SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The volume of a conical water tank is determined using the standard formula V = ⅓π r²h, where V represents volume, r is the radius of the base, and h is the height. By substituting the given values of V = 46,200 m³, r = 70 m, and π = ²²⁄₇ into the formula, we set up the following equation:

46200 = ⅓ × ²²⁄₇ × 70 × 70 × h

To simplify the right side of the equation, the 7 in the denominator cancels one factor of 70 to become 10:

46200 = ⅓ × 22 × 10 × 70 × h

46200 = ¹⁵⁴⁰⁰⁄₃ × h

Next, isolate h by multiplying both sides by 3 and dividing by 15,400:

h = (46200 × 3)/(15400)

Since 46,200 divided by 15,400 equals exactly 3, the calculation reduces to:

h = 3 × 3

h = 9 m

Explanation

To determine the combined ratio A:B:C from the given ratios A:B and B:C, the value of the common term B must be synchronized. This can be achieved using the following formula for combining ratios a:b and c:d:

A:B:C = (a × c) : (b × c) : (b × d)

Given: A:B = 6:8 B:C = 5:10

Substituting these values (a=6, b=8, c=5, d=10) into the formula: A:B:C = (6 × 5) : (8 × 5) : (8 × 10) A:B:C = 30 : 40 : 80

To simplify the result, divide each term by the greatest common divisor, which is 10: A:B:C = 3 : 4 : 8

Alternatively, simplifying the individual ratios first gives A:B = 3:4 and B:C = 1:2. Multiplying the terms of the second ratio by 4 to match the value of B in the first ratio results in B:C = 4:8. Combining these yields the same continuous ratio of 3:4:8.

Explanation

The expression is simplified using the BODMAS rule, which dictates the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, evaluate the terms inside the square bracket. In the sub-expression (70 - 60 ÷ 2 of 6), the of operation is performed first, yielding 2 × 6 = 12. The division follows as 60 ÷ 12 = 5, and subtraction gives 70 - 5 = 65. In the sub-expression (20 - 10) ÷ 10, the bracket is solved first to get 10, followed by division 10 ÷ 10 = 1. Summing these results gives 65 + 1 = 66.

Second, simplify the remaining part of the expression. In 600 ÷ 25 of (2 × 3), the bracket is solved first as 6, then the of operation yields 25 × 6 = 150. Performing the division gives 600 ÷ 150 = 4. This result is multiplied by the next term (2 of 3), which is 6, resulting in 4 × 6 = 24. Finally, calculate 2 of 5 = 10.

Combining the simplified components results in 66 + 24 + 10 = 100.

Explanation

Let the speed of the boat in still water be x km/h and the speed of the stream be y km/h. The upstream speed is U = x - y and the downstream speed is D = x + y.

From the first condition, the boat travels 78 km upstream and 78 km downstream in 32 hours: (78)/(U) + (78)/(D) = 32 Dividing by 2: (39)/(U) + (39)/(D) = 16 — (Equation 1)

From the second condition, the boat travels 15 km upstream and 52 km downstream in 9 hours: (15)/(U) + (52)/(D) = 9 — (Equation 2)

To solve these equations, multiply Equation 1 by 4 and Equation 2 by 3 to align the downstream coefficients: 4 × ((39)/(U) + (39)/(D)) = 64 ⟹ (156)/(U) + (156)/(D) = 64 3 × ((15)/(U) + (52)/(D)) = 27 ⟹ (45)/(U) + (156)/(D) = 27

Subtracting the two results: (111)/(U) = 37 ⟹ U = ¹¹¹⁄₃₇ = 3 km/h

Substitute U = 3 into Equation 2: ¹⁵⁄₃ + (52)/(D) = 9 ⟹ 5 + (52)/(D) = 9 ⟹ (52)/(D) = 4 ⟹ D = 13 km/h

The speed of the boat in still water x is the average of the downstream and upstream speeds: x = (D + U)/(2) = (13 + 3)/(2) = 8 km/h

The distance covered by the boat in still water in 12 hours is: Distance = x × Time = 8 × 12 = 96 km

Explanation

In triangle ACB, since circle C touches circles A and B externally, the distances between the centers are equal to the sum of their radii. Therefore, AC = 2 + r and BC = 1 + r. The problem states that ∠ ACB = 90° and the distance between centers A and B is AB = 5 cm. Applying the Pythagorean theorem to the right-angled triangle ACB:

AC² + BC² = AB²

Substituting the expressions for the sides:

(2 + r)² + (1 + r)² = 5²

Expanding the terms:

4 + 4r + r² + 1 + 2r + r² = 25

Combining like terms and simplifying the equation:

2r² + 6r + 5 = 25 2r² + 6r - 20 = 0 r² + 3r - 10 = 0

Factoring the quadratic equation gives (r + 5)(r - 2) = 0. Solving for r yields r = -5 or r = 2. Since a radius must be a positive length, the value of r is 2 cm.

