SSC CGL Tier 1 2014 Quantitative Aptitude Questions with Solutions

Explanation

The relationship between the Selling Price (SP) and the Marked Price (MP) after successive discounts is given by the formula SP = MP × (1 - d₁) × (1 - d₂). Converting the given discount percentages into fractions simplifies the calculation: 12½% = ⅛ and 10% = ⅒.

The multipliers for the discounts are (1 - ⅛) = ⅞ and (1 - ⅒) = ⁹⁄₁₀. Substituting the given Selling Price of Rs. 6,300 into the equation:

6,300 = MP × ⅞ × ⁹⁄₁₀

6,300 = MP × ⁶³⁄₈₀

To find the Marked Price, rearrange the equation:

MP = (6,300 × 80)/(63)

MP = 100 × 80 = 8,000

Explanation

The total sale (revenue) is the product of the ticket price and the number of visitors. The relationship between percentage changes is given by the formula for successive changes: Total % Change = x + y + ˣʸ⁄₁₀₀, where x is the percentage change in price and y is the percentage change in the number of visitors.

Given that the discount is 20%, we have x = -20. The total increase in sales is 28%. Substituting these values into the formula:

28 = -20 + y + (-20 × y)/(100)

28 + 20 = y - 0.2y

48 = 0.8y

y = (48)/(0.8) = 60

Therefore, the percentage of increase in the number of visitors is 60%.

Explanation

A proportion exists between two ratios a:b and c:d if they are equal, which is mathematically expressed as (a)/(b) = (c)/(d) or ad = bc. In the case of 12:9 = 16:12, we can simplify both ratios to their lowest terms. Dividing the terms of the first ratio by their greatest common divisor, 3, we get ¹²⁄₉ = ⁴⁄₃. Similarly, dividing the terms of the second ratio by their greatest common divisor, 4, we get ¹⁶⁄₁₂ = ⁴⁄₃. Alternatively, using the product of extremes and means, 12 × 12 = 144 and 9 × 16 = 144. Since the products are equal, the ratios form a correct proportion.

Explanation

The volume of a hemispherical bowl is given by the formula V_bₒ_wₗ = ⅔ π R³, where R is the radius of the sphere. The volume of a conical vessel is calculated using V_cₒₙₑ = ⅓ π r² h, where r = 6 cm and h = 12 cm. The problem states that 8 bowlfuls are required to fill the conical vessel, which gives the relation 8 × V_bₒ_wₗ = V_cₒₙₑ.

Substituting the known values into the equation: 8 × ⅔ π R³ = ⅓ π (6)² (12) ¹⁶⁄₃ π R³ = ⅓ π × 36 × 12

By canceling (π)/(3) from both sides, the equation simplifies to: 16 R³ = 36 × 12 16 R³ = 432 R³ = ⁴³²⁄₁₆ R³ = 27

Taking the cube root of both sides, we get R = 3 cm. Therefore, the radius of the sphere is 3 cm.

Explanation

Let the common perimeter for all three shapes be P. For a given perimeter, the area of a regular polygon increases as the number of sides increases, and the circle (which can be considered a polygon with infinite sides) provides the maximum possible area.

For a circle with circumference P, 2π r = P, so r = (P)/(2π). The area C = π r² = π ((P)/(2π))² = (P²)/(4π) ≈ (P²)/(12.57).

For a square with perimeter P, the side s = (P)/(4). The area S = s² = ((P)/(4))² = (P²)/(16).

For an equilateral triangle with perimeter P, the side a = (P)/(3). The area T = √3{4} a² = √3{4} ((P)/(3))² = √3 P²{36} ≈ (1.732 P²)/(36) ≈ (P²)/(20.78).

Comparing the denominators, 12.57 < 16 < 20.78. Since the numerators are identical, a smaller denominator results in a larger value. Thus, C > S > T is the correct relationship.

