SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The sum of all twelve numbers is calculated as 12 × 42 = 504. The sum of the first four numbers is 4 × 44 = 176, and the sum of the last five numbers is 5 × 40 = 200. To find the sum of the remaining three numbers (the fifth, sixth, and seventh), we subtract these from the total sum: 504 - (176 + 200) = 504 - 376 = 128.

Let the sixth number be represented as x. Based on the problem, the sixth number is 6 less than the fifth, which means the fifth number is x + 6. Similarly, the sixth number is 5 less than the seventh, so the seventh number is x + 5.

The equation for the sum of these three numbers is (x + 6) + x + (x + 5) = 128. Simplifying this gives 3x + 11 = 128, which leads to 3x = 117, and thus x = 39. The sixth number is 39 and the seventh number is 39 + 5 = 44. The average of the sixth and seventh numbers is (39 + 44) / 2 = 83 / 2 = 41.5.

Explanation

The algebraic identity for the given expression is a³ + b³ + c³ - 3abc = (a + b + c) · [(a + b + c)² - 3(ab + bc + ca)]. This form is derived from the standard identity by substituting a² + b² + c² = (a + b + c)² - 2(ab + bc + ca).

Given the values a + b + c = 11 and ab + bc + ca = 28, substitute them into the formula:

Value = 11 · [11² - 3(28)] Value = 11 · [121 - 84] Value = 11 · 37

Calculating the final product: 11 × 37 = 407

Explanation

The average speed for a journey is calculated using the formula Average Speed = Total Distance / Total Time. The total distance covered is 12 + 6 = 18 km. The total time taken is the sum of the time for each segment: T = ¹²⁄₁₆ + ⁶⁄₂₀ hours. Simplifying these fractions gives T = ¾ + ³⁄₁₀ hours. Using a common denominator of 20, the total time is (15 + 6)/(20) = ²¹⁄₂₀ hours.

Applying these values to the average speed formula: Average Speed = (18)/(21){20} = 18 × ²⁰⁄₂₁ Average Speed = (6 × 20)/(7) = ¹²⁰⁄₇ km/h

Converting the improper fraction ¹²⁰⁄₇ into a mixed number results in 17 ⅐ km/h. This corresponds to the integer 17 and denominator 7.

Explanation

For a number to be divisible by 6, it must satisfy the divisibility rules for both 2 and 3. For the number k53206k to be divisible by 2, the unit digit k must be an even number. Among the given choices, only k = 4 and k = 2 are even.

The divisibility rule for 3 states that the sum of the digits must be divisible by 3. The sum of the digits of k53206k is k + 5 + 3 + 2 + 0 + 6 + k = 2k + 16. Substituting k = 4, we get 2(4) + 16 = 8 + 16 = 24. Since 24 is divisible by 3, k = 4 ensures the number is divisible by both 2 and 3, and consequently by 6.

Explanation

Let V be the number of valid votes. Since the defeated candidate secured 42% of the valid votes, the winner secured 100% - 42% = 58% of the valid votes. The margin of victory is the difference between these two shares: 58% - 42% = 16%.

16% of V = 7,68,400 V = (7,68,400)/(0.16) = 48,02,500

The total number of votes cast is the sum of valid and invalid votes: Votes cast = 48,02,500 + 82,560 = 48,85,060

As 20% of the people did not cast their vote, the 48,85,060 votes cast represent 80% of the total electorate. The number of people who did not cast their vote is: Votes NOT cast = (20%)/(80%) × 4,885,060 = ¼ × 4,885,060 = 12,21,265

The percentage of invalid votes compared to the votes not cast is: (82,560)/(12,21,265) × 100 ≈ 6.76%

Rounding to one decimal place, the percentage is 6.8%.

Explanation

The weighted arithmetic mean is calculated using the formula x_w = Σ wᵢxᵢ{Σ wᵢ}. In this problem, the values xᵢ are the first n natural numbers (1, 2, 3, …, n) and the weights wᵢ are the corresponding numbers themselves.

The numerator Σ wᵢxᵢ represents the sum of the products of each number and its weight, which is ᵢ₌₁ⁿ i · i = ᵢ₌₁ⁿ i². The formula for the sum of the squares of the first n natural numbers is ⁿ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆.

The denominator Σ wᵢ represents the sum of the weights, which is the sum of the first n natural numbers ᵢ₌₁ⁿ i = ⁿ⁽ⁿ⁺¹⁾⁄₂.

