SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

Using the trigonometric identity Cosec² θ = 1 + Cot² θ, we substitute the given value Cot θ = 21/20:

Cosec² θ = 1 + (²¹⁄₂₀)²

Cosec² θ = 1 + ⁴⁴¹⁄₄₀₀

Cosec² θ = (400 + 441)/(400) = ⁸⁴¹⁄₄₀₀

Taking the square root of both sides gives Cosec θ = √(⁸⁴¹⁄₄₀₀) = ²⁹⁄₂₀.

Alternatively, this can be solved using the properties of a right-angled triangle. Since Cot θ = (Base)/(Perpendicular) = ²¹⁄₂₀, we can identify the sides as Base = 21 and Perpendicular = 20. The Hypotenuse is calculated using the Pythagorean theorem: H = √(21² + 20²) = √(441 + 400) = √(841) = 29. The value of Cosec θ is defined as the ratio of Hypotenuse to Perpendicular, which is ²⁹⁄₂₀.

Explanation

For an isosceles triangle with two equal sides a = 20 cm and a base b = 30 cm, the altitude h drawn to the base bisects it into two equal segments of 15 cm each. Using the Pythagorean theorem on one of the resulting right-angled triangles, the height is: h = √(a² - (ᵇ⁄₂)²) h = √(20² - 15²) = √(400 - 225) = √(175) h = √(25 × 7) = 5√7 cm

The area of the triangle is calculated using the formula: Area = ½ × base × height Area = ½ × 30 × 5√7 = 15 × 5√7 = 75√7 cm²

Alternatively, Heron's formula can be applied. First, find the semi-perimeter s: s = (20 + 20 + 30)/(2) = 35 Area = √(s(s-a)(s-a)(s-b)) Area = √(35(35-20)(35-20)(35-30)) Area = √(35 × 15 × 15 × 5) Area = 15 √(175) = 15 × 5√7 = 75√7 cm²

Explanation

To find the quotient, we perform the division of the dividend 2143 by the divisor 38.

We first divide the first three digits of the dividend, 214, by 38. Since 38 × 5 = 190 and 38 × 6 = 228, the first digit of the quotient is 5. Subtracting 190 from 214 leaves a remainder of 24.

Next, we bring down the last digit of the dividend, 3, to the remainder 24, making the new number 243. We then divide 243 by 38. Since 38 × 6 = 228 and 38 × 7 = 266, the second digit of the quotient is 6. Subtracting 228 from 243 leaves a final remainder of 15.

The division process can be expressed by the formula Dividend = (Divisor × Quotient) + Remainder, which in this case is 2143 = (38 × 56) + 15. Therefore, the integer quotient is 56.

Explanation

The time taken by B to complete the job alone is calculated using the work rate formula 1/T_B = 1/T_A₊_B - 1/T_A. Given that A and B together take 15 days and A alone takes 20 days, the equation becomes 1/T_B = 1/15 - 1/20. Using the least common multiple of 60, the calculation yields 1/T_B = (4 - 3) / 60 = 1/60, which means B would take 60 days to complete the entire job alone. To find the time required for B to complete half of the job, we divide the total time by 2, resulting in 60 / 2 = 30 days.

Explanation

In a rectangle, the diagonal forms the hypotenuse of a right-angled triangle with the two adjacent sides. By applying the Pythagoras theorem, d² = l² + w², where d is the diagonal, l is the length, and w is the width. Given d = 51 cm and one side w = 24 cm, the other side l is calculated as:

l = √(51² - 24²) l = √((51 + 24)(51 - 24)) l = √(75 × 27) l = √2025 = 45 cm

The area of the rectangle is the product of its length and width:

Area = l × w = 45 × 24 = 1080 sq cm.

