expansion
C. Repeating decimal expansion
D. Both B and C
Answer : D

Question. Which of the following options is INCORRECT?
A. The number of terms in the A.P. 3, 6, 9, 12,...., 111 is 37.
B. If the first three terms of an A.P. are x – 1, x + 1 and 2x + 3, then the value of x is 0.
C. The sum of first 'n' natural numbers is [n(n + 1)/2]²
D. None of these
Answer : C

Question. In the given figure, AD ⊥ BC, BE ⊥ AC, CF ⊥ AB, then AF² + BD² + CE² = 

A. OA² + OB² + OC²
B. OD² + OE² + OF²
C. AB² + BC² + AC²
D. AE² + BF² + CD²
Answer : D

Question. Solve : 1/2(x+2y) + 5/3(3x - 2y) = 3/2 and 5/4(x + 2y) - 3/5(3x - 2y) = 61/60
A. x = 2/3, y = 1/3
B. x = 1/2, y = 5/4
C. x = 5/4, y = 1/2
D. x = 1/3, y = 2/3
Answer : B

Question. Divide 56 into four parts which are in A.P. such that the ratio of product of extremes to the product of means is 5 : 6.
A. 8, 12, 16, 20
B. 10, 14, 18, 22
C. 12, 16, 18, 20
D. None of these
Answer : A

Question. Simplify :(a + 1/b)m x (a - 1/b)n / (b + 1/a)m x (b - 1/a)n
A. (a/b)^m-n
B. (a/b)^m+n
C. (b/a)^m/n 
D. (b/a)^mn
Answer : B

Question. If ABCD is a field in the form of a quadrilateral whose sides are indicated in the figure (not drawn to scale) and ∠DAB = 90°, then find the area of the field.

A. 298 m²
B. 186 m²
C. 306 m²
D. 195 m²
Answer : C

Question. Study the question and the statements given below and decide which of the statements is/are necessary to answer the question?
What is the capacity of the cylindrical tank?
I. The area of the base is 61600 sq. cm.
II. The height of the tank is 1.5 times the radius.
III. The circumference of base is 880 cm.
A. Only I and II
B. Only II and III
C. Only I and III
D. Only II and either I or III
Answer : D

Direction (a - b) : The given bar graph shows the area under the rice crop during different years in India. Study the graph carefully and answer the following questions.

Question a. What is the average area (in hectares) under the rice crop during the years 2005-2008?
A. 2000000
B. 2500000
C. 4500000
D. 3000000
Answer : A

Question b. Find the difference between the area under the rice crop during 2006-2008 and 2008-2010.
A. 375000
B. 450000
C. 750000
D. 75000
Answer : C

Question. If A + B = 90°, then

A. tan A
B. cot A
C. sec² A
D. cosec²A
Answer : A

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

Answer : D

EVERYDAY MATHEMATICS

Question. 5 cans of lemon drink and 7 packets of litchi juice cost ₹ 50 while 7 cans of lemon drink and 5 packets of litchi juice cost ₹ 46. Calculate the cost of one can of lemon drink.
A. ₹ 8
B. ₹ 12
C. ₹ 3
D. ₹ 5
Answer : C

Question. In a marriage ceremony of her daughter Poonam, Ashok has to make arrangements for the accommodation of 150 persons. For this purpose, he plans to build a conical tent in such a way that each person has 4 sq. metres of the space on ground and 20 cubic metres of air to breath. What should be the height of the conical tent?
A. 20 m
B. 15 m
C. 12 m
D. 30 m
Answer : B

Question. A person had ₹ 21900. He lent a part of it at 5% and the remaining at 1(2/3)% simple interest for one year. If the total interest (annually) was ₹ 584, find the sum lent at 5%.
A. ₹ 15330
B. ₹ 6750
C. ₹ 6570
D. ₹ 13530
Answer : C

Question. Two stations are 110 km apart on a straight line. One train starts from A at 7 a.m. and moves towards B at 20 km/h while another train starts from B at 8 a.m. and travels towards A at a speed of 25 km/h. At what time will they meet?
A. 9:00 a.m.
B. 10:30 a.m.
C. 10:00 a.m.
D. 11:00 a.m.
Answer : C

Question. Anurag’s annual income is ₹ 636000. He spends 22% of his monthly income on paying bills, 18% on household items, 12% on paying his children’s fee and 4% he donates to a charity. If two-fifth of the remaining amount he invests in mutual funds, then what is the amount left with him every month?
A. ₹ 17850
B. ₹ 12162
C. ₹ 9328
D. ₹ 13992
Answer : D