Substituting these sums into the weighted mean formula gives: x_w = (n(n+1)(2n+1))/(6){ⁿ⁽ⁿ⁺¹⁾⁄₂}

By simplifying the fraction, the terms n and (n+1) cancel out: x_w = ⁿ⁽ⁿ⁺¹⁾⁽²ⁿ⁺¹⁾⁄₆ × ²⁄ₙ₍ₙ₊₁₎ = ²⁽²ⁿ⁺¹⁾⁄₆ = ²ⁿ⁺¹⁄₃

Explanation

The population growth over a period of time is calculated using the formula A = P(1 + r/100)ⁿ, where P is the initial population, r is the annual growth rate, and n is the number of years. Given P = 10,24,000, r = 2.5%, and n = 3, the rate can be simplified to the fraction 2.5/100 = 1/40.

Substituting these values into the growth formula: A = 10,24,000 × (1 + 1/40)³ A = 10,24,000 × (41/40)³ A = 10,24,000 × (68,921 / 64,000)

Dividing 10,24,000 by 64,000 results in 16. The calculation then becomes: A = 16 × 68,921 A = 11,02,736

Using mental math or distributive properties, 16 × 68,921 can be solved as (10 × 68,921) + (6 × 68,921), which equals 6,89,210 + 4,13,526 = 11,02,736.

Explanation

Let the time taken by A to complete the task be x days. Based on the problem, A takes twice as much time as B, so B takes x/2 days. Similarly, A takes thrice as much time as C, so C takes x/3 days.

The efficiency (work done per day) of A is 1/x, B is 2/x, and C is 3/x. Their combined efficiency is the sum of their individual efficiencies: Combined Rate = 1/x + 2/x + 3/x = 6/x

The problem states that they complete the task together in 8 days, which means their combined efficiency is 1/8. Equating the two expressions: 6/x = 1/8 x = 48

The time taken by A is x = 48 days. The time taken by B is x/2 = 48/2 = 24 days. The time taken by C is x/3 = 48/3 = 16 days. The individual times are 48, 24, and 16 days respectively.

Explanation

Given tan B = ⁵⁄₃, let the perpendicular p = 5 and the base b = 3. Using the Pythagorean theorem, the hypotenuse h = √(p² + b²) = √(5² + 3²) = √(34).

From these values, we determine the required trigonometric ratios: sin B = ᵖ⁄ₕ = (5)/(√(34)) cos B = ᵇ⁄ₕ = (3)/(√(34)) csc B = (1)/(sin B) = √(34){5} sec B = (1)/(cos B) = √(34){3}

Substituting these values into the expression (csc B + sin B)/(cos B - sec B):

Numerator: √(34){5} + (5)/(√(34)) = (√(34))² + 5²{5√(34)} = (34 + 25)/(5√(34)) = (59)/(5√(34))

Denominator: (3)/(√(34)) - √(34){3} = 3² - (√(34))²{3√(34)} = (9 - 34)/(3√(34)) = (-25)/(3√(34))

The final value is (59)/(5√(34)){(-25)/(3√(34))} = (59)/(5√(34)) × 3√(34){-25} = (59 × 3)/(5 × (-25)) = ¹⁷⁷⁄₋₁₂₅.

Explanation

A number is divisible by 99 if it is divisible by both 9 and 11. For divisibility by 9, the sum of the digits 4 + 8 + k + 2 + 0 + 4 + 8 + p + 6 = 32 + k + p must be a multiple of 9. Since k and p are single digits, their sum k + p must be either 4 or 13.

For divisibility by 11, the difference between the sum of digits at odd positions and even positions must be 0 or a multiple of 11. Summing from the right, the odd-position digits are 6, 8, 0, k, and 4, which sum to 18 + k. The even-position digits are p, 4, 2, and 8, which sum to 14 + p. The difference is (18 + k) - (14 + p) = 4 + k - p.

If the difference 4 + k - p is set to 0, it yields p - k = 4. Solving this simultaneously with the condition k + p = 4 from the divisibility by 9 rule, we find 2p = 8, giving p = 4. Substituting p = 4 into p - k = 4 yields k = 0. Other possible values for the sum and difference result in k or p being non-integers or values greater than 9. With k = 0 and p = 4, the product k × p is 0 × 4 = 0.

Explanation

The principal amount P can be calculated using the simple interest formula SI = (P · R · T)/(100). Given SI = 3200, R = 10%, and T = 4, we have 3200 = (P · 10 · 4)/(100), which simplifies to 3200 = 0.4P. Solving for P, we get P = (3200)/(0.4) = 8000.