Explanation

The Marked Price (MP) is determined by increasing the Cost Price (CP) by the markup percentage. Given CP = Rs 7500 and a markup of 60%: MP = CP × (1 + ⁶⁰⁄₁₀₀) = 7500 × 1.6 = 12000

The Selling Price (SP) is obtained after applying a 10% discount on the Marked Price: SP = MP × (1 - ¹⁰⁄₁₀₀) = 12000 × 0.9 = 10800

Alternatively, using a single step with multipliers: SP = CP × 1.60 × 0.90 SP = 7500 × 1.44 = 10800

Explanation

Let the number to be added to each of the four numbers be x. For the resulting numbers to be in proportion, the ratio of the first two numbers must equal the ratio of the last two numbers. The condition is expressed as:

(94 + x)/(24 + x) = (100 + x)/(26 + x)

To solve for x, cross-multiply the terms: (94 + x)(26 + x) = (100 + x)(24 + x)

Expanding both sides of the equation: 2444 + 94x + 26x + x² = 2400 + 100x + 24x + x²

Simplify the equation by canceling x² from both sides and combining like terms: 2444 + 120x = 2400 + 124x

Isolate the variable x by moving the terms: 124x - 120x = 2444 - 2400 4x = 44 x = 11

By adding 11 to each number, we get 105, 35, 111, and 37. The ratio ¹⁰⁵⁄₃₅ = 3 and the ratio ¹¹¹⁄₃₇ = 3. Since the ratios are equal, the numbers are in proportion.

Explanation

Let x be the average after 16 matches. Since the average increased by 3 runs after the 16th match, the average for the first 15 matches was x - 3. The total runs scored after 16 matches is given by 16x. This total can also be calculated by adding the runs scored in the 16th match to the total runs from the previous 15 matches:

16x = 15(x - 3) + 81

Expanding the equation to solve for x: 16x = 15x - 45 + 81 16x - 15x = 36 x = 36

This result can also be found by considering the distribution of runs. The 81 runs scored in the 16th match contributed to the new average and also provided the 3-run increase for each of the 15 previous matches. Therefore, the average after the 16th match is: 81 - (15 × 3) 81 - 45 = 36

Explanation

The Cost Price (CP) of 12 bananas is Rs 60, so the CP of a single banana is 60 / 12 = Rs 5. The Selling Price (SP) of 5 bananas is Rs 32, which means the SP of a single banana is 32 / 5 = Rs 6.4.

The gain per banana is calculated as SP - CP = 6.4 - 5 = Rs 1.4.

To calculate the gain percentage, use the formula: Gain% = (Gain)/(CP) × 100 Gain% = (1.4)/(5) × 100 = 1.4 × 20 = 28%

Alternatively, equate the quantity of bananas using the LCM of 12 and 5, which is 60. CP for 60 bananas = 60 × 5 = Rs 300 SP for 60 bananas = 12 × 32 = Rs 384 Gain% = (384 - 300)/(300) × 100 = ⁸⁴⁄₃ = 28%

Explanation

Let the radius of the circle be r. Since O is the center and R, P, Q are points on the circle, we have OR = OQ = OP = r. It is given that QS = OR, which implies QS = r.

In △ OQS, since OQ = QS = r, it is an isosceles triangle. Therefore, the base angles are equal: ∠ QOS = ∠ QSR = 30°

In △ OQS, the exterior angle ∠ OQP at vertex Q is equal to the sum of the interior opposite angles: ∠ OQP = ∠ QOS + ∠ QSR = 30° + 30° = 60°

In △ OPQ, since OP = OQ = r, it is an isosceles triangle. Therefore: ∠ OPQ = ∠ OQP = 60°

Since R, O, S is a line segment, ∠ POR is the exterior angle of △ OPS at vertex O. The exterior angle of a triangle is equal to the sum of the two opposite interior angles: ∠ POR = ∠ OPS + ∠ PSO ∠ POR = 60° + 30° = 90°

Explanation

Let the amount paid to labourer B be x. Since A is paid 180 percent of B, the amount paid to A is 1.8x. The total daily payment for both is Rs. 1120.

x + 1.8x = 1120 2.8x = 1120 x = (1120)/(2.8) x = ¹¹²⁰⁰⁄₂₈ x = 400

Explanation

The average speed for the entire journey is calculated as Total Distance / Total Time = 99 km / 9 hours = 11 km/hr. By applying the rule of alligation, the ratio of the time spent travelling on foot to the time spent on the bicycle is determined by the difference between the individual speeds and the mean speed.

Ratio of time (foot : bicycle) = (18 - 11) : (11 - 9) = 7 : 2.

Given the total time of the journey is 9 hours, the time spent travelling on foot corresponds to the 7 parts of the 7:2 ratio. Since the sum of the ratio parts 7 + 2 = 9 matches the total hours, the man travelled for 7 hours on foot. The distance travelled on foot is therefore Speed × Time = 9 km/hr × 7 hours = 63 km.

