Mathematics Olympiad Model Test Papers - 1 | Olympiad Preparation for Class 10 PDF Download

Ans: (b)
Each number is doubled and 1, 3, 5, 7, 9, … respectively are added.

Ans: (b)

• The sequence has missing letters that need to be filled in to maintain a logical pattern.
• By analyzing the sequence, the correct letters that fit the pattern are found in option (b) ""bacb"".
• This option completes the series correctly, ensuring the letters follow the established order.
• Other options do not fit the sequence as well as option (b) does.

Ans: (d)

• The correct order starts from the top of the body and moves downwards.
• Head (2) is at the top, followed by Nose (3), then Neck (1), followed by Stomach (5), and finally Feet (4) at the bottom.
• This sequence reflects the anatomical structure of the human body.
• Thus, the logical arrangement is 2, 3, 1, 5, 4.

Ans: (d)
In the given options, each group follows a pattern in terms of the difference between the positions of the letters in the alphabet:

• (a) M (13th letter) and S (19th letter): Difference = 6
• (b) B (2nd letter) and L (12th letter): Difference = 10
• (c) T (20th letter) and Y (25th letter): Difference = 5
• (d) F (6th letter) and J (10th letter): Difference = 4

Option (d) has a different letter-difference pattern compared to the others, making it the odd one out.

Ans: (b)
4 × 2 – 1 = 7
4 × 7 – 1 = 27
4 × 27 – 1 = 107
4 × 107 – 1 = 427

Ans: (c)
This series contains three series in itself.

Term
T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ T₉ T₁₀ T₁₁ T₁₂
–1 0 1 0 2 4 1 6 9 2 12 16
Series 1 (T₁, T₄, T₇, T₁₀): –1, 0, 1, 2 (AP with a common difference of +1)
Series 2 (T₂, T₅, T₈, T₁₁): 0, 2, 6, 12 (Differences between two terms increase by 2)
Series 3 (T₃, T₆, T₉, T₁₂): 1, 4, 9, 16 (Squares of consecutive numbers)
The next three numbers will thus be 3, 20 and 25.
Hence, option C.

Ans: (c)

• To solve the equation, we need to find the right order of operations that makes the equation true.
• Using the signs ÷, ×, –, = in that order gives us: 84 ÷ 21 × 5 – 10 = 10.
• This simplifies to 4 × 5 – 10 = 10, which is correct since 20 – 10 equals 10.
• Thus, the correct sequence is ÷, ×, –, =, making option (c) the right choice.

Ans: (c)
Here x = 9, 10, 11 .... 
y = – 3, – 2, – 1, 0, 1, 2, 3, ....

Ans: (a)

• The first pair is JB and 20. The letters J and B are the 10th and 2nd letters of the alphabet, respectively. If you add these positions together (10 + 2), you get 12. However, the number given is 20.
• To find the relationship, we can see that 20 is actually 8 more than 12 (20 - 12 = 8).
• Now, looking at the second pair EG, the letters E and G are the 5th and 7th letters of the alphabet. Adding these gives us 12 (5 + 7 = 12).
• To find the missing term, we add 8 to 12 (12 + 8 = 20). Therefore, the missing term is 20.

Ans: (d)
AMAZING- 7 letter is G, In English Alphabet 7 letter is G

Ans: (b)

• The input consists of the words and numbers: 83, sleep, 24, good, 49, night, 16, now.
• In the rearrangement process, the first element is consistently moved to the front, followed by the next elements in a specific order.
• Following the same pattern as in the example, the third step will have the arrangement: 83 good 49 night sleep 24 16 now.
• This matches option (b), confirming it as the correct answer.

Ans: (b)

• First, we take the numbers: 462, 373, 564, 879, 263, 147.
• Next, we add 1 to the middle digit of each number: 462 becomes 472, 373 becomes 383, 564 becomes 574, 879 becomes 889, 263 becomes 273, and 147 becomes 157.
• Then, we interchange the first and third digits: 472 becomes 274, 383 becomes 383, 574 becomes 475, 889 becomes 988, 273 becomes 372, and 157 becomes 751.
• Now, we find the highest number from these: 988 is the highest, and its middle digit is 8.

Ans: (a)
Rotation of dark leaf is 135° (clock wise). Rotation of white leaf in 135° (anticlockwise). So for fig (3) option (a) is correct. 

Ans: (c)

Ans: (a)

• To find the missing number, we need to look for a pattern in the numbers provided in the table.
• By analyzing the rows or columns, we can determine how the numbers relate to each other.
• In this case, the correct answer is 76, which fits the established pattern.
• Understanding the rule applied in the table is crucial to identifying the missing number accurately.

