CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set I

LOGICAL REASONING

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 fig. (4).

Answer : C

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

Answer : B

Question. Which of the following is the ninth to the right of the seventeenth from the right end in the given arrangement?
M 4 E T % J 9 I B @ U 8 © N # W F 1 V 7 ★ 2 A H 3 Y 5 $ 6 K
A. A
B. %
C. N
D. ©
Answer : A

Question. Select the figure which satisfies the same condition of placement of the dots as in Fig. (X).

Answer : C

Question. Find the missing number, if a certain rule is followed row-wise or column-wise.

A. 15
B. 18
C. 20
D. 25
Answer : D

Question. Select the figure from the options which will replace the (?) so that a series is formed.

Answer : D

Question. In the following figure, the triangle represents the politicians; the circle represents the graduates and the rectangle represents the members of Parliament.

Who among the following are the graduate members of parliament but not politicians?
A. B
B. D
C. C
D. F
Answer : C

Question. In a certain code language, when did you come is written as ti na ki ja, will you come again is written as na pa sa ja and she will go is written as pa da ra.
How is again written in that code language?
A. na
B. sa
C. ja
D. da
Answer : B

Question. Two positions of a dice are shown below. Find the number of dots on the face opposite to the face having four dots.

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

Question. If 'P + Q' me ans 'P i s the brothe r of Q', 'P × Q' means 'P is the father of Q' and 'P – Q' means 'P is the sister of Q', then which of the following represents 'S is the niece of T'?
A. T × M + S – K
B. K – S × M + T
C. T + M × S – K
D. T × S + M – K
Answer : C

Question. From amongst six boys A, B, C, D, E and F and five girls P, Q, R, S and T, a team of six is to be selected under the following conditions.
(1) A and D have to be together.
(2) C cannot go with S.
(3) S and T have to be together.
(4) B cannot be teamed with E.
(5) D cannot go with P.
(6) B and R have to be together.
(7) C and Q have to be together.
If there be five boys in the team, then the alone girl member is
A. P
B. Q
C. R
D. S
Answer : B

Question. Select the figure from the options, which when placed in the blank space of Fig. (X) would complete the pattern.

Answer : D

Question. In the following 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 operation of numbers progress from left to right.
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 the perfect square, the even number is to be subtracted from the perfect square.
(iv) If an odd number is followed by a prime odd number, the first number is to be divided by the second number.
(v) If an odd number is followed by an even number, the even number is to be subtracted from the odd number.
14     11     9
104     k   11
If k is the resultant of the first row, what is the resultant of the second row?
A. 1331
B. 132
C. 5
D. 11
Answer : D

Question. How many pairs of letters are there in the word VENGEANCE which have the same number of letters between them as in the English alphabet?
A. One
B. Two
C. Three
D. More than three
Answer : D

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

Answer : D

MATHEMATICAL REASONING

Question. If a and b are the zeros of the quadratic polynomial f(x) = kx² + 4x + 4 such that a² + b² = 24, find the values of k.
A. −1, 2/3
B. 1, 2/3
C. −1, 3/2
D. 1, 3/2
Answer : A

Question. Solve the following system of linear equations:
7x - 2y / xy = 5, 8x + 7y / xy = 15, x ≠ 0, y ≠ 0
A. x = 1, y = –1
B. x = 1, y = 2
C. x = 1, y = 1
D. x = 1, y = –2
Answer : C

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

Question. The volume of a cuboid is 1536 m3. Its length is 16 m, and its breadth and height are in the ratio 3 : 2. Find the breadth and height of the cuboid respectively.
A. 12 m, 8 m
B. 15 m, 6 m
C. 21 m, 14 m
D. 18 m, 12 m
Answer : A

Question. Which of the following steps of construction is INCORRECT while drawing a tangent to a circle of radius 6 cm and making an angle of 30° with a line passing through the centre.
Steps of construction :
Step I : Draw a circle with centre O and radius 3 cm.
Step II : Draw a radius OA of this circle and produce it to B.
Step III : Construct an angle ∠AOP equal to the complement of 30° i.e. equal to 60°.
Step IV : Draw perpendicular to OP at A which intersects OA produced at Q
Clearly, PQ is the desired tangent such that ∠OQP = 30°.
A. Both step I and step IV
B. Only step III
C. Both step III and step IV
D. Only step I
Answer : A

Question. In the given figure, AD and BE are perpendiculars to BC and AC respectively. Consider the following relations :

