CBSE Class 10 Mathematics IMO Olympiad MCQs with Answers Set E

MATHEMATICS

Question. If one of the zeros of a quadratic polynomial of the form x² + ax + b is negative of the other, then it ______.
A. Has no linear term and the constant term is negative.
B. Has no linear term and the constant term is positive.
C. Can have a linear term but the constant term is negative.
D. Can have a linear term but the constant term is positive.
Answer : A

Question. Solve for x and y :
(1+b/b) x + (1+a/a) y = b - a, x,b - 4y/a = 5 ; ab ≠ 0.
A. x = –a, y = b
B. x = b², y = a²
C. x = a², y = b²
D. x = b, y = –a
Answer : D

Question. In the given figure, ABC is a triangle right angled at B and BD ⊥ AC. If AD = 4 cm and CD = 5 cm, find BD and AB respectively.

A. 2√5 cm, 3√5 cm
B. 3√5 cm, 6 cm
C. 2√5cm, 6 cm
D. 3√5cm, 8 cm
Answer : C

Question. In DABC right angled at B, BC = 5 cm and AC – AB = 1 cm.
Evaluate 1+ sin C / cos C
A. 5/13
B. 13
C. 12/13
D. 5
Answer : D

Question. Which of the following statements is correct ?
A. sin (A + B) = sin A + sin B
B. The value of sin q increases as q decreases
C. The value of cos q increases as q increases
D. None of these
Answer : D

Question. The following table shows the daily pocket allowance given to the children of a multistorey building. The mean of the pocket allowance is ` 18. Find out the missing frequency.

A. 8
B. 16
C. 12
D. 4
Answer : A

Question. Three years ago, the average age of Latika, Garima and Megha was 27 years and that of Garima and Megha 5 years ago was 20 years. Latika’s present age is _______.
A. 30 years
B. 36 years
C. 40 years
D. 46 years
Answer : C

Question. Find the mode of the following frequency distribution :

A. 20.33
B. 30.12
C. 33.33
D. 60.43
Answer : C

Question. A small scale industry produces a certain number of items per day. The cost of production of each item (in rupees) was calculated to be 74 minus twice the number of articles produced in a day. On a particular day, the total cost of production was ₹ 540. Which of the following equations represent how to find the number of items produced on that day?
A. 74 + 2x = 540
B. x² + 74x – 540 = 0
C. 74 – 2x = 540
D. x² – 37x + 270 = 0
Answer : D

Question. The sum of first n terms of an A.P. is given by (n² + 8n).
Find the 12^th term of the A.P. Also find the n^th term of the A.P.
A. 31, 2n + 9
B. 31, 2n + 7
C. 30, 2n + 6
D. 30, 2n + 8
Answer : B

Question. In the given figure, PQ is the chord of circle and PT is the tangent at P such that ∠QPT = 60°. Then ∠PRQ is ________.

A. 135°
B. 150°
C. 120°
D. 110°
Answer : C

Question. In the given figure, arcs are drawn by taking vertices A, B and C of an equilateral triangle of side 10 cm to intersect the sides BC, CA and AB at their respective mid-points D, E and F. Find the area of the shaded region. [Use π = 3.14]

A. 39.25 cm²
B. 48.50 cm²
C. 78.50 cm²
D. 28.25 cm²
Answer : A

Question. Two cleanliness hoardings are put on two poles of equal heights standing on either side of a roadway 50 m wide between the poles. The elevations of the tops of the poles from a point between them are 60° and 30°. Find the height of the pole.
A. 50√3m
B. (25/3)√3m
C. 25√3m
D. (25/2)√3m
Answer : D

Question. Beena gave a simple multiplication question to her students. But one student reversed the digits of both numbers and carried out the multiplication and found that the product was exactly the same as the one expected by Beena. Which one of the following pairs of numbers will fit in the description of the question?
A. 14, 22
B. 13, 62
C. 19, 33
D. 42, 28
Answer : B

Question. For what values of k will the following pair of linear equations have infinitely many solutions?
2x + 3y = 4 and (k + 2) x + 6y = 3k + 2
A. 1
B. –1
C. 2
D. –2
Answer : C

Question. The values of l for which the quadratic equation x² + 5lx + 16 = 0 has no real root is _____.
A. l > 8
B. l < –5
C. − 8/5 < l < 8/5
D. − 8/5 ≤ l < 0
Answer : C

Question. A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Six seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.
A. 3 seconds
B. 6 seconds
C. 9 seconds
D. 5 seconds
Answer : A

Question. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is odd ?
A. 1/2
B. 3/4
C. 3/8
D. 1/4
Answer : D

Question. Find the median of the following data :

A. 51.5
B. 25.5
C. 28.5
D. 31
Answer : B

Question. A number is selected at random from the numbers :
5, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 24, 25, 25, 27, 30, 30, 36, 37, 37, 39, 40, 40, 46.
Find the probability that the selected number is a prime number.
A. 0
B. 1/8
C. 1/6
D. 1/12
Answer : B

Question. The value of expression

A. – 0.2
B. 0.9
C. 1.27
D. – 0.06
Answer : A

Question. There are two circles intersecting each other. Another smaller circle with centre O, is lying between the common region of two larger circles. Centres of the circle (i.e., A, O and B) are lying on a straight line. AB = 16 cm and the radii of the larger circles are 10 cm each. What is the area of the smaller circle ?