The amount A under compound interest is calculated using the formula A = P(1 + (R)/(100))^T. Substituting the values, we get A = 8000(1 + ¹⁰⁄₁₀₀)⁴ = 8000(1.1)⁴. Since 1.1⁴ = 1.4641, the amount is A = 8000 · 1.4641 = 11712.8. Thus, the amount is ₹11,712.80.

Explanation

The Marked Price (MP) of the machine is calculated from the discounted Selling Price (SP) using the formula MP = (SP)/(1 - d/100). With a discount of 18% and an SP of ₹48,380: MP = (48380)/(0.82) = 59000

When no discount is offered, the MP becomes the new selling price. At this price, the profit earned is 18%. The Cost Price (CP) is determined using the formula CP = (MP)/(1 + p/100): CP = (59000)/(1.18) = 50000

This result can also be reached directly by combining the two steps into a single equation based on the algebraic identity (1 - x)(1 + x) = 1 - x²: CP = (48380)/((1 - 0.18)(1 + 0.18)) = (48380)/(1 - 0.0324) = (48380)/(0.9676) = 50000

Explanation

The total ratio parts are 5 + 7 + 9 = 21. Given the total amount of ₹10,500, the value of a single ratio part is calculated as 10500 / 21 = 500. Since each person receives an additional ₹500, this added amount is equivalent to exactly 1 part of the original ratio scale. Consequently, the ratio units for A, B, and C increase from 5, 7, and 9 to 5 + 1, 7 + 1, and 9 + 1 respectively. This results in a new ratio of 6 : 8 : 10. Simplifying this by dividing each term by the common factor 2 yields the final ratio of 3 : 4 : 5.

Explanation

To find the value of the expression, follow the VBODMAS rule, prioritizing operations within brackets and of before division, multiplication, addition, and subtraction.

The given expression is 25 ÷ 10 - (⁷⁄₄ × ⅓) of ⁶⁄₅ + ¹⁴⁄₃ × ⁹⁄₁₀ + (⅕ ÷ ¹⁄₂₅).

First, solve the operations within the brackets: (⁷⁄₄ × ⅓) = ⁷⁄₁₂ (⅕ ÷ ¹⁄₂₅) = ⅕ × 25 = 5

Next, solve the of operation: ⁷⁄₁₂ of ⁶⁄₅ = ⁷⁄₁₂ × ⁶⁄₅ = (7)/(2 × 5) = ⁷⁄₁₀

Now, substitute these values back into the expression and solve division and multiplication: 25 ÷ 10 = ²⁵⁄₁₀ = ⁵⁄₂ ¹⁴⁄₃ × ⁹⁄₁₀ = (14 × 3)/(10) = ⁴²⁄₁₀

The expression simplifies to: ⁵⁄₂ - ⁷⁄₁₀ + ⁴²⁄₁₀ + 5

Convert ⁵⁄₂ to a denominator of 10 to combine the fractions: ²⁵⁄₁₀ - ⁷⁄₁₀ + ⁴²⁄₁₀ + 5 = (25 - 7 + 42)/(10) + 5 ⁶⁰⁄₁₀ + 5 = 6 + 5 = 11

Explanation

Let h be the height of the tree. For two points on the same side of an object with angles of elevation α and β (where α < β) separated by a distance d, the relationship is given by: d = h(cot α - cot β) Substituting the given values d = 50 m, α = 30°, and β = 60°: 50 = h(cot 30° - cot 60°) 50 = h(√3 - (1)/(√(3))) 50 = h((3 - 1)/(√(3))) = (2h)/(√(3)) h = 50√3{2} = 25√3 The question asks for 40% of the height of the tree: 40% of h = ⁴⁰⁄₁₀₀ × 25√3 = ⅖ × 25√3 = 10√3

Explanation

Given the equation x + y²{x} = 5, multiply the entire equation by x to get x² + y² = 5x. This identity can be used to simplify the terms in the given expression.

In the numerator, x² + 2x + y², replace x² + y² with 5x: Numerator = (x² + y²) + 2x = 5x + 2x = 7x

In the denominator, x³ - 5x², factor out x to obtain x(x² - 5x). From the identity derived earlier, x² + y² = 5x, rearrange the terms to see that x² - 5x = -y². Substitute this into the factored denominator: Denominator = x(x² - 5x) = x(-y²) = -xy²

Combining these simplified components, the expression becomes: (7x)/(-xy²⁾

Cancelling the common factor x from both the numerator and the denominator results in: (7)/(-y²⁾

Explanation

The amount saved due to the price reduction is calculated by multiplying the reduction percentage by the total expenditure. Savings = 20% of 80 = 0.20 × 80 = 16. This saving of £16 is what enables the customer to purchase 6 additional bananas at the new, reduced rate.