Explanation

The discount amount is the difference between the list price and the price paid by the customer: Discount = 440 - 396 = 44 The discount rate is calculated using the formula: Discount Rate = (Discount)/(List Price) × 100 Substituting the given values into the formula: Discount Rate = ⁴⁴⁄₄₄₀ × 100 = ⅒ × 100 = 10%

Explanation

Let Nisha buy n oranges of each variety. The cost price of the first variety is ½ rupee per orange, and the cost price of the second variety is ⅓ rupee per orange. For a total of 2n oranges, the total cost price is n · ½ + n · ⅓ = ⁵ⁿ⁄₆ rupees.

To find the cost price for one dozen (12 oranges), we set 2n = 12, which gives n = 6. Substituting this into the total cost price formula, the cost price for 12 oranges is (5 · 6)/(6) = 5 rupees.

To achieve a profit of 20%, the selling price is calculated using the formula SP = CP × (1 + (Profit%)/(100)). Substituting the cost price of 5 rupees and the profit margin of 20%, we get SP = 5 × (1 + ²⁰⁄₁₀₀) = 5 × 1.2 = 6 rupees.

Explanation

If a value is R% more than a second value, the percentage by which the second value is less than the first is calculated using the formula [R / (100 + R)] × 100%. Given that A's salary is 50% more than B's, we substitute R = 50 into the formula. This results in [50 / (100 + 50)] × 100 = (50 / 150) × 100. Simplifying the fraction gives (1 / 3) × 100, which equals 33 ⅓%.

Alternatively, if we assume B's salary is 100 units, then A's salary is 150 units (since it is 50% higher). The difference between their salaries is 50 units. To find the percentage by which B's salary is less than A's, we compare this difference to A's salary: (50 / 150) × 100. This simplifies to 100 / 3, resulting in 33 ⅓%.

Explanation

When two individuals move toward each other from two different points, their relative speed is the sum of their individual speeds. In this case, the relative speed is 4 km/hr + 6 km/hr = 10 km/hr.

The time taken to meet is calculated by dividing the total distance between them by their relative speed. Time = Distance / Relative Speed = 20 km / 10 km/hr = 2 hours.

Since both A and B start walking toward each other at 7:00 a.m., they will meet 2 hours later. Meeting time = 7:00 a.m. + 2 hours = 9:00 a.m.

Explanation

The speed of an individual is the product of the number of steps taken in a given time and the distance covered in each step. Let the distance of one step of the policeman be Lₚ and that of the thief be Lₜ. According to the problem, 5 Lₚ = 7 Lₜ, which implies the ratio of the lengths of their steps is Lₚ / Lₜ = 7/5.

In the time the thief takes 10 steps, the policeman takes 8 steps. Let nₚ = 8 and nₜ = 10. The ratio of their speeds is calculated as:

Speedₚₒₗᵢ_cₑₘₐₙ{Speedₜₕᵢₑ_f} = (nₚ × Lₚ)/(nₜ × Lₜ) = ⁸⁄₁₀ × ⁷⁄₅

Speedₚₒₗᵢ_cₑₘₐₙ{Speedₜₕᵢₑ_f} = ⁵⁶⁄₅₀ = ²⁸⁄₂₅

Thus, the ratio of the speeds of the policeman and the thief is 28:25.

Explanation

The slant height l of the cone is calculated using the Pythagorean theorem because the height, radius, and slant height form a right-angled triangle, where l = √(r² + h²). Substituting the given base radius r = 7 cm and height h = 24 cm, we find l = √(7² + 24²) = √(49 + 576) = √(625) = 25 cm. The area of the iron sheet required to prepare the cone corresponds to its total surface area, which includes the curved surface area and the base area, given by the formula A = π r(l + r). Substituting the known values into the formula yields A = ²²⁄₇ × 7 × (25 + 7) = 22 × 32 = 704 cm².

Explanation

A prime number is defined as a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. The sequence of prime numbers begins with 2, which is followed by 3, 5, 7, 11, and so forth. Because 2 is the first natural number to satisfy the criteria of having exactly two divisors and being greater than 1, it is the least prime number. It is also the only even prime number in the set of all primes.

