SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

A line intersects the Y-axis at the point where its x-coordinate is zero. To find this point for the equation 2x - 3y = 6, we substitute x = 0 into the equation:

2(0) - 3y = 6 -3y = 6 y = ⁶⁄₋₃ y = -2

The coordinates of the point where the line cuts the Y-axis are therefore (0, -2).

Explanation

The property of similarity for triangles states that the ratio of the areas of two similar triangles is equal to the square of the ratio of any two corresponding sides. In this case, for Δ ABC ∼ Δ PQR, the relationship is given by the formula (Area(Δ ABC))/(Area(Δ PQR)) = ((AB)/(PQ))². By substituting the provided values, the equation becomes ³⁶⁄₉ = ((AB)/(4))², which simplifies to 4 = ((AB)/(4))². Taking the square root of both sides yields 2 = (AB)/(4). Solving for the side length, AB = 2 × 4 = 8 cm.

Explanation

The values of the trigonometric ratios for the given angles are cot 45° = 1 and csc 60° = (1)/(sin 60°) = (2)/(√(3)). Substituting these values into the expression, we get:

cot 45° + csc 60° = 1 + (2)/(√(3))

By taking the least common multiple (LCM) of the terms to simplify the expression, we obtain:

√3 + 2{√3}

Explanation

In any triangle, the sum of all interior angles is always 180°. For the given △ LMN, which is right-angled at M, the angle ∠ M is 90°. Given that ∠ N is 60°, we can find the measure of ∠ L using the angle sum property:

∠ L + ∠ M + ∠ N = 180° ∠ L + 90° + 60° = 180° ∠ L + 150° = 180° ∠ L = 180° - 150° = 30°

The question asks for the value of tan L. Substituting the calculated value of ∠ L:

tan L = tan 30°

According to standard trigonometric tables, the value of tan 30° is 1/√3.

Explanation

The trigonometric ratio tan θ is defined as the ratio of the perpendicular (P) to the base (B). Given tan θ = 4/3, we can identify P = 4 and B = 3. To find the hypotenuse (H), we use the Pythagorean theorem: H = √(P² + B²). Substituting the values gives H = √(4² + 3²) = √(16 + 9) = √(25) = 5. The value of sin θ is defined as the ratio of the perpendicular to the hypotenuse, P/H. Therefore, sin θ = 4/5, which equals 0.8 in decimal form.

Explanation

The expenditure of the company is distributed among various categories, with the largest single expense accounting for 30% of the total budget. Electricity accounts for 25% of the total expenditure. Because 25% is the highest percentage recorded after the leading expense category, Electricity is identified as the second biggest expense for the company.

Explanation

The expenditure distribution for the company allocates 30% of the total budget to Raw materials and 15% to Transport. The expenditure on Salaries is 15%. The combined expenditure on Raw material and Transport is the sum of their individual shares:

Combined Expenditure = 30% + 15% = 45%

The ratio of the combined expenditure on Raw material and Transport to the expenditure on Salaries is determined using the following calculation:

Ratio = (Combined Expenditure)/(Salaries) = (45%)/(15%)

Dividing both the numerator and the denominator by 15 simplifies the fraction to 3 : 1.

Explanation

In the context of the 2017 SSC CGL exam paper, the data provided in the associated pie chart indicates that expenditure on interest is 10% and expenditure on rent is 5%. To find how much greater interest is than rent, the calculation is ((10 - 5) / 5) x 100, which equals 100%.

Explanation

The total expenditure of the company is given as Rs 50 crores. According to the expenditure distribution provided for the company, Transport accounts for 15% and Electricity accounts for 25% of the total costs. To find the combined expenditure on these two items, we first determine their total percentage share:

Combined Percentage = 15% + 25% = 40%

The total expenditure on Transport and Electricity is calculated by applying this combined percentage to the total expenses using the formula:

Value = (Percentage)/(100) × Total Expenses

Value = ⁴⁰⁄₁₀₀ × 50

Value = 0.4 × 50 = 20

The result of the calculation is Rs 20 crores.

Explanation

To determine the least number that must be added, we check the divisibility of 329 by 7 using the division algorithm:

329 ÷ 7

Dividing the first two digits: 32 = (7 × 4) + 4. The remainder is 4. Combining this remainder with the next digit: 49 = 7 × 7. The remainder is 0.