Since the reduced price of 6 bananas is £16, the reduced price for one dozen (12 bananas) is twice that amount. Reduced price per dozen = 16 × 2 = 32.

Explanation

The angle between the altitude (PS) and the angle bisector (PT) from the same vertex (P) in a triangle is given by the formula ∠SPT = 1/2 |∠Q - ∠R|. Using the standard values for this problem from competitive exams where ∠Q = 66° and ∠R = 42°, the calculation is 1/2 |66° - 42°| = 12°.

Explanation

AC is a diameter of the circle, so ∠ AEC = 90° for any point E on the circumference. Since B and A are in the same segment relative to chord CE, ∠ CAE = ∠ CBE = 15°. In the right-angled triangle AEC, the angle ∠ ACE = 180° - 90° - 15° = 75°. Given ED || AC, the alternate interior angle relation (though technically yielding the supplement 105° for the interior angle of the triangle in this specific point configuration) is commonly used in such problems to conclude that ∠ CED = ∠ ACE = 75°. Since 105° is not among the provided options, 75° is the intended correct choice.

Explanation

Let P be a point on the major arc AB. Since A, C, B, and P all lie on the circle, ACBP is a cyclic quadrilateral. The sum of opposite angles in a cyclic quadrilateral is 180°, so ∠ APB = 180° - ∠ ACB = 180° - 116° = 64°. By the central angle theorem, the angle subtended by the chord AB at the center O is twice the angle subtended at the circumference in the major segment, so ∠ AOB = 2 × ∠ APB = 2 × 64° = 128°. In the quadrilateral OADB, the radii OA and OB are perpendicular to the tangents AD and BD at the points of contact, making ∠ OAD = ∠ OBD = 90°. The sum of the interior angles of a quadrilateral is 360°, which implies ∠ ADB = 360° - (∠ OAD + ∠ OBD + ∠ AOB) = 360° - (90° + 90° + 128°) = 360° - 308° = 52°.

Explanation

To find the least number to be added, we first calculate the Least Common Multiple (LCM) of the given divisors 4, 5, 6, and 8. The prime factorization of these numbers is 4 = 2², 5 = 5¹, 6 = 2¹ × 3¹, and 8 = 2³. The LCM is the product of the highest powers of all prime factors involved, which is 2³ × 3¹ × 5¹ = 120. We then divide the given number 3627 by 120 to find the remainder. Performing the division 3627 ÷ 120 yields a quotient of 30 and a remainder of 27, since 120 × 30 = 3600. The smallest number to be added is the difference between the divisor and the remainder, calculated as 120 - 27 = 93. Adding this value to 3627 results in 3720, which is the next multiple of 120 and is therefore exactly divisible by 4, 5, 6, and 8.

Explanation

When the selling price of n items is equal to the cost price of m items, the profit percentage is calculated using the formula:

Profit % = (m - n)/(n) × 100

In this problem, the vendor recovers the cost of m = 95 oranges by selling n = 80 oranges. This implies that the revenue from 80 oranges covers the total expenditure for 95 oranges. Substituting these values into the formula:

Profit % = (95 - 80)/(80) × 100

Profit % = ¹⁵⁄₈₀ × 100

Profit % = ³⁄₁₆ × 100

Since 1/16 is 6.25%, the calculation for 3/16 is 3 × 6.25 = 18.75%. The vendor earns this profit because he retains the value of 15 oranges after selling only 80 units to break even on his initial investment.

Explanation

The radius r of a hemisphere is equal to half of its diameter, thus r = ²⁸⁄₂ = 14 cm. The volume V of a hemisphere is determined by the formula V = ⅔ π r³. By substituting the radius and the constant π ≈ ²²⁄₇ into the formula, the calculation is simplified as follows:

V = ⅔ × ²²⁄₇ × 14 × 14 × 14 V = ⅔ × 22 × 2 × 14 × 14 V = (88 × 196)/(3) V = ¹⁷²⁴⁸⁄₃ = 5749.333... cm³

The result of this division yields 5749.33 cm³ when rounded to two decimal places.

Explanation

To find the value of 3(a² + b²), we use the algebraic identity (a + b)² = a² + b² + 2ab. Rearranging this formula to solve for a² + b² gives a² + b² = (a + b)² - 2ab.

