SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The area of an equilateral triangle with side a = 24 cm is given by the formula A_T = √3{4} a². Substituting the given values, A_T = (1.732)/(4) × 24² = 1.732 × 144 = 249.408 cm².

The radius r of the inscribed circle in an equilateral triangle is r = (a)/(2√(3)). For a side of 24 cm, r = (24)/(2√(3)) = (12)/(√(3)) = 4√3 cm. The area of this inscribed circle is A_C = π r² = ²²⁄₇ × (4√3)² = ²²⁄₇ × 48 = ¹⁰⁵⁶⁄₇ ≈ 150.857 cm².

The area of the remaining portion of the triangle is the difference between the area of the triangle and the area of the circle: 249.408 - 150.857 = 98.551 cm². This value is approximately 98.5 cm².

Explanation

The length of the shadow s of a pole with height h when the Sun is at an elevation θ is determined by the formula s = h cot θ. Given the pole height h = 15 m, we calculate the lengths of the shadows for both angles of elevation.

When the elevation of the Sun is 30°, the shadow length s₁ is 15 cot 30°. Since cot 30° = √3, the length s₁ = 15√3 m.

When the elevation of the Sun increases to 60°, the shadow length s₂ is 15 cot 60°. Since cot 60° = (1)/(√(3)), the length s₂ = (15)/(√(3)). Multiplying the numerator and denominator by √3 to rationalize it gives s₂ = 15√3{3} = 5√3 m.

The difference between the lengths of the shadows is s₁ - s₂ = 15√3 - 5√3 = 10√3 m.

Explanation

When interest is compounded semi-annually, the annual interest rate is halved and the number of compounding periods is doubled. Given the principal P = Rs. 12500, the annual rate R = 12%, and the time T = 1 year, the rate per six-month period is r = 12% / 2 = 6%, and the number of periods is n = 2.

The interest earned (CI) is calculated using the formula: CI = P [(1 + r/100)ⁿ - 1] CI = 12500 [(1 + 6/100)² - 1] CI = 12500 [(1.06)² - 1] CI = 12500 [1.1236 - 1] CI = 12500 × 0.1236 CI = 1545

Alternatively, the interest for the first 6 months is 6% of 12500, which equals Rs. 750. For the second 6 months, the interest is calculated on the amount at the end of the first period, which is 12500 + 750 = 13250. The interest for the second period is 6% of 13250, which equals Rs. 795. The total interest earned over the year is 750 + 795 = 1545.

Explanation

To find the smallest value of α such that α, β > 0, we consider the equations for the given trigonometric ratios:

tan(α - β) = 1 ⟹ α - β = 45°, 225°, … or 180° n + 45°. sec(α + β) = 2/√3 ⟹ cos(α + β) = √3/2 ⟹ α + β = 30°, 330°, 390°, … or 360° m ± 30°.

Since α and β are positive, it must hold that α + β > α - β (because β > 0) and α + β > β - α (because α > 0). This implies α + β > |α - β|.

Taking the smallest positive value for the difference, α - β = 45°. For this value, α + β must be greater than 45°. The smallest value in the set {30°, 330°, 390°, …} that satisfies α + β > 45° is 330°.

Solving the resulting system of linear equations: α + β = 330° α - β = 45°

Adding the two equations: 2α = 330° + 45° = 375° α = 375°/2 = 187.5°

This value is expressed in the options as 187½ or 187.5. Subtracting the equations gives 2β = 285°, which yields β = 142.5°. Since both α and β are positive and satisfy the original equations, the result is consistent.

Explanation

The Uber Cup, officially known as the World Team Championships for Women, is the premier international badminton tournament contested by women's national teams. The competition was established in 1956 after being proposed by Betty Uber, a former British badminton player who sought to create a women's equivalent to the men's Thomas Cup. Organized by the Badminton World Federation (BWF), the event is held biennially and remains one of the most prestigious team titles in global badminton, determining the world champion nation in the women's category.