Explanation

Let a = x - y, b = y - z, and c = z - x. Summing these three variables gives a + b + c = (x - y) + (y - z) + (z - x) = 0. According to the algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca), if the sum of three variables is zero, the sum of their cubes is equal to three times their product (a³ + b³ + c³ = 3abc).

Applying this identity to the numerator of the given expression, (x - y)³ + (y - z)³ + (z - x)³ simplifies to 3(x - y)(y - z)(z - x). Substituting this simplified form into the original expression yields (3(x - y)(y - z)(z - x))/((x - y)(y - z)(z - x)). Upon canceling the common terms (x - y)(y - z)(z - x) from the numerator and the denominator, the expression evaluates to 3.

Explanation

The single equivalent discount for two successive discounts of x% and y% is calculated using the formula x + y - ˣʸ⁄₁₀₀. By substituting the given values x = 10 and y = 15 into this expression, the calculation becomes 10 + 15 - (10 × 15)/(100). This simplifies to 25 - 1.5, resulting in a single equivalent discount of 23.5%. This method effectively combines the two percentage reductions into one cumulative rate relative to the original price.

Explanation

Let the efficiency of the slower pipe be x. Since the faster pipe fills the tank two and a half times as quickly, its efficiency is 2.5x. The combined efficiency of both pipes working together is x + 2.5x = 3.5x.

The total capacity of the tank is the product of the combined efficiency and the time taken to fill it together. Total capacity = 3.5x × 20 = 70x.

The time taken by the faster pipe alone to fill the tank is calculated by dividing the total capacity by its individual efficiency. Time = 70x / 2.5x = 70 / 2.5 = 28 minutes.

This can also be expressed using ratios. The ratio of efficiencies of the faster pipe to the slower pipe is 2.5:1, which simplifies to 5:2. The combined efficiency is 5 + 2 = 7 units per minute. Total work is 7 × 20 = 140 units. The time taken by the faster pipe is 140 / 5 = 28 minutes.

Explanation

In congruent triangles Δ PQR and Δ ABC, the corresponding parts are equal (CPCT). Given the correspondence PQR ↔ ABC, the side PQ corresponds to side AB, and the angle ∠ R corresponds to angle ∠ C.

First, calculate the measure of ∠ C in Δ ABC using the angle sum property of a triangle: ∠ A + ∠ B + ∠ C = 180° 70° + 50° + ∠ C = 180° 120° + ∠ C = 180° ∠ C = 60°

From the congruence relation Δ PQR ≅ Δ ABC: PQ = AB = 6 cm ∠ R = ∠ C = 60°

Therefore, the condition QP = 6 cm and ∠ R = 60° is true.

Explanation

Let the number of cups of tea be t and the number of cups of coffee be c. Since Rama bought drinks for 7 friends, the total number of cups is t + c = 7. The total cost of these drinks is given by the equation 30t + 40c = 240, where 30 and 40 are the prices of tea and coffee respectively. From the first equation, we can express the number of coffee cups as c = 7 - t. Substituting this into the cost equation yields 30t + 40(7 - t) = 240. Expanding the terms gives 30t + 280 - 40t = 240, which simplifies to -10t = 240 - 280, or -10t = -40. Dividing both sides by -10, we find t = 4. Thus, she purchased 4 cups of tea.

Explanation

Let R be the radius of the larger circle and r be the radius of the smaller circle. The area of the annular portion between these two concentric circles is given by the formula A = π R² - π r² = π(R² - r²).

A chord of the larger circle that is tangent to the smaller circle is bisected at the point of tangency. Since the chord length is 20 cm, each half of the chord measures 10 cm. A right-angled triangle is formed by the radius of the smaller circle r, the half-chord length of 10 cm, and the radius of the larger circle R as the hypotenuse. According to the Pythagorean theorem, the relationship is R² = r² + 10², which simplifies to R² - r² = 100. Substituting this value into the area formula, we get A = π(100) = 100π cm².

Explanation

The expression (1)/(a²) + (1)/(b²) + (2)/(ab) is in the form of the standard algebraic identity x² + y² + 2xy = (x + y)², where x = ¹⁄ₐ and y = (1)/(b).