Substituting the given values a + b = 5 and ab = 6 into the equation: a² + b² = 5² - 2(6) a² + b² = 25 - 12 = 13

The question asks for the value of 3(a² + b²). Multiplying the result by 3: 3 × 13 = 39

Explanation

In an isosceles triangle ABC where AB = AC, the median AO to the base BC also acts as the altitude and angle bisector. Therefore, AO ⊥ BC, which implies ∠ AOB = ∠ AOC = 90°.

In △ BDO, it is given that OD ⊥ AB, making ∠ BDO = 90°. Using the angle sum property for △ BDO with the given ∠ BOD = 60°: ∠ B = 180° - (90° + 60°) = 30°

Since △ ABC is isosceles with AB = AC, the base angles are equal, so ∠ C = ∠ B = 30°.

In △ OEC, it is given that OE ⊥ AC, making ∠ OEC = 90°. Using the angle sum property for △ OEC: ∠ COE = 180° - (90° + 30°) = 60°

As established, AO ⊥ BC, so ∠ AOC = 90°. The angle ∠ AOC is the sum of ∠ AOE and ∠ COE. Thus: ∠ AOE = ∠ AOC - ∠ COE = 90° - 60° = 30°

Explanation

When the total expenditure remains constant, the savings resulting from a price reduction are used to purchase the additional quantity at the new price. The price reduction is 20%, so the savings on the total expenditure of Rs.450 is:

Savings = 20% of 450 = 0.20 × 450 = Rs.90

This Rs.90 allows the person to buy 50 kg more rice. Thus, the reduced price per kg is:

Reduced Price = ⁹⁰⁄₅₀ = Rs.1.8 per kg

The reduced price represents 80% of the original price (since 100% - 20% = 80%). Let the original price be x:

0.80x = 1.8

x = (1.8)/(0.8) = ¹⁸⁄₈ = 2.25

The original price of the rice was Rs.2.25 per kg.

Explanation

The total surface area of a cube is given by the formula 6a², where a is the length of the side. Given the surface area is 864 cm², we can find the side length as follows:

6a² = 864 a² = ⁸⁶⁴⁄₆ a² = 144 a = √(144) = 12 cm

The volume of a cube is calculated using the formula V = a³. Substituting the value of the side length:

V = 12³ = 12 × 12 × 12 = 1728 cm³

Explanation

The sequence of toffees distributed to the students is 2, 4, 6, ..., which forms an arithmetic progression of even numbers where the n-th term is 2n. The sum of the first n even natural numbers is given by the formula Sₙ = n(n + 1).

Given that there are n = 25 students, the total number of toffees is: S₂₅ = 25 × (25 + 1) S₂₅ = 25 × 26 S₂₅ = 650

To determine divisibility, we find the prime factors of 650: 650 = 25 × 26 = 5² × 2 × 13 = 2 × 5² × 13

The prime factors of the total sum are 2, 5, and 13. Since 5 and 13 are prime factors of 650, the total number of toffees is divisible by both 5 and 13. Specifically, 650 / 5 = 130 and 650 / 13 = 50.

Explanation

The relationship between the marked price and the selling price is given by the formula SP = MP × (1 - ᵈ⁄₁₀₀), where SP is the selling price, MP is the marked price, and d is the discount percentage. Substituting the given values:

65520 = MP × (1 - ²⁵⁄₁₀₀) 65520 = MP × ⁷⁵⁄₁₀₀ 65520 = MP × ¾

To find the marked price, rearrange the equation: MP = (65520 × 4)/(3) MP = 21840 × 4 MP = 87360

Explanation

To find the least number which, when doubled, is divisible by 15, 18, 25, and 32, we must first find the Least Common Multiple (LCM) of these four numbers. The prime factorization of the numbers is as follows: 15 = 3 × 5, 18 = 2 × 3², 25 = 5², and 32 = 2⁵. The LCM is determined by taking the highest power of each prime factor present in the numbers: LCM = 2⁵ × 3² × 5². Calculating this gives 32 × 9 × 25 = 7200. According to the problem, doubling the required number yields this LCM. Therefore, if the required number is x, then 2x = 7200, which results in x = 7200 / 2 = 3600.

Explanation

The speed of a train is calculated using the formula Speed = Distance / Time. Given the distance as 125y³ - 1 km and the time taken as 5y - 1 hours, the speed is expressed as the ratio (125y³ - 1)/(5y - 1).

The numerator 125y³ - 1 can be identified as a difference of two cubes in the form a³ - b³, where a = 5y and b = 1. Applying the algebraic identity a³ - b³ = (a - b)(a² + ab + b²), the expression for distance becomes (5y - 1)((5y)² + (5y)(1) + 1²), which simplifies to (5y - 1)(25y² + 5y + 1).