Explanation

Onions store their food reserves in the form of sugars, specifically fructans, within their fleshy leaf scales. Unlike most other plants that store energy as starch, onions contain very little to no starch, and cellulose serves only as a structural component of the cell walls.

Explanation

Let the required number be x. The problem states that ⅘ of the number exceeds ¾ of the number by 8. This relationship is represented by the equation:

⅘x - ¾x = 8

To solve for x, find the common denominator for the fractions, which is 20:

(16x - 15x)/(20) = 8

ˣ⁄₂₀ = 8

x = 8 × 20

x = 160

Using the fractional difference method, the difference between the two parts is ⅘ - ¾ = (16 - 15)/(20) = ¹⁄₂₀. Since ¹⁄₂₀ of the number is equal to 8, the total value is 8 × 20 = 160.

Explanation

The Marked Price (MP) of the shirt is the total price before the discount was applied, which is found by adding the Selling Price (SP) and the discount amount. MP = SP + Discount = 546 + 109 = 655

The rate of discount is calculated as the percentage of the discount relative to the Marked Price using the following formula: Rate of Discount = (Discount)/(Marked Price) × 100

Substituting the given values into the formula: Rate of Discount = ¹⁰⁹⁄₆₅₅ × 100 ≈ 16.64%

This result is approximately 16%.

Explanation

Let the two numbers be x and 5x based on the given ratio 1:5. The product of these numbers is given as 320.

x × 5x = 320 5x² = 320 x² = ³²⁰⁄₅ = 64 x = √(64) = 8

The two numbers are 8 and 5 × 8 = 40. The squares of these numbers are 8² = 64 and 40² = 1600. The difference between the squares is:

1600 - 64 = 1536

Alternatively, using the expression for the difference of squares: (5x)² - x² = 25x² - x² = 24x² Substituting x² = 64: 24 × 64 = 1536

Explanation

The average age is calculated by dividing the total age of all individuals by the total number of individuals. The total age of boys is bx and the total age of girls is ay, making the sum bx + ay, while the total population is x + y. Therefore, the average age is (bx + ay) / (x + y).

Explanation

The total cost price for the man includes the initial purchase price and the expenditure on repairs. The cost price per machine is calculated by subtracting the profit per machine from the selling price per machine: CPᵤₙᵢₜ = SPᵤₙᵢₜ - Profitᵤₙᵢₜ = 300 - 150 = 150 rupees.

For all 30 machines, the total cost price is 30 × 150 = 4500 rupees. Given that the initial purchase price for the 30 defective machines was 1000 rupees, the amount spent on repairs is the difference between the total cost price and the purchase price: 4500 - 1000 = 3500 rupees.

Explanation

Given that x is 5 times longer than y, we can establish the relationship as x = 5y. To find the percentage by which y is less than x, we calculate the difference between the two values and divide it by the reference value x.

The difference between the values is x - y. Substituting x = 5y, the difference is 5y - y = 4y.

The percentage by which y is less than x is given by the formula: Percentage = (x - y)/(x) × 100

Substituting the known relationship: Percentage = (5y - y)/(5y) × 100 Percentage = (4y)/(5y) × 100 Percentage = ⅘ × 100 = 80%

Since the value of y is 20 percent of x, it is 80 percent less than x.

Explanation

The total amount required for the journey is calculated by first determining the quantity of fuel needed and then multiplying it by the cost per litre.

Fuel required (litres) = (Total Distance)/(Mileage) Fuel required = (540 km)/(45 km/litre) = 12 litres

Total cost (Rupees) = Fuel required × Cost per litre Total cost = 12 litres × 20 Rs./litre = 240 Rupees

Explanation

The relationship between the principal P and the final amount A under compound interest is given by the formula A = P(1 + ʳ⁄₁₀₀)ⁿ, where r is the annual interest rate and n is the number of years. In this case, A = 6655, r = 10%, and n = 3. Substituting these values into the formula gives 6655 = P(1 + ¹⁰⁄₁₀₀)³, which simplifies to 6655 = P(¹¹⁄₁₀)³. Calculating the cube of the fraction, we get 6655 = P × ¹³³¹⁄₁₀₀₀. To find the principal, we rearrange the equation to P = (6655 × 1000)/(1331). Since 1331 × 5 = 6655, the calculation simplifies to P = 5 × 1000 = 5000.