Explanation

In problems where the difference between each divisor and its corresponding remainder is constant, the required number is found using the formula LCM(divisors) - k, where k is that constant difference. Calculating the differences for the given sets: 12 - 8 = 4, 18 - 14 = 4, 36 - 32 = 4, and 45 - 41 = 4. The constant difference is k = 4.

The Least Common Multiple (LCM) of the divisors 12, 18, 36, and 45 is determined by their prime factors. The prime factorizations are 12 = 2² × 3, 18 = 2 × 3², 36 = 2² × 3², and 45 = 3² × 5. Taking the highest power of each prime factor present in these numbers, the LCM = 2² × 3² × 5, which simplifies to 4 × 9 × 5 = 180.

The final answer is obtained by subtracting the constant difference from the calculated LCM: 180 - 4 = 176. This number satisfies the condition of leaving the specified remainders when divided by each of the given divisors.

Explanation

Since the filling rate is constant, the time taken is directly proportional to the volume of the pool being filled. The remaining portion of the pool to be filled is 1 - ⅗ = ⅖.

Using the unitary method or direct proportion, if ⅗ of the capacity takes 8 hours, then the time x required to fill the remaining ⅖ capacity is calculated as: x = 8 × (2/5)/(3/5) = 8 × ⅔ = ¹⁶⁄₃ hours

To convert ¹⁶⁄₃ hours into hours and minutes, we perform the division: ¹⁶⁄₃ = 5 ⅓ hours Since 1 hour equals 60 minutes, the fractional part is: ⅓ × 60 = 20 minutes

The additional time required to finish filling the pool is 5 hours 20 minutes.

Explanation

The relationship between efficiency and the time taken to complete a piece of work is inversely proportional. Let the efficiency of P be 100%. Since Q is 50% more efficient than P, the efficiency of Q is 150%. The ratio of the efficiency of P to Q is 100:150, which simplifies to 2:3.

Because efficiency and time are inversely proportional (E₁ × T₁ = E₂ × T₂), the ratio of the time taken by P to the time taken by Q is the inverse of their efficiency ratio, which is 3:2. Given that P takes 9 days to complete the work, we can set up the proportion where 3 units of time equal 9 days. This means 1 unit equals 3 days. Therefore, the time taken by Q, which is represented by 2 units, is 2 × 3 = 6 days.

Explanation

The total work is equated using the product of the number of workers and the time taken: 16M × 15 = 24C × 20, where M is the efficiency of a man and C is the efficiency of a child. This gives 240M = 480C, which simplifies to 1M = 2C, meaning one man is as efficient as two children. The total work can be expressed as 480 units (in terms of child-days). The efficiency of the required group of 8 men and 8 children is calculated by converting men into children: 8M + 8C = 8(2C) + 8C = 16C + 8C = 24C. The time required to complete the work is (Total Work)/(Combined Efficiency) = (480C)/(24C) = 20 days. Since the problem defines that 24 children take 20 days, the equivalent group of 8 men and 8 children also takes exactly 20 days.

Explanation

To find the better investment, calculate the yield for each: the first stock yields 4/120 = 1/30 or approximately 3.33%, while the second stock yields 3/80 = 0.0375 or 3.75%. Since the second stock offers a higher rate of return on the investment, it is the better choice.

Explanation

Let the initial price of the item be 100. A raise of 20% increases the price by 0.20 × 100 = 20, making the new price 120. The raised portion is specifically the 20 units added to the original price. A discount of 25% is then applied only to this raised portion, which is calculated as 0.25 × 20 = 5. The final price of the item is the price after the raise minus this discount, resulting in 120 - 5 = 115. Comparing this final price to the initial price of 100, the net effect is an increase of 15, which is a 15% increase.

