SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

A number is divisible by 6 if it is divisible by both 2 and 3. The given number 249k876 ends in the digit 6, which is even, ensuring it is divisible by 2 for any digit k. For the number to be divisible by 3, the sum of its digits must be a multiple of 3.

The sum of the digits is 2 + 4 + 9 + k + 8 + 7 + 6 = 36 + k. For 36 + k to be divisible by 3, k must be a multiple of 3 because 36 is already divisible by 3. The possible values for the digit k are 0, 3, 6, and 9. To find the least value of k, we select the smallest number from this set, which is 0. Substituting k = 0 results in the number 2490876, which is divisible by both 2 and 3.

Explanation

The simple interest is calculated using the formula SI = (P × R × T)/(100), where P is the principal, R is the rate of interest, and T is the time period. According to the problem, the simple interest on ₹7,200 for 3 years at 16% is equal to the simple interest on ₹9,600 for 4 years at x%. Equating the two expressions for simple interest:

(7200 × 16 × 3)/(100) = (9600 × x × 4)/(100)

By canceling the denominator 100 and the zeros from the principals on both sides, we get:

72 × 16 × 3 = 96 × 4 × x

Simplifying the equation:

72 × 48 = 384 × x

3456 = 384x

x = ³⁴⁵⁶⁄₃₈₄

x = 9

Alternatively, dividing both sides by 2400 and simplifying the ratios directly, we have 3 × 16 × 3 = 4 × x × 4, which leads to 144 = 16x, and thus x = 9.

Explanation

The cost price (CP) of the sugar is calculated as 44.8 / 1.12 = 40 per kg. To achieve an overall profit of 5 percent on the initial investment, the total revenue from selling the remaining 4/5 of the sugar must be 1.05 * 40 = 42 per kg of the original amount. Dividing the target revenue by the remaining quantity (0.8 kg) gives a required selling price of 42 / 0.8 = 52.5 per kg.

Explanation

For a system of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have no solution, the lines must be parallel and not coincident. This condition is satisfied when the ratios of the coefficients of x and y are equal, but not equal to the ratio of the constant terms: (a₁)/(a₂) = (b₁)/(b₂) ≠ (c₁)/(c₂)

Comparing the given equations kx - 15y + 7 = 0 and 7x - 21y - 19 = 0 with the standard form, we have a₁ = k, b₁ = -15, c₁ = 7, a₂ = 7, b₂ = -21, and c₂ = -19.

Substituting the values into the ratio of the coefficients: ᵏ⁄₇ = ⁻¹⁵⁄₋₂₁

Simplifying the right-hand side by dividing both terms by -3: ᵏ⁄₇ = ⁵⁄₇

Solving for k yields k = 5. Verifying the second part of the condition, we see that ⁵⁄₇ ≠ ⁷⁄₋₁₉, confirming the system has no solution when k = 5.

Explanation

The average number of cars manufactured in the year 2022 is determined by calculating the sum of the production quantities for all car types in that year and dividing by the total number of car types. Based on the table, the production figures for the year 2022 are 3551 for Car X, 3020 for Car Y, and 2600 for Car Z.

Total production in 2022 = 3551 + 3020 + 2600 = 9171

Since there are three types of cars (X, Y, and Z), the average is:

Average = ⁹¹⁷¹⁄₃ = 3057

Explanation

The perimeter of a circle, also known as the circumference, is given by the formula P = 2π r, where r is the radius of the circle. The length of an arc that subtends an angle θ at the center of the circle is calculated using the formula L = (θ)/(360°) × 2π r.

To find the angle at which the arc length is exactly half the perimeter, we set up the equation L = ½P. Substituting the respective formulas, we get (θ)/(360°) × 2π r = ½ × 2π r. By dividing both sides of the equation by 2π r, the relation simplifies to (θ)/(360°) = ½. Solving for θ gives θ = (360°)/(2) = 180°. This angle represents a straight line across the center, dividing the circle into two equal semicircles.

Explanation

The total capacity of the tank is determined by taking the Least Common Multiple (LCM) of the time taken by each pipe: LCM(15, 25, 40) = 600 units. Based on this capacity, the individual efficiencies of the pipes are calculated as follows: Efficiency of pipe A = 600 / 15 = 40 units/hr Efficiency of pipe B = 600 / 25 = 24 units/hr Efficiency of pipe C = - (600 / 40) = -15 units/hr (negative because it is an emptying pipe)

When all three pipes are opened together, the net efficiency is 40 + 24 - 15 = 49 units/hr. In the first 5 hours, the volume of water filled is 49 × 5 = 245 units. The remaining capacity to be filled is 600 - 245 = 355 units. After pipe A is closed, only pipes B and C remain open, with a combined efficiency of 24 - 15 = 9 units/hr. The time required to fill the remaining part of the tank is 355 / 9 hours, which simplifies to 39⁴⁄₉ hours.

