CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set J

LOGICAL REASONING

Question. If 'A D B' means 'A is the brother of B', 'A # B' means 'A is the father of B' and 'A * B' means 'A is the mother of B'. Then, which of the following means 'R is the son of M'?
A. M D R # S
B. M # S * R
C. M # S # R
D. M * R # S
Answer : D

Question. Count the number of triangles formed in the given figure.

A. 11
B. 18
C. 17
D. None of these
Answer : D

Question. Find the missing number.

A. 5
B. 6
C. 8
D. 12
Answer : A

Question. Kiara is standing at the centre of a circular field.
Her son is sitting to the East of her on the boundary of the field. Kiara goes towards her son and gives him some chips, then she turns to right and walks along the boundary of the field equal to three-fourth of the length of the circular field. She turns back and walk (1/8)th of the boundary of the circular field. In which direction is she from the starting point?
A. North-East
B. North-West
C. South-East
D. North
Answer : B

Question. The following letters are coded as follows.
Letters B V M L D P A E R
Codes 9 2 1 7 5 3 6 4 8
While coding the given letters following conditions are also to be observed.
(i) If the first as well as the last letters are vowels, both are coded as $.
(ii) If the first letter is vowel and last letter is consonant, both are coded as %.
(iii) If the first letter is a consonant and last is a vowel, their codes are to be interchanged.
UBMEPD
A. 91%%43
B. 19%43%
C. 914%3%
D. %9143%
Answer : D

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

Answer : D

Question. Three different positions of the same die are given. select a figure from the options which will be formed when the die is unfolded.

Answer : D

Question. If 'P' denotes '+', 'Q' denotes '×', 'R' denotes '÷', 'S' denotes '–' and '=' denotes 'greater than', then which of the following statements is true?
A. 4 P 8 R 2 S 1 Q 6 = 1
B. 3 S 1 Q 8 P 6 R 2 = 0
C. 8 R 2 S 3 Q 4 P 6 = 2
D. 9 P 2 Q 6 S 4 R 2 = 21
Answer : A

Question. Select the correct mirror image of Fig. (X) along the mirror PQ.

Answer : A

Question. In a group of six women Priya, Sanchi, Trishu, Parul, Shikha and Anjali, four are table tennis players, four are postgraduates in economics, one is postgraduate in commerce and three are bank employees. Priya and Shikha are among the bank employees while Anjali and Trishu are unemployed. Trishu and Sanchi are among the table tennis players. Anjali, Shikha, Trishu and Sanchi are postgraduates in economics of whom two are bank employees. If Parul is postgraduate in commerce who among the following is definitely both a table tennis player and a bank employee?
A. Sanchi
B. Priya
C. Anjali
D. Trishu
Answer : A

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

Question. This question is based on the six numbers given below :
271 361 912 714 459 187
If the first and the second digits of each number are interchanged and if the third digit of each number is placed between these two digits, then which number will be the third number from the top, if the new numbers are arranged in the descending order?
A. 187
B. 271
C. 459
D. 361
Answer : D

Question. Study the following information and answer the question that follows.
'P © Q' means 'P is neither greater than nor smaller than Q'.
'P # Q' means 'P is either greater than or equal to Q'.
'P $ Q' means 'P is either smaller than or equal to Q'.
'P % Q' means 'P is greater than Q'.
'P @ Q' means 'P is smaller than Q'.
If the given statements are true, then which of the following two conclusions I and II given below is/are definitely true?
Statements : V $ T, T # R, R @ B
Conclusions : I. V @ B II. B % T
A. Only I
B. Only II
C. Both I and II
D. Neither I nor II
Answer : D

Question. Which of the following figures represents the same condition of placement of dots as in Fig. (X)?

Answer : A

Question. Which of the following Venn diagrams best describes the relationship amongst, 'Milk, Apple and Fruit'?

Answer : C

MATHEMATICAL REASONING

Question. Which of the following statements is INCORRECT?
A. The rational form of 17 6¯ 53/3. is .
B. 0.423442344423... is a rational number.
C. The equivalent form of 16 + 2 9¯ is 19/1.
D. √25 + √64 is a rational number.
Answer : B

Question. The equations representing the given graph is ____.

A. 7x + 2y = 11; y – 2x = 3
B. 2x + 7y = 11; 5x + (35y/2) = 25
C. 3x – 7y = 10; 8y – 6x = 4
D. 3x – 4y = 1; 8y – 6x = 4
Answer : D

Question. In an A.P., Sm = n and Sn = m also m > n, find the sum of first (m – n) terms.
A. (m - n)(m + 2n)/m
B. (m + n)(2m + n)/m
C. (m - n)(m + 2n)/n
D. (m - n)(2m + n)/n
Answer : A

Question. The mean of the following frequency distribution is 180 cm. Find the missing frequency f.

