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Logical Reasoning | |
Mathematical Reasoning | |
Everyday Mathematics | |
Achievers Section |
Note: The questions provided in this document are similar to the questions that were asked in the actual Olympiad exam. So, we recommend you study these for your Olympiad preparation
Q1: A man travels 30 m towards South, then he takes turn and travels 10 m towards East. He takes turn and travels 23 m towards North, and finally taking turn travels 14 m towards East. In which direction and how far is he now from the starting point?
(a) 15 m, North-West
(b) 17 m, North-East
(c) 25 m, South-East
(d) 20 m, South-West
Ans: (c)
Also, FD = BC = 10 m
∴ FE = FD + DE = (10 + 14) m = 24 m
In ΔAFE, using Pythagoras theorem,
So, the man is at 25 m in the South-East direction from the starting point.
Q2: The following question consists of a set of three figures X, Y and Z showing a sequence of folding a piece of paper. Figure Z shows the manner in which the folded paper has been cut. Select a figure from the options which would most closely resemble the unfolded form of Figure Z.
(a)
(b)
(c)
(d)
View AnswerAns: (c)
Q3: In the following question, two sequences of numbers are provided. You need to calculate the resulting number for each sequence based on the specified rules, and then answer the question below the sequences. The operations on the numbers are performed from left to right.
Rules:
(i) If an odd number is followed by another composite odd number, they should be added.
(ii) If an even number is followed by an odd number that is not a perfect square, they should be added.
(iii) If an even number is followed by a perfect square, the even number should be subtracted from the perfect square.
(iv) If an odd number is followed by a prime odd number, the first number should be divided by the second number.
(v) If an odd number is followed by an even number, the second number should be subtracted from the first number.
12, 64, 17
20, p, 16
If p is the result of the first sequence, what is the result of the second sequence?
(a) 8
(b) 63
(c) 73
(d) 79
Ans: (c)
Q4: Find the median of P, Q, R and S, where P is the mean of the first 21 natural numbers. Q is the mean of the squares of the first 7 natural numbers. R is the sum of the first 10 odd natural numbers and S is the sum of the first 10 even natural numbers.
(a). 60
(b) 50
(c) 40
(d) 30
Ans: (a)
Q = (7 × 8 × 15) ÷ (7 × 6) = 20
R = 102 = 100
S = 10 × 11 = 110
Median of the value 11, 20, 100, 121 is (100 + 20)/2 = 60.
Q5: There is a specific connection between the two numbers surrounding : :. Determine this relationship and choose the number that will substitute the "?".
197 : 227 :: ? : 291
(a) 257
(b) 220
(c) 260
(d) 259
Ans: (a)
Q6: Calculate the mean of a grouped data in which the sum of the frequencies is 40 and the sum of the products of the frequency and mid-values of each class is 1600.
(a) 50
(b) 40
(c) 30
(d) 20
Ans: (b)
Q7: Identify the number that is absent in the following sequence:
2, 17, 82, ?, 626
(a) 17
(b) 84
(c) 257
(d) 625
Ans: (c)
Q8: Study the given Venn diagram carefully and answer the question that follows:
How many Dancers are Actors but not Singers?
(a) 15
(b) 2
(c) 18
(d) 8
Ans: (d)
Since, only 8 is common to circle and triangle and not in rectangle.
Q9: The range of the scores is 30, 24, 37, 52, 28 and x, where p < 24 is 30. Find the value of p.
(a) 21
(b) 31
(c) 22
(d) 28
Ans: (c)
Given, 30, 24, 37, 52, 28 and p
Range = Maximum value - Minimum value
30 = 52 - p
p = 52 - 30
p = 22
Q10: Select a figure from the options which satisfies the same conditions of placement of the dots as in the given figure.
(a)
(b)
(c)
(d)
Ans: (b)
Q11: Find the missing number, if same rule is followed in all the three figures.
(a) 45
(b) 68
(c) 36
(d) 80
Ans: (b)
The rule followed is :
(7 × 5) – (6 + 4) = 35 – 10 = 25;
(8 × 6) – (4 + 2) = 48 – 6 = 42
Similarly, (10 × 8) – (5 + 7) = 80 – 12 = 68
Q12: Three positions of a dice are shown below. Find the number on the face opposite to the face having number ‘4’.
(a) 6
(b) 5
(c) 2
(d) 3
Ans: (b)
The numbers on opposite faces are : (1, 2), (3, 6) and (4, 5).
Q13: If ‘A’ represents ‘×’, ‘B’ represents ‘÷’, ‘C’ represents ‘+’, and ‘D’ represents ‘–’, what is the result of 275B25D12A2C70?