(i) DADC ~ DBEC
(ii) CB × CE = CB × CD
(iii) DABC ~ DDCE
(iv) CD × DB = CA × DE
Which of the following options hold?
A. (i), (ii) and (iii) are true
B. Both (ii) and (iii) are true
C. Both (i) and (iv) are true
D. None of these
Answer : D

Question. Which of the following are terminating decimals?
(i) 23/8
(ii) 125/441
(iii) 67/50
(iv) 96/41
(v) 81/5
(vi) 2⁶/5⁶
A. (i), (ii), (iii), (iv)
B. (i), (iii), (v), (vi)
C. (i), (ii), (v), (vi)
D. All of these
Answer : B

Question. Find the value of p, if mean of the following data is 6.

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

Question. The common root of the quadratic equations x2 – 3x + 2 = 0 and 2x2 – 5x + 2 = 0 is
A. 2
B. 1
C. –2
D. 1/2
Answer : A

Question. From a point on the ground, 15 m away from the root of a tree, the angle of elevation of the top of the tree is 30°. Find the ratio of the distance of the point from the top of the tree to the height of the tree. (Take √3 =1.732 ).
A. 1 : 2
B. 2 : 1
C. 3 : 1
D. 1: 3
Answer : B

Question. Find the area of the triangle whose vertices are A (− 3/2. 3), B(6, –2) and C(–3, 4).
A. 0 sq. units
B. 5√2sq. units
C. 9 sq. units
D. 15 sq. units
Answer : A

Question. What must be added to the polynomia lf(x) = x⁴ + 2x³ – 2x² + x – 1 so that the resulting polynomial is exactly divisible by x² + 2x – 3?
A. x – 2
B. 2x – 1
C. –x + 2
D. –2x + 2
Answer : A

Question. In the given figure (not drawn to scale) AB and CD are two diameters of a circle with centre O, perpendicular to each other and OC is the diameter of the smaller circle. If OC = 10 cm, then find the area of the shaded region. (Take π = 3.14)

A. 136.2 cm²
B. 133.6 cm²
C. 135.5 cm²
D. 134.5 cm²
Answer : C

Question. Solve :
x - a / x b +  x - b / x - a = a/b + b/a (x ≠ a, b, a ≠ b)
A. 1, a – b
B. 0, a + b
C. 0, a – b
D. 1, a + b
Answer : B

Question. Which of the following is TRUE?
A. y = mx represents a line passing through the origin.
B. Any line parallel to x-axis is y = b.
C. The point (2, 3) lies on the line 3y = (5/2) x + 4.
D. All of these
Answer : D

Question. The sum of an A.P. whose first term is a, second term is b and the last term is c, is equal to
A. (a + c)(b + c - 2a) / 2(b - a)
B. (a - c)(b + c - 2a) / 2(b - a)
C. (a + c)(b - c + 2a) / 2(b - a)
D. None of these
Answer : A

Question. The length of a line segment is 10 units. If one end is at (2, –3) and the abscissa of the second end is 10, then its ordinate is
A. 3
B. –3
C. –9
D. Both A and C
Answer : D

Question. If two positive integers m and n are expressible in the form m = pq³ and n = p³q², where p, q are prime numbers, then HCF (m, n) =
A. pq
B. pq²
C. p³q³
D. p²q³
Answer : B

Question. ΔABC is right angled at C and DE ⊥ AB. Find the length of AE and DE respectively.

A. (15/13)cm, (36/13)cm
B. (36/13)cm, (13/12)cm
C. (15/13)cm, (1/3)cm
D. (36/13)cm, (1/13)cm
Answer : A

Question. A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 13 cm, 14 cm and 15 cm and the parallelogram stands on the base 14 cm, find the height of the parallelogram.
A. 6 cm
B. 8 cm
C. 7 cm
D. 14 cm
Answer : A

EVERYDAY MATHeMATICS

Question. The cost of 2 sarees and 4 shirts is ₹ 16000. The cost of 1 saree is same as the cost of 6 shirts. The cost of 12 such shirts is
A. ₹ 12000
B. ₹ 24000
C. ₹ 48000
D. ₹ 25000
Answer : A

Question. 85 L 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 L
B. 6.5 L
C. 7.25 L
D. 8 L
Answer : A

Question. A person invested in all ₹ 2600 at 4 %, 6 % and 8 % per annum simple interest. At the end of the year, he got the same interest in all the three cases.
The money invested at 4 % is
A. ₹ 200
B. ₹ 600
C. ₹ 800
D. ₹ 1200
Answer : D