Explanation

Let S be the current sum of the ages of the two brothers. In 10 years, the sum of their ages increases by 20 because each brother ages by 10 years. According to the problem, this future sum is double the current sum, giving the relation S + 20 = 2S, which simplifies to S = 20.

The difference between their ages D is given as 8. The ages of two individuals can be found using the sum and difference formulas where the younger age is (S - D)/(2) and the elder age is (S + D)/(2). Substituting the known values, the younger brother's age is (20 - 8)/(2) = 6 years and the elder brother's age is (20 + 8)/(2) = 14 years. The ratio of the younger brother's age to the elder brother's age is 6:14, which simplifies to 3:7.

Explanation

For a sequence of n successive odd integers, the arithmetic mean is equal to the middle term. The average of the five scores is 185 / 5 = 37. In a set of 5 consecutive odd integers, the 3rd term represents the middle value. Since the common difference between successive odd integers is 2, the highest term, which is the 5th term, is calculated by adding two increments of 2 to the middle term. Computing 37 + (2 × 2) results in 41.

Explanation

The area A of an equilateral triangle with side length s is given by the formula A = √3{4}s². If the side is decreased by 2 cm, the new side length is s - 2, and the new area is A' = √3{4}(s - 2)².

The problem states that the decrease in area is 4√3 cm². Setting up the equation: √3{4}s² - √3{4}(s - 2)² = 4√3

Dividing both sides by √3{4} simplifies the equation: s² - (s - 2)² = 16

Expanding the term (s - 2)²: s² - (s² - 4s + 4) = 16 s² - s² + 4s - 4 = 16 4s = 20 s = 5

The length of each side of the triangle is 5 cm.

Explanation

When a rectangular sheet is rolled along its length to form a cylinder, the length of the sheet becomes the circumference of the cylinder's base, and the breadth of the sheet becomes the height of the cylinder.

Given length L = 12 cm and breadth B = 5 cm. Circumference 2π r = 12 ⟹ r = (12)/(2π) = (6)/(π) cm. Height h = 5 cm.

The volume of the cylinder is given by the formula V = π r² h. Substituting the values: V = π × ((6)/(π))² × 5 V = π × (36)/(π²) × 5 V = (180)/(π) cm³

Explanation

The total purchase amount is calculated by summing the individual prices of all the items bought by the customer. Total Amount = 650 + 500 + 65 = 1215 The merchant offers a flat discount of 10% on the items in the store. To find the total discount obtained by the customer, we apply this percentage to the total sum of the purchases. Total Discount = Total Amount × ¹⁰⁄₁₀₀ Total Discount = 1215 × 0.10 = 121.50

Explanation

To find the value of 8x³ + 27y³, use the algebraic identity for the sum of cubes: a³ + b³ = (a + b)³ - 3ab(a + b)

Let a = 2x and b = 3y. The expression to be evaluated is: 8x³ + 27y³ = (2x)³ + (3y)³

Substituting a and b into the identity: 8x³ + 27y³ = (2x + 3y)³ - 3(2x)(3y)(2x + 3y) 8x³ + 27y³ = (2x + 3y)³ - 18xy(2x + 3y)

Using the values 2x + 3y = 11/2 and xy = 5/6, substitute them into the formula: 8x³ + 27y³ = (11/2)³ - 18(5/6)(11/2)

Calculate the cube of the first term: (11/2)³ = 1331 / 8

Calculate the second term: 18 × (5/6) × (11/2) = 15 × (11/2) = 165 / 2

To subtract the terms, convert them to a common denominator of 8: 165/2 = (165 × 4) / (2 × 4) = 660 / 8

Perform the subtraction: 8x³ + 27y³ = 1331/8 - 660/8 = 671 / 8

Explanation

From x⁴ + 1/x⁴ = 119, we find x² + 1/x² = 11, which leads to (x - 1/x)² = 9, so x - 1/x = ±3. Using the identity x³ - 1/x³ = (x - 1/x)³ + 3(x - 1/x), substituting 3 gives 36 and substituting -3 gives -36, resulting in ±36.