Explanation

The given equation x² - 4x + 1 = 0 can be rearranged by dividing both sides by x, which yields x - 4 + ¹⁄ₓ = 0, or x + ¹⁄ₓ = 4.

The expression to be evaluated is (x⁶ + 1)/(x³), which simplifies to x³ + (1)/(x³).

For any expression of the form x + ¹⁄ₓ = k, the value of x³ + (1)/(x³) is given by the formula k³ - 3k. Substituting k = 4 into this formula:

x³ + (1)/(x³) = 4³ - 3(4) x³ + (1)/(x³) = 64 - 12 = 52

Explanation

To find the value of (a + 2)² + (1)/((a + 2)²), we transform the given equation (a - 2) + (1)/((a + 2)) = -1 by adding 4 to both sides. This modification allows us to express the equation in terms of (a + 2): (a - 2) + 4 + (1)/(a + 2) = -1 + 4 (a + 2) + (1)/(a + 2) = 3

By letting x = a + 2, the equation takes the form x + ¹⁄ₓ = 3. We utilize the algebraic identity for the square of a sum, (x + ¹⁄ₓ)² = x² + (1)/(x²) + 2, to solve for the required expression. Squaring both sides of x + ¹⁄ₓ = 3 gives: (x + ¹⁄ₓ)² = 3² x² + (1)/(x²) + 2 = 9 x² + (1)/(x²) = 9 - 2 = 7

Substituting (a + 2) back for x, we determine that (a + 2)² + (1)/((a + 2)²) = 7.

Explanation

To find the value of x, square both sides of the equation √(1 + ˣ⁄₁₄₄) = ¹³⁄₁₂ to remove the square root:

1 + ˣ⁄₁₄₄ = ¹⁶⁹⁄₁₄₄

Subtract 1 from both sides to isolate the term with x:

ˣ⁄₁₄₄ = ¹⁶⁹⁄₁₄₄ - 1

ˣ⁄₁₄₄ = (169 - 144)/(144)

ˣ⁄₁₄₄ = ²⁵⁄₁₄₄

By comparing the numerators of the fractions on both sides, we find x = 25.

This problem follows a standard pattern where √(1 + (x)/(n²)) = (n + 1)/(n). In such cases, the value of x is always the difference of the squares of the numerator and the denominator:

x = 13² - 12² = 169 - 144 = 25

Explanation

In a triangle, the incenter is the point where the internal angle bisectors intersect. The angle subtended by any side of the triangle at the incenter is always equal to 90° plus half of the vertex angle opposite to that side.

For Δ PQR with incenter O, the relation between ∠ POR and its opposite vertex angle ∠ PQR is: ∠ POR = 90° + ½ ∠ PQR

Given that ∠ POR = 115°, we substitute this value into the formula: 115° = 90° + ½ ∠ PQR 115° - 90° = ½ ∠ PQR 25° = ½ ∠ PQR ∠ PQR = 25° × 2 ∠ PQR = 50°

Explanation

The area of a triangle A is related to its inradius r and semi-perimeter s by the formula A = r · s. The semi-perimeter is defined as half of the total perimeter P, which can be expressed as s = (P)/(2). By substituting the expression for the semi-perimeter into the area formula, we obtain A = r · (P)/(2).