Question. The average score of Sehwag in 10 innings was 77 runs. In the 11th inning he had scored zero runs. The overall average score of Sehwag in all the 11 innings was
A. 77 runs
B. 27 runs
C. 11 runs
D. 70 runs
Answer : D

Question. 8% of the voters in an election did not cast their votes. In the election, there were only two candidates. The winner by obtaining 48% of the total votes defeated his contestant by 1100 votes. The total number of voters in the election was
A. 2100
B. 22000
C. 23500
D. None of these
Answer : D

Question. P, Q and R completed a piece of work costing ₹ 1800. P worked for 6 days, Q for 4 days and R for 9 days. If their daily wages are in the ratio 5 : 6 : 4, how much amount will be received by P?
A. ₹ 800
B. ₹ 600
C. ₹ 900
D. ₹ 750
Answer : B

Question. A shopkeeper bought pens at the rate of 8 pens for ₹ 34 and sold them at 12 pens for ₹ 57. The number of pens, that he should sell to have a profit of ₹ 900, must be
A. 1400
B. 1600
C. 1800
D. 2000
Answer : C

Question. A train when moves at an average speed of 40 km/hr, reaches its destination on time. When its average speed becomes 35 km/hr, then it reaches its destination 15 minutes late. Find the length of journey.
A. 30 km
B. 40 km
C. 70 km
D. 80 km
Answer : C

Question. A circular swimming pool is surrounded by a concrete wall 4 ft. wide. If the area of the concrete wall surrounding the pool is 11/25 that of the pool, then the radius of the pool is
A. 8 ft
B. 16 ft
C. 20 ft
D. 30 ft
Answer : C

Question. There are total 100 coins consisting of 20 paise, 50 paise and ₹ 1 in the ratio of 7 : 8 : 5. What is the number of coins of 50 paise if the difference between the amount yielded by 20 paise and ₹ 1 coins is ₹ 18?
A. 32
B. 40
C. 26
D. 56
Answer : B

ACHIEVERS SECTION

Question. In the given figure (not drawn to scale), TB is a tangent to the circle with centre at O. TO meets the circle at C and A. D is another point on the circle and ∠BTC = 22°.

Find :
(i) ∠OAB (ii) ∠ABT (iii) ∠BDC
      (i)       (ii)         (iii)
A. 124°    34°         34°
B. 34°      34°        124°
C. 34°     124°       34°
D. 34°     124°       38°
Answer : C

Question. From a pack of 52 cards, two red kings and two black queens are removed. Now, a card is drawn at random. Then match the columns
              Column I                     Column II
(i) Probability of getting              (p) 1
a heart is 2
(ii) Probability of getting             (q) 1
a black card is 24
(iii) Probability of getting a         (r) 1
king is 6
(iv) Probability of getting a        (s) 1
face card is 4
A. (i) → (s), (ii) → (p), (iii) → (r), (iv) → (q)
B. (i) → (p), (ii) → (s), (iii) → (r), (iv) → (q)
C. (i) → (s), (ii) → (p), (iii) → (q), (iv) → (r)
D. (i) → (p), (ii) → (s), (iii) → (q), (iv) → (r)
Answer : C

Question. Study the statements carefully.
Statement I : The given pair of equations –x + 2y
+ 2 = 0 and 2x – y – 4 = 0 has no solution
Statement II : Any pair of equations ax + by + c = 0,
mx + ny + p = 0 has no solution iff
a/m = b/n = c/p
A. Both Statement I and Statement II are true.
B. Both Statement I and Statement II are false.
C. Statement I is true, Statement II is false.
D. Statement I is false, Statement II is true.
Answer : B

Question. The points A(–5, 5), B(1, –3) and C(4, –3) are shown in the given figure.

Find :
(a) the coordinates of the midpoint of AC.
(b) the length of AB.
           (a)                  (b)
A. (− 1/2, 1)           9 units
B. (− 1/2, 1)           10 units
C. (− 1/2, 1)           9 units
D. (− 1/2, 1)           10 units
Answer : B

Question. Which of the following is INCORRECT?
A. 2(cos²45° + tan260°) – 6(sin²45° – tan²30°) = 6
B. sin²45° + cos²45° / tan60° = 1/3
C. (cosecθ – sinθ) (secθ – cosθ) (tanθ + cotθ) = 1/2
D. 1 - sinθ / 1+sinθ = (secθ - tanθ)²
Answer : C