By substituting these values, the expression simplifies to: (¹⁄ₐ + (1)/(b))²

To combine the terms within the parentheses, we find a common denominator: ¹⁄ₐ + (1)/(b) = (b + a)/(ab)

The problem states that a + b = 1. Substituting this value into the numerator of the fraction gives: (1)/(ab)

Finally, squaring this value to complete the calculation for the original expression: ((1)/(ab))² = (1)/(a²b²)

Explanation

The volume V of a cylinder is given by the formula V = π r² h, where r is the radius and h is the height. The problem states the radius r is 4 units. The diameter is twice the radius, so d = 2 × 4 = 8 units. Since the height is equal to the diameter, h = 8 units. Substituting these values into the formula gives V = π × 4² × 8. Calculating the squares and products results in V = π × 16 × 8 = 128π cubic units.

Explanation

The debt of ₹9,810 due in 1 year represents the total amount (A) consisting of the principal (Present Worth) and the simple interest for one year. We first calculate the Present Worth (PW) of the debt at the start of the year using the standard formula:

PW = (A)/(1 + RT)

Substituting the given values A = 9810, R = 9% = 0.09, and T = 1 year:

PW = (9810)/(1 + (0.09 × 1)) = (9810)/(1.09) = 9000

The trader chooses to settle the account after 6 months (0.5 years). The cash payment required at that time is the value of the Present Worth plus the interest it would accrue over those 6 months:

Payment = PW × (1 + R × Tₛₑₜₜₗₑ)

Payment = 9000 × (1 + 0.09 × 0.5)

Payment = 9000 × (1 + 0.045) = 9000 × 1.045 = 9405

The trader should pay ₹9,405 to settle the account.

Explanation

The given expression (tan(x))/(sec(x)) simplifies to sin(x) because (sin(x)/cos(x))/(1/cos(x)) = sin(x). Substituting the given value, we find sin(x) = ½.

The expression to be evaluated is (sin(x) + cos(x))². Expanding this using the algebraic identity (a+b)² = a² + b² + 2ab and the trigonometric identity sin²(x) + cos²(x) = 1, we get: (sin(x) + cos(x))² = sin²(x) + cos²(x) + 2sin(x)cos(x) = 1 + sin(2x)

For the principal value where sin(x) = ½, the angle x is 30°. Substituting this value into the simplified formula for the expression: 1 + sin(2 · 30°) = 1 + sin(60°) = 1 + √3{2} = 2 + √3{2}

Explanation

In a circle, a perpendicular drawn from the center to a chord bisects the chord. This creates a right-angled triangle where the radius r is the hypotenuse, the perpendicular distance from the center d is one leg, and half the length of the chord c/2 is the other leg.

According to the Pythagoras theorem: r² = (c/2)² + d²

Given the chord length c = 50 cm and the perpendicular distance d = 21 cm: Half-chord length = 50 / 2 = 25 cm

Substituting the values into the formula: r² = 25² + 21² r² = 625 + 441 r² = 1066 r = √(1066) cm

Explanation

A tangent to a circle is always perpendicular to the radius at the point of contact. This property creates a right-angled triangle where the radius r and the tangent length L are the legs, and the distance from the center to the external point d is the hypotenuse. According to the Pythagorean theorem:

r² + L² = d²

Substituting the given values r = 8 cm and d = 17 cm into the equation:

8² + L² = 17² 64 + L² = 289 L² = 289 - 64 L² = 225 L = √(225) = 15 cm

The calculation is also simplified by recognizing the Pythagorean triplet (8, 15, 17), which directly identifies the length of the tangent as 15 cm.

Explanation

The mixture is formed by combining three varieties of tea in the ratio 1:1:2. Let the price of the third variety be x per kg. Using the weighted average formula for the price of a mixture:

Pₘᵢₓ = (w₁ P₁ + w₂ P₂ + w₃ P₃)/(w₁ + w₂ + w₃)

Substituting the given values: 350 = (1(250) + 1(300) + 2(x))/(1 + 1 + 2)

Solving the equation for x: 350 = (250 + 300 + 2x)/(4) 350 × 4 = 550 + 2x 1400 = 550 + 2x 2x = 1400 - 550 2x = 850 x = 425

The price of the third variety of tea powder is ₹425/kg.