(a) –57
(b) 69
(c) 57
(d) 71
Ans: (c)
Q14: Which of the following figures is not exactly embedded in the given figure as one of its parts?
(a)
(b)
(c)
(d)
Ans: (d)
Q15: In a row of girls, Kavya is 10th from the left end while Radha is 12th from the right end. Now, both of them interchange their positions such that Radha becomes 18th from the right end. How many girls are there in the row?
(a) 27
(b) 29
(c) 30
(d) 28
Ans: (a)
Q16: What is the difference between the range and the mode of the data given below?
6, 25, 30, 32, 40,18, 22, 65, 23, 24, 24, 45
(a) 24
(b) 35
(c) 59
(d) 45
Ans: (b)
Arranging in the ascending order:
6, 18, 22, 23, 24, 24, 25, 30, 32, 40, 45, 65
Range = 65 - 6 = 59
Mode = 24
Difference = 59 - 24 = 35
Q17: If the difference between the semi-perimeter and the sides of a triangle △ABC are 8 cm, 7 cm, and 5 cm respectively, then find the area of the triangle.
(a) 20√7 cm2
(b) 10√10 cm2
(c) 20√14 cm2
(d) 140 cm2
Ans: (c)
Q18: If 5x - 5x-1 = 20, what is the value of (2x) x?
(a) 2
(b) 16
(c) 4
(d) 5
Ans: (b)
Q19: Find the value of m, if x + 4 is a factor of the polynomial x2 + 3x + m.
(a) 4
(b) -4
(c) 2
(d) -2
Ans: (b)
Let p(x) = x2 + 3x + m
Since (x + 4) or (x - (-4)} is a factor of p(x).
∴ p (-4) = 0
(-4)2 + 3( -4) + m = 0
16 - 12 + m = 0
m = -4
Q20: What is the value of x (y - z)3 + y (z - x)3 + z (x - y)3, when it is given that x + y + z = 0?
(a) 0
(b) y + z
(c) 1
(d) (z + x)2
Ans: (a)
Now, x + y + z = 0
⇒ x = -y - z, y = - x - z, z = - x - y
x (y - z)3 + y (z - x)3 + z (x - y)3
= (- y - z) (y - z)3 + (- z - x) (z - x)3 + (- x - y) (x - y)3
= - (y + z) (y - z)3 - (z + x) (z - x)3 - (x + y) (x - y)3
= - [(y2 - z2) (y - z)2 + (z2 - x2) (z - x)2 + (x2 - y2) (x - y)2]
= - [(y2 - z2) (y2 - 2yz + z2) + (z2 - x2) (z2 - 2xz + x2) + (x2 - y2) (x2 - 2xy + y2)]
= - [(y4 - z4) - 2yz (y2 - z2) + (z4 - x4) - 2xz (z2 - x2) + (x4 - y4) - 2xy (x2 - y2)]
= 2yz (y2 - z2) + 2xz (z2 - x2) + 2xy (x2 - y2)
= 2 (y3z - yz3 + z3x - x3z + x3y - xy3)
= 2 [x3 (y - z) + y3 (z - x) + z3 (x - y)]
= 2 [- (y - z) (z - x) (x - y). (x + y + z)]
= 0 (As, x + y + z = 0)
Q21: A die is rolled 125 times. The outcomes are recorded as follows:
Find the probability of getting a prime number.
(a) 82/125
(b) 72/125
(c) 58/125
(d) 13/25
Ans: (b)
Q22: Which of the following is not a factor of x4 - 3x2 + 2?
(a) x - 1
(b) x - 2
(c) x + 1
(d) x + √2
Ans: (b)
Q23: If the average of the data 12, 9, 18, (x + 2), 30, 31, 8, x + 4, 39, 34 is 22.9, then find the value of x.
(a) 22
(b) 23
(c) 20
(d) 21
Ans: (d)
Q24: If x2021 + 2021 is divided by (x + 1), what is the remainder?
(a) 2020
(b) 2019
(c) 1
(d) 2022
Ans: (a)
Let p(x) = x2021 + 2021
p (-1) = (-1)2021 + 2021
= -1 + 2021 = 2020
Q25: Which of the following pairs does not satisfy the equation 2x - y = 6?
(a) (1, –4)
(b) (2, –2)
(c) (5, 4)
(d) (7, –8)
Ans: (d)
Q26: A design on a floor is made up of triangular tiles. The sides of each triangle are 26 cm, 28 cm, and 30 cm. Find the cost of polishing all 180 tiles on the floor at the rate of ₹1.25 per cm².
(a) ₹67,920
(b) ₹99,480
(c) ₹89,460
(d) ₹79,560
Ans: (b)
Q27: If (x + 1) (x + 3) (x + 5) (x + 7) + p is a perfect square, then p = _________.