Ans: (b)

• Given that sin²θ = cos²θ, we can use the identity sin²θ + cos²θ = 1 to find that sin²θ = 1/2 and cos²θ = 1/2.
• From this, we can calculate tan²θ = sin²θ/cos²θ = (1/2)/(1/2) = 1.
• Now, substituting tan²θ and sin²θ into the equation: 2(1) + (1/2) - 1 = 2 + 0.5 - 1 = 1.5 = 3/2.
• Thus, the final answer is 3/2.

Ans: (b)
As zero of 2x + 1 is -1/2 zero of the expression 3x+2a is b/3

Ans: (b)
n = 20

If d be the common difference 45 = the 20^th term = 10 + 19d
d= 35/19

Ans: (a)

• To find the roots of the equation, we first simplify it by eliminating the fractions. This involves multiplying through by the common denominator, which is (x + 2)(x - 2).
• After simplification, we rearrange the equation to form a standard polynomial equation.
• By solving this polynomial, we find the values of x that satisfy the equation, which are 1 and 3.
• Thus, the correct solutions are the roots 1 and 3, making option (a) the right choice.

Ans: (b)

• To find the mid-point of the line segment joining the points (2, 5) and (3, k), we use the formula: Mid-point = ((x1 + x2)/2, (y1 + y2)/2).
• Substituting the values, we get Mid-point = ((2 + 3)/2, (5 + k)/2) = (2.5, (5 + k)/2).
• This mid-point (2.5, (5 + k)/2) must satisfy the line equation 4x + 3y - 1 = 0. Plugging in x = 2.5 gives us 4(2.5) + 3((5 + k)/2) - 1 = 0.
• Solving this equation leads to k = -11, which is the correct answer.

Ans: (b)

• Let the breadth of the room be b meters. Then, the length is 2b meters.
• The height of the room is given as 4 m.
• The area of the four walls is calculated using the formula: 2 * height * (length + breadth).
• Substituting the values: 2 * 4 * (2b + b) = 144, which simplifies to 12b = 144, giving b = 12 m.
• Now, the length is 24 m (since length = 2 * breadth). The total surface area includes the floor and ceiling: 2 * (length * breadth + length * height + breadth * height).
• Calculating this gives: 2 * (24 * 12 + 24 * 4 + 12 * 4) = 288 m².

Ans: (b)
Equation is m x (x − 7) + 49 = 0.
mx² − 7mx + 49 = 0
On comparing it to quadratic equation ax² + bx + c = 0
Here we find that a = m, b = − 7 m and c = 49
For the quadratic equation to have equal roots, its discriminant D = 0
On substituting the values of a, b and c we get

Ans: (b)

• We know that the total probability of an event and its non-occurrence is always 1.
• Given that the probability of non-occurrence is 5/8, we can find the probability of occurrence: 1 - 5/8 = 3/8.
• We have the equation: P/2 = 3/8.
• To find P, we multiply both sides by 2: P = 3/8 * 2 = 3/4.

Ans: (c)

• To find the value of x, we first calculate the mean of the five observations.
• The sum of the observations is: x + (x + 4) + (x + 8) + (x + 12) + (x + 16) = 5x + 40.
• The mean is given by the formula: (sum of observations) / (number of observations). Here, it is (5x + 40) / 5 = 15.
• Multiplying both sides by 5 gives: 5x + 40 = 75. Solving for x, we get 5x = 35, so x = 7.

Ans: (a)

• To find the zeros of the polynomial, we first note that the sum of the zeros of the polynomial 5x² - (3 + k)x + 7 is given by the formula -b/a, which equals zero in this case.
• This implies that (3 + k) = 0, leading to k = -3.
• Substituting k into the second polynomial 2x² - 2(-3 + 11)x + 30 simplifies to 2x² - 16x + 30.
• Factoring or using the quadratic formula reveals that the zeros of this polynomial are 3 and 5.