Explanation

Let the two numbers be 3x and 4x, where x represents their highest common factor (HCF). The Least Common Multiple (LCM) of two numbers expressed as a ratio a:b is the product of the ratio terms and their HCF, such that LCM = a × b × x. Substituting the given values into this formula, we have 3 × 4 × x = 180, which simplifies to 12x = 180. Dividing both sides by 12, we find x = 15. The second number in the ratio is 4x, so its value is 4 × 15 = 60.

Explanation

The change in the mean age is determined by the net change in the sum of all ages divided by the total number of teachers. When a 60-year-old teacher is replaced by a 30-year-old teacher, the total sum of ages decreases by 60 - 30 = 30 years. With the total number of teachers remaining constant at 30, the change in the mean age is calculated as ⁻³⁰⁄₃₀ = -1 year. This reduction in the total sum results in the mean age of the group decreasing by exactly 1 year.

Explanation

The sum of the ages of A, B, and C is calculated as 3 × 84 = 252 years. When D joins the group, the total sum of the ages of A, B, C, and D becomes 4 × 80 = 320 years. The age of D is found by taking the difference between these two sums, which is 320 - 252 = 68 years. E's age is 4 years more than D, so E is 68 + 4 = 72 years old. The sum of the ages of the group consisting of B, C, D, and E is 4 × 78 = 312 years. By comparing the two four-person groups, we see that (A + B + C + D) - (B + C + D + E) = 320 - 312, which simplifies to A - E = 8. Substituting the value of E into the equation, A - 72 = 8, we find that the age of A is 80 years.

Explanation

The cost price (CP) of the article can be calculated using the formula CP = (SP × 100)/(100 - Loss%) . Given that the selling price (SP) is 450 and the loss is 10%, the calculation is:

CP = (450 × 100)/(100 - 10) = (450 × 100)/(90) = 500

Now, if the article is sold for 540, the new selling price is 540. Since the selling price is greater than the cost price, there is a gain. The gain percentage is calculated using the formula:

Gain% = (New SP - CP)/(CP) × 100

Substituting the values:

Gain% = (540 - 500)/(500) × 100 = ⁴⁰⁄₅₀₀ × 100 = 8%

Explanation

Let the initial amount of money be x. The loss is 20 ½%, which is equivalent to 20.5%. The money remaining after this loss is 100% - 20.5% = 79.5% of x.

The man then spends 80% of this remainder, which leaves him with 20% of that specific amount. Given that the final amount remaining is 159, we can express the situation with the following equation: x × (79.5)/(100) × ²⁰⁄₁₀₀ = 159

To simplify the calculation, express 79.5% as a fraction and 20% as ⅕: x × ¹⁵⁹⁄₂₀₀ × ⅕ = 159 x × ¹⁵⁹⁄₁₀₀₀ = 159

Solving for x: x = (159 × 1000)/(159) x = 1000

Explanation

The average pass rate is calculated by dividing the total number of students who passed by the total number of students who appeared across both years.

Total number of students who appeared = 80 + 60 = 140

Number of students who passed in the first year = 80 × 60% = 80 × ⁶⁰⁄₁₀₀ = 48

Number of students who passed in the second year = 60 × 80% = 60 × ⁸⁰⁄₁₀₀ = 48

Total number of students who passed = 48 + 48 = 96

The average pass percentage is given by the formula: Average Pass Rate = (Total Passed)/(Total Appeared) × 100

Substituting the values: Average Pass Rate = ⁹⁶⁄₁₄₀ × 100 = ²⁴⁄₃₅ × 100 = ⁴⁸⁰⁄₇% = 68 ⁴⁄₇%

Explanation

The speed of sound is determined by the formula v = ᵈ⁄ₜ, where d is the distance and t is the time interval. First, the distance must be converted from kilometers to meters to match the units of the options: 1.34 km = 1.34 × 1000 = 1340 meters. Substituting the given values into the formula, we get v = (1340 m)/(4 sec). Dividing 1340 by 4 yields a speed of 335 m/sec.