Explanation

To simplify the expression [(⅙÷ ⅑× ⅔)× 18 + 5], follow the BODMAS rule which dictates the order of operations as Brackets, Division, Multiplication, and Addition.

First, evaluate the inner parentheses. According to the order of operations, division is performed before multiplication: ⅙ ÷ ⅑ = ⅙ × 9 = ⁹⁄₆ = ³⁄₂

Next, complete the calculation within the parentheses by multiplying the result with the next fraction: ³⁄₂ × ⅔ = 1

Now, substitute this result back into the main expression inside the square brackets. Perform the multiplication before the addition: 1 × 18 + 5 18 + 5 = 23

Explanation

The total profit is the sum of the profits earned on each individual item. The profit for an item is calculated using the formula: Profit = Cost Price × (Profit%)/(100).

For the Mobile, the profit is 20,000 × ¹⁰⁄₁₀₀ = 2,000. For the Laptop, the profit is 72,000 × ²⁰⁄₁₀₀ = 14,400. For the Mobile covers, the cost price is given per unit, and assuming one unit was sold, the profit is 300 × ¹²⁄₁₀₀ = 36.

To find the total profit earned by the shopkeeper, add the individual profits together: 2,000 + 14,400 + 36 = 16,436.

Explanation

The area of a sector of a circle is given by the formula A = (θ)/(360°) × π r², where A is the area, θ is the central angle in degrees, and r is the radius. Given that the area A = 16π and the radius r = 8, we can substitute these values into the formula:

16π = (θ)/(360°) × π × (8)²

Dividing both sides by π simplifies the equation to:

16 = (θ × 64)/(360°)

We can further simplify by dividing both sides by 16:

1 = (θ × 4)/(360°)

Rearranging the equation to solve for θ:

4θ = 360° θ = (360°)/(4) = 90°

The central angle of the sector is 90°.

Explanation

For the isosceles triangle △ ABC with sides a = 20 cm, b = 20 cm, and c = 10 cm, the semi-perimeter s is calculated as s = (20 + 20 + 10)/(2) = 25 cm. Using Heron's formula: Area(△ ABC) = √(s(s-a)(s-b)(s-c)) = √(25(25-20)(25-20)(25-10)) Area(△ ABC) = √(25 · 5 · 5 · 15) = 5 · 5√(15) = 25√(15) Substituting the given value √(15) = 3.87: Area(△ ABC) = 25 · 3.87 = 96.75 cm²

For the right-angled triangle △ PQR, with hypotenuse 13 cm and one perpendicular side 12 cm, the other perpendicular side is found via the Pythagorean theorem: Base = √(13² - 12²) = √(169 - 144) = √(25) = 5 cm Area(△ PQR) = ½ · base · height = ½ · 5 · 12 = 30 cm²

The difference in area between the two triangles is 96.75 - 30 = 66.75 cm².

Explanation

Total wages are calculated as the product of the number of employees and the wage per head. Let the initial number of employees be x and the initial wage per head be y. The initial total wage is xy.

When two quantities forming a product change by a% and b%, the net percentage change in their product is given by the formula: Net Change % = a + b + ᵃᵇ⁄₁₀₀

Given an increase of 20% in employees (a = +20) and a decrease of 18% in wages per head (b = -18), substitute these values into the formula: Net Change % = 20 + (-18) + (20 × (-18))/(100) Net Change % = 2 - ³⁶⁰⁄₁₀₀ Net Change % = 2 - 3.6 Net Change % = -1.6%

The negative result indicates a decrease. Therefore, the total wages decrease by 1.6%.

Explanation

The net profit or loss percentage when an item is subjected to a markup and then a discount can be determined using the successive percentage formula Net% = x + y + ˣʸ⁄₁₀₀.

In this case, the markup percentage x = 20 and the discount percentage y = -30 (since a discount reduces the price). Substituting these values into the formula:

Net% = 20 - 30 + (20 × (-30))/(100)

Net% = -10 - ⁶⁰⁰⁄₁₀₀

Net% = -10 - 6 = -16%

The negative result indicates a loss of 16%. This can also be verified by assuming a cost price of 100. A 20% markup results in a marked price of 120. Applying a 30% discount on 120 gives a discount value of 120 × 0.30 = 36. The selling price is then 120 - 36 = 84. Since the selling price is 16 less than the cost price of 100, the loss is 16%.