To find the perimeter, the formula can be rearranged as: P = (2A)/(r)

Given that the area A = 34 sq. cm and the inradius r = 4 cm, the calculation is as follows: P = (2 × 34)/(4) P = ⁶⁸⁄₄ P = 17 cm

Explanation

Given the trigonometric equation tanθ + cotθ = 2, we can express cotθ as the reciprocal of tanθ, giving tanθ + (1)/(tanθ) = 2. To solve for tanθ, let x = tanθ. The equation becomes x + ¹⁄ₓ = 2, which leads to x² - 2x + 1 = 0. Factoring this quadratic equation gives (x - 1)² = 0, which implies x = 1. Thus, tanθ = 1 and cotθ = ¹⁄₁ = 1. Using the algebraic identity that any non-zero real number raised to the power of zero is 1, we evaluate the expression tan⁰θ + cot⁰θ = 1⁰ + 1⁰ = 1 + 1 = 2.

Explanation

The work done is constant and is proportional to the product of the number of men and the time taken, according to the formula M₁ D₁ = M₂ D₂. Let the initial number of men be x. We are given that these men can complete the work in D₁ = 60 days. When the number of men increases by 6, the new number of men becomes M₂ = x + 6, and the work is finished 20 days earlier, making the new time D₂ = 60 - 20 = 40 days.

Plugging these values into the formula: x × 60 = (x + 6) × 40 60x = 40x + 240 20x = 240 x = 12

The initial number of men working is 12.

Explanation

Given the equations a² = b + c, b² = c + a, and c² = a + b, we can evaluate the expression by manipulating each term to have a common denominator. For the first term, multiply the numerator and denominator by a: ¹⁄ₐ₊₁ = ᵃ⁄ₐ₍ₐ₊₁₎ = (a)/(a²+a)

Substituting the given identity a² = b + c into the denominator: (a)/(a²+a) = (a)/(b+c+a) = (a)/(a+b+c)

Similarly, for the second and third terms, multiply by b/b and c/c respectively and substitute the given identities b² = c + a and c² = a + b: (1)/(b+1) = (b)/(b²+b) = (b)/(c+a+b) = (b)/(a+b+c) (1)/(c+1) = (c)/(c²+c) = (c)/(a+b+c) = (c)/(a+b+c)

Summing these terms inside the parentheses: (a)/(a+b+c) + (b)/(a+b+c) + (c)/(a+b+c) = (a+b+c)/(a+b+c) = 1

The value of the full expression is 3 × (1) = 3. Alternatively, by symmetry, we can assume a = b = c. Then a² = 2a, which gives a = 2. Substituting a=b=c=2 into the expression yields 3 × (⅓ + ⅓ + ⅓) = 3 × 1 = 3.

Explanation

In triangle ABC, P is the midpoint of BC because CP = PB. Since triangles ABP and ABC share the same altitude from vertex A to the base BC, the ratio of their areas is equal to the ratio of their bases. Thus, Area(ABP) = (BP/BC) × Area(ABC). Substituting the values, Area(ABP) = (1/2) × 10.8 = 5.4 square cm.

In triangle ABP, point Q lies on AB such that 2AQ = QB. This implies that AB = AQ + QB = AQ + 2AQ = 3AQ, or AQ = (1/3)AB. Triangles APQ and ABP share the same altitude from vertex P to the base AB. Therefore, the ratio of their areas is proportional to the lengths of their bases along AB. Area(APQ) = (AQ/AB) × Area(ABP). Substituting the calculated area, Area(APQ) = (1/3) × 5.4 = 1.8 square cm.

Explanation

The height of a tower h can be determined from the height of a building H and the angles of depression to its top ₁ and bottom ₂ using the formula h = H (1 - (tan ₁)/(tan ₂)). Given H = 60 m, ₁ = 30°, and ₂ = 60°, we substitute these values into the equation:

h = 60 × (1 - (tan 30°)/(tan 60°))

h = 60 × (1 - 1/√3{√3})

h = 60 × (1 - ⅓)

h = 60 × ⅔

h = 40 m

Explanation

Let the radius of the larger circle be R = 12 cm and the radius of the inner concentric circle be r cm. The concentric circle divides the total area of the larger circle into two equal parts, which means the area of the inner circle is exactly half the area of the larger circle. Using the area formula A = π r², the relationship is given by:

π r² = ½ π R²

r² = R²{2}

r = (R)/(√(2))

Substituting the given value R = 12 cm into the equation:

r = (12)/(√(2))

By rationalizing the denominator, the expression becomes:

r = 12 × √2{2} = 6√2 cm.