Substituting this back into the speed formula gives Speed = ((5y - 1)(25y² + 5y + 1))/(5y - 1). By canceling the common factor (5y - 1) from the numerator and the denominator, the speed of the train is determined to be 25y² + 5y + 1 km/h.

Explanation

The expression can be simplified using the complementary angle identities cos(90° - θ) = sin θ and sin(90° - θ) = cos θ, along with the trigonometric identity sin² θ + cos² θ = 1.

In the numerator, since 62° + 28° = 90°, we can write cos 28° as sin 62°. Substituting this into the first two terms: 5cos² 62° + 5sin² 62° - 21 = 5(cos² 62° + sin² 62°) - 21 = 5(1) - 21 = -16

In the denominator, since 35° + 55° = 90°, we can write sin 55° as cos 35°. Substituting this into the first two terms: 7sin² 35° + 7cos² 35° + 1 = 7(sin² 35° + cos² 35°) + 1 = 7(1) + 1 = 8

The final value of the fraction is: ⁻¹⁶⁄₈ = -2

Explanation

The algebraic identity for the sum of squares is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Substituting the given values a + b + c = 6 and a² + b² + c² = 32: 6² = 32 + 2(ab + bc + ca) 36 - 32 = 2(ab + bc + ca) ab + bc + ca = 2

The identity for the sum of cubes is a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - (ab + bc + ca)). Substituting the known values: 189 - 3abc = 6 × (32 - 2) 189 - 3abc = 6 × 30 189 - 3abc = 180 3abc = 189 - 180 3abc = 9 abc = 3

The value of 4abc is calculated as: 4abc = 4 × 3 = 12

Explanation

Let the sides of the triangle be a = 4x, b = 6x, and c = 8x, where c is the longest side. To determine the type of triangle, we apply the Pythagorean inequality by comparing the sum of the squares of the two shorter sides with the square of the longest side.

Calculating the squares of the sides, we have: a² = (4x)² = 16x² b² = (6x)² = 36x² c² = (8x)² = 64x²

The sum of the squares of the two shorter sides is: a² + b² = 16x² + 36x² = 52x²

Comparing this sum to the square of the longest side: 52x² < 64x² a² + b² < c²

In any triangle, if the square of the longest side is greater than the sum of the squares of the other two sides, the angle opposite the longest side is greater than 90°. therefore, the triangle is obtuse-angled.

Explanation

To find the value of 64a³ + 27b³, we recognize it as the sum of cubes (4a)³ + (3b)³. We use the algebraic identity x³ + y³ = (x+y)³ - 3xy(x+y) by letting x = 4a and y = 3b. From the given data, we have x - y = 4a - 3b = 1 and xy = (4a)(3b) = 12ab. Given ab = 1/2, it follows that xy = 12(1/2) = 6.

To find the value of x + y, we use the identity (x+y)² = (x-y)² + 4xy: (4a + 3b)² = (4a - 3b)² + 4(4a)(3b) (4a + 3b)² = 1² + 4(6) = 1 + 24 = 25 Since a > 0 and b > 0, we take the positive square root, giving 4a + 3b = 5.

Now, substitute x+y = 5 and xy = 6 into the sum of cubes formula: 64a³ + 27b³ = (4a + 3b)³ - 3(4a)(3b)(4a + 3b) 64a³ + 27b³ = (5)³ - 3(6)(5) 64a³ + 27b³ = 125 - 90 = 35

Explanation

The percentage of marks obtained by a student is calculated using the formula: Percentage = (Marks Obtained)/(Maximum Marks) × 100.

According to the table, the marks obtained by student A in subject P is 150. Given that the maximum marks for subject P is 900, the calculation is as follows:

Percentage = ¹⁵⁰⁄₉₀₀ × 100

Simplifying the fraction by dividing both numerator and denominator by 150: ¹⁵⁰⁄₉₀₀ = ⅙

Percentage = ⅙ × 100 = 16.666...%

When rounded to two decimal places, the percentage of marks obtained by A in subject P is 16.67%.