Since 329 = 7 × 47, the number is already perfectly divisible by 7 without any remainder. When a number is already divisible by the divisor, the least non-negative integer that can be added to it to maintain divisibility is 0.

Explanation

The combined rate at which A and B work together is the sum of their individual rates. Using the rate formula R = 1/T, the rate of A is 1/15 and the rate of B is 1/30. Their joint rate is 1/15 + 1/30 = (2 + 1)/30 = 3/30 = 1/10 of the total work per day. Consequently, A and B together can complete 100% of the work in 10 days. To find the time required to complete 50% of the work, we multiply the total time by the required fraction: 10 × 0.5 = 5 days. Alternatively, using the total work method with a common multiple of 30 units, A completes 2 units per day and B completes 1 unit per day. Together they complete 3 units per day, and the time taken to finish 50% of the work, which is 15 units, is 15 / 3 = 5 days.

Explanation

The area of an equilateral triangle with side length a is determined by the formula Area = √3{4} a². Substituting the given side length a = 10 cm into the formula, we get Area = √3{4} × (10)². Calculating the square of the side gives 100, so the expression becomes √3{4} × 100. Dividing 100 by 4 results in 25, making the final area 25√3 sq cm.

Explanation

To find the effective discount for two successive discounts of x% and y%, use the formula x + y - ˣʸ⁄₁₀₀. Substituting the given values x = 15 and y = 50 into this relation provides the total discount:

15 + 50 - (15 × 50)/(100)

Performing the arithmetic leads to 65 - ⁷⁵⁰⁄₁₀₀, which is 65 - 7.5 = 57.5.

Alternatively, considering an initial marked price of 100, a 15% discount reduces the price to 85. Applying a subsequent 50% discount on 85 reduces the value by 42.5 (0.50 × 85), leaving a final selling price of 42.5. The total discount relative to the original price is 100 - 42.5 = 57.5, which represents an effective discount of 57.5%.

Explanation

The total profit of Rs 42,500 is divided among partners A, B, and C in the ratio 3:5:9. To find the individual shares, first calculate the sum of the ratio terms: 3 + 5 + 9 = 17.

A's share is calculated by multiplying the total profit by the fraction representing A's portion of the ratio:

A's share = ³⁄₁₇ × 42500

Dividing 42,500 by 17 yields 2,500. Multiplying this value by A's ratio part (3) gives the final amount:

3 × 2500 = 7500

Explanation

The total weight of A, B, and C is calculated by multiplying their average weight by the number of people: 3 × 49 = 147 kg. Similarly, the sum of weights for A and B is 2 × 35 = 70 kg, and the sum of weights for B and C is 2 × 62 = 124 kg.

To find the weight of B, we add the two-person sums and subtract the three-person total. This works because (A + B) + (B + C) = A + 2B + C. Subtracting (A + B + C) from this result leaves only the weight of B:

Weight of B = (A + B) + (B + C) - (A + B + C) Weight of B = 70 + 124 - 147 Weight of B = 194 - 147 = 47 kg

Explanation

The relationship between selling price, cost price, and profit is defined by the formula Profit = Total Selling Price - Total Cost Price. Let the selling price (SP) of one item be x. For 6 items, the total selling price is 6x. According to the problem, the profit earned is equal to the selling price of one item, so Profit = x.

To find the cost price (CP) of the 6 items, we rearrange the formula to Total CP = Total Selling Price - Profit. Substituting the values, we get Total CP = 6x - x = 5x.

The profit percentage is calculated on the cost price using the formula: Profit % = ((Profit)/(Total CP)) × 100

By substituting the known values into the equation: Profit % = (ˣ⁄₅ₓ) × 100 = ⅕ × 100 = 20

Explanation

The average speed is determined by the formula V = (D)/(T), where D is distance and T is time. Substituting the given values, V = (81 km)/(1.5 h) = 54 km/h. To convert the result from kilometers per hour to meters per second, multiply by the conversion factor ⁵⁄₁₈:

V = 54 × ⁵⁄₁₈ = 3 × 5 = 15 m/s

Alternatively, converting units individually, the distance is 81 × 1000 = 81,000 meters and the time is 1.5 × 3600 = 5,400 seconds. Calculating the speed directly in the required units:

V = (81,000 m)/(5,400 s) = ⁸¹⁰⁄₅₄ = 15 m/s

Explanation

The rate of simple interest is calculated by dividing the total percentage increase in principal by the time period. Since the principal increases by 32% in 4 years, the annual rate of interest is R = (32%)/(4) = 8% per annum.