Explanation

Using the complementary angle identity cos θ = sin(90° - θ), the first equation cos x = sin y implies x + y = 90°.

Using the identity tan θ = cot(90° - θ), the second equation cot(x - 40°) = tan(50° - y) can be rewritten as: cot(x - 40°) = cot(90° - (50° - y)) x - 40° = 40° + y x - y = 80°

Now, solve the system of linear equations:

1. x + y = 90°
2. x - y = 80°

Adding the two equations: 2x = 170° x = 85°

Substituting x = 85° into the first equation: 85° + y = 90° y = 5°

Explanation

The combined rate of Ram (R) and Hari (H) is R + H = (12 kg)/(2 days) = 6 kg/day. In the first 5 days, they worked together and cut a total of 5 × 6 = 30 kg of nuts. The remaining quantity of nuts to be cut is 58 - 30 = 28 kg. Since Ram took 8 additional days to cut the rest, Ram's individual rate is R = (28 kg)/(8 days) = 3.5 kg/day.

To find Hari's rate, we subtract Ram's rate from the combined rate: H = 6 - 3.5 = 2.5 kg/day. The time taken by Hari to cut 10 kg of nuts is calculated by dividing the target quantity by his daily rate: (10 kg)/(2.5 kg/day) = 4 days.

Explanation

The relation between the Marked Price (MP) and the Cost Price (CP) is given by the formula:

(MP)/(CP) = (100 + Gain%)/(100 - Discount%)

Substituting the given values where CP = 720, Gain = 15%, and Discount = 10%:

(MP)/(720) = (100 + 15)/(100 - 10)

(MP)/(720) = ¹¹⁵⁄₉₀

MP = ¹¹⁵⁄₉₀ × 720

MP = 115 × 8

MP = 920

Explanation

Let the total amount be T. The combined share of A and B expressed as a fraction of the total is 1/2 + 1/3 = 5/6. Since the amount is fully distributed among A, B, and C, the fraction received by C is 1 - 5/6 = 1/6. Given that C receives ₹ 1200, we can set up the equation T/6 = 1200, which results in a total amount T = 7200. A receives 1/2 of this total amount, so A's share is 7200 / 2 = 3600.

Explanation

The gain percentage is calculated using the formula Profit % = (SP - CP)/(CP) × 100. Based on the problem, the selling price SP is ⁸⁄₅ of the cost price CP, which can be represented as SP = ⁸⁄₅ CP. By substituting this value into the formula, we obtain (8)/(5)CP - CP{CP} × 100. This simplifies to (3)/(5)CP{CP} × 100 = ⅗ × 100 = 60%. Using a ratio method, if the cost price is assumed to be 5 units, the selling price becomes 8 units. The gain is 3 units, and the gain percentage is calculated as ⅗ × 100, which equals 60%.

Explanation

If the salary of Manoj is x% less than that of Subhash, then the percentage by which Subhash's salary is more than Manoj's is given by the formula:

Percentage More = (x)/(100 - x) × 100%

Given x = 40, substituting the value into the formula:

Percentage More = (40)/(100 - 40) × 100 = ⁴⁰⁄₆₀ × 100

Percentage More = ⅔ × 100 = 66 ⅔%

Alternatively, let the salary of Subhash be 100. Since Manoj's salary is 40% less, Manoj's salary is 100 - 40 = 60. The difference between their salaries is 40. To find how much Subhash's salary is more than Manoj's in percentage, we compare the difference to Manoj's salary: ⁴⁰⁄₆₀ × 100, which simplifies to 66.66% or 66 ⅔%.

Explanation

To find the least number that must be subtracted from 18265 to make it a perfect square, we use the long division method for calculating square roots. First, group the digits from right to left in pairs: 1, 82, 65.