Explanation

The distance d between the house and the station can be calculated using the relationship between speed, distance, and the difference in time. When walking at v₁ = 5 km/hr, the person is 7 minutes late, and at v₂ = 6 km/hr, the person is 5 minutes early. The total time difference Δ t between these two scenarios is 7 + 5 = 12 minutes. To maintain consistency in units, this time is converted to hours: Δ t = 12/60 = 1/5 hours.

Using the standard formula for distance when two different speeds result in a known time difference: d = (v₁ × v₂)/(v₂ - v₁) × Δ t

Substituting the given values into the formula: d = (5 × 6)/(6 - 5) × ¹²⁄₆₀ d = ³⁰⁄₁ × ⅕ d = 6 km

This can also be verified using the time equation t = d/v. The difference in time taken at the two speeds is ᵈ⁄₅ - ᵈ⁄₆ = ¹²⁄₆₀. Solving for d gives (6d - 5d)/(30) = ⅕, which results in ᵈ⁄₃₀ = ⅕, leading to d = 6 km.

Explanation

The total amount A after compound interest is calculated as A = P + CI, where P is the principal and CI is the compound interest. Here, A = 1800 + 378 = 2178. The formula for the amount under compound interest is A = P(1 + r/100)ⁿ, where r is the annual interest rate and n is the time in years.

Substituting the given values into the formula: 2178 = 1800(1 + 10/100)ⁿ 2178 / 1800 = (1.1)ⁿ 1.21 = (1.1)ⁿ

Recognizing that 1.21 = 1.1², the equation becomes (1.1)² = (1.1)ⁿ. Comparing the powers on both sides, n = 2.

Explanation

The product can be efficiently calculated using the algebraic identity (x + a)(x - b) = x² + x(a - b) - ab. By taking a base of 200, the expression is rewritten as:

204 × 197 = (200 + 4)(200 - 3)

Applying the identity where x = 200, a = 4, and b = 3:

= 200² + 200(4 - 3) - (4 × 3) = 40000 + 200(1) - 12 = 40000 + 200 - 12 = 40200 - 12 = 40188

This can also be verified by using the distributive property: 200 × 197 + 4 × 197 = 39400 + 788 = 40188.

Explanation

The expression is simplified using the power of a power rule, which states that (aᵐ)ⁿ = aᵐ^·ⁿ. The n-th root of a value, ⁿ√a, can be rewritten as the fractional exponent a^¹^⁄^ₙ. Applying these properties to ((ⁿ√(x²))^ⁿ^⁄^₂)², the expression becomes (((x²)^¹^⁄^ₙ)^ⁿ^⁄^₂)². Multiplying the exponents sequentially yields: 2 · ¹⁄ₙ · ⁿ⁄₂ · 2 The product of the first three terms, 2 · ¹⁄ₙ · ⁿ⁄₂, simplifies to 1. This leaves x¹ as the term inside the outermost parentheses. Finally, raising x¹ to the power of 2 results in x².

Explanation

To find the value of n, express all terms in the equation as powers of base 3 using the exponential laws aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ. The equation is interpreted as (√3)⁵ × 9² = 3ⁿ × 3√3.

The term (√3)⁵ is rewritten as (3¹^/²)⁵ = 3⁵^/². The term 9² is rewritten as (3²)² = 3⁴. On the right side, the term 3√3 is rewritten as 3¹ × 3¹^/² = 3³^/².

Substituting these values into the equation: 3⁵^/² × 3⁴ = 3ⁿ × 3³^/²

Applying the product rule to combine terms on each side: 3⁽⁵^/²⁺⁴⁾ = 3⁽ⁿ⁺³^/²⁾ 3¹³^/² = 3ⁿ⁺³^/²

By equating the exponents since the bases are the same: 13/2 = n + 3/2 n = 13/2 - 3/2 n = 10/2 n = 5

Explanation

The given expression is p(p² + 3p + 3). By distributing p into the terms inside the parentheses, we get p³ + 3p² + 3p. This expression is very similar to the binomial expansion of (p + 1)³, which is p³ + 3p² + 3p + 1.