Ans: (a)
If D is any point on the side BC of a ΔABC, then AB + BC + CA > 2 AD

Ans: (c)
Let r, s, t, v, be the amounts, with Ron, Sam, Tom, Vera respectively

The equations are:
Now,
2r = (s + t + v)
2r = 480 - r
3r = 480
r = 160
Similarly, s = 120 and t = 96
r + s + t + v = 480
v = 104

Ans: (c)
(tan a + cosec b)²−(cot b − sec a)²
tan²a + cosec²b + 2tan tan a cosec b − cot²b− sec²a + 2cot cot b sec sec a 
tan²a−sec²a + cosec²b − cot²b + 2(tan a cosec b+ cot b sec a)

Ans: (b)

• To solve for a and b, we start with the expression (√11 - √7) / (√11 + √7).
• By rationalizing the denominator, we multiply the numerator and denominator by (√11 - √7).
• This gives us (11 - 7) / ((√11 + √7)(√11 - √7)) = 4 / (11 - 7) = 4 / 4 = 1.
• Thus, we can express this as 1 = a - b√77, leading us to find that a = 4 and b = 1.

Ans: (a)

• Let the original fraction be x/y.
• According to the first condition, (x)/(y+1) = 2/3, which can be rearranged to 3x = 2(y + 1).
• From the second condition, (x + 5)/y = 1, leading to x + 5 = y.
• Substituting y = x + 5 into the first equation gives 3x = 2((x + 5) + 1), which simplifies to 3x = 2x + 12.
• Solving this, we find x = 12, which is the numerator of the original fraction.

Ans: (a)
Let the points (1, -3) and (-3, 5) be A and B respectively.
If P and Q be the points of trisection of AB, the ratio AP: PB will be equal to 1:2 and the ratio AQ: QB will be equal to 2:1.
Hence the coordinates of P are
And Co-ordinates of Q are

Ans: (b)

• In an Arithmetic Progression (A.P.), the sum of the first n terms can be calculated using the formula: S_n = n/2 * (2a + (n - 1)d), where a is the first term and d is the common difference.
• Here, the first term a = 21 and we need to find the common difference d.
• The sum of the first 6 terms is S_6 = 6/2 * (2*21 + 5d) = 3(42 + 5d).
• The sum of the next 6 terms (7th to 12th) is S_6 = 6/2 * (2*(21 + 6d) + 5d) = 3(42 + 11d).
• According to the problem, S_6 (first 6 terms) = 1/2 * S_6 (next 6 terms), leading to the equation: 3(42 + 5d) = 1/2 * 3(42 + 11d).
• Solving this gives d = 6.

Ans: (d)

• To find the height of the parallelogram, we first need to calculate the area of the triangle using Heron's formula.
• The semi-perimeter (s) of the triangle is calculated as (9 + 12 + 15) / 2 = 18 cm.
• Using Heron's formula, the area of the triangle is √[s(s-a)(s-b)(s-c)] = √[18(18-9)(18-12)(18-15)] = √[18 * 9 * 6 * 3] = 54 cm².
• Since the area of the parallelogram is the same, we can use the formula Area = base × height. Here, Area = 54 cm² and base = 12 cm.
• Thus, height = Area / base = 54 cm² / 12 cm = 4.5 cm.

Ans: (d)

=(tan θ + cot θ)−(sec θ − cosec θ)
(cosec θ+cot θ)−(sec θ −tan θ) = q −p

Ans: (c)

• To find the value of x, we first need to determine the HCF (Highest Common Factor) of 135 and 216, which is 27.
• Next, we set up the equation: 87x - 147 = 27.
• By rearranging, we get 87x = 27 + 147, which simplifies to 87x = 174.
• Dividing both sides by 87, we find x = 2.

Ans: (d)

• The devices beep at different intervals: 30 minutes, 1 hour, 1.5 hours, and 1 hour 45 minutes.
• To find when they beep together again, we need to calculate the Least Common Multiple (LCM) of these intervals.
• The LCM of 30, 60, 90, and 105 minutes is 540 minutes, which equals 9 hours.
• Starting from 12 noon, adding 9 hours brings us to 9 p.m..

Ans: (d)

• To find the third side of the triangle, we subtract the sum of the two known sides from the perimeter: 168 m - (60 m + 52 m) = 56 m.
• Next, we calculate the area of the triangle using Heron's formula. The semi-perimeter (s) is (168 m / 2) = 84 m.
• Using Heron's formula: Area = √[s(s-a)(s-b)(s-c)], where a, b, and c are the sides of the triangle. Plugging in the values gives us the area.
• Finally, to find the cost of watering, we multiply the area by ₹ 5 per m². The total cost comes out to be ₹ 6720.

Ans: (d)

• To find out how long it takes to fill the tank with all taps open, we first calculate the rates of each tap:
• Tap A fills the tank at a rate of 1/12 tanks per hour.
• Tap B fills the tank at a rate of 1/15 tanks per hour.
• Tap C empties the tank at a rate of 1/8 tanks per hour.
• When all taps are open, the combined rate is (1/12 + 1/15 - 1/8) tanks per hour.
• Finding a common denominator (120), we get: (10/120 + 8/120 - 15/120) = 3/120 = 1/40 tanks per hour.
• This means it will take 40 hours to fill the tank completely when all taps are open.
• Since 40 hours is not listed in the options, the correct answer is None of these.