Explanation

The effective rate of compound interest for two years at a rate of r% per annum is given by the formula R = r + r + (r²)/(100). For a rate of 15%, the effective interest rate is 15 + 15 + (15²)/(100) = 30 + 2.25 = 32.25%.

Given that the total compound interest earned is Rs 20640, we use the relation CI = P × (R)/(100) to calculate the sum invested (P). Substituting the values into the equation:

20640 = P × (32.25)/(100) P = (20640 × 100)/(32.25) P = (2064000)/(32.25) P = 64000

Explanation

The given equation is ⁵ˣ⁄₂ - (7(6x - 3)/(2)){4} = ⅝. First, simplify the term inside the bracket by expressing 6x - ³⁄₂ as (12x - 3)/(2). Substituting this back into the equation, we get ⁵ˣ⁄₂ - (7(12x - 3)/(2)){4} = ⅝, which further simplifies to ⁵ˣ⁄₂ - (84x - 21)/(8) = ⅝.

To clear the denominators, multiply the entire equation by the common denominator 8. This yields 4(5x) - (84x - 21) = 5. Expanding the terms results in 20x - 84x + 21 = 5. Combining the x terms gives -64x + 21 = 5. Subtracting 21 from both sides leads to -64x = 5 - 21, which simplifies to -64x = -16. Solving for x gives x = ⁻¹⁶⁄₋₆₄, which reduces to x = ¼.

Explanation

The relationship between the given terms is defined by the algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b). Substituting the given values a³ - b³ = 91 and a - b = 1 into this formula yields:

91 = (1)³ + 3ab(1) 91 = 1 + 3ab 90 = 3ab ab = 30

Explanation

Let the required fraction be x. According to the problem statement, the difference between the fraction and twice its reciprocal is 7/15. This relationship can be expressed by the equation x - 2(1/x) = 7/15.

Substituting x = 5/3 into the equation: LHS = 5/3 - 2(3/5) LHS = 5/3 - 6/5

To solve this, we take the least common multiple of the denominators 3 and 5, which is 15: LHS = (25 - 18) / 15 LHS = 7/15

Since the calculated value matches the given difference of 7/15, the fraction 5/3 is correct. Alternatively, the condition leads to the quadratic equation 15x² - 7x - 30 = 0. Factoring this as (3x - 5)(5x + 6) = 0 yields the positive root x = 5/3.

Explanation

The sum of the first n terms of an arithmetic progression (AP) is calculated using the formula Sₙ = ⁿ⁄₂(a + l), where n is the number of terms, a is the first term, and l is the last term. In this problem, the given values are n = 17, a = -20, and l = 28. Substituting these values into the formula, we obtain S₁₇ = ¹⁷⁄₂(-20 + 28). Calculating the sum within the parentheses results in 8. The equation then simplifies to S₁₇ = ¹⁷⁄₂ × 8, which can be further reduced to 17 × 4 by dividing 8 by 2. Performing the final multiplication gives a total sum of 68. This direct formulaic approach is the standard method for finding the sum of a finite arithmetic sequence when the first term, last term, and total number of terms are provided.

Explanation

The reflection of a point (x, y) in a vertical line x = k results in a point (x', y') where the y-coordinate remains unchanged (y' = y) and the x-coordinate is determined by the formula x' = 2k - x. This is because the line x = k must be the midpoint of the segment connecting the original point and its reflection.

Given the point (-3, 1) and the line of reflection x = -2, we have x = -3, y = 1, and k = -2. Applying the formula for the x-coordinate:

x' = 2(-2) - (-3) x' = -4 + 3 = -1

Since the y-coordinate remains the same, y' = 1. Therefore, the reflected point is (-1, 1).

Explanation

The centroid G(x, y) of a triangle with vertices A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) is defined by the average of the coordinates of its vertices, expressed by the formulas x = (x₁ + x₂ + x₃)/(3) and y = (y₁ + y₂ + y₃)/(3). Given the centroid is (2, 2), vertex A is (7, -1), and vertex B is (1, 2), we can substitute these values into the formulas to find the coordinates of vertex C(x₃, y₃). For the x-coordinate, the equation is 2 = (7 + 1 + x₃)/(3), which simplifies to 6 = 8 + x₃, giving x₃ = -2. For the y-coordinate, the equation is 2 = (-1 + 2 + y₃)/(3), which simplifies to 6 = 1 + y₃, giving y₃ = 5.