Explanation

In a right-angled triangle ABC with ∠ B = 90°, the condition tan A = 1 implies that A = 45°. To find the value of 4 sin A cos A, we can use the trigonometric identity sin 2A = 2 sin A cos A. The expression can be rewritten as 2(2 sin A cos A), which simplifies to 2 sin 2A. Substituting A = 45° into this identity gives 2 sin(2 × 45°) = 2 sin 90°. Since sin 90° = 1, the result is 2 × 1 = 2. Alternatively, by direct substitution of the values sin 45° = (1)/(√(2)) and cos 45° = (1)/(√(2)), the expression 4 sin A cos A becomes 4 × (1)/(√(2)) × (1)/(√(2)) = 4 × ½ = 2.

Explanation

To find the value of x³ - (1)/(x³), we first determine the value of x - ¹⁄ₓ using the algebraic identity (x - ¹⁄ₓ)² = x² + (1)/(x²) - 2. Substituting the given value x² + (1)/(x²) = 83, we get:

(x - ¹⁄ₓ)² = 83 - 2 = 81

Since x > 1, it follows that x > ¹⁄ₓ, making the difference x - ¹⁄ₓ positive. Taking the square root of both sides, we find x - ¹⁄ₓ = √(81) = 9.

Now, we use the identity for the difference of cubes: x³ - (1)/(x³) = (x - ¹⁄ₓ)³ + 3(x - ¹⁄ₓ). Substituting the value x - ¹⁄ₓ = 9:

x³ - (1)/(x³) = 9³ + 3(9) x³ - (1)/(x³) = 729 + 27 = 756

Explanation

In a circle, the radius drawn to the point of contact is perpendicular to the tangent. Therefore, ∠ APO = 90° and ∠ AQO = 90°. In the quadrilateral APOQ, the sum of the interior angles must be 360°. This leads to the relationship ∠ PAQ + ∠ APO + ∠ POQ + ∠ AQO = 360°. Substituting the known values, we get 40° + 90° + ∠ POQ + 90° = 360°.

Simplifying this, ∠ PAQ + ∠ POQ = 180°, which confirms that the angle between the two tangents and the angle subtended by the radii at the center are supplementary. By substituting the given angle, ∠ POQ = 180° - 40° = 140°.

Explanation

To find the combined ratio A:B:C, the value of the common term B must be identical in both given ratios. The ratio A:B = 6:8 can be simplified to 3:4 by dividing both terms by 2. We are also given B:C = 7:10. The values for B in these ratios are 4 and 7, respectively.

The Least Common Multiple (LCM) of 4 and 7 is 28. We adjust both ratios so that B equals 28: A:B = 3 × 7 : 4 × 7 = 21:28 B:C = 7 × 4 : 10 × 4 = 28:40

By combining these results, we obtain the ratio A:B:C = 21:28:40.

Explanation

The circumference of the base of a cylinder is given by C = 2π r. Substituting the given values: 44 = 2 × ²²⁄₇ × r 44 = ⁴⁴⁄₇ × r r = (44 × 7)/(44) = 7 cm

The volume of a cylinder is calculated using the formula V = π r² h. Substituting r = 7 cm and h = 25 cm: V = ²²⁄₇ × 7² × 25 V = 22 × 7 × 25 V = 154 × 25 = 3850 cm³

To find the capacity in liters, we use the conversion 1000 cm³ = 1 liter: Capacity = ³⁸⁵⁰⁄₁₀₀₀ = 3.85 liters

Explanation

The average return of a portfolio is calculated using the weighted average return formula, where the total return is the sum of the products of each stock's weight and its individual return:

Rₚ = (w_A × r_A) + (w_B × r_B) + (w_C × r_C) + (w_D × r_D)

Substituting the given percentages and returns:

Rₚ = (0.20 × 6%) + (0.30 × 8%) + (0.20 × 4%) + (0.30 × -5%)

Calculating each component:

Stock A: 0.20 × 6 = 1.2% Stock B: 0.30 × 8 = 2.4% Stock C: 0.20 × 4 = 0.8% Stock D: 0.30 × -5 = -1.5%

Summing these values:

Rₚ = 1.2% + 2.4% + 0.8% - 1.5% Rₚ = 4.4% - 1.5% = 2.9%

Explanation

To simplify the expression 13 - [7 - (4 + 7) - {7 - (7 - 4 + 7)} + 14], we follow the BODMAS rule, which dictates the order of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction. We begin with the innermost round brackets.