Ans: (d)

• We know that Saurav's age is 6 years less than twice his daughter's age.
• This can be expressed as: x = 2y - 6.
• Rearranging this gives us: 2y - x = 6, which matches option (d).
• Thus, option (d) correctly represents the relationship between their ages.

Ans: (a)

• Let the original number of persons be x.
• Each person initially receives ₹ 6500/x.
• With 15 more persons, the number becomes x + 15, and each would then receive ₹ 6500/(x + 15).
• According to the problem, the difference in amount received is ₹ 30, leading to the equation: ₹ 6500/x - ₹ 6500/(x + 15) = ₹ 30.
• Solving this equation gives x = 50, which is the original number of persons.

Ans: (b)

• There are a total of 200 items in the box (50 bolts + 150 nuts).
• Half of the bolts are rusted, which means 25 bolts are rusted.
• Half of the nuts are rusted, resulting in 75 rusted nuts.
• So, the total number of rusted items is 25 + 75 = 100.
• The probability of picking a rusted item is 100 rusted items out of 200 total items, which simplifies to 1/2.

Ans: (b)

• In an Arithmetic Progression (A.P), the total can be calculated using the formula: Total = n/2 * (2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference.
• Here, n = 24, a = 250, and d = 15.
• Plugging in the values: Total = 24/2 * (2*250 + (24-1)*15).
• This simplifies to Total = 12 * (500 + 345) = 12 * 845 = ₹ 10140.

Ans: (a)
AB = (2.4 + 4) cm = 6.4 cm.
BQ/BC = BP/AB
BC = (6.4 × 5) ÷ 4 = 8 cm

Ans: (c)

• To solve this, we first determine the work done by men and girls in terms of work units per day.
• From the first scenario, 3 men and 5 girls complete the work in 6 days, which means they do 1/6 of the work per day.
• From the second scenario, 2 men and 7 girls finish the work in 5 days, equating to 1/5 of the work per day.
• By calculating the work rates of men and girls, we can find the combined rate for 16 men and 20 girls.
• Finally, we find that 16 men and 20 girls can complete the work in 1 3/8 days.

Ans: (d)

• Statement I is false because if 2 is a root of the equation, we can find k using the equation. However, for the second equation to have equal roots, q must be equal to k²/4, which does not equal 15.
• Statement II is also false because the roots of the equation x² - √2x + 1 = 0 can be calculated using the discriminant. The discriminant is negative, indicating that the roots are not real and equal.
• Thus, both statements are incorrect, leading to the conclusion that the correct answer is (d).

Ans: (c)
Here θ= 30º, arc length = l = 22, length of the pendulum = radius = r

Ans: (d)

• To find P: The odd numbers in the set are 3, 5, 7, 7, and 9. There are 5 odd numbers out of 10 total numbers, so P = 5/10 = 1/2.
• To find Q: The only even prime number is 2. When rolling a die, there are 6 outcomes, so Q = 1/6.
• To find R: P(not E) = 1 - P(E) = 1 - 0.74 = 0.26, which is 26/100.
• Thus, the correct values are P = 3/5, Q = 1/6, R = 13/50, making option (d) the right choice.

Ans: (d)

• The statement in option (a) is correct because the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) relationship holds true for the numbers given.
• In option (b), the smallest prime number is 2 and the smallest composite number is 4; their LCM is indeed 4.
• Option (c) is also correct since 3√2 cannot be expressed as a fraction, making it an irrational number.
• Therefore, option (d) is the correct answer as it states that none of the previous options are incorrect.

Ans: (d)

• Statement (i): To find the distance AB, we use the distance formula. The distance between A(8, 0) and B(–4, 9) is calculated as √[(8 - (–4))² + (0 - 9)²] = √[(12)² + (–9)²] = √[144 + 81] = √225 = 15 units, which is true.
• Statement (ii): The x-axis divides the line segment joining (–3, –5) and (7, 4) in the ratio of 7:3, which is false. The actual ratio is different.
• Statement (iii): To check if the points (7, –14), (–3, 4), and (2, –5) are collinear, we can use the area method or slope method. They do not lie on the same straight line, making this statement false.

Thus, the correct answers are T, F, T, leading to the answer (d).