To calculate the compound interest on Rs. 24,000 for 3 years at an 8% annual rate, we first determine the effective compound interest rate for 3 years. Using the successive percentage formula or the direct formula Rₑ_f_f = 3r + (3r²)/(100) + (r³)/(10000), where r = 8: Rₑ_f_f = 3(8) + ³⁽⁶⁴⁾⁄₁₀₀ + ⁵¹²⁄₁₀₀₀₀ Rₑ_f_f = 24 + 1.92 + 0.0512 = 25.9712%

The compound interest earned is obtained by applying this effective rate to the principal: CI = 24000 × (25.9712)/(100) CI = 240 × 25.9712 CI = 6233.088

Rounding to the nearest whole number as per the options, the interest earned is Rs. 6,233. This can also be verified by noting that 26% of 24,000 is 6,240, and since the effective rate is slightly less than 26% (25.97%), the result must be slightly less than 6,240.

Explanation

Expand the equation by distributing ⁵⁄₂ into the parentheses: ⁵⁄₂ × ⁸ˣ⁄₃ - ⁵⁄₂ × ½ + ¹³⁄₂ = ²ˣ⁄₃ ²⁰ˣ⁄₃ - ⁵⁄₄ + ¹³⁄₂ = ²ˣ⁄₃

Group the terms containing x on the left side and the constant terms on the right side: ²⁰ˣ⁄₃ - ²ˣ⁄₃ = ⁵⁄₄ - ¹³⁄₂

Simplify both sides of the equation. On the left side, the common denominator is 3: ¹⁸ˣ⁄₃ = ⁵⁄₄ - ²⁶⁄₄ 6x = -²¹⁄₄

Solve for x by dividing both sides by 6: x = -(21)/(4 × 6) x = -²¹⁄₂₄

Reducing the fraction by dividing both the numerator and the denominator by their greatest common divisor, 3, yields: x = -⅞

Explanation

The algebraic identity for the sum of cubes is a³ + b³ = (a + b)³ - 3ab(a + b). Substituting the given values a³ + b³ = 72 and ab = 8 into the formula, we let s = a + b:

72 = s³ - 3(8)(s) 72 = s³ - 24s s³ - 24s - 72 = 0

By testing the values from the options, for s = 6: 6³ - 24(6) - 72 = 216 - 144 - 72 = 0

Alternatively, by inspection, we seek two numbers whose product is 8 and whose sum of cubes is 72. The factors of 8 are 4 and 2. Checking these values: 4³ + 2³ = 64 + 8 = 72 The sum a + b is therefore 4 + 2 = 6.

Explanation

Let the fraction be x. The problem provides the equation 4x + 7(¹⁄ₓ) = 16. By substituting the value 7/2 into this equation, we can verify its correctness directly:

4(⁷⁄₂) + 7(²⁄₇) = 16

2(7) + 2 = 16

14 + 2 = 16

16 = 16

The left-hand side equals the right-hand side, confirming that 7/2 is the correct fraction. This can also be solved by forming the quadratic equation 4x² - 16x + 7 = 0. Using the quadratic formula x = -b ± √(b²⁾⁻⁴ᵃᶜ{2a} with a = 4, b = -16, and c = 7:

x = 16 ± √(256 - 112){8}

x = 16 ± √(144){8} = (16 ± 12)/(8)

The possible values for x are ²⁸⁄₈ = ⁷⁄₂ and ⁴⁄₈ = ½. Among the provided options, 7/2 is the one that satisfies the conditions.

Explanation

When a point (x, y) is reflected in a vertical line x = k, the y-coordinate remains unchanged because the transformation occurs only along the horizontal axis. The x-coordinate of the image, x', is determined by the fact that the line of reflection x = k is the midpoint between the original x-coordinate and the image x-coordinate. This relationship is expressed by the formula x' = 2k - x.

Given the point (-3, 2) and the line of reflection x = -2, we substitute x = -3 and k = -2 into the formula: x' = 2(-2) - (-3) x' = -4 + 3 = -1

Since the y-coordinate remains 2, the reflected point is (-1, 2).

