CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set K

LOGICAL REASONING

Question. If in a certain language, blemish is coded as aodphvg, then how will chapter be coded in that language?
A . debovtd
B . bkzsshq
C . bgAqmfp
D . cahtpre
Answer : D

Question. Three positions of a dice are shown below. Find the number of dots on the face opposite to the face bearing 3 dots.

A . 5
B . 6
C . 4
D . C annot be determined
Answer : D

Question. A family consists of six members P, Q, R, X, Y and Z. Q is the son of R but R is not the mother of Q. P and R is a married couple. Y is the brother of R. X is the daughter of P. Z is the brother of P. Which of the following is a pair of brothers?
A . R and Y
B . Q and X
C . P and Z
D . P and X
Answer : A

Question. Find the missing number, if a certain rule is followed row-wise or column-wise.
1296      1​​​​2        2304
961         2         1089
441         ?         900
A . 0
B . 2
C . 3
D . 9
Answer : B

Question. Given an input line of words and numbers, a machine rearranges them following a particular rule in each step. The following is an illustration of an input and steps of rearrangement.
Input : talk seven 37 48 given 83 likely 62
Step I : 37 talk seven 48 given 83 likely 62
Step II : 37 talk 48 seven given 83 likely 62
Step III : 37 talk 48 seven 62 given 83 likely
Step IV : 37 talk 48 seven 62 likely given 83
Step V : 37 talk 48 seven 62 likely 83 given
Step V is the last step for this input.
Following the above rule, if step II for an input is "23 working 48 32 park blossom 26 garden", then what will be its fifth step?
A . 23 working 26 park 48 32 blossom garden
B . 23 working 26 park 32 48 blossom garden
C . 23 working 26 32 park 48 blossom garden
D . 23 working 26 48 park 32 blossom garden
Answer : D

Question. Which of the following Venn diagrams best represents the relationship amongst, "Professors, Doctors, Cardiologists"?

Answer : D

Question. Which of the following is the twelfth to the left of the twenty-first from the left end in the given arrangement?
B 4 @ D A © 7 9 F % 2 R 5 H 6 E ★ N $ 1 U W 3 P T 8 δ V # Y I
A . R
B . 1
C . 5
D . F
Answer : D

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

A . 50
B . 52
C . 54
D . None of these
Answer : A

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

Answer : A

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 a 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 a perfect square, then 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 second one is to be subtracted from the first one.
10 15 5
14 11 p
If p is the resultant of the first row, then what will be resultant of the second row?
A . 6
B . 81
C . 9
D . None of these
Answer : A

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

Answer : D

Question. Which of the following figures will complete the pattern in Fig. (X)?

A. T
B. S
C. S
D. S
Answer : D

Question. Rahul starts from point A and travels 4 km in East direction to reach point B, now he takes turn and travels 3 km in North direction to reach point C. Calculate the shortest distance between point A and C and in which direction is he from the starting point?
A. 5 km, North-East
B. 4 km, North-East
C. 5 km, South-West
D. None of these
Answer : D

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

Answer : A

Question. There is a set of three figures X, Y, and Z which shows 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 shows the unfolded form of Fig. (Z).

Answer : C

MATHEMATICAL REASONING

Question. If sin θ + cos θ = a, then find the value of sin⁶ θ + cos⁶ θ.
A. 3 - 4(a² + 1)2/4
B . 4 - 3(a² - 1)2/4
C . 4 - 3(a² + 1)2/4
D . 3 - 4 (a² - 1)2/4
Answer : B

Question. In the given figure, TBP and TCQ are tangents to the circle, whose centre is O. Also, ∠PBA = 60° and ∠ACQ = 70°. Find ∠BAC and ∠BTC.

A . 45°, 60°
B . 80°, 30°
C . 60°, 90°
D . 50°, 80°
Answer : D

Question. The length of the hypotenuse of a right angled triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. Find the length of these sides.
A . 15 cm, 20 cm
B . 20 cm, 25 cm
C . 10 cm, 15 cm
D . 5 cm, 10 cm
Answer : A

Question. Three positive integers a₁, a₂ and a₃ are in A.P. such that a₁ + a₂ + a₃ = 33 and a₁ × a₂ × a₃ = 1155. Find the values of a₁, a₂ a₃.
A . 15, 20, 17
B . 10, 11, 12
C . 7, 11, 15
D . 7, 15, 20
Answer : A

Question. Which of the following is a rational number?
A . S um of (2 + 3) and its reciprocal
B . S quare root of 18
C . S quare root of 7 + 4 3
D . None of these
Answer : A