Explanation

The expression to be evaluated is cos B(sec B - tan B). Using the trigonometric identities sec B = (1)/(cos B) and tan B = (sin B)/(cos B), we can simplify the expression by distributing cos B:

cos B(sec B - tan B) = cos B · (1)/(cos B) - cos B · (sin B)/(cos B)

Upon simplification, the cos B terms cancel out in both parts of the expression:

1 - sin B

We are given that sin B = ¹⁵⁄₁₇. Substituting this value into the simplified expression gives:

1 - ¹⁵⁄₁₇ = (17 - 15)/(17) = ²⁄₁₇

Explanation

The single equivalent discount for two successive discounts of x% and y% is given by the formula D = x + y - ˣʸ⁄₁₀₀.

Substituting the given values x = 10% and y = 16% into the formula: D = 10 + 16 - (10 × 16)/(100) D = 26 - ¹⁶⁰⁄₁₀₀ D = 26 - 1.6 D = 24.4%

Alternatively, if the initial price is 100, the first discount of 10% reduces the price to 90. The second discount of 16% is then applied to 90, which is 90 × 0.16 = 14.4. The total discount is 10 + 14.4 = 24.4, which represents a single discount of 24.4% on the original price of 100.

Explanation

According to the mid-point theorem in geometry, joining the mid-points of the sides of a triangle divides the original triangle into four smaller triangles of equal area. Consequently, the area of △ PQR is one-fourth the area of the equilateral triangle ABC.

The area of an equilateral triangle with side length s is calculated using the formula: Area = √3{4}s²

Substituting the given side length s = 11 cm for △ ABC: Area(△ ABC) = √3{4} × 11² = 121√3{4} cm²

Since △ PQR is formed by joining the mid-points P, Q, and R, its area is: Area(△ PQR) = ¼ × Area(△ ABC) = ¼ × 121√3{4} = 121√3{16} cm²

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. In this case, the relative speed of P and Q is 10 m/s + 8 m/s = 18 m/s.

The time taken for them to meet for the first time on a circular track is given by the formula Time = (Distance)/(Relative Speed). The distance to be covered relatively to meet is the total length of the circular track.

Substituting the given values into the formula: Time = ¹⁶²⁰⁄₁₈ = 90 seconds

Explanation

The ratio of ingredients A and B in the solution is constant across both containers. In the first container, the ratio of A to B is given by 252 : 441. To simplify this ratio, we divide both numbers by their greatest common divisor, 63:

Ratio A:B = ²⁵²⁄₆₃ : ⁴⁴¹⁄₆₃ = 4 : 7

The second container has a total volume of 1188 litres. According to the ratio 4 : 7, the total number of parts in the solution is 4 + 7 = 11. To find the volume of one part, we divide the total quantity by the total number of parts:

1 part = ¹¹⁸⁸⁄₁₁ = 108 litres

The quantity of ingredient B in the second container is represented by 7 parts of the ratio. Thus, the total volume of B is:

Quantity of B = 7 × 108 = 756 litres

Explanation

The total length of the segment PQ is fixed at 10 cm. Since R is a point on PQ, the relationship between the segments is PQ = PR + QR. Initially, with PR = 6 cm, the length of QR is: QR = PQ - PR = 10 - 6 = 4 cm

When the length of PR is increased by 5%, the new length PR' is calculated as: PR' = 6 + (0.05 × 6) = 6 + 0.3 = 6.3 cm

Since the points P and Q are fixed, the total distance PQ remains 10 cm. Therefore, the new length QR' is: QR' = PQ - PR' = 10 - 6.3 = 3.7 cm

The absolute decrease in the length of QR is 4 - 3.7 = 0.3 cm. To find the percentage decrease relative to the original length of QR, we use the formula: Percentage Decrease = (Decrease)/(Original Length) × 100 Percentage Decrease = (0.3)/(4) × 100 = 7.5%

Explanation

The net work done per minute is the difference between the filling rate and the emptying rate. Let the total capacity of the tank be the Least Common Multiple (LCM) of 5 and 6, which is 30 units. The filling pipe adds 30 / 5 = 6 units per minute, and the emptying pipe removes 30 / 6 = 5 units per minute. When both pipes are opened simultaneously, the net filling rate is 6 - 5 = 1 unit per minute. The time required to fill the tank is the total capacity divided by the net rate, calculated as 30 / 1 = 30 minutes. Using the work rate formula 1/T = 1/x - 1/y, where x = 5 and y = 6, we get 1/T = 1/5 - 1/6 = (6 - 5) / 30 = 1/30, which gives T = 30 minutes.