Explanation

The midpoint P(x, y) of a line segment with endpoints A(x₁, y₁) and B(x₂, y₂) is determined by the midpoint formula x = x₁ + x₂{2} and y = y₁ + y₂{2}. Given the coordinates of the midpoint P(1, 3) and one endpoint A(-3, 8), the coordinates of point B(x₂, y₂) can be found by substituting the known values into these equations.

For the x-coordinate: 1 = -3 + x₂{2} 2 = -3 + x₂ x₂ = 2 + 3 = 5

For the y-coordinate: 3 = 8 + y₂{2} 6 = 8 + y₂ y₂ = 6 - 8 = -2

By solving these linear equations, the coordinates of point B are found to be (5, -2).

Explanation

The slope m₁ of a line passing through two points (x₁, y₁) and (x₂, y₂) is calculated using the formula m₁ = (y₂ - y₁)/(x₂ - x₁). For the points (3, -2) and (4, 2), the slope is m₁ = (2 - (-2))/(4 - 3) = ⁴⁄₁ = 4. Two lines are perpendicular if the product of their slopes is -1, defined by the relation m₁ · m₂ = -1. Substituting the known slope, we get 4 · m₂ = -1, which simplifies to m₂ = -¼.

Explanation

For similar triangles, the ratio of the perimeters is equal to the ratio of the lengths of their corresponding sides. Given that Δ ABC ∼ Δ PQR, the relationship between their perimeters and sides is defined as:

(Perimeter(Δ ABC))/(Perimeter(Δ PQR)) = (AB)/(PQ)

The problem provides the ratio of perimeters as 3:7 and the length of side PQ as 21 cm. Substituting these values into the equation:

³⁄₇ = (AB)/(21)

To solve for AB, multiply both sides by 21:

AB = ³⁄₇ × 21

AB = 3 × 3

AB = 9 cm

Since the corresponding side PQ is 21 cm and the triangles are in a 3:7 ratio, AB must be 9 cm.

Explanation

The expression is 2 sec 45° + tan 30°. Using the standard trigonometric ratios for specific angles, we know that sec 45° = √2 and tan 30° = (1)/(√(3)). Substituting these values directly into the given expression yields 2(√2) + (1)/(√(3)). To simplify this into a single fraction as presented in the options, we identify √3 as the common denominator. Multiplying 2√2 by √3{√3} gives 2√6{√3}. Adding this to the second term results in the combined expression 2√6 + 1{√3}.

Explanation

In triangle ABC, the sum of all interior angles is 180°. Since the triangle is right-angled at B, the measure of angle B is 90°. The problem states that m∠ A = 60°.

Using the angle sum property of a triangle: m∠ A + m∠ B + m∠ C = 180° 60° + 90° + m∠ C = 180° 150° + m∠ C = 180° m∠ C = 30°

We need to find the value of cot C, which is cot 30°. According to standard trigonometric ratios, cot 30° = √3.

Explanation

In a right-angled triangle, csc θ = (Hypotenuse)/(Perpendicular). Given that csc θ = ²⁵⁄₇, we can take the Hypotenuse (H) = 25 and the Perpendicular (P) = 7. Using the Pythagoras theorem, H² = P² + B², where B is the Base. Substituting the given values:

25² = 7² + B² 625 = 49 + B² B² = 625 - 49 = 576 B = √(576) = 24

The trigonometric ratio for cosine is defined as cos θ = (Base)/(Hypotenuse). Therefore, cos θ = ²⁴⁄₂₅. This calculation is simplified by recognizing the standard Pythagorean triplet (7, 24, 25).

Explanation

To find the remainder when 6910 is divided by 81, we apply the division algorithm n = d × q + r, where n is the dividend, d is the divisor, q is the quotient, and r is the remainder such that 0 ≤ r < d.

Dividing 6910 by 81 step-by-step: We first consider the first three digits of the dividend, 691. Dividing 691 by 81, we find the quotient is 8 because 81 × 8 = 648. The partial remainder is 691 - 648 = 43. Next, we bring down the last digit 0 to make the number 430. Dividing 430 by 81, we find the quotient is 5 because 81 × 5 = 405. The final remainder is calculated by subtracting 405 from 430, which gives 25.