Question. In a cyclic quadrilateral ABCD, ∠A = (x + 2)°, ∠B = (y + 3)°, ∠C = (3y + 8)° and ∠D = (4x – 8)°. Find the smallest and the largest angle.
A . 48°, 143°
B . 37°, 132°
C . 37°, 143°
D . 20°, 132°
Answer : B

Question. Area of a triangle whose sides are 18 cm, 24 cm and 30 cm is 36 k cm2. Find the value of k.
A . 8
B . 4
C . 6
D . 10
Answer : D

Question. A child's game has 8 triangles of which 3 are blue and rest are red and 10 squares of which 6 are blue and rest are red. One piece is lost at random. Find the probability that it is a
(i) Red Triangle (ii) Blue Square
       (i)            (ii)
A . 5/18          1/9
B . 4/9            5/9
C . 6/9            4/9
D . 5/18          1/3
Answer : D

Question. On dividing 6x³ + 8x² – 3x + 8 by a polynomial g(x), the quotient and remainder were 3x + 4 and 6x + 20, respectively. Find g(x).
A . 2x – 3
B . 2x² + 4
C . 3x² – 4
D . 2x² – 3
Answer : D

Question. If P(9a – 2, – b) divides the line segment joining A(3a + 1, – 3) and B(8a, 5) in the ratio 3 : 1, then the values of a and b respectively are
A . –1, –3
B . –3, 1
C . 1, –3
D . 1, 3
Answer : B

Question. Find the mean, mode and median of the following data.

A . 60.72, 61, 61
B . 60.72, 62, 61
C . 61.72, 61, 62
D . 61.72, 61, 61
Answer : B

Question. Select the INCORRECT match.
A. cos² θ + 1/1 + cot² θ = 1
B. (1 + tan² θ) (1 + sin θ) (1 – sin θ) = 1
C. tan θ + sin θ / tan θ sin θ = sec θ - 1 / sec θ + 1
D. sin³ θ + cos³ θ / sin θ + cos θ + sin θ cos θ = 1
Answer : A

Question. Equilateral triangles are drawn on the sides of a right triangle, then the area of the triangle on the hypotenuse is equal to of the areas of the triangles on the other two sides.
A . P roduct
B . S um
C . D ifference
D . None of these
Answer : A

Question. Which of the following system of equations has no solution?
A . 3x – y = 2, 9x – 3y = 6
B . 4x – 7y + 28 = 0, 5y – 7x + 9 = 0
C . 3x – 5y – 11 = 0, 6x – 10y – 7 = 0
D . None of these
Answer : B

Question. An elastic belt is placed round the rim of a pulley of radius 5 cm. One point on the belt is pulled directly away from the centre O of the pulley until it is at P, 10 cm away from O as shown in the figure. Find the length of the belt that is in contact with the rim of the pulley. Also, find the area of the shaded region.

A. (π/20)cm, (3/25)cm (3√3 − π) cm²
B. (25π/3)cm, 1/3 (3√3 − π) cm²
C. (π/3)cm.25/3 (2√3 − π) cm²
D . (20π/3)cm, 25/3 (3√3 − π) cm²
Answer : B

Question. In the given figure, O is the centre of the circle. Find the value of x − y − z / 20°.

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

Question. PQRST is a regular pentagon and bisector of ∠TPQ meets SR at L. If bisector of ∠SRQ meets PL at M, find ∠RML.
A . 36°
B . 38°
C . 26°
D . 28°
Answer : B

Question. Find the number of zeroes of f(x), in each case.

     (i)     (ii)    (iii)    (iv)
A. 1       2       1       4
B. 1       2       4       1
C. 4       2       1       4
D. 1       2       3       4
Answer : D

Question. If roots of the equation (a² + b²)x² – 2(ac + bd)x + (c² + d²) = 0 are equal, then bc – ad = .
A. 1
B. 0
C. –1
D. 2
Answer : C

Question. The line segment joining the midpoints of the diagonals of a trapezium are parallel to each of the parallel sides and is equal to the difference of these sides.
A . 1/2
B . 2/3
C . 1/4
D . 3/4
Answer : D

EVERYDAY MATHeMATICS

Question. The cash difference between the selling prices of an article at a profit of 4% and 6% is ₹ 3. The ratio of the two selling prices is
A . 51 : 52
B . 52 : 53
C . 51 : 53
D . 52 : 55
Answer : B

Question. A certain number of tennis balls were purchased for ₹ 450. Five more balls could have been purchased in the same amount if each ball was cheaper by ₹ 15.
The number of balls purchased was
A . 10
B . 15
C . 20
D . 25
Answer : D

Question. Two,tanks are of the same capacity. The dimensions of the first tank are 12 cm × 8 cm × 4 cm. The second,tank has a square base with depth 6 cm, then find the side of the square.
A . 12 cm
B . 6 cm
C . 8 cm
D . 10 cm
Answer : C