(a) 16
(b) 39
(c) 64
(d) 36
Ans: (a)
(x+ 1) (x + 3) (x+5) (x+7)+p=(x2+8x+7) (x2+8x+15)+p
= (a + 7) (a + 15) + p (Let us assume that x2 + 8x = a)
= a2 + 22a + 105 + p
= a2 + 2 (a) (11) + (105 + p) is perfect square.
Q28: A copper wire, 3 mm in diameter, is wound around a cylinder whose length is 1.2 m and diameter is 10 cm, so as to cover the curved surface of the cylinder. Find the length of the wire (Use π = 3.14).
(a) 125.6 m
(b) 1256 m
(c) 12.56 m
(d) 1.256 m
Ans: (a)
Q29: Determine the square of (n + 1) if 11√n = √343 + √112
(a) 8
(b) 7
(c) 64
(d) 49
Ans: (c)
Q30: The expression (a + b)(a - b)(a2 - ab + b2)(a2 + ab + b2) simplifies to what?
(a) a6 + b6
(b) a6 - b6
(c) a3 - b3
(d) a3 + b3
Ans: (b)
Q31: The compound interest on a sum for 2 years is ₹832 when compounded annually, and the simple interest on the same sum at the same rate for the same time period is ₹800. What is the rate of interest?
(a) 12%
(b) 10%
(c) 6%
(d) 8%
Ans: (d)
Q32: Which of the following is a factor of x (y2 - z2) + y (z2 - x2) + z (x2 - y2)?
(a) (x + y + z)
(b) (x + y - z)
(c) (x + y)
(d) (x - y)
Ans: (d)
Let f(x) = x (y2- z2) + y (z2- x2) + Z (x2 - y2)
Put x = y
f(y)=y (y2-z2)+ y (z2- y2) + z (y2- y2)
= y (y2-z2) - y (y2- z2) + 0
= 0
x - y is a factor of f(x).
Q33: Factorize: a3(b - c)3 + b3(c - a)3 + c3(a - b)3
(a) 3abc
(b) 3abc(a - b)(b - c)(c - a)
(c) a3 + b3 + c3 - 3abc
(d) 0
Ans: (b)
Q34: Find the equation of a line passing through point (-2, 3) and perpendicular to 9x + 7y + 3 = 0.
(a) 7x - 9y + 41 = 0
(b) 8x - 9y + 41 = 0
(c) 7x - 8y + 31 = 0
(d) 7x + 9y + 31 = 0
Ans: (a)
Here, (x1, y1) = (-2, 3), a = 9 and b = 7.
∴ Equation of the line perpendicular to 9x + 7y + 3 = 0 and passing through (-2, 3) is
b (x - x1) - a (y - y1) = 0.
That is, 7(x + 2) - 9(y - 3) = 0
7x - 9y + 41 = 0
Hence, the required equation of the line is 7x - 9y + 41 = 0.
Q35: Find the mid-point of the line segment joining the points (2, -7) and (8, -4).
(a) (2, -3.5)
(b) (5, -4.5)
(c) (5, -5.5)
(d) (2, -4.5)
Ans: (c)
Let A (2, -7) and B (8, -4) be the given points and M be the mid-point of AB
Q36: A and B are friends. A is 5 years older than B. B’s sister C is half the age of B, and A’s father D is 8 years older than twice the age of B. If D is currently 48 years old, what are the present ages of A, B, and C (in years)?
(a) 10, 20, 35
(b) 25, 15, 10
(c) 25, 20, 10
(d) 25, 20, 40
Ans: (c)
Q37: Muskan possesses a total of ₹ (2y3 + y2 - 2y - 1). She purchases 2y + 1 pairs of shoes for herself. What is the price of each pair of shoes?
(a) (y + 1)(2y + 1)
(b) (y - 1)(2y + 1)
(c) (y - 1)(y + 1)
(d) (3y + 1)(2y + 1)
Ans: (c)
Q38: The average monthly salary of 10 members of a group is ₹2890. If one more member whose monthly salary is ₹3000 joins the group, then find the average monthly salary of all members of the group.
(a) ₹2800
(b) ₹2940
(c) ₹2900
(d) ₹3000
Ans: (c)
Q39: In a debate competition, there are 35 students participating, consisting of 13 girls and the remaining are boys. If one student is chosen as the winner, what is the probability that the winner is a boy?
(a) 8/25
(b) 8/33
(c) 17/33
(d) 22/35
Ans: (d)
Q40: If a pipe can fill a cistern in 6 hours and another pipe can empty the cistern in 15 hours, then what part of the cistern will be filled in 1 hour, if both pipes are opened simultaneously?