First, we calculate the values inside the parentheses: 4 + 7 = 11 7 - 4 + 7 = 3 + 7 = 10

Substituting these back into the expression, we get: 13 - [7 - 11 - {7 - 10} + 14]

Next, we solve the curly braces: {7 - 10} = -3

Updating the expression again: 13 - [7 - 11 - (-3) + 14] 13 - [7 - 11 + 3 + 14]

Now, we simplify the terms inside the square brackets. Performing addition and subtraction from left to right: 7 - 11 = -4 -4 + 3 = -1 -1 + 14 = 13

Finally, we substitute the value of the square brackets back into the main expression: 13 - 13 = 0

Explanation

The discount on the sari is the difference between the marked price and the actual selling price. Discount = Marked Price - Selling Price = 10,490 - 9,441 = 1,049 The discount percentage is calculated relative to the marked price using the formula: Discount Percentage = (Discount / Marked Price) × 100 Substituting the given values: Discount Percentage = (1,049 / 10,490) × 100 Discount Percentage = (1 / 10) × 100 = 10%

Explanation

The rate of the leak is the difference between the filling rate of the pipe and the net filling rate when the leak is active. Using the standard work rate formula: (1)/(Tₗₑₐₖ₎ = (1)/(T_fᵢₗₗ₎ - (1)/(Tₙₑₜ₎ Substitute the given values T_fᵢₗₗ = 12 hours and Tₙₑₜ = 18 hours: (1)/(Tₗₑₐₖ₎ = ¹⁄₁₂ - ¹⁄₁₈ Finding a common denominator of 36: (1)/(Tₗₑₐₖ₎ = (3 - 2)/(36) = ¹⁄₃₆ Tₗₑₐₖ = 36 hours

Explanation

Let the total number of voters be V. According to the problem, 80% of the voters cast their votes, and 5% of those votes were invalid, meaning 95% of the polled votes were valid. The winning candidate received 70% of the valid votes, while the losing candidate received the remaining 30%. The majority of 36,480 votes represents the difference between the winner's and the loser's shares, which is 40% (70% - 30%) of the valid votes.

The relationship can be expressed by the following equation: V × ⁸⁰⁄₁₀₀ × ⁹⁵⁄₁₀₀ × ⁴⁰⁄₁₀₀ = 36,480

Simplifying the decimals: V × 0.80 × 0.95 × 0.40 = 36,480 V × 0.304 = 36,480

To find V: V = (36,480)/(0.304) V = 1,20,000

Explanation

The length of a direct common tangent to two circles is calculated using the formula L = √(d² - (r₁ - r₂)²), where d is the distance between the centers of the circles, and r₁ and r₂ are their respective radii.

In this problem, the centers are M and N with d = MN = 16 cm. The radii are r₁ = 12 cm and r₂ = 8 cm. Since the distance between the centers (16 cm) is less than the sum of the radii (12 + 8 = 20 cm) but greater than the difference of the radii (12 - 8 = 4 cm), the circles intersect at two points, and QR represents the direct common tangent.

Substituting the given values into the formula: QR = √(16² - (12 - 8)²) QR = √(256 - 4²) QR = √(256 - 16) QR = √(240)

To simplify √(240), we look for the largest perfect square factor: QR = √(16 × 15) QR = 4√(15) cm

Explanation

The average score of the group is determined using the weighted average formula A = (n₁ A₁ + n₂ A₂)/(n₁ + n₂). In this case, the first group has n₁ = 12 students with an average score A₁ = 62, and the second group consists of the remaining n₂ = 30 - 12 = 18 students with an average score A₂ = 76.

To simplify the calculation, the ratio of the number of students in the two groups, 12:18, can be reduced to 2:3. The average of the entire group is calculated as:

Average = ((2 × 62) + (3 × 76))/(2 + 3)

Average = (124 + 228)/(5)

Average = ³⁵²⁄₅ = 70.4

This method calculates the mean score for the total group of 30 students by taking the sum of all individual scores and dividing it by the total number of students.