Explanation

Ramesh's profit arises from both the price markup and the use of a false weight. The cost price of 1 kg (1000 g) of rice is ₹30. Since he provides only 800 g to the customer, his actual cost for the transaction is: Actual CP = ³⁰⁄₁₀₀₀ × 800 = 24 He sells this 800 g quantity for ₹36, pretending it is 1 kg. The profit earned on this transaction is the difference between the selling price and his actual cost: Profit = 36 - 24 = 12 The actual profit percentage is calculated by dividing this profit by the cost of the goods actually delivered: Profit % = ( ¹²⁄₂₄ ) × 100 = 50% This can also be calculated using the ratio method. The price ratio (SP/CP) is 36/30 = 6/5 and the weight ratio (Nominal/Actual) is 1000/800 = 5/4. Multiplying these ratios gives the final ratio of ⁶⁄₅ × ⁵⁄₄ = ⁶⁄₄ = ³⁄₂. A ratio of 3/2 indicates a profit of 1 unit on a cost of 2 units, which is 50%.

Explanation

For a number to be divisible by 5, its last digit y must be either 0 or 5. However, the number must also be divisible by 8, which implies it must be an even number. Therefore, y must be 0.

The divisibility rule for 8 states that the last three digits of the number must be divisible by 8. In the number 62684xy, the last three digits are 4x0. We need to find a digit x such that the three-digit number 4x0 is divisible by 8.

We can express 4x0 as 400 + 10x. Since 400 is already divisible by 8 (400 = 8 × 50), the term 10x must also be divisible by 8. Among the possible single-digit values for x, if x = 4, then 10x = 40, and 40 is divisible by 8 (40 = 8 × 5). Thus, the last three digits become 440, and 440 ÷ 8 = 55. This confirms that x = 4 and y = 0 satisfy both divisibility conditions.

Explanation

For a system of linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to have no solution, the lines must be parallel and distinct. This occurs when the following condition is satisfied: (a₁)/(a₂) = (b₁)/(b₂) ≠ (c₁)/(c₂)

From the given equations, we have: a₁ = 1, b₁ = -a, c₁ = 2 - a a₂ = 1 - a, b₂ = 6, c₂ = a + b

First, equate the ratios of the coefficients of x and y: (1)/(1 - a) = ⁻ᵃ⁄₆ 6 = -a(1 - a) 6 = -a + a² a² - a - 6 = 0 (a - 3)(a + 2) = 0

This yields a = 3 or a = -2. Matching these with the provided options, we use a = 3.

Next, check the condition for the constant terms using a = 3: (b₁)/(b₂) ≠ (c₁)/(c₂) ⁻³⁄₆ ≠ (2 - 3)/(3 + b) -½ ≠ (-1)/(3 + b) 3 + b ≠ 2 b ≠ -1

When a = 3, the ratio of the coefficients is -1/2. For the system to have no solution, the ratio of the constants must not equal -1/2, which happens when b ≠ -1. Hence a = 3, b ≠ -1 (option B).

Explanation

The problem states that the time taken to row 10 km downstream is equal to the time taken to row 4 km upstream. Since time is constant, the ratio of the downstream speed D to the upstream speed U is equal to the ratio of the distances: D : U = 10 : 4 = 5 : 2. Let D = 5k and U = 2k.

The total time for the 30 km round trip is 14 hours. This is the sum of the time taken to go downstream and the time taken to return upstream: (30)/(D) + (30)/(U) = 14 Substituting the values of D and U: ³⁰⁄₅ₖ + ³⁰⁄₂ₖ = 14 ⁶⁄ₖ + ¹⁵⁄ₖ = 14 ²¹⁄ₖ = 14 k = ²¹⁄₁₄ = ³⁄₂

Using k = ³⁄₂, we find D = 5 × ³⁄₂ = ¹⁵⁄₂ km/h and U = 2 × ³⁄₂ = 3 km/h. The speed of the stream v is half the difference between the downstream and upstream speeds: v = (D - U)/(2) = (15)/(2) - 3{2} = (9)/(2){2} = ⁹⁄₄

Explanation

The remainder of the expression (17¹³ - 21) ÷ 18 can be calculated efficiently using the properties of modular arithmetic. We evaluate each part of the expression relative to the divisor 18.

For the first term, 17¹³ 18, we note that 17 ≡ -1 18. Substituting this value into the power expression, we get: 17¹³ ≡ (-1)¹³ 18 Since the exponent 13 is odd, (-1)¹³ = -1. Therefore, 17¹³ ≡ -1 18.