A. 4π cm²
B. 2π cm²
C. (4/π)cm²
D. (π/4)cm²
Answer : A

Question. If the points A(– 2, 1), B(a, b) and C(4, –1) are collinear and a – b = 1, find the values of a and b respectively.
A. 1, 0
B. 1, –1
C. 0, 1
D. –1, 1
Answer : A

Question. A circular paper is folded along its diameter, then again it is folded to form a quadrant. Then it is cut as shown in the figure, after it the paper was reopened in the original circular shape. Find the ratio of the original paper to that of the remaining paper?

(The shaded portion is cut off from the quadrant. The radius of quadrant OAB is 5 cm and radius of each semicircle is 1 cm)
A. 25 : 16
B. 25 : 9
C. 20 : 9
D. None of these
Answer : A

Question. There are 100 apples in a box. 20 of them are rotten. At random, two apples are taken one by one consecutively without replacement. What is the probability that both of them are good ?
A. 316/495
B. 19/495
C. 16/99
D. 32/99
Answer : A

Question. Which of the following statements is INCORRECT ?
(i) In order to divide a line segment internally in the ratio m : n, both m and n are real numbers.
(ii) A pair of tangents can be constructed to a circle inclined at an angle of 165°.
A. Only (i)
B. Only (ii)
C. Both (i) and (ii)
D. Neither (i) nor (ii)
Answer : A

Question. If a and b be two zeros of the quadratic polynomial p(x) = 2x² – 3x + 7, evaluate 1/2α-3 + 1/δ - 3
A. − 3/14
B. 3/7
C. − 5/4
D. 3/14
Answer : A

Question. √5 − 3− 2 is _____.
A. A rational number
B. A natural number
C. Equal to zero
D. An irrational number
Answer : D

Question. In figure, the line segment LM is parallel to side XZ of ΔXYZ and it divides the triangle into two parts of equal areas. Find the ratio XL/XY

A. √2 −1: √2
B. √2 +1: √2
C. 1− √2 : 2
D. 2 − √2 : √2
Answer : A

Question. Two ships are sailing in the sea on the either side of the lighthouse, the angles of depression of two ships as observed from the top of the lighthouse are 60° and 45° respectively. If the distance between the ships is 200 (√3+1/√3) metres, find the height of the lighthouse.
A. (100/√3)m
B. 100 m
C. 200 m
D. (1+√3) m
Answer : C

Question. A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment.
A. 4.50 m
B. 2.125 m
C. 1.125 m
D. 3.25 m
Answer : C

Question. In the given figure, if ∠POQ = 130°, then ∠SOR is equal to

A. 50°
B. 45°
C. 35°
D. 55°
Answer : A

Question. In the given figure, XW is a tangent to the circle with centre O at X and YZW is a straight line. Find the value of y.

A. 30°
B. 35°
C. 40°
D. 50°
Answer : A

Question. If the common difference of an A.P. is 5, then a₁₈ – a₁₃ = _____.
A. 5
B. 20
C. 30
D. 25
Answer : D

Question. If the points A(1, –2), B(2, 3), C(–3, 2) and D(– 4, –3) are the vertices of parallelogram ABCD, then taking AB as the base, find the height of this parallelogram.
A. (√26/24)units
B. (14/√13)units
C. (24/√26)units
D. (√13/14)units
Answer : C

Question. If the point P(x, 2) divides the line segment joining the points A(12, 5) and B(4, – 3) in the ratio k : 1, then find the value of x.
A. 8
B. 67/8
C. 67/3
D. 9
Answer : D

Question. ABC is a right-angled triangle, right angled at A. A circle is inscribed in it. The lengths of the two sides containing the right angle are 5 cm and 12 cm. Find the radius of the circle.
A. 1/2 cm
B. 13 cm
C. 2 cm
D. 10 cm
Answer : C

Question. In figure, ABCDEF is any regular hexagon with different vertices A, B, C, D, E and F as the centres of circles with same radius ‘r’ units are drawn. Find the area of the shaded portion.