Explanation

The perpendicular from the center of a circle to a chord bisects the chord. Let the distances from the center O to the chords AB and CD be x and y respectively. Given the lengths AB = 24 cm and CD = 18 cm, the half-lengths of the chords are 12 cm and 9 cm. In the right-angled triangles formed by the radius r, the half-chord, and the distance from the center, we apply the Pythagorean theorem:

r² = x² + 12² r² = y² + 9²

Equating the two expressions for r² gives x² + 144 = y² + 81, which simplifies to y² - x² = 63. If the chords are on opposite sides of the center, the distance between them is x + y = 21. Using the algebraic identity y² - x² = (y - x)(y + x), we substitute the values:

63 = (y - x)(21) y - x = 3

We now have a system of linear equations: y + x = 21 and y - x = 3. Adding these gives 2y = 24, so y = 12 cm, and x = 9 cm. Substituting x = 9 back into the radius formula:

r² = 9² + 12² = 81 + 144 = 225 r = √(225) = 15 cm

Explanation

Let u be the speed of the boat in still water and v be the speed of the stream. When the distance traveled is constant, the ratio of the speed of the boat in still water to the speed of the stream can be calculated using the times taken for upstream (Tᵤₚ) and downstream (T_dₒ_wₙ) travel.

The relationship is given by the formula: u / v = (Tᵤₚ + T_dₒ_wₙ) / (Tᵤₚ - T_dₒ_wₙ)

Substituting the given values, where Tᵤₚ = 36 hours and T_dₒ_wₙ = 18 hours: u / v = (36 + 18) / (36 - 18) u / v = 54 / 18 u / v = 3 / 1

Alternatively, since distance D is constant, we have: D = (u + v) × 18 D = (u - v) × 36 (u + v) × 18 = (u - v) × 36 u + v = 2(u - v) u + v = 2u - 2v 3v = u u / v = 3 / 1

Explanation

The initial ratio of men to women is given as 5 : 3. Since the number of men does not change when additional women are employed, we can represent the number of men as 5x and the initial number of women as 3x. After adding 150 women, the new number of women becomes 3x + 150. The new ratio of men to women is 5 : 4.

Since the ratio parts for men remain the same (5 parts) in both the old and new ratios, the increase in the ratio parts for women represents the additional employees. The women's part increases from 3 to 4, which is an increase of 4 - 3 = 1 part. This 1 part corresponds to the 150 additional women. Therefore, 1 part = 150. The number of men is represented by 5 parts, which is calculated as 5 × 150 = 750.

Explanation

The ratio of the total surface area to the curved surface area of a cylinder is determined by the formula (2π r(h + r))/(2π rh), where r is the radius and h is the height. This expression simplifies to (h + r)/(h) by canceling the common terms 2π r. Given that the diameter is 28 cm, the radius r is half of that, which is 14 cm. Substituting the radius 14 cm and the height h = 12 cm into the simplified ratio gives (12 + 14)/(12). This calculation results in ²⁶⁄₁₂, which simplifies to 13:6 when divided by the common factor of 2.

Explanation

By definition, two triangles are congruent (≅) if they have the exact same shape and size. Congruence implies that all corresponding side lengths are equal and all corresponding interior angles are equal.

Because the area of a triangle is a function of its side lengths (for example, using Heron's formula or the ½ × base × height formula), any two congruent triangles must possess the same area. Given that Δ EFG ≅ Δ HIJ, it follows that Area(Δ HIJ) = Area(Δ EFG). Therefore, since the area of Δ EFG is 124 cm², the area of Δ HIJ is also 124 cm².

Explanation

Given the equation a + b = 2c, multiply both sides of the expression by the term (a - b): (a + b)(a - b) = 2c(a - b) Applying the algebraic identity for the difference of two squares, x² - y² = (x + y)(x - y), the left side simplifies to a² - b². Distributing the term on the right side gives: a² - b² = 2ac - 2bc To align the equation with the given expression, rearrange the terms by adding b² and 2bc to both sides: a² + 2bc = b² + 2ac

Explanation

For two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to be parallel, the ratio of their coefficients must satisfy (a₁)/(a₂) = (b₁)/(b₂). Substituting the coefficients from the given equations 4x + (k - 2)y + 3 = 0 and (k - 2)x + 9y - 5 = 0, we have:

(4)/(k - 2) = (k - 2)/(9)

Rearranging the equation leads to (k - 2)² = 36. Taking the square root gives k - 2 = ± 6, which results in k = 8 or k = -4. Since the problem states k > 0, we take k = 8. To find the final value, substitute k = 8 into the expression k² + 6:

8² + 6 = 64 + 6 = 70

Explanation

First, simplify the term within the parentheses by finding a common denominator: 5 - ¹⁵⁄₁₆ = (5 × 16 - 15)/(16) = (80 - 15)/(16) = ⁶⁵⁄₁₆.

Next, convert the mixed fraction 1¹²⁄₁₃ into an improper fraction: 1¹²⁄₁₃ = (1 × 13 + 12)/(13) = ²⁵⁄₁₃.