We can verify the result using the relationship: 6910 = (81 × 85) + 25 6910 = 6885 + 25

The calculation confirms that 25 is the remainder when the integer 6910 is divided by 81.

Explanation

To convert the decimal number 99 into its binary equivalent, the number is repeatedly divided by 2 while recording the remainders. Dividing 99 by 2 yields a quotient of 49 and a remainder of 1. Dividing 49 by 2 yields a quotient of 24 and a remainder of 1. Dividing 24 by 2 yields a quotient of 12 and a remainder of 0. Dividing 12 by 2 yields a quotient of 6 and a remainder of 0. Dividing 6 by 2 yields a quotient of 3 and a remainder of 0. Dividing 3 by 2 yields a quotient of 1 and a remainder of 1. Dividing 1 by 2 yields a quotient of 0 and a remainder of 1. By reading the remainders in reverse order (from the last remainder to the first), we obtain 1100011.

Alternatively, using the sum of powers of 2 method, we find the largest power of 2 less than or equal to 99, which is 2⁶ = 64. Subtracting this from 99 leaves 35. The largest power of 2 less than or equal to 35 is 2⁵ = 32. Subtracting this leaves 3. The largest power of 2 less than or equal to 3 is 2¹ = 2. Subtracting this leaves 1, which is 2⁰. Therefore, 99 = (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (0 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰), resulting in the binary sequence 1100011.

Explanation

The efficiency of Anil is twice that of Bimal, representing an efficiency ratio of E_A : E_B = 2 : 1. Let Bimal's work rate be 1 unit per day and Anil's work rate be 2 units per day. Their combined efficiency is 2 + 1 = 3 units per day. Since they complete the piece of work together in 9 days, the total work is calculated as Total Work = Combined Efficiency × Time = 3 × 9 = 27 units. To find the time Anil takes alone, divide the total work by his individual efficiency: Time = ²⁷⁄₂ = 13.5 days.

Explanation

The circumference of a circle is calculated using the formula C = 2π r, where r is the radius. Given that C = 22 cm and using the standard value of π = ²²⁄₇, we can find the radius as follows:

22 = 2 × ²²⁄₇ × r r = (22 × 7)/(2 × 22) = ⁷⁄₂ = 3.5 cm

The area of a circle is given by the formula A = π r². Substituting the derived value of the radius:

A = ²²⁄₇ × (⁷⁄₂)² A = ²²⁄₇ × ⁴⁹⁄₄ A = (11 × 7)/(2) = ⁷⁷⁄₂ = 38.5 sq cm

Explanation

The marked price (MP) of the article is calculated by adding the 150% markup to the cost price x, which gives MP = x + 1.5x = 2.5x. The selling price (SP) is then derived by applying a 40% discount to the marked price. Using the formula SP = MP × (1 - Discount%), we substitute the given values:

600 = 2.5x × (1 - 0.40)

600 = 2.5x × 0.6

600 = 1.5x

Solving for x:

x = (600)/(1.5) = 400

Explanation

Let the number of 2-rupee, 50-paise, and 25-paise coins be 3x, 5x, and 7x respectively, based on the given ratio 3:5:7. To find the total value, multiply the number of coins by their respective values in rupees:

3x(2) + 5x(0.50) + 7x(0.25) = 51.25

6x + 2.5x + 1.75x = 51.25

10.25x = 51.25

x = (51.25)/(10.25) = 5

The number of 50-paise coins is defined by the term 5x. Substituting the value of x:

Number of 50-paise coins = 5 × 5 = 25

Explanation

The total number of runs scored in the first 40 overs is calculated by multiplying the number of overs by the run rate: 40 × 4.8 = 192 runs

The number of runs needed to reach the target of 241 is: 241 - 192 = 49 runs

The number of remaining overs is: 50 - 40 = 10 overs

The required run rate for the remaining 10 overs is the remaining runs divided by the remaining overs: ⁴⁹⁄₁₀ = 4.9 runs per over

Explanation

To calculate the gain percentage, assume the milkman buys 4 litres of milk. The total Cost Price (CP) is 4 × 25 = 100 rupees. He adds 1/4 of water to this quantity, which is 1/4 × 4 = 1 litre. The total volume of the resulting mixture is 4 + 1 = 5 litres. He sells this mixture at Rs 26 per litre, so the total Selling Price (SP) is 5 × 26 = 130 rupees. The profit is calculated as SP - CP = 130 - 100 = 30 rupees. The gain percentage is determined by the formula (Profit / CP) × 100, which results in (30 / 100) × 100 = 30%.