(a) 1/40
(b) 1/30
(c) 1/35
(d) 1/10
Ans: (d)
As given, a pipe can fill a cistern in 6 hours.
So, in 1 hour the pipe can fill 1/6 part of the cistern.
Now, another pipe can empty the cistern in 15 hours.
So, in 1 hour the pipe can empty 1/15 part of the cistern.
∴ Part filled in 1 hour =
Q41: An advertisement board is shaped like an isosceles triangle with two sides measuring 10 m and the base measuring 12 m. Calculate the expense for painting it at a rate of ₹2.25 per m2.
(a) ₹98
(b) ₹112
(c) ₹48
(d) ₹108
Ans: (d)
Q42: Vinay and Vijay begin their journey simultaneously from City A to City B, which is located 60 km away. Vinay moves at a speed that is 4 km/hour less than Vijay. After reaching City B, Vijay immediately returns and encounters Vinay 12 km from City B. What is Vinay's speed?
(a) 8 km/hour
(b) 12 km/hour
(c) 16 km/hour
(d) 10 km/hour
Ans: (a)
Q43: In a prayer assembly, students are organized in rows to create a perfect square. There are 49 students who are not included in this arrangement. Given that the total number of students in the school is 2450, what is the number of students in each row?
(a) 51
(b) 50
(c) 39
(d) 49
Ans: (d)
Q44: 60 kg of a specific type of rice priced at ₹ 32 per kg is combined with 48 kg of a different type of rice, and the resulting mixture is sold for ₹ 28 per kg. If there is neither profit nor loss from this new selling price, what is the cost (per kg) of the second type of rice?
(a) ₹ 30
(b) ₹ 25
(c) ₹ 23
(d) ₹ 26
Ans: (c)
Q45: A spherical ball of lead, 3 cm in diameter, is melted and recast into three spherical balls. The diameters of two of these balls are 1.5 cm and 2 cm, respectively. What is the diameter of the third ball?
(a) 2.66 cm
(b) 2.5 cm
(c) 3 cm
(d) 3.5 cm
Ans: (b)
Q46: Harry is given 3 co-ordinates (x + y, x - y), (2x + y, 2x - y), (x - y, x + y). He needs to find a fourth co-ordinate such that on joining all the 4 points the figure thus obtained is parallelogram:
(a) (y, -y)
(b) (-y, y)
(c) (-x, x)
(d) (-x, y)
Ans: (b)
Let A (x + y, x - y), B (2x + y,2x - y), C (x - y, x + y) and D (p, q) be the vertices of a parallelogram.
The midpoints of diagonals AC and BD have the same coordinates.
⇒ q = y
Hence the vertex is (-y, y).
Q47: A line is perpendicular to 3x − 8y + 7 = 0, and it makes an intercept of 13 with the x-axis. Find the equation of the line.
(a) 7x + 3y - 77 = 0
(b) 8x + 3y - 104 = 0
(c) 5x + 3y − 55 = 0
(d) 3x + 8y - 88 = 0
Ans: (b)
Slope of the line perpendicular to 3x - 8y + 7 = 0 is
Given that, x-intercept of the required line is 13
It passes through (13, 0).
Hence, the required line is
Q48: A triangle is formed between the coordinate axes and the straight line mx + ny + p = 0. The line makes an equal intercept on the x and y-axis. Which of the following is true?
(a) | a | = | b |
(b) | a | = | c |
(c) | b | = | c |
(d) | a | = | d |
Ans: (a)
The equation of line AB, i.e., mx + ny + p = 0 in intercept form is mx + ny = -p
Δ AOB is isosceles Δ as OA = OB, i.e., x - intercept = y - intercept
Q49: The sum of the digits of a two-digit number is 9. If 27 is added to it, the digits of the number get reversed. The number is:
(a) 16
(b) 36
(c) 48
(d) 54
Ans: (b)
Let the digit at unit's place be y and the digit at ten's place be x.
So, the number will be 10x + y.
According to the first condition-
x + y = 9 ....(1)
According to the second condition-
10y + x = 10x + y + 27
9y = 9x + 27
y = x + 3 ...(2)
Put the value of y in (2), we have
x + x + 3 = 9
x = 3
x + y = 9
3 + y = 9
y = 6
∴ The number is 36.
Q50: If the point F lies in between M and N and C is the midpoint of MF which of the following is true?
(a) MC + FN = MN
(b) MF + CF = MN
(c) MC + CF = MN
(d) CF + CN = MN
Ans: (d)
From the figure given above,
CF + FN = CN .......... (1).
and MC + CF + FN = MN
So, MC + CN = MN (From equation ... 1)
Since, C is the mid-point of MF, MC = MF.
Hence, option d is the correct answer.
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