Explanation

The slope m of a linear equation in the standard form Ax + By = C is determined by the formula m = -(A)/(B). For the given equation ax - 4y = -6, the coefficient of x is A = a and the coefficient of y is B = -4. Substituting these into the formula yields: m = -ᵃ⁄₋₄ = ᵃ⁄₄ The problem states that the slope is -³⁄₂. By setting the calculated slope equal to the given value, we get: ᵃ⁄₄ = -³⁄₂ Multiplying both sides by 4 to isolate a: a = -³⁄₂ × 4 a = -6

Explanation

The expression tan 6° tan 36° tan 84° tan 54° tan 45° can be simplified using the complementary angle identity tan(90° - θ) = cot θ and the reciprocal relation tan θ · cot θ = 1. Together, these imply that if A + B = 90°, then tan A · tan B = 1.

Identify the complementary pairs in the given expression: For the first pair, 6° + 84° = 90°, which gives tan 6° · tan 84° = tan 6° · cot 6° = 1. For the second pair, 36° + 54° = 90°, which gives tan 36° · tan 54° = tan 36° · cot 36° = 1.

The final term is tan 45°, which is a standard trigonometric value equal to 1.

Substituting these results into the product: (tan 6° · tan 84°) · (tan 36° · tan 54°) · tan 45° = 1 · 1 · 1 = 1

Explanation

In triangle ABC, since DE is parallel to BC, triangle ADE is similar to triangle ABC by the Angle-Angle (AA) similarity criterion. The ratio of the corresponding sides of these similar triangles is given by AD/AB. Given the ratio AD:DB = 1:2, we can express the total length of side AB as AD + DB, which equals 1 + 2 = 3 units. Consequently, the side ratio AD/AB is 1/3.

The property of similar triangles states that the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, the ratio of the area of triangle ADE to the area of triangle ABC is (1/3)² = 1/9.

Given that the area of triangle ABC is 45 sq cm, the area of triangle ADE is calculated as 1/9 × 45 = 5 sq cm. The area of the quadrilateral BDEC is the area remaining when triangle ADE is removed from triangle ABC. Thus, the area of quadrilateral BDEC = Area(Δ ABC) - Area(Δ ADE) = 45 - 5 = 40 sq cm.

Explanation

To find the value of the expression √2 sec 45° + (1)/(√(3)) tan 30°, we substitute the standard trigonometric ratios. The value of sec 45° is √2 because it is the reciprocal of cos 45° = (1)/(√(2)). The value of tan 30° is (1)/(√(3)).

Substituting these values into the given expression: √2 × √2 + (1)/(√(3)) × (1)/(√(3))

Simplifying the terms: (√2)² + ((1)/(√(3)))² = 2 + ⅓

Calculating the sum: ⁶⁄₃ + ⅓ = ⁷⁄₃

Explanation

In the right-angled triangle XYZ, the angle at Y is 90°, which makes ZX the hypotenuse. Given that m∠ Z = 30° and the hypotenuse ZX = 9 cm, we can find the length of the adjacent side YZ using the cosine trigonometric ratio.

The formula for the cosine of an angle in a right triangle is: cos θ = (Adjacent side)/(Hypotenuse)

Applying this to ∠ Z: cos 30° = (YZ)/(ZX)

Substituting the known values cos 30° = √3{2} and ZX = 9: √3{2} = (YZ)/(9)

Solving for YZ: YZ = 9 × √3{2} = 9√3{2} cm

Explanation

In a right-angled triangle, cos θ is the ratio of the base to the hypotenuse. Given cos θ = ¹⁵⁄₁₇, we can consider the base b = 15 and the hypotenuse h = 17. Using the Pythagorean theorem, the perpendicular p is calculated as p = √(h² - b²). Substituting the given values, p = √(17² - 15²) = √(289 - 225) = √(64) = 8. The trigonometric function csc θ is defined as the ratio of the hypotenuse to the perpendicular, which is ʰ⁄ₚ. Substituting the values, csc θ = ¹⁷⁄₈.