Question. Puneet decides to attend some urgent meeting on a particular day in the month of December 20XX. If 1st December 20XX falls on Saturday, then what is the probability that the day chosen by him to attend the meeting is a Saturday?
A. 2/15
B. 1/6
C. 1/10
D. None of these
Answer : D

Question. Marbles of diameter 1.4 cm are dropped into a cylindrical beaker containing some water and are fully submerged. The diameter of the beaker is 7 cm. How many marbles have been dropped in it, if the water rises by 5.6 cm?
A. 350
B. 250
C. 200
D. 150
Answer : D

Question. A man arranges to pay off a debt of ₹ 3600 by 40 annual installments which are in A.P. When 30 of the installments were paid, he died leaving one-third of the debt unpaid. The value of 8th installment is
A. ₹ 35
B. ₹ 50
C. ₹ 65
D. None of these
Answer : C

Question. A leak at the bottom of a tank can empty the full tank in 6 hours. An inlet pipe fills water at the rate of 4 litres per minute. When the tank is full, the inlet is opened and due to the leak, the tank gets empty in 8 hours. The capacity of the tank is
A. 5260 litres
B. 5760 litres
C. 5846 litres
D. 6970 litres
Answer : B

Question. In a year, the mean of the annual incomes of Amit and Somya was ₹ 3800. The mean of the annual incomes of Miara and Somya was ₹ 4800 and the mean of the annual incomes of Miara and Amit was ₹ 5800. What is the mean of the incomes of the three?
A. ₹ 4800
B. ₹ 5600
C. ₹ 4200
D. ₹ 5000
Answer : A

ACHIEVERS SECTION

Question. Read the statements carefully and state 'T' for true and 'F' for false. 
(P) The point (7/4, 7/8) divides the line segment joining the points (4, –1) and (–2, 4) internally in the ratio3 : 5.
(Q) The point (5, 0) on y-axis is equidistant from (–1, 2) and (3, 4).
(R) The points (8, 1), (3, – 4) and (2, –5) are collinear.
(S) The centroid of the triangle whose vertices are
(1, 4), (–1, –1) and (3, –2) is (1, 1/3)
    (P)   (Q)    (R)     (S)
A. F      F        F       T
B. T      T        F       F
C. T      F        T       T
D. T      T        T       T
Answer : C

Question. Match the division of polynomials given in Column-I with their respective remainders given in Column -II and select the correct option.

A. (i)→(q), (ii)→(p), (iii)→(r), (iv)→(s)
B. (i)→(p), (ii)→(r), (iii)→(s), (iv)→(q)
C. (i)→(r), (ii)→(p), (iii)→(s), (iv)→(q)
D. (i)→(r), (ii)→(q), (iii)→(p), (iv)→(s)
Answer : C

Question. Read the statements carefully and select the correct option.
Statement-I : The graphical representation of 2x + y = 6 and 2x – y + 2 = 0 will be a pair of parallel lines.
Statement-II : When k = –1, then linear equations 5x + ky = 4 and 15x + 3y = 12 have infinitely many solutions.
A. Both Statement-I and Statement-II are true.
B. Statement-I is true but Statement-II is false.
C. Statement-I is false but Statement-II is true.
D. Both Statement-I and Statement-II are false.
Answer : D

Question. The given figure (not drawn to scale) shows a running track surrounding a grassed enclosure PQRSTU.
The enclosure consists of rectangle PQST with two semicircular ends.

(i) Calculate the area of the grassed enclosure.
(ii) Given that the track is of constant width 7 m, calculate the outer perimeter ABCDEF of the track.
[Take π = 22/7]
           (i)               (ii)
A. 17850 m²       532 m
B. 17850 m²       664 m
C. 12320 m²       532 m
D. None of these
Answer : B

Question. Fill in the blanks.
(i) In two similar triangles, ABC and PQR, if their corresponding altitudes AD and PS are in the ratio 4 : 9, then ar(DABC): ar(DPQR) = _____.
(ii) Area of an equilateral triangle described on the side of a square is ______ the area of equilateral triangle described on its diagonal.
(iii) The altitude of an equilateral triangle with side 'a' equals ______ .
       (P)               (Q)           (R)
A. 16 : 81           Twice         a/2
B. 4 : 9               Half           2a
C. 16 : 81           Half           √3a/2
D. 4: 9               Twice         √3a/2
Answer : C