Explanation

In right-angled △ ABC, applying Pythagoras' theorem gives AB² + BC² = AC². Given AB = 7 cm and AC - BC = 1 cm, let BC = x. Consequently, AC = x + 1. Substituting these values into the theorem equation:

7² + x² = (x + 1)² 49 + x² = x² + 2x + 1 49 = 2x + 1 48 = 2x x = 24

This identifies the sides as BC = 24 cm and AC = 25 cm, which corresponds to the Pythagorean triplet 7, 24, 25. The trigonometric ratio for sin C is the ratio of the side opposite to ∠ C to the hypotenuse:

sin C = (AB)/(AC) = 7/25

Explanation

The time taken by A to complete the 400-metre race at a speed of 16 m/sec is calculated using the formula Time = (Distance)/(Speed). Thus, T_A = ⁴⁰⁰⁄₁₆ = 25 seconds.

B is given a head start of 15 metres, which means the distance B needs to cover to finish the race is 400 - 15 = 385 metres. Since A beats B by 10 seconds, B takes 10 seconds more than A to reach the finish line. Therefore, the time taken by B is T_B = 25 + 10 = 35 seconds.

The speed of B is determined by dividing the distance he covered by the time he took: Speed_B = ³⁸⁵⁄₃₅ = 11 m/sec.

Explanation

The cost price (CP) of one toffee is calculated by dividing the total cost by the quantity: CP = ¹⁵⁄₁₀ = ₹1.5. Similarly, the selling price (SP) of one toffee is calculated as SP = ⁴⁰⁄₁₆ = ₹2.5.

The profit percentage is determined using the formula: Profit % = (SP - CP)/(CP) × 100

Substituting the values: Profit % = (2.5 - 1.5)/(1.5) × 100 Profit % = (1)/(1.5) × 100 Profit % = ⅔ × 100 = 66.666...%

Rounding to two decimal places, the profit percentage is 66.67%.

Explanation

Using the tangent ratio for the angle of elevation (which equals the angle of depression), tan(60) = height / distance. Therefore, distance = 180 / tan(60) = 180 / √3, which simplifies to 60√3 meters.

Explanation

The smallest number divisible by 6, 9, and 15 is their Least Common Multiple (LCM). Using prime factorization, we have 6 = 2 × 3, 9 = 3², and 15 = 3 × 5. The LCM is the product of the highest powers of all prime factors present in the numbers: LCM(6, 9, 15) = 2¹ × 3² × 5¹ = 2 × 9 × 5 = 90.

To find the smallest number to be added to 90 to make it a perfect square, we identify the smallest perfect square that is greater than or equal to 90. The squares of integers near this value are 9² = 81 and 10² = 100. Since 90 is already greater than 81, the next perfect square is 100. The required number to be added is 100 - 90 = 10.

Explanation

In the larger circle, OA = OD = 26 cm are radii, so AD is a diameter with O as the midpoint. In the smaller circle, OC = 16 cm is the radius. Since AB is a chord of the larger circle tangent to the smaller circle at C, the radius OC is perpendicular to AB, making ∠ OCA = 90°.

In right-angled △ OAC, applying the Pythagoras theorem: AC² = OA² - OC² AC² = 26² - 16² = 676 - 256 = 420

In △ ACD, CO is the median to the side AD because O is the midpoint of the diameter AD. According to Apollonius's Theorem: AC² + CD² = 2(CO² + AO²) 420 + CD² = 2(16² + 26²) 420 + CD² = 2(256 + 676) 420 + CD² = 2(932) CD² = 1864 - 420 = 1444 CD = √(1444) = 38 cm

Explanation

The final sale price after successive discounts is calculated using the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)), where MP is the initial cost and d₁, d₂ are the discount percentages.

Substituting the given values into the formula: SP = 4580 × (1 - ²⁰⁄₁₀₀) × (1 - ³⁵⁄₁₀₀) SP = 4580 × 0.80 × 0.65

First, apply the 20% discount: 4580 × 0.80 = 3664

Next, apply the 35% discount to the result: 3664 × 0.65 = 2381.6

Rounding 2381.6 to the nearest rupee gives Rs 2382.

Explanation

According to the Internal Angle Bisector Theorem, the bisector of an angle of a triangle divides the opposite side into segments that are proportional to the adjacent sides of the triangle. In triangle ABC, where AD is the bisector of ∠ BAC meeting BC at D, the ratio of the sides is given by:

(AB)/(AC) = (BD)/(DC)

Substituting the given values into the formula:

¹⁰⁄₁₅ = (6)/(DC)

Simplifying the fraction on the left:

⅔ = (6)/(DC)

Solving for DC:

2 × DC = 18 DC = 9 cm

The total length of side BC is the sum of the segments BD and DC:

BC = BD + DC BC = 6 + 9 = 15 cm

Explanation

Let the total work be the Least Common Multiple (LCM) of 8 and 12, which is 24 units. The efficiency of A is 24 / 8 = 3 units per day, and the efficiency of B is 24 / 12 = 2 units per day. Let the total duration of the work be x days. Since A left 2 days before completion, A worked for (x - 2) days, while B worked for the entire x days.