Question. The average age of a husband and his wife was 23 years at the time of their marriage. After five years of their marriage they have a one-year old child. The average age of the family now is
A . 19 years
B . 23 years
C . 28.5 years
D . 29.3 years
Answer : B

Question. Priyanshu cover a certain distance between his home to school by cycle. Having an average speed of 30 km/hr, he is late by 20 mins. However, with a speed of 40 km/hr, he reaches his house 10 mins earlier. Find the distance between his house and school.
A . 50 km
B . 60 km
C . 30 km
D . 55 km
Answer : A

Question. Madan pays income tax at the rate of 10%. If his income is increased by 10% and his tax rate increases to 15%, then his net income after paying tax would increase by ₹ 350. What is Madan's income ?
A . ₹ 8,000
B . ₹ 10,000
C . ₹ 12,000
D . ₹ 14,000
Answer : A

Question. Four milkmen rented a pasture. A grazed 24 cows for 3 months; B grazed 10 cows for 5 months; C grazed 35 cows for 4 months and D grazed 21 cows for 3 months. If A's share of rent is ₹ 720, find the total rent of the field.
A . ₹ 4280
B . ₹ 2240
C . ₹ 3250
D . ₹ 3500
Answer : B

Question. Sneh's present age is 1/6 th of her father's present age. Sneh's father's age will be twice of Vimal's age after 10 years. If Vimal's eighth birthday was celebrated two years before, then what is Sneh's present age ?
A . 6(2/3)years
B . 24 years
C . 30 years
D . None of these
Answer : B

Question. Gold is 19 times as heavy as water and copper is 9 times as heavy as water. In what ratio should these be mixed to get an alloy 15 times as heavy as water?
A . 1 : 1
B . 3 : 1
C . 1 : 2
D . 3 : 2
Answer : B

Question. Amit gets pocket money from his father every day. Out of the pocket money, he saves ₹ 2.75 on first day and on each succeeding day he increases his saving by 25 paise. Find the amount saved by Amit on 14th day.
A . ₹ 6
B . ₹ 12
C . ₹ 8
D . ₹ 10
Answer : B

ACHIEVERS SECTION

Question. A manufacturer of TV sets produced 600 units in the 3rd year and 700 units in the 7th year. Assuming that the production increases uniformly by a fixed number every year, find
(i) the production in the first year
(ii) the production in the 10th year
(iii) the total production in 7 years.
           (i)                   (ii)                  (iii)
A . 500 units        600 units         5000 units
B . 475 units        800 units         4500 units
C . 550 units        775 units         4375 units
D . 600 units        800 units         6000 units
Answer : D

Question. State 'T' for true and 'F' for false.
(i) Area enclosed by two concentric circles with radius R and r respectively such that R > r is p(R² – r²).
(ii) The lengths of tangents drawn from an external point to a circle are not equal.
(iii) There is one and only one tangent at any point on the circumference of a circle.
(iv) Ratio of the area of the sector of a circle with central angle 90° to the area of that circle is 1 : 4.
      (i)      (ii)    (iii)      (iv)
A . F        F        F        F
B . F        T        F        F
C . T        F        T        T
D . T        T        T        T
Answer : A

Question. Which of the following options hold?
Statement I : The equation √3x² − 2√2x − 2√3 = 0 has real and equal roots.
Statement II : If the difference between the roots of the quadratic equation x² + kx + 12 = 0 is 1, then the value of k is 6.
A . Both Statement I and Statement II are true.
B . Statement I is true but Statement II is false.
C . S tatement I is false but Statement II is true.
D . B oth Statement I and Statement II are false.
Answer : C

Question. Match the following columns.

A . (i) - (R), (ii) - (Q), (iii) - (P)
B . (i) - (R), (ii) - (P), (iii) - (Q)
C . (i) - (P), (ii) - (Q), (iii) - (R)
D . (i) - (Q), (ii) - (P), (iii) - (R)
Answer : D

Question. Fill in the blanks.
If a pair of linear equations a1x + b1y + c1 = 0 and a²x + b²y + c² = 0, then
(i) a¹/a² = b¹/b² ≠ c¹/c² ⇒ The pair of linear equations is P .
(ii) a¹/a² ≠ b¹/b² ⇒ The pair of linear equations is Q .
(iii) a¹/a² = b¹/b² = c¹/c² ⇒ The pair of linear equations is R .
               P                      Q                     R
A . Inconsistent       Consistent        Consistent
B . Inconsistent       Inconsistent      Consistent
C . Inconsistent       Inconsistent      Inconsistent
D . Inconsistent       Consistent        Inconsistent
Answer : D