For the second term, we simplify 21 relative to 18: 21 ≡ 3 18

Now, combining these results back into the original expression: 17¹³ - 21 ≡ -1 - 3 18 17¹³ - 21 ≡ -4 18

In modular arithmetic, a negative remainder is converted to a positive one by adding the divisor. Adding 18 to the result: -4 + 18 = 14

Thus, the final remainder is 14.

Explanation

The expression is simplified using the BODMAS rule, which dictates the order of operations as Division, Multiplication, Addition, and then Subtraction.

First, perform the divisions: 16 ÷ 2 = 8 15 ÷ 3 = 5

Substitute these values back into the expression: 8 - 7 × 5 + 5 × 5 + 4 × 3 - 10

Next, perform the multiplications: 7 × 5 = 35 5 × 5 = 25 4 × 3 = 12

Substitute these values back into the expression: 8 - 35 + 25 + 12 - 10

Now, group the positive and negative terms together to perform addition and subtraction: Sum of positive terms: 8 + 25 + 12 = 45 Sum of negative terms: -35 - 10 = -45

The final calculation is: 45 - 45 = 0

Explanation

The curved surface area (CSA) of a cylinder is defined by the formula CSA = 2π rh, where r is the radius and h is the height. Given that the curved surface area is 880 cm², we can substitute the known values into the formula using π = 22/7:

880 = 2 × ²²⁄₇ × rh

To find the product of the radius and height (rh), rearrange the equation to isolate the term rh:

880 = ⁴⁴⁄₇ × rh

rh = (880 × 7)/(44)

rh = 20 × 7

rh = 140 cm²

Explanation

The minute hand of a clock covers a complete circle, which represents 2π radians, in 60 minutes. To find the angle turned in a specific period, we calculate the portion of the full rotation that corresponds to the elapsed time.

The angular speed of the minute hand is determined by the formula: Angular speed = (Total Angle)/(Total Time) = (2π)/(60) = (π)/(30) radians per minute

To find the angle θ turned in 25 minutes, we multiply the angular speed by the time duration: θ = 25 × (π)/(30)

By simplifying the expression and dividing both 25 and 30 by their greatest common divisor, which is 5, we arrive at the result: θ = (5π)/(6) radians

This calculation confirms the total angular displacement of the minute hand over the specified 25-minute interval.

Explanation

The average score of the entire class is calculated using the weighted average formula A = (n₁x₁ + n₂x₂)/(n₁ + n₂). In this problem, n₁ = 12 students have an average score of x₁ = 62, and the remaining n₂ = 30 - 12 = 18 students have an average score of x₂ = 72.

To simplify the calculation, the ratio of the number of students can be used: n₁ : n₂ = 12 : 18, which reduces to 2 : 3. Applying these values to the weighted average formula:

A = (2 × 62 + 3 × 72)/(2 + 3)

A = (124 + 216)/(5)

A = ³⁴⁰⁄₅ = 68

Alternatively, the total score of the class is the sum of the scores of both groups. The first group contributes 12 × 62 = 744 points, and the second group contributes 18 × 72 = 1296 points. The total sum of scores is 744 + 1296 = 2040. Dividing this total by the 30 students in the class results in 2040 / 30 = 68.

Explanation

Substitute m = a sec A and y = b tan A into the given expression. For the first two terms, b² m² - a² y² = b² (a sec A)² - a² (b tan A)² = a² b² sec² A - a² b² tan² A = a² b² (sec² A - tan² A). Using the fundamental trigonometric identity sec² A - tan² A = 1, this portion of the expression simplifies to a² b².

The third term is evaluated as (a² y²)/(b² m²) = (a² (b tan A)²)/(b² (a sec A)²) = (a² b² tan² A)/(a² b² sec² A) = (tan² A)/(sec² A). Converting these to sine and cosine, we get (sin² A / cos² A)/(1 / cos² A) = sin² A. Combining this with the final term of the expression, we have sin² A + cos² A. Applying the identity sin² A + cos² A = 1, the complete expression sums to a² b² + 1.

Explanation

When two pipes P and Q fill a tank in time t₁ and t₂ respectively, the time taken to fill the tank when both are opened together is given by the formula T = (t₁ × t₂) / (t₁ + t₂).