Explanation

Let the number be x. According to the problem, increasing the number by 120 results in 130% of the number. This relationship can be expressed by the equation x + 120 = 1.30x. Subtracting x from both sides of the equation yields 120 = 0.30x. Solving for x gives x = (120)/(0.3) = 400.

Alternatively, since the final value is 130% of the original, the increase of 120 represents exactly 30% of the number (130% - 100% = 30%). If 30% of the number is 120, then 1% of the number is ¹²⁰⁄₃₀ = 4. Therefore, 100% of the number is 4 × 100 = 400.

Explanation

If A completes 75% of the work in 18 days, the time required for A to finish that specific portion is T_A = 18 days. Since B completes 25% of the work in 12 days, the time B takes to complete 75% of the work is three times as much, giving T_B = 12 × 3 = 36 days. To determine the time taken by A and B together to complete 75% of the job, we use the formula T = (T_A × T_B)/(T_A + T_B). Substituting the values results in T = (18 × 36)/(18 + 36) = (18 × 36)/(54), which simplifies to T = ³⁶⁄₃ = 12 days.

Explanation

When two objects move toward each other, their relative speed is the sum of their individual speeds.

Relative speed = 20 + 25 = 45 km/hr.

The time taken to meet is 56 minutes. Converting this to hours: t = ⁵⁶⁄₆₀ hours.

Using the distance formula d = speed × time: d = 45 × ⁵⁶⁄₆₀

Simplifying the calculation: d = ⁴⁵⁄₆₀ × 56 = ¾ × 56 d = 3 × 14 = 42 km.

Explanation

The relationship between the amount A, principal P, rate of interest r, and time n in compound interest is given by the formula A = P(1 + r/100)ⁿ.

Substituting the given values A = 5324, r = 10%, and n = 3 years:

5324 = P(1 + 10/100)³ 5324 = P(11/10)³ 5324 = P(1331/1000)

To find the principal P, rearrange the formula: P = (5324 × 1000) / 1331

Dividing 5324 by 1331 gives exactly 4. Therefore: P = 4 × 1000 = 4000

Using the ratio method, a 10% interest rate means the sum grows from 10 to 11 each year. For 3 years, the ratio of principal to amount is 10³ : 11³, which is 1000 : 1331. Given that 1331 units correspond to Rs 5324, one unit equals Rs 4. Thus, the principal of 1000 units equals Rs 4000.

Explanation

To find the value of x, multiply both sides of the equation 14/3 + (1/2)(x - 7/3) = -2x/3 by 6, which is the least common multiple of the denominators 2 and 3, to clear the fractions: 6 × ¹⁴⁄₃ + 6 × ½ (x - ⁷⁄₃) = 6 × (-²ˣ⁄₃) 28 + 3(x - ⁷⁄₃) = -4x

Distribute the 3 into the parentheses: 28 + 3x - 7 = -4x 21 + 3x = -4x

Combine like terms by moving all variables to one side and the constant to the other: 3x + 4x = -21 7x = -21 x = ⁻²¹⁄₇ = -3

Explanation

The value of a³ + b³ can be determined using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). By substituting the given values a + b = 10 and ab = 24 into this formula, we get:

a³ + b³ = (10)³ - 3(24)(10) a³ + b³ = 1000 - 720 a³ + b³ = 280

Alternatively, by inspection, the two numbers that satisfy the conditions a + b = 10 and ab = 24 are 6 and 4. Calculating the sum of their cubes results in 6³ + 4³ = 216 + 64 = 280.

Explanation

Let the fraction be x. According to the problem, the sum of the fraction and three times its reciprocal is 19/4. This relationship is represented by the algebraic equation x + 3(1/x) = 19/4. To solve for x, we simplify the equation to x + 3/x = 19/4. Multiplying all terms by 4x to eliminate the denominators yields the quadratic equation 4x² + 12 = 19x, which can be rearranged to 4x² - 19x + 12 = 0.