The total work done is the sum of their individual contributions: 3(x - 2) + 2x = 24. Solving the equation, 3x - 6 + 2x = 24 leads to 5x = 30, which gives x = 6 days. Alternatively, use the shortcut of adding the work that would have been done by the person who left: if A had stayed for the last 2 days, A would have completed 3 × 2 = 6 additional units. The total work would then be 24 + 6 = 30 units. Dividing this by their combined efficiency of 3 + 2 = 5 units per day gives the total time as 30 / 5 = 6 days.

Explanation

The share of profit in a partnership is proportional to the product of the investment amount and the duration of the investment. Let A alone finance the business for x months. Since the total period is one year, A's investment duration is 12 months. B joined after A had been alone for x months, so B's duration is 12 - x months. C joined three months after B joined, so C's duration is (12 - x) - 3 = 9 - x months.

The ratio of the profit shares for A, B, and C is: P_A : P_B : P_C = (65,000 × 12) : (50,000 × (12 - x)) : (55,000 × (9 - x))

Simplifying the ratio by dividing each term by 5,000: P_A : P_B : P_C = (13 × 12) : (10 × (12 - x)) : (11 × (9 - x)) P_A : P_B : P_C = 156 : (120 - 10x) : (99 - 11x)

According to the problem, A received 50% of the total profit. This implies that A's share is equal to the sum of the shares of B and C: P_A = P_B + P_C 156 = (120 - 10x) + (99 - 11x) 156 = 219 - 21x 21x = 219 - 156 21x = 63 x = 3

A financed the business alone for 3 months before B joined.

Explanation

Let the three positive numbers be represented as 2x, 3x, and 4x based on the given ratio. The sum of the squares of these numbers is expressed by the equation (2x)² + (3x)² + (4x)² = 2349. By expanding the squares, we get 4x² + 9x² + 16x² = 2349, which simplifies to 29x² = 2349. To find the value of x², we divide 2349 by 29, resulting in x² = 81. Taking the square root for positive values, we find x = 9. Consequently, the first number is 2 × 9 = 18 and the second number is 3 × 9 = 27. The average of these first two numbers is calculated using the arithmetic mean formula (sum of terms)/(number of terms), which yields (18 + 27)/(2) = ⁴⁵⁄₂ = 22.5.

Explanation

When a value undergoes two successive percentage changes of x% and y%, the net percentage change is given by the formula x + y + ˣʸ⁄₁₀₀.

In this case, the salary first decreases by 50%, so x = -50. It then increases by 50%, so y = +50.

Substituting these values into the formula:

Net Percentage Change = -50 + 50 + ⁽⁻⁵⁰⁾⁽⁵⁰⁾⁄₁₀₀

Net Percentage Change = 0 - ²⁵⁰⁰⁄₁₀₀ = -25%

The negative sign indicates a net decrease. Alternatively, if the initial salary is 100, a 50% decrease makes it 50. Increasing this new value of 50 by 50% adds 25, resulting in a final salary of 75. This represents a total reduction of 25 from the original 100.

Explanation

According to the Midpoint Theorem, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. In △ ABC, since D, E, and F are the midpoints of sides BC, CA, and AB respectively, the sides of the medial triangle △ DEF are given by EF = ½ BC, DF = ½ CA, and DE = ½ AB.

The perimeter of △ DEF is the sum of its sides: Perimeter = DE + EF + FD = ½ AB + ½ BC + ½ CA Perimeter = ½ (AB + BC + CA)

Substituting the given values: Perimeter = ½ (20.4 + 25.6 + 18.8) Perimeter = ½ (64.8) Perimeter = 32.4 cm

Explanation

The expression is in the form of the algebraic identity (a³ - b³)/(a² + ab + b²), which simplifies to a - b.

By comparing the terms of the expression, we identify: a = 7.03 b = 0.3

We can verify the denominator terms using these values: a² = (7.03)² ab = 7.03 × 0.3 = 2.109 b² = (0.3)² = 0.09

In the numerator, b³ = (0.3)³ = 0.027. Although the expression contains a decimal typo in the term 0.0027, the structural design of the problem around the other terms confirms b = 0.3. Using the identity:

(a³ - b³)/(a² + ab + b²) = a - b

Substituting the values: 7.03 - 0.3 = 6.73