Now, multiply the two resulting fractions: ⁶⁵⁄₁₆ × ²⁵⁄₁₃. To simplify the calculation, divide both the numerator 65 and the denominator 13 by their greatest common divisor, which is 13. This gives: ⁵⁄₁₆ × ²⁵⁄₁ = ¹²⁵⁄₁₆.

Finally, convert the improper fraction ¹²⁵⁄₁₆ back into a mixed fraction. Dividing 125 by 16 gives a quotient of 7 (16 × 7 = 112) and a remainder of 13 (125 - 112 = 13). This results in 7¹³⁄₁₆.

Explanation

Let the capacity of the cistern be the least common multiple of the filling times 8 minutes and 10 minutes, which is 40 units. The normal efficiency of the first pipe is 40 / 8 = 5 units per minute, and the normal efficiency of the second pipe is 40 / 10 = 4 units per minute.

Due to the obstruction, the flow rates are reduced. The efficiency of the first pipe becomes 5 × ⅝ = ²⁵⁄₈ units per minute, and the efficiency of the second pipe becomes 4 × ⅗ = ¹²⁄₅ units per minute. The combined efficiency during the obstruction is ²⁵⁄₈ + ¹²⁄₅ = (125 + 96)/(40) = ²²¹⁄₄₀ units per minute.

The obstruction was removed, and the cistern was filled in the next 3 minutes. During these 3 minutes, both pipes worked at their normal combined efficiency of 5 + 4 = 9 units per minute. The amount of water filled during this period is 9 × 3 = 27 units.

The amount of water filled while the obstruction was present is 40 - 27 = 13 units. To find the time taken before the full flow began, we divide this remaining work by the combined obstructed efficiency: Time = (13)/(221){40} = (13 × 40)/(221) = ⁴⁰⁄₁₇ = 2⁶⁄₁₇ minutes.

Explanation

To find the value of a² + b², we square both given equations and add them together.

Expanding a²: a² = (xcosθ + ysinθ)² = x²cos²θ + y²sin²θ + 2xycosθ

Expanding b²: b² = (xsinθ - ycosθ)² = x²sin²θ + y²cos²θ - 2xysinθ

Now, adding a² and b²: a² + b² = (x²cos²θ + y²sin²θ + 2xysinθ) + (x²sin²θ + y²cos²θ - 2xysinθ)

The cross terms +2xysinθ and -2xysinθ cancel each other out. Grouping the x² and y² terms gives: a² + b² = x²(cos²θ + sin²θ) + y²(sin²θ + cos²θ)

Using the fundamental trigonometric identity sin²θ + cos²θ = 1, we get: a² + b² = x²(1) + y²(1) = x² + y²

Explanation

For a system of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have an infinite number of solutions, the lines must be coincident. This occurs when the ratios of the coefficients of x, y, and the constant terms are all equal: (a₁)/(a₂) = (b₁)/(b₂) = (c₁)/(c₂).

Given the equations 4x + 12y + 36 = 0 and 5x + ky + 45 = 0, we identify the coefficients as a₁ = 4, b₁ = 12, c₁ = 36 and a₂ = 5, b₂ = k, c₂ = 45. Substituting these values into the condition for infinite solutions, we get ⅘ = ¹²⁄ₖ = ³⁶⁄₄₅. Simplifying the ratio of the constant terms, ³⁶⁄₄₅ = ⅘, which confirms the consistency of the system. To find the value of k, we set the ratios of the x and y coefficients equal: ⅘ = ¹²⁄ₖ. Solving for k through cross-multiplication, we have 4k = 60, which gives k = 15.

Explanation

In a right-angled triangle ABC where ∠ C = 90° and CM ⊥ AB, the altitude CM divides the hypotenuse AB into two segments AM and MB. According to the Geometric Mean Theorem (property of altitude to the hypotenuse), the square of the altitude is equal to the product of the segments of the hypotenuse.

First, calculate the length of the segment AM: AM = AB - BM = 18 cm - 6 cm = 12 cm

Now, apply the formula for the altitude CM: CM² = AM · BM CM² = 12 · 6 CM² = 72 CM = √(72) CM = √(36 · 2) CM = 6√2 cm

Explanation

According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is a multiple of 9. For the number 57x716, the sum of the digits is calculated as 5 + 7 + x + 7 + 1 + 6 = 26 + x. To make this sum divisible by 9, the value of 26 + x must be a multiple of 9. The smallest multiple of 9 greater than or equal to 26 is 27. Setting 26 + x = 27 gives x = 1. Since x must be a single digit, and 1 is the smallest non-negative integer satisfying the condition, the least value for x is 1.