A. 2πr² sq. units
B. 4πr² sq. units
C. πr² sq. units
D. 6πr² sq. units
Answer : A

Question. A cylindrical pipe has inner diameter of 4 cm and water flows through it at the rate of 20 m per minute.
How long would it take to fill a conical tank, with diameter of base as 80 cm and depth 72 cm ?
A. 5 minutes
B. 3 minutes 56 seconds
C. 4 minutes 20 seconds
D. 4 minutes 48 seconds
Answer : D

Question. If sum of the squares of zeros of the polynomial 6x² + x + k is 25/36, find k.
A. 2
B. –2
C. 1
D. –1
Answer : B

Question. In the Maths Olympiad in a school, two representatives from two teams, while solving a quadratic equation, committed the following mistakes :
(i) One of them made a mistake in the constant term and got the roots as 5 and 9.
(ii) The other committed an error in the coefficient of x and he got the roots as 12 and 4. In the meantime, they realised that they were wrong and together they managed to get it right. Find the right quadratic equation.
A. x² + 4x + 14 = 0
B. 2x² + 7x – 24 = 0
C. x² – 14x + 48 = 0
D. 3x² – 17x + 52 = 0
Answer : C

Question. What must be added to the polynomial 3l⁴ + 5l³ – 7l² + 5l + 3 so that the resulting polynomial is exactly divisible by l² + 3l + 1 ?
A. –3l + 1
B. –3l – 1
C. 3l + 1
D. 3l – 1
Answer : B

Question. In the given figure, DE || BC. If DE : BC = 3 : 5, find the ratio of the area of DADE to the area of trapezium BCED.

A. 9 : 16
B. 25 : 9
C. 16 : 9
D. 9 : 25
Answer : A

Question. If cos θ - sin θ / cos θ + sin θ = 1 - √3 / 1 + √3 and 0° < θ < 90°, then find the angle θ.
A. 30°
B. 60°
C. 90°
D. 45°
Answer : B

Question. Evaluate : 4 (sin⁴ 30° + cos⁴ 60°) – 3 (cos² 45° – sin² 90°) + (sin² 60° + sin² 45°)
A. 3(1/4)
B. 1/4
C. 3/4
D. 7/16
Answer : A

ACHIEVERS SECTION

Question. Which of the following options hold ?
Statement 1 : If p, q, r and s are real numbers such that pr = 2 (q + s), then atleast one of the equations x² + px + q = 0 and x² + rx + s = 0 has real roots.
Statement 2 : If a, b, c are distinct real numbers, then the equation (x – a) (x – b) + (x – b) (x – c) + (x – c) (x – a) = 0 has real and distinct roots.
A. Statement 1 is true, Statement 2 is false
B. Statement 1 is false, Statement 2 is true
C. Both Statement 1 and Statement 2 are true
D. Both Statement 1 and Statement 2 are false
Answer : C

Question. Rakesh has to buy a TV. He can buy TV either making cash down payment of ₹ 14000 at once or by making 12 monthly instalments as below :
₹ 1500 (1^st month), ₹ 1450 (2^nd month), ₹ 1400 (3^rd month), ₹ 1350 (4^th month), ........
Each instalment except the first is ₹ 50 less than the previous one.
Find (i) Amount of the instalment paid in the 9^th month.
(ii) Total amount paid in 12 instalments.
(iii) How much extra he has to pay in addition to the amount of cash down payment ?
        (i)                       (ii)                  (iii)
A. ₹ 1100             ₹ 16700             ₹ 900
B. ₹ 1200             ₹ 14700             ₹ 600
C. ₹ 1100             ₹ 14700             ₹ 700
D. ₹ 1100             ₹ 14700             ₹ 900
Answer : C

Question. Find the mode of the distribution from the following table :

A. 48.904
B. 50.909
C. 62.804
D. 64.324
Answer : B

Question. Arrange the given steps in correct order while constructing a triangle similar to a given triangle ABC with its sides equal to 3/4 of the corresponding sides of the triangle ABC (i.e., of scale factor 3/4).
1. Locate 4 points B₁, B₂, B₃ and B₄ on BX so that BB₁ = B₁B₂ = B₂B₃ = B₃B₄.
2. Draw a line through C′ parallel to the line CA to intersect BA at A′.
3. Draw any ray BX making an acute angle with BC on the side opposite to the vertex A.
4. Join B₄C and draw a line through B₃ parallel to B₄C to intersect BC at C′.
Then, ΔA′BC′ is the required triangle.
A. 3, 4, 2, 1
B. 2, 4, 3, 1
C. 3, 1, 4, 2
D. 2, 4, 1, 3
Answer : C

Question. Match the columns :

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