A. 10
B. 8
C. 12
D. 15
Answer : A

Directions (a - b) : Study the following table and answer the questions that follow.

Results of half-yearly and annual examinations of class X in a school

Question a. What is the total number of students who failed in either of the two exams but not both?
A. 94
B. 90
C. 47
D. None of these
Answer : A

Question b. The students who passed in half-yearly but failed in annual exams are approximately what percent of total number of students?
A. 10%
B. 12%
C. 18%
D. 15%
Answer : B

Question. The circumference of the base of a cylindrical vessel is 132 cm and its height is 25 cm. How many litres of water can it hold?
A. 34.11 L
B. 45.40 L
C. 24.65 L
D. 34.65 L
Answer : D

Question. In the given figure, AE is the bisector of the exterior ∠CAD meeting BC produced at E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE.

A. 12 cm
B. 16 cm
C. 20 cm
D. 18 cm
Answer : D

Question. Let ABC be a right triangle in which AB = 3 cm, BC = 4 cm and ∠B = 90°. BD is the perpendicular from B on AC. The circle through B, C, D is drawn.
Given below are the steps of construction of a pair of tangents from A to this circle. Which of the following steps is INCORRECT?
Step I : Draw DABC and perpendicular BD from B on AC.
Step II : Draw a circle with BC as diameter. This circle will pass through D.
Step III : Let O be the mid-point of BC. Join AO.
Step IV : Draw a circle with AO as diameter. This circle cuts the circle drawn in step II at B and P. Join AO, AP and AB are desired tangents drawn from A to the circle passing through B, C and D.
A. Only step I
B. Only step II
C. Only step III
D. Only step IV
Answer : D

Question. For which of the following system of equations, x = 6, y = –4 is the solution?
(i) 1/2x - 1/y = - 1 and 1/x + 1/2y + 8
(ii) 2/x + 2/3y = 1/6 and 3/x + 2/y = 0
A. Only (I)
B. Only (II)
C. Both (I) and (II)
D. Neither (I) nor (II)
Answer : B

Question. If p and q are zeroes of the quadratic polynomial 2x² + 2(m + n)x + m² + n², find the quadratic polynomial whose zeroes are (p + q)² and (p – q)².
A. x² + 2 mnx + (m² + n²)²
B. x² – 4 mnx – (m² – n²)²
C. x² – 2 mnx – (m² – n²)²
D. x² + 4 mnx – (m² – n²)²
Answer : B

Question. A point O is taken inside an equilateral ΔABC. If OL ⊥ BC, OM ⊥ AC and ON ⊥ AB such that OL = 14 cm, OM = 10 cm and ON = 6 cm, then find the area of ΔABC.

A. 300√3 cm²
B. 200√3 cm²
C. 300 cm²
D. 325√2 cm²
Answer : A

Question. If the sum of the zeroes of the quadratic polynomial f(t) = kt² + 2t + 3k is equal to their product, then find the value of k.
A. − 2/3
B. 2/3
C. 1/3
D. −1/3
Answer : A

Question. The radii of two concentric circles are 16 cm and 10 cm. AB is a diameter of the bigger circle. BD is tangent to the smaller circle touching it at D. Find the length of AD.
A. 3√130 cm
B. 2√139 cm
C. 2√130 cm
D. 4√139 cm
Answer : B

Question. The probability of getting a bad egg from a lot of 400 eggs is 0.035. Find the number of bad eggs in the lot. Also find the probability of getting a good egg.
A. 7, 0.965
B. 14, 0.965
C. 14, 0.65
D. 7, 0.65
Answer : B

Question. If cosec θ – sin θ = m and sec θ – cos θ = n, then find the value of (m² n)^2/3 + (mn²)^2/3.
A. 0
B. sin q
C. 1
D. cos q
Answer : C

Question. Which of the following rational numbers have non-terminating decimal expansion?
A. 144/225
B. 25/36
C. 49/256
D. 7/250
Answer : B

Question. If x be the area of a right angled DABC in which ∠ABC = 90° and BC = b, then the length of the altitude BN on the hypotenuse AC is ________.
A. 2b/√b⁴ + 4x²
B. 2bx/√b⁴ - 4x²
C. 4bx/√b² + 4x²
D. 2bx/√b⁴ + 4x²
Answer : D

Question. Which of the following is/are correct?
A. Product of two irrational numbers is always an irrational number.
B. The sum of two irrational numbers is always an irrational number.
C. Sum of rational and irrational number is always an irrational number.
D. None of these
Answer : C

Question. Find the area of the shaded portion in the given figure, if E is the midpoint of AB, AGE & BEF are quadrants and DOC is a semicircle.