Substituting the given values t₁ = 36 minutes and t₂ = 45 minutes:

T = (36 × 45) / (36 + 45) T = 1620 / 81 T = 20 minutes

Alternatively, using the LCM method, let the capacity of the tank be the least common multiple of 36 and 45, which is 180 units. The efficiency (work per minute) of pipe P is 180 / 36 = 5 units per minute. The efficiency of pipe Q is 180 / 45 = 4 units per minute. The combined efficiency of both pipes is 5 + 4 = 9 units per minute. The total time taken to fill the tank is the total capacity divided by the combined efficiency: 180 / 9 = 20 minutes.

Explanation

The trade discount is calculated on the total list price of the items purchased. The dealer bought 6 smart phones at a rate of Rs. 25,000 each.

Total List Price = Number of units × Price per unit Total List Price = 6 × 25,000 = 1,50,000

The manufacturer offers a trade discount of 30% on this total amount.

Trade Discount = Total List Price × Discount Rate Trade Discount = 1,50,000 × ³⁰⁄₁₀₀ Trade Discount = 1,500 × 30 = 45,000

Thus, the total amount of trade discount allowed to the dealer is Rs. 45,000.

Explanation

The mean proportional between two numbers a and b is calculated using the geometric mean formula √(a × b). For the given values 12.96 and 0.16, the mean proportional is √(12.96 × 0.16).

To solve this efficiently, calculate the square root of each factor individually using the property √(x × y) = √x × √y. Since 36² = 1296, the square root of 12.96 is 3.6. Similarly, since 4² = 16, the square root of 0.16 is 0.4.

Multiplying these two square roots gives the final result: 3.6 × 0.4 = 1.44

Alternatively, by converting the decimals into fractions: √(¹²⁹⁶⁄₁₀₀ × ¹⁶⁄₁₀₀) = √(1296) × √(16){√(10000)} = (36 × 4)/(100) = ¹⁴⁴⁄₁₀₀ = 1.44

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Since Δ RST is similar to Δ ABC, the ratio of their perimeters corresponds to the ratio of the side RS to the side AB. The relationship is given by the formula:

(Perimeter of Δ RST)/(Perimeter of Δ ABC) = (RS)/(AB)

Substituting the known values into the equation:

²⁶⁄₃₉ = (RS)/(24)

The fraction ²⁶⁄₃₉ can be simplified by dividing both the numerator and the denominator by their greatest common divisor, 13:

⅔ = (RS)/(24)

To find the value of RS, multiply both sides by 24:

RS = ⅔ × 24

RS = 2 × 8

RS = 16 cm

Explanation

According to the angle sum property of a triangle, the sum of all three interior angles must be exactly 180°. To determine if a set of angles can form a triangle, we apply the formula ∠ A + ∠ B + ∠ C = 180°. For the set {30°, 60°, 90°}, the sum is calculated as 30° + 60° + 90° = 180°. Because this total is exactly equal to 180°, the angles satisfy the requirement for forming a triangle. Specifically, this set represents a right-angled triangle, where the 90° angle is the largest and the two acute angles, 30° and 60°, are complementary.

Explanation

The simplification of the expression (g-h)² - (g+h)² + 4gh utilizes the standard algebraic identities for the square of a difference and the square of a sum. Specifically, (g-h)² = g² - 2gh + h² and (g+h)² = g² + 2gh + h².

Substituting these identities into the original expression results in: (g² - 2gh + h²) - (g² + 2gh + h²) + 4gh

Distributing the negative sign through the second parenthetical term leads to: g² - 2gh + h² - g² - 2gh - h² + 4gh

By combining like terms, the g² and -g² terms eliminate each other, as do the h² and -h² terms. This leaves the expression as: -2gh - 2gh + 4gh

Combining the remaining terms yields: -4gh + 4gh = 0

Explanation

For the first 2 years, the simple interest is calculated using the formula I = (P × R × T)/(100), where P is the principal, R is the rate of interest, and T is the time. Substituting the given values P = 10000, R = 15, and T = 2: I₁ = (10000 × 15 × 2)/(100) = 3000

As per the agreement for the extension, this interest amount is added to the initial principal to form the new principal for the subsequent two years. Pₙₑ_w = 10000 + 3000 = 13000

The interest for the next 2 years is calculated on this updated principal at the same simple interest rate of 15% per annum: I₂ = (13000 × 15 × 2)/(100) = 3900

The total amount paid by Rajesh at the end of 4 years is the sum of the principal for the second period and the interest accrued during that period: Total Amount = 13000 + 3900 = 16900

Explanation

The area of a sector of a circle can be calculated using the formula A = ½ r l, where r is the radius and l is the length of the arc. This formula is derived from the relation between the sector area and the total area of the circle, where the sector area is proportional to the arc length relative to the circumference.