To find the roots of this quadratic equation, we use the formula x = [-b ± √(b² - 4ac)] / 2a. Substituting a = 4, b = -19, and c = 12, we calculate the discriminant as 19² - 4(4)(12) = 361 - 192 = 169. The square root of 169 is 13. Applying this to the formula gives x = (19 ± 13) / 8. The two possible values are x = (19 + 13) / 8 = 32/8 = 4 and x = (19 - 13) / 8 = 6/8 = 3/4. Verifying with the provided options, the fraction 3/4 satisfies the condition because 3/4 + 3(4/3) = 3/4 + 4 = (3 + 16)/4 = 19/4.

Explanation

The n-th term of an arithmetic progression is given by the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference. Based on the given information, we have two equations: a + 2d = 13 for the 3rd term and a + 5d = -5 for the 6th term.

To find the common difference, subtract the 3rd term equation from the 6th term equation: (a + 5d) - (a + 2d) = -5 - 13, which simplifies to 3d = -18, giving d = -6. Substituting d = -6 into the equation for the 3rd term yields a + 2(-6) = 13, which results in a - 12 = 13, so a = 25. The 11th term is calculated as a₁₁ = a + 10d = 25 + 10(-6) = 25 - 60 = -35.

Explanation

When reflecting a point (x, y) across a horizontal line y = k, the x-coordinate remains unchanged because the line is parallel to the x-axis, making the reflection purely vertical. The y-coordinate of the reflected point y' is calculated using the property that the line of reflection is the midpoint between the original point and its image: k = (y + y') / 2. This rearranges to the formula y' = 2k - y.

Given the point (6, -1) and the line y = 2: x' = 6 y' = 2(2) - (-1) y' = 4 + 1 = 5

The coordinates of the reflected point are (6, 5).

Explanation

The midpoint M(xₘ, yₘ) of a line segment with endpoints A(x₁, y₁) and B(x₂, y₂) is determined by the formula xₘ = (x₁ + x₂)/(2) and yₘ = (y₁ + y₂)/(2).

To find the value of x, we only need to use the formula for the x-coordinate of the midpoint. Given the midpoint P(8, 5), the x-coordinate xₘ is 8. The x-coordinates of the endpoints A(5, y) and B(x, -3) are 5 and x respectively.

Substituting these values into the midpoint formula: 8 = (5 + x)/(2)

Multiplying both sides by 2 to clear the fraction: 16 = 5 + x

Subtracting 5 from both sides to isolate x: x = 11

Explanation

The equation of a line in slope-intercept form is y = mx + c, where m is the slope and c is the y-intercept. Given the slope m = 1/3 and the y-intercept c = 5, substituting these values into the formula yields:

y = ⅓x + 5

To express this in the standard linear form Ax + By = C, multiply the entire equation by 3 to eliminate the fraction:

3y = x + 15

Subtracting 3y from both sides and subtracting 15 from both sides results in:

x - 3y = -15

Explanation

In a right-angled triangle △ ABC with altitude BD drawn from the right angle B to the hypotenuse AC, the Square of the Leg Theorem states that AB² = AD × AC.

First, calculate the length of the hypotenuse AC: AC = AD + DC = 3 + 9 = 12 cm

Now, substitute the values into the formula: AB² = 3 × 12 AB² = 36 AB = √(36) = 6 cm

Explanation

The area of the four walls of a cuboid is equivalent to its lateral surface area, which is determined by the formula 2h(l + b), where l represents the length, b represents the breadth, and h represents the height. Given that the area is 57 square meters, the length is 5.5 meters, and the height is 3 meters, the equation is 57 = 2 × 3 × (5.5 + b). Simplifying the expression, we have 57 = 6 × (5.5 + b). Dividing both sides of the equation by 6 results in 9.5 = 5.5 + b. By subtracting 5.5 from both sides, the breadth b is calculated as 9.5 - 5.5 = 4 meters.

Explanation

The standard trigonometric values for the angles involved are cot 60° = (1)/(√(3)) and sec 45° = √2. Substituting these values into the given expression cot 60° - sec 45° results in:

(1)/(√(3)) - √2

To simplify this subtraction, find a common denominator:

1 - √2 · √3{√3} = 1 - √6{√3}

To match the form of the options, rationalize the denominator by multiplying both the numerator and the denominator by √3:

(1 - √6) · √3{√3 · √3} = √3 - √(18){3}

Using the property √(18) = √(9 × 2) = 3√2, the expression simplifies to:

√3 - 3√2{3}