Explanation

The sum of the squares of real numbers is equal to zero if and only if each individual term is zero. Applying this property to the equation (x - 2)² + (y + 3)² + (z - 15)² = 0, we can set each squared expression to zero:

x - 2 = 0 ⟹ x = 2 y + 3 = 0 ⟹ y = -3 z - 15 = 0 ⟹ z = 15

Now substitute these values into the required expression x + y + z - 5:

2 + (-3) + 15 - 5 = 2 - 3 + 15 - 5 = 9

Explanation

To find the Least Common Multiple (LCM) of 64 and 56, we can utilize the prime factorization method. First, express each number as a product of its prime factors: 64 = 2⁶ and 56 = 2³ × 7. The LCM is determined by taking the highest power of each prime factor that appears in either number. In this case, the prime factors involved are 2 and 7. The highest power of 2 is 2⁶ and the highest power of 7 is 7¹. Therefore, LCM = 2⁶ × 7¹ = 64 × 7 = 448.

Alternatively, the relationship between the Least Common Multiple and the Highest Common Factor (HCF) can be used, defined by the formula LCM(a, b) = (a × b)/(HCF(a, b)). The HCF of 64 and 56 is 8, as it is the largest integer that divides both numbers without a remainder. Applying the formula, LCM = (64 × 56)/(8) = 64 × 7 = 448. Both methods consistently yield the same result.

Explanation

The time taken by A and B together to complete the job is Tₐ_b = 6.75 days, which can be written in fractional form as 27/4 days. The time taken by A alone is Tₐ = 9 days. Using the work-rate formula where the sum of individual rates equals the combined rate:

1/Tₐ + 1/T_b = 1/Tₐ_b

Substituting the known values into the equation:

1/9 + 1/T_b = 1/(27/4)

1/9 + 1/T_b = 4/27

To solve for the rate of B, subtract the rate of A from the combined rate:

1/T_b = 4/27 - 1/9

By converting 1/9 to a fraction with a denominator of 27, we get 3/27. The calculation becomes:

1/T_b = 4/27 - 3/27 = 1/27

Since the rate of work for B is 1/27 of the job per day, the total time required for B to finish the job alone is the reciprocal of the rate, which is 27 days.

Explanation

The surface area A of a sphere is calculated using the formula A = 4π r², where r is the radius. Substituting the given surface area of 154 sq cm and taking π = 22/7, we have 4 × (22/7) × r² = 154. By rearranging the terms to solve for r², we get r² = (154 × 7) / (4 × 22). Since 154 divided by 22 is 7, the expression simplifies to r² = (7 × 7) / 4, or r² = 49/4. Taking the square root of both sides gives r = 7/2 = 3.5 cm. The diameter of the sphere is twice the radius, so d = 2 × 3.5 = 7 cm.

Explanation

The total Marked Price (MRP) of the items purchased is the sum of the price of 1 shirt and 3 pairs of trousers: Total MRP = 100 + (3 × 300) = 100 + 900 = 1000

The discount amount for the shirt is 10% of 100, which equals 10. The total discount for 3 pairs of trousers is 20% of the total cost of the trousers, which is 20% of 900: Total Discount on Trousers = ²⁰⁄₁₀₀ × 900 = 180

The total discount amount received by Pritam is the sum of the individual discounts: Total Discount = 10 + 180 = 190

The effective discount percentage is determined by the ratio of the total discount to the total MRP: Effective Discount % = (Total Discount)/(Total MRP) × 100 Effective Discount % = ¹⁹⁰⁄₁₀₀₀ × 100 = 19%

Explanation

Let B's wealth be represented by B. The problem states that A's wealth A = ⁵⁄₇ B and C's wealth C = ¹⁰⁄₇ B. To determine the ratio of C's wealth to A's wealth, we express the relationship as (C)/(A). By substituting the given values, we get (10)/(7) B{⁵⁄₇ B}. In this expression, the common variable B and the denominator 7 in both the numerator and the denominator cancel out, simplifying the calculation to ¹⁰⁄₅. Reducing this fraction yields 2/1, which corresponds to the ratio 2:1.

Explanation

For n consecutive odd numbers with an average A, the largest number in the set can be calculated using the formula A + (n - 1). In this problem, the average A is 64 and the number of terms n is 4. Substituting these values into the formula gives 64 + (4 - 1) = 64 + 3 = 67.