Given: Radius r = 12 cm Arc length l = 7 cm

Substituting these values into the formula: A = ½ × 12 cm × 7 cm A = 6 cm × 7 cm A = 42 cm²

Explanation

Let the initial income be 100. Since Kalyan spends 80% of his income, his initial expenditure is 80 and his initial savings are 100 - 80 = 20.

When the income increases by 15%, the new income becomes 115. The expenditure increases by 5%, so the new expenditure is 80 + (5% of 80) = 80 + 4 = 84.

The new savings are calculated as New Income - New Expenditure, which is 115 - 84 = 31. The increase in savings is 31 - 20 = 11.

To find the percentage increase in savings, use the formula: Percentage Increase = (Change in Savings / Original Savings) × 100 Percentage Increase = (11 / 20) × 100 = 55%

Explanation

To calculate the overall profit percentage, assume the cost price of 1 kg (1000 gm) is 1000 units. The seller claims to sell 1000 gm at a 20% profit, making the selling price 1200 units. However, the seller actually provides only 950 gm.

The actual cost price to the seller for the quantity delivered is the cost of 950 gm, which is 950 units. The revenue generated from this transaction is 1200 units.

Total Profit = Selling Price - Actual Cost Price Total Profit = 1200 - 950 = 250 units

Profit percentage is calculated on the actual cost incurred: Profit % = (Profit)/(Actual Cost Price) × 100 Profit % = ²⁵⁰⁄₉₅₀ × 100 Profit % = ²⁵⁄₉₅ × 100 Profit % = ⁵⁄₁₉ × 100 = ⁵⁰⁰⁄₁₉%

Explanation

The given expression follows the algebraic identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. By substituting a = 1.5, b = 2.5, and c = 3.5, the numerator of the expression matches this identity perfectly:

Numerator: 1.5² + 2.5² + 3.5² + 2(1.5)(2.5) + 2(2.5)(3.5) + 2(1.5)(3.5) = (1.5 + 2.5 + 3.5)²

The denominator of the expression is 1.5 + 2.5 + 3.5. Therefore, the simplification of the expression is:

((1.5 + 2.5 + 3.5)²)/(1.5 + 2.5 + 3.5) = 1.5 + 2.5 + 3.5

By adding the terms: 1.5 + 2.5 + 3.5 = 4.0 + 3.5 = 7.5

Explanation

Let the cost of one biscuit be B, one chocolate be C, and one cup of coffee be K. Based on the given information, we can form the following system of linear equations: 2B + C = 69 (Equation 1) 2C + 3K = 127 (Equation 2) 3B + 4C + 2K = 229 (Equation 3)

Adding the three equations together: (2B + C) + (2C + 3K) + (3B + 4C + 2K) = 69 + 127 + 229 5B + 7C + 5K = 425

To find the value of 5B + 5C + 5K, we first need to determine the cost of a chocolate (C). From Equation 1, we have B = (69 - C) / 2. From Equation 2, we have K = (127 - 2C) / 3. Substituting these into Equation 3: 3((69 - C) / 2) + 4C + 2((127 - 2C) / 3) = 229 Multiplying the entire equation by 6 to eliminate denominators: 9(69 - C) + 24C + 4(127 - 2C) = 1374 621 - 9C + 24C + 508 - 8C = 1374 1129 + 7C = 1374 7C = 245 C = 35

Now, substituting C = 35 into the summed equation 5B + 7C + 5K = 425: 5B + 5C + 5K + 2C = 425 5(B + C + K) + 2(35) = 425 5(B + C + K) + 70 = 425 5(B + C + K) = 425 - 70 5(B + C + K) = 355

Explanation

The average of the squares of the first n natural numbers is given by the formula (n + 1)(2n + 1) / 6. Substituting n = 47, we get (48 * 95) / 6, which simplifies to 8 * 95 = 760.

Explanation

The total distance covered by Akshara and Siddharth when they meet at R is twice the distance from P to Q, which is 144 km. Since they travel for the same amount of time, the distance Akshara travels (PR) is calculated as (Speed of Akshara / Sum of speeds) * Total distance, which is (8 / (8 + 10)) * 144 = 64 km.

Explanation

Let the initial number of gold coins be x and non-gold coins be 4x. After adding 20 gold coins, the ratio becomes (x + 20) / 4x = 2/3, which gives x = 12. The new total number of coins is (12 + 20) gold + 48 non-gold, which equals 80 coins.