By adding and subtracting 1, the expression can be rewritten as (p³ + 3p² + 3p + 1) - 1, which simplifies to (p + 1)³ - 1.

Given that p = 99, we substitute this value into the simplified formula: (99 + 1)³ - 1 = 100³ - 1 1,000,000 - 1 = 999,999

Explanation

The expression sin² θ + cos² θ is a fundamental trigonometric identity that is always equal to 1 for any angle θ. Since 1 is provided as option A, it is the mathematically correct answer for the question as stated. While option B corresponds to the expansion of sin⁶ θ + cos⁶ θ, based on the literal text of the question, A is correct.

Explanation

The trigonometric equation tan θ + cot θ = 2 involves a function and its reciprocal. By substituting cot θ = (1)/(tan θ), the equation becomes tan θ + (1)/(tan θ) = 2. For any positive real number x, the equation x + ¹⁄ₓ = 2 has only one solution, which is x = 1. This can be shown by rearranging the equation into the quadratic form x² - 2x + 1 = 0, which simplifies to (x - 1)² = 0. Thus, tan θ = 1. Within the specified range 0 < θ < 90°, the tangent function is equal to 1 only when θ = 45°. Substituting this value back into the original expression confirms that tan 45° + cot 45° = 1 + 1 = 2.

Explanation

The expression 1 - (cos² A)/(1 - sin A) is simplified using the trigonometric identity cos² A = 1 - sin² A. Substituting this identity into the numerator, the expression becomes: 1 - (1 - sin² A)/(1 - sin A)

Applying the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b), the numerator 1 - sin² A can be factored into (1 - sin A)(1 + sin A). This substitution yields: 1 - ((1 - sin A)(1 + sin A))/(1 - sin A)

By cancelling the common factor (1 - sin A) from the numerator and the denominator, the expression simplifies to: 1 - (1 + sin A)

Distributing the negative sign through the parentheses results in: 1 - 1 - sin A

Performing the final subtraction leads to the result: -sin A

Explanation

The expression is based on the algebraic identity (a + b)² + (a - b)² = 2(a² + b²). Let a = 941 and b = 149. The given mathematical expression can be represented as:

((a + b)² + (a - b)²)/(a² + b²)

Substituting the identity into the numerator, the expression becomes:

(2(a² + b²))/(a² + b²)

The term (a² + b²) in the numerator cancels with the term (a² + b²) in the denominator:

(2(941² + 149²))/(941² + 149²) = 2

This result is independent of the specific numerical values of a and b, as long as the denominator is non-zero. By applying this simplification, the complex calculation reduces directly to the integer 2.

Explanation

Let h be the height of the tower. When the sun's altitude (angle of elevation) changes from ₁ to ₂, the change in the length of the shadow d is given by the formula d = h(cot₂ - cot₁).

Substituting the given values d = 40 m, ₂ = 30°, and ₁ = 60° into the formula:

40 = h(cot 30° - cot 60°)

40 = h(√3 - (1)/(√(3)))

40 = h((3 - 1)/(√(3)))

40 = (2h)/(√(3))

Rearranging to solve for h:

2h = 40√3

h = 20√3 m

Explanation

The height of the tower h is related to the length of its shadow s and the sun's altitude θ through the trigonometric relationship tan θ = h/s. Substituting the given shadow length of 9 m and the sun's altitude of 30° into the formula:

tan 30° = ʰ⁄₉

Using the trigonometric constant tan 30° = 1/√3, the equation becomes:

(1)/(√(3)) = ʰ⁄₉

Solving for h by multiplying both sides by 9:

h = (9)/(√(3))

To simplify the expression, the denominator is rationalized by multiplying both the numerator and the denominator by √3:

h = 9 × √3{√3 × √3} = 9√3{3} = 3√3 m.