Explanation

In a rhombus PQRS, the diagonals PR and QS are perpendicular bisectors of each other. This geometric property means that any point equidistant from the vertices P and R must lie on the perpendicular bisector of the line segment PR. Given the condition OP = OR, the point O must lie on the diagonal QS. Since O is an interior point of the rhombus situated on the diagonal QS, the points S, O, and Q are collinear. In geometry, the angle formed by three collinear points where the vertex O lies between S and Q is a straight angle. A straight angle has a measure of 180°, thus ∠ SOQ = 180°.

Explanation

To find the single equivalent discount for two successive discounts of d₁% and d₂%, the standard formula is D = d₁ + d₂ - (d₁ × d₂)/(100).

In this problem, the first discount d₁ is 40% and the second discount d₂ is 20%. Substituting these values into the formula gives:

D = 40 + 20 - (40 × 20)/(100)

D = 60 - ⁸⁰⁰⁄₁₀₀

D = 60 - 8 = 52%

Alternatively, if we assume the marked price of the goods is 100 units, the first discount of 40% reduces the price to 60 units. The second discount of 20% is then applied to the remaining 60 units, which equals 12 units (0.20 × 60 = 12). The final selling price becomes 60 - 12 = 48 units. The total discount is the difference between the initial price and the final price, which is 100 - 48 = 52 units, or 52%.

Explanation

In the given frequency distribution, the marks categories (0, 10, 20, 30, 40 and above) indicate that the maximum marks for the subject are 50.

To find the number of students eligible for higher studies, we first calculate the qualifying marks based on the 40% requirement: Qualifying Marks = 40% of Total Marks Qualifying Marks = ⁴⁰⁄₁₀₀ × 50 = 20

According to the table, the number of students who obtained 20 and above marks in Chemistry is 66. Therefore, 66 students are eligible to pursue higher studies in Chemistry.

Explanation

The given equation 8a³ + b³ + 27c³ = 18abc can be rewritten in the standard algebraic form x³ + y³ + z³ = 3xyz by substituting x = 2a, y = b, and z = 3c. This leads to (2a)³ + (b)³ + (3c)³ = 3(2a)(b)(3c).

According to the algebraic identity x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx), the condition x³ + y³ + z³ = 3xyz is satisfied if either x + y + z = 0 or x = y = z.

Applying the condition x + y + z = 0 to the variables 2a, b, and 3c, we get: 2a + b + 3c = 0

Rearranging the terms to match the given options: b + 3c = -2a

Explanation

Let the original cost price of the article be 100. The first dealer makes a profit of 30%, so the price at which the second dealer buys the article is 100 + 30 = 130. The final selling price of the article, after passing through the second dealer, reflects a total profit of 40% on the original cost price. Therefore, the final selling price is 100 + 40 = 140.

The profit made by the second dealer is the difference between their selling price and their buying price, which is 140 - 130 = 10. To find the profit percentage for the second dealer, we calculate the profit relative to their own cost price:

Profit percentage = (Profit)/(Cost Price for second dealer) × 100

Profit percentage = ¹⁰⁄₁₃₀ × 100 = ¹⁰⁰⁄₁₃

By dividing 100 by 13, we get 7 as the quotient and 9 as the remainder. Thus, the profit percentage is 7⁹⁄₁₃%.

Explanation

To find the value of sin⁴ A + cos⁴ A, we use the identity sin⁴ A + cos⁴ A = (sin² A + cos² A)² - 2 sin² A cos² A. Since sin² A + cos² A = 1, this simplifies to 1 - 2 sin² A cos² A.

First, square the given equation sin A + cos A = (1)/(2√(2)): (sin A + cos A)² = ((1)/(2√(2)))² sin² A + cos² A + 2 sin A cos A = ⅛ 1 + 2 sin A cos A = ⅛ 2 sin A cos A = ⅛ - 1 = -⅞ sin A cos A = -⁷⁄₁₆

Now, substitute the value of sin A cos A into the identity for sin⁴ A + cos⁴ A: sin⁴ A + cos⁴ A = 1 - 2 (sin A cos A)² sin⁴ A + cos⁴ A = 1 - 2 (-⁷⁄₁₆)² sin⁴ A + cos⁴ A = 1 - 2 (⁴⁹⁄₂₅₆) sin⁴ A + cos⁴ A = 1 - ⁴⁹⁄₁₂₈ sin⁴ A + cos⁴ A = (128 - 49)/(128) = ⁷⁹⁄₁₂₈