A. 154 cm²
B. 416 cm²
C. 284 cm²
D. 546 cm²
Answer : D

EVERYDAY MATHeMATICS

Question. Six years hence, a man’s age will be three times the age of his son and three years ago he was nine times as old as his son. The present age of the man is _____.
A. 28 years
B. 30 years
C. 32 years
D. 34 years
Answer : B

Question. A person earns 15% on an investment but losses 10% on another investment. If the ratio of the two investments be 3 : 5, then what is the gain or loss on the two investments taken together?
A. 1.625% loss
B. 13.125% gain
C. 13.125% loss
D. None of these
Answer : D

Question. Amit can complete 1/4 th of a work in 10 days, Varun can complete 1/3 rd of the work in 13 days and Gaurav can complete 40% of the work in 15 days. Who will complete the work first, if they start working together?
A. Amit
B. Varun
C. Gaurav
D. All the three will complete at same time
Answer : C

Question. A shopkeeper allows a discount of 10% on the marked price of an item but charges a sales tax of 8% on the discounted price. If the customer pays ₹ 680.40 as the price including the sales tax, then what is the marked price of the item?
A. ₹ 630
B. ₹ 700
C. ₹ 780
D. None of these
Answer : B

Question. A father said to his son, "I was as old as you are at present at the time of your birth." If the father's age is 38 years now, the son's age five years back was ______.
A. 14 years
B. 19 years
C. 12 years
D. 17 years
Answer : A

Question. 85 kg of a mixture contains milk and water in the ratio 27 : 7. How much more water is to be added to get a new mixture containing milk and water in the ratio 3 : 1 ?
A. 5 kg
B. 6.5 kg
C. 7.25 kg
D. 8 kg
Answer : A

Question. A bank offers 5% p.a. compound interest calculated on half-yearly basis. A customer deposits ₹ 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is ________.
A. ₹ 120
B. ₹ 121
C. ₹ 122
D. ₹ 123
Answer : B

Question. Amar has some notes of ₹ 10 and ₹ 20. The total number of notes are 70 and the total amount of money with him is ₹ 1,050. Find the number of notes of ₹ 10 and ₹ 20 with him.
A. 30 and 40
B. 35 and 35
C. 40 and 30
D. 45 and 25
Answer : B

Question. Two pens and three pencils cost ₹ 86. Four pens and a pencil cost ₹ 112. Find the cost of a pen and that of a pencil respectively.
A. ₹ 12, ₹ 25
B. ₹ 25, ₹ 12
C. ₹ 14, ₹ 22
D. ₹ 22, ₹ 14
Answer : B

Question. In a camp, there is a meal for 120 men or 200 children.
If 150 children have taken the meal, how many men will be catered to with the remaining meal ?
A. 20
B. 30
C. 40
D. 50
Answer : B

ACHIEVERS SECTION

Question. State 'T' for true and 'F' for false.
I. A sequence is an A.P., if and only if the sum of its n terms is of the form An2 + Bn, where A and B are constants.
II. If 18, a, b, –3 are in A.P., then a + b = 15.
III. If a, c, b are in A.P., then 2c = a + b.
IV. The nth term from the end of an A.P. is the (m – n + 1)th term from the beginning, where m terms are in A.P.
      I        II      III        IV
A. F        F       T        T
B. T        T       T        F
C. F        T       T        F
D. T        T       T        T
Answer : D

Question. Consider the following statements.
Statement I : In the given figure, O is the centre of the circle with D, E and F as mid points of AB, BO and OA, respectively. If ∠DEF is 30°, then ∠ACB is 60°.

Statement II : Angle subtended by an arc at the centre is twice the angle subtended by it on the remaining part of the circle.
Which of the following options hold?
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 : A

Question. Fill in the blanks.
(i) P and Q are the end points of the diameter of a circle, having its centre at R. If the coordinates of P and Q are respectively (–3, 3) and (5, 1), then the coordinates of R are_____.
(ii) If the point C(–1, 2) divides internally the line segment joining A(2, 5) and B in the ratio 3 : 4, then the coordinates of B are______.
(iii) If the points A(1, 2), B(0, 0) and C(a, b) are collinear, then____.
(iv) The points (–4, 0), (4, 0) and (0, 3) are the vertices of a/an____.
        (i)           (ii)           (iii)                    (iv)
A. (1, 2)     (–5, –2)     2b = a       Isosceles triangle
B. (2, 1)     (–5, –2)     2a = b       Right angled triangle
C. (1, 2)     (–5, –2)     2a = b       Isosceles triangle
D. (2, 4)     (–5, 2)       2b = a       Right angled triangle
Answer : C

Question. Match the following:

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

Question. A die is thrown twice. What is the probability that
(i) 5 will not come up either time?
(ii) 5 will come up atleast once?
         (i)            (ii)
A. 24/36        11/36
B. 25/36        13/36
C. 11/36        25/36
D. 25/36        11/36
Answer : C