This relationship exists because the average of an even number of consecutive odd integers is the arithmetic mean of the two middle terms. With an average of 64, the two central odd numbers must be 64 - 1 = 63 and 64 + 1 = 65. Following the sequence of consecutive odd numbers, the full set is 61, 63, 65, and 67. Therefore, the largest number among them is 67.

Explanation

To find the cost price (CP) when the selling price (SP) is Rs 960 and the gain is 20%, we use the formula: CP = SP × (100)/(100 + Gain%) CP = 960 × ¹⁰⁰⁄₁₂₀ = 8 × 100 = Rs 800

When the new selling price is Rs 1120, the profit is calculated as: Profit = New SP - CP = 1120 - 800 = Rs 320

The profit percentage is then: Profit% = (Profit)/(CP) × 100 = ³²⁰⁄₈₀₀ × 100 = ³²⁰⁄₈ = 40%

A faster way is to use the direct ratio: if 120% of CP equals 960, then 1% of CP equals 960/120 = 8. For a selling price of 1120, the corresponding percentage is 1120 / 8 = 140%. This represents a profit of 140% - 100% = 40%.

Explanation

Let the total wealth of the man be W. The amount willed to charity is 25% of W, which is 0.25W. The remaining wealth willed to the family is 100% - 25% = 75% of W, which is 0.75W. The question asks for the family's share expressed as a percentage of the charity's share. This is calculated using the formula: Percentage = (Family Wealth)/(Charity Wealth) × 100 Substituting the values: Percentage = (0.75W)/(0.25W) × 100 Percentage = 3 × 100 = 300

Explanation

Since the time taken in both scenarios is the same, distance is directly proportional to speed (d ∝ s). The ratio of the actual speed to the hypothetical speed is 9 : 15, which simplifies to 3 : 5. Because time is constant, the ratio of the distances travelled is also 3 : 5.

Let the actual distance be 3x and the hypothetical distance be 5x. According to the problem, the difference between these distances is 3 km. We can set up the equation 5x - 3x = 3, which simplifies to 2x = 3. Solving for x gives x = 1.5. The actual distance travelled by the person is 3x, which is 3 × 1.5 = 4.5 km.

Explanation

The amount A received after two years with successive interest rates r₁ and r₂ is calculated using the formula A = x(1 + r₁{100})(1 + r₂{100}). Given the first year rate is 5 percent and the second year rate is 15 percent, the equation becomes:

9660 = x(1 + ⁵⁄₁₀₀)(1 + ¹⁵⁄₁₀₀)

9660 = x(²¹⁄₂₀)(²³⁄₂₀)

9660 = x(⁴⁸³⁄₄₀₀)

To solve for x, rearrange the terms:

x = (9660 × 400)/(483)

Since 9660 divided by 483 equals 20, the calculation simplifies to:

x = 20 × 400 = 8000

The initial amount invested by Ganesh was Rs 8000.

Explanation

The given equation is (4(2x)/(5) - ³⁄₂){3} + ⁷⁄₅ = ³⁷⁄₅. To solve for x, first subtract ⁷⁄₅ from both sides of the equation: (4(2x)/(5) - ³⁄₂){3} = (37 - 7)/(5) = ³⁰⁄₅ = 6. Multiply both sides by 3 to eliminate the denominator, resulting in 4(²ˣ⁄₅ - ³⁄₂) = 18. Dividing both sides by 4 gives ²ˣ⁄₅ - ³⁄₂ = ¹⁸⁄₄ = 4.5. To isolate the term with x, add ³⁄₂ (which is 1.5) to both sides: ²ˣ⁄₅ = 4.5 + 1.5 = 6. Multiplying the equation by 5 yields 2x = 30, and dividing by 2 gives the final value x = 15.

Explanation

To find the value of a³ - b³, use the algebraic identity a³ - b³ = (a - b)³ + 3ab(a - b). Substitute the given values a - b = 4 and ab = -3 directly into the formula:

a³ - b³ = (4)³ + 3(-3)(4) a³ - b³ = 64 - 36 a³ - b³ = 28

Alternatively, the identity a³ - b³ = (a - b)(a² + ab + b²) can be used. First, find a² + b² using the relation (a - b)² = a² + b² - 2ab. Substituting the values gives 4² = a² + b² - 2(-3), which simplifies to 16 = a² + b² + 6, resulting in a² + b² = 10. Finally, a³ - b³ = 4(10 + (-3)) = 4(7) = 28.