Explanation

First, we apply basic trigonometric identities for angles in different quadrants: cos(90° - θ) = sin θ sin(90° + θ) = cos θ sin²(90° + θ) = (cos θ)² = cos² θ cos²(90° - θ) = (sin θ)² = sin² θ

Substituting these into the numerator of the expression: Numerator = 4sin θ cos² θ - 4cos θ sin² θ Numerator = 4sin θ cos θ (cos θ - sin θ)

Now, simplify the denominator: cos((180° + θ)/(2)) = cos(90° + (θ)/(2)) = -sin((θ)/(2))

Since the options provided are constants, the value of the expression must be independent of θ. We can determine this constant by substituting a specific value for θ. Let θ = 90°: Numerator = 4sin 90° cos² 90° - 4cos 90° sin² 90° = 4(1)(0)² - 4(0)(1)² = 0 Denominator = cos((180° + 90°)/(2)) = cos(135°) = -(1)/(√(2))

The value of the expression at θ = 90° is: (0)/(-1/√(2)) = 0

Testing another value, such as θ = 45°: Numerator = 4sin 45° cos² 45° - 4cos 45° sin² 45° Since sin 45° = cos 45° = (1)/(√(2)), the terms are identical: Numerator = 4((1)/(√(2)))(½) - 4((1)/(√(2)))(½) = 0 Denominator = cos(112.5°) ≠ 0

Since the expression evaluates to 0 for these points, the constant value of the expression is 0.

Explanation

The given equation is a³ + 3a² + 3a = 7. By adding 1 to both sides of the equation, we can complete the cube on the left side:

a³ + 3a² + 3a + 1 = 7 + 1

Using the algebraic identity (x + y)³ = x³ + 3x²y + 3xy² + y³, we can identify the left side as the expansion of (a + 1)³:

(a + 1)³ = 8

Taking the cube root of both sides gives:

a + 1 = 2 a = 1

We are required to find the value of a² + 2a. Substituting a = 1 into this expression:

1² + 2(1) = 1 + 2 = 3

Explanation

Simple Interest (SI) is the difference between the Amount and the Principal. SI = 1230 - 1040 = 190 The time T is 1.5 years, or ³⁄₂ years.

Using the formula for the rate of interest: R = (SI × 100)/(P × T) R = (190 × 100)/(1040 × 1.5) R = ¹⁹⁰⁰⁰⁄₁₅₆₀ R = ¹⁹⁰⁰⁄₁₅₆

Dividing both the numerator and the denominator by 4: R = ⁴⁷⁵⁄₃₉

To convert this to a mixed fraction: 475 = 39 × 12 + 7 R = 12⁷⁄₃₉%

Explanation

The Arithmetic Mean - Harmonic Mean (AM-HM) inequality states that for any set of positive real numbers, the arithmetic mean is greater than or equal to the harmonic mean. For the three positive numbers x, y, and z, the inequality is expressed as:

(x + y + z)/(3) ≥ (3)/(1){x + (1)/(y) + (1)/(z)}

Given the constraint x + y + z = 1, we can substitute this value into the numerator of the arithmetic mean:

⅓ ≥ (3)/(1){x + (1)/(y) + (1)/(z)}

To find the least value of the expression, we rearrange the inequality to solve for the sum of the reciprocals:

¹⁄ₓ + (1)/(y) + (1)/(z) ≥ 3 × 3

¹⁄ₓ + (1)/(y) + (1)/(z) ≥ 9

The minimum value of 9 is achieved when x = y = z. Since their sum is 1, this occurs when x = y = z = 1/3. Substituting these values back into the expression gives 3 + 3 + 3 = 9.

Explanation

The mean proportional of two numbers x and y is defined as the square root of their product, given by the formula √(x · y). Substituting the given values 8a² and 18 / a⁴ into this formula, we get √(8a² · (18)/(a⁴)).

Simplifying the expression inside the square root, the product is (144a²)/(a⁴), which reduces to (144)/(a²). Taking the square root of this fraction yields √(144){√(a²)}. Since a is a positive number, the square root of a² is a and the square root of 144 is 12. Therefore, the mean proportional is 12 / a.

Explanation

Using componendo and dividendo on the given equation, we get (2 sec α) / (2 tan α) = (7 + 4) / (7 - 4), which simplifies to sec α / tan α = 11/3. Since sec α / tan α = (1 / cos α) / (sin α / cos α) = 1 / sin α = cosec α, the value is 11/3.