Explanation

Expanding the left-hand side of the given equation (½(a - b))² + ab = p(a + b)², we get ¼(a - b)² + ab. Using the algebraic identity (a - b)² = a² - 2ab + b², the expression can be rewritten as ¼(a² - 2ab + b²) + ab.

To simplify, we find a common denominator for the terms on the left: (a² - 2ab + b² + 4ab)/(4) = (a² + 2ab + b²)/(4)

Recognizing that a² + 2ab + b² = (a + b)², the equation becomes: ((a + b)²)/(4) = p(a + b)²

Since it is given that a ≠ -b, the term (a + b)² is non-zero. Dividing both sides of the equation by (a + b)², we find p = ¼.

Explanation

To find the value of x² - y², we utilize the algebraic identity x² - y² = (x + y)(x - y).

First, determine the value of (x + y) by adding the two given equations: (999x + 888y) + (888x + 999y) = 1332 + 555 1887x + 1887y = 1887 Dividing the entire equation by 1887 yields x + y = 1.

Next, determine the value of (x - y) by subtracting the second equation from the first: (999x + 888y) - (888x + 999y) = 1332 - 555 111x - 111y = 777 Dividing the entire equation by 111 yields x - y = 7.

Substituting these results into the identity: x² - y² = (x + y)(x - y) = 1 × 7 = 7

Explanation

In similar triangles, the ratios of corresponding sides are equal. Based on the given angle equalities m∠ ACB = m∠ MNP and m∠ ABC = m∠ MPN, the vertices correspond as A ↔ M, B ↔ P, and C ↔ N. This establishes the similarity △ ABC ∼ △ MPN.

The ratio of corresponding sides is determined by: (AB)/(MP) = (BC)/(PN)

Substituting the given values into the formula: (2.25)/(4.5) = (BC)/(7.5)

Simplifying the ratio on the left side: ½ = (BC)/(7.5)

Solving for BC: BC = (7.5)/(2) = 3.75 cm

Explanation

In an equilateral triangle ABC with side length s, the mid-points D, E, and F of sides AB, BC, and AC create segments of length s/2. Specifically, BD = ½AB = s/2 and BE = ½BC = s/2. According to the Mid-point Theorem, the segment joining the mid-points of two sides of a triangle is parallel to the third side and half its length. Therefore, DF ∥ BC and DF = ½BC = s/2, while EF ∥ AB and EF = ½AB = s/2. In the quadrilateral BEFD, opposite sides are parallel (DF ∥ BE and EF ∥ BD), which defines a parallelogram. Since all four sides are equal in length (BD = BE = EF = DF = s/2), the quadrilateral is a rhombus. Because the internal angle ∠ B is 60° (not 90°), it cannot be a square or a rectangle.

Explanation

In a parallelogram, opposite angles are equal, so ∠ A = ∠ C. In a cyclic quadrilateral, the sum of opposite angles is 180°, so ∠ A + ∠ C = 180°. Substituting the first property into the second, we get ∠ A + ∠ A = 180°, which simplifies to 2∠ A = 180°. Therefore, ∠ A = 90°. This result implies that any cyclic parallelogram must be a rectangle.

Explanation

In △ OAC, the sides OA and OC are equal as they are radii of the same circle. This makes △ OAC an isosceles triangle, where the angles opposite to the equal sides are also equal, so ∠ OAC = ∠ OCA = 20°. Using the angle sum property of a triangle, the central angle ∠ AOC = 180° - (∠ OAC + ∠ OCA) = 180° - (20° + 20°) = 140°.

For a point B lying on the minor arc AC, ∠ ABC subtends the major arc AC. According to circle geometry theorems, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circumference. Therefore, ∠ ABC is half of the reflex angle ∠ AOC. Calculating the reflex angle, we get 360° - 140° = 220°. Consequently, the magnitude of ∠ ABC = ½ × 220° = 110°.