Explanation

Let the fraction be x. According to the problem, the sum of twice the fraction and three times its reciprocal is 29/3. This relationship is represented by the equation 2x + ³⁄ₓ = ²⁹⁄₃. To solve for x, multiply the entire equation by the common denominator 3x to clear the fractions:

3x(2x) + 3x(³⁄ₓ) = 3x(²⁹⁄₃) 6x² + 9 = 29x

Rearranging the terms into a standard quadratic equation ax² + bx + c = 0 gives: 6x² - 29x + 9 = 0

Using the quadratic formula x = -b ± √(b²⁾⁻⁴ᵃᶜ{2a}: x = 29 ± √((-29)²⁾⁻⁴⁽⁶⁾⁽⁹⁾{2(6)} x = 29 ± √(841 - 216){12} x = 29 ± √(625){12} x = (29 ± 25)/(12)

This yields two possible values for x: x = (29 + 25)/(12) = ⁵⁴⁄₁₂ = ⁹⁄₂ x = (29 - 25)/(12) = ⁴⁄₁₂ = ⅓

Substituting x = 9/2 back into the original condition, we get 2(9/2) + 3(2/9) = 9 + 2/3 = 29/3, which satisfies the problem statement.

Explanation

The common difference d of an arithmetic progression is the change in term value divided by the change in its position, expressed as d = (aₙ - aₘ)/(n - m). Using the given 12ᵗʰ term a₁₂ = 5 and 7ᵗʰ term a₇ = -15:

d = (5 - (-15))/(12 - 7) = ²⁰⁄₅ = 4

To find the 16ᵗʰ term a₁₆, we apply the common difference to the 12ᵗʰ term using the relation a₁₆ = a₁₂ + (16 - 12)d:

a₁₆ = 5 + 4(4) a₁₆ = 5 + 16 = 21

Explanation

When a point (x, y) is reflected in a horizontal line y = k, the x-coordinate remains unchanged because the reflection occurs vertically. The line y = k acts as the midpoint between the original y-coordinate and the reflected y-coordinate y'. This relationship is expressed by the formula y' = 2k - y.

Given the point (4, -3) and the line of reflection y = -2, we substitute x = 4, y = -3, and k = -2 into the transformation equations. The x-coordinate of the reflection is x' = 4. For the y-coordinate, the calculation is y' = 2(-2) - (-3), which simplifies to y' = -4 + 3 = -1. The resulting reflected point is (4, -1).

Explanation

The distance d between two points (x₁, y₁) and (x₂, y₂) is determined by the distance formula d = √((x₂ - x₁)² + (y₂ - y₁)²). Substituting the given coordinates (2, 7) and (k, -5) with the distance 13, the equation becomes:

13 = √((k - 2)² + (-5 - 7)²)

Squaring both sides to eliminate the square root:

13² = (k - 2)² + (-12)² 169 = (k - 2)² + 144 169 - 144 = (k - 2)² 25 = (k - 2)²

Taking the square root of both sides gives two possible equations for k:

k - 2 = 5 or k - 2 = -5 k = 7 or k = -3

Comparing these results with the provided options, k = 7 is the matching value.

Explanation

A line perpendicular to the general form Ax + By + C = 0 is represented by the equation Bx - Ay + k = 0. Given the line 5x + 3y = 6, any line perpendicular to it will have the form 3x - 5y = k.

The Y-intercept of a line is the value of y when x = 0. The problem states that the Y-intercept is -3, which means the line passes through the point (0, -3). Substituting these coordinates into the perpendicular equation:

3(0) - 5(-3) = k 0 + 15 = k k = 15

Substituting the value of k back into the equation 3x - 5y = k results in 3x - 5y = 15.

Explanation

In △ ABC, since DE is parallel to BC, the smaller triangle △ ADE is similar to the larger triangle △ ABC by the AA (Angle-Angle) similarity criterion, as they share ∠A and have equal corresponding angles.

The ratio of the sides of the two triangles is AD/AB. Given the ratio AD : DB = 2 : 5, the side AB = AD + DB = 2 + 5 = 7 units. Thus, the ratio of corresponding sides AD/AB = 2/7.

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides: Area(△ ADE) / Area(△ ABC) = (AD/AB)² = (2/7)² = 4/49

Given that the area of △ ADE is 8 cm², we can find the area of △ ABC: 8 / Area(△ ABC) = 4/49 Area(△ ABC) = (8 × 49) / 4 = 98 cm²

The area of the quadrilateral BDEC is the difference between the area of the large triangle and the small triangle: Area(BDEC) = Area(△ ABC) - Area(△ ADE) = 98 - 8 = 90 cm²

Explanation

The standard trigonometric values for 30° are sin 30° = ½ and cos 30° = √3{2}. Substituting these values into the expression sin 30° - √2 cos 30° gives:

½ - √2 × √3{2}

Simplifying the second term by multiplying the square roots in the numerator:

½ - √(2 × 3){2} = ½ - √6{2}

Combining the terms over the common denominator results in:

1 - √6{2}

Explanation

In the right-angled triangle Δ XYZ, where ∠ Y = 90°, the side ZX acts as the hypotenuse and YZ is the side adjacent to ∠ Z. The relationship between these sides is defined by the cosine trigonometric ratio:

cos(Z) = (YZ)/(ZX)

Given that m∠ Z = 60° and the hypotenuse ZX = 9√3 cm, the formula is applied as follows:

cos(60°) = (YZ)/(9√(3))

Substituting the standard trigonometric value cos(60°) = ½ into the equation:

½ = (YZ)/(9√(3))

To solve for YZ, multiply both sides by 9√3:

YZ = 9√3{2} cm

The length of the side YZ is determined by applying the basic trigonometric property of a right-angled triangle.

Explanation

In a right-angled triangle, the secant of an angle θ is defined as the ratio of the hypotenuse to the base. Given sec θ = ¹³⁄₁₂, we can represent the hypotenuse h as 13 and the base b as 12. According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides: h² = p² + b², where p is the perpendicular or opposite side.

Substituting the known values, we get 13² = p² + 12², which simplifies to 169 = p² + 144. Solving for p, we find p² = 169 - 144 = 25, so p = 5. The sine of an angle θ is defined as the ratio of the perpendicular side to the hypotenuse. Therefore, sin θ = ᵖ⁄ₕ = ⁵⁄₁₃.

Explanation

To convert the decimal number 101 to binary, we represent it as a sum of powers of 2. The largest power of 2 less than 101 is 2⁶ = 64. Subtracting this from 101 leaves 37. The largest power of 2 less than 37 is 2⁵ = 32. Subtracting 32 from 37 leaves 5. The largest power of 2 less than 5 is 2² = 4. Subtracting 4 from 5 leaves 1, which is 2⁰.

Mathematically, 101 = 64 + 32 + 4 + 1. Expressing this in terms of powers of 2 gives: 101 = (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (0 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰).

By taking the coefficients of these powers from 2⁶ down to 2⁰, we obtain the binary sequence 1100101. Alternatively, using the repeated division-by-2 method, the remainders generated are 1, 0, 1, 0, 0, 1, 1 when read from the last quotient to the first remainder, confirming the result.

Explanation

Let the total work to be painted be the Least Common Multiple (LCM) of the given durations 42, 21, and 7, which is 42 units. The efficiency, or work done per day, for each person is calculated by dividing the total work by the number of days they take.

The efficiency of A is 42 / 42 = 1 unit per day. The efficiency of B is 42 / 21 = 2 units per day. The combined efficiency of A, B, and C working together is 42 / 7 = 6 units per day. To determine the efficiency of C alone, subtract the efficiencies of A and B from the combined efficiency: 6 - (1 + 2) = 3 units per day. The number of days required for C to complete the painting job alone is the total work divided by C's daily efficiency: 42 / 3 = 14 days.

Explanation

The circumference of a circle is given by the formula C = 2π r. Substituting the given value of 26.4 cm and using π = ²²⁄₇, we get:

2 × ²²⁄₇ × r = 26.4

⁴⁴⁄₇ × r = 26.4

r = (26.4 × 7)/(44)

r = 0.6 × 7 = 4.2 cm

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

A = ²²⁄₇ × (4.2)²

A = ²²⁄₇ × 4.2 × 4.2

A = 22 × 0.6 × 4.2

A = 13.2 × 4.2 = 55.44 sq cm