The largest square less than or equal to 1 is 1² = 1. Subtracting 1 from 1 leaves 0. Bring down the next pair, 82. Double the current quotient (1) to get 2. We need to find a digit x such that 2x × x ≤ 82. Testing values, 23 × 3 = 69 while 24 × 4 = 96. Therefore, the next digit in the quotient is 3. Subtracting 69 from 82 leaves a remainder of 13.

Now, bring down the last pair, 65, to get 1365. Double the current quotient (13) to get 26. We need to find a digit x such that 26x × x ≤ 1365. Testing values, 265 × 5 = 1325 and 266 × 6 = 1596. The correct digit is 5. Subtracting 1325 from 1365 leaves a remainder of 40.

This remainder of 40 indicates that 18265 is 40 greater than the nearest perfect square, which is 135² = 18225. Thus, 40 is the least number that must be subtracted from 18265 to obtain a perfect square.

Explanation

Let the total work be the least common multiple of the given durations, which is LCM(36, 24, 18) = 72 units. The daily efficiency of the pairs is calculated by dividing the total work by their respective time taken. Thus, the efficiency of A + B is 72 / 36 = 2 units/day, the efficiency of B + C is 72 / 24 = 3 units/day, and the efficiency of A + C is 72 / 18 = 4 units/day.

By adding these efficiencies together, we get the combined efficiency of all pairs: (A + B) + (B + C) + (A + C) = 2 + 3 + 4 = 9 units/day. This sum represents 2(A + B + C), so the combined efficiency of A, B, and C working together is 9 / 2 = 4.5 units/day. To find the time required for the three of them to finish the work together, divide the total work by their combined efficiency: 72 / 4.5 = 16 days.

Explanation

A parallelogram is divided into two congruent triangles by its diagonal. The area of the parallelogram is therefore equal to twice the area of the triangle formed by its two adjacent sides and the diagonal.

Given the sides of the triangle are a = 21 cm, b = 20 cm, and c = 29 cm, we check if they form a Pythagorean triplet. Applying the Pythagorean theorem: 20² + 21² = 400 + 441 = 841 29² = 841

Since a² + b² = c², the triangle is a right-angled triangle with the sides of 20 cm and 21 cm acting as the base and height.

Area of the triangle = ½ × base × height = ½ × 20 × 21 = 210 sq. cm.

Area of the parallelogram = 2 × Area of the triangle = 2 × 210 = 420 sq. cm.

Explanation

The relationship between profit percentage (P), mark-up percentage (M), and discount percentage (D) is given by the formula P = M - D - (M × D)/(100). This formula derived from successive percentage changes represents how the cost price is increased by the mark-up and subsequently reduced by a discount to result in the final profit margin. Given that the dealer marks the goods 20% above the cost price (M = 20) and achieves a profit of 8% (P = 8), we substitute these values into the equation:

8 = 20 - D - (20 × D)/(100)

8 = 20 - D - 0.2D

1.2D = 20 - 8

1.2D = 12

D = (12)/(1.2) = 10%

Explanation

Let the two numbers be x and y. According to the given condition, the sum of the two numbers is thrice their difference. This relationship can be expressed algebraically as:

x + y = 3(x - y)

Expanding the equation gives:

x + y = 3x - 3y

By rearranging the terms to bring x on one side and y on the other, we get:

y + 3y = 3x - x

4y = 2x

To determine the ratio of the two numbers, we calculate the value of x/y:

x/y = 4/2

x/y = 2/1

The ratio of the numbers is therefore 2:1.

Explanation

The teacher's age can be determined by comparing the total age of the group before and after the teacher is included. The total age of the 36 students is calculated by multiplying the number of students by their average age: 36 × 14 = 504 years. When the teacher is added to the group, the total number of people becomes 37, and the new average age increases by one year to 14 + 1 = 15 years. The total age of the 37 people is 37 × 15 = 555 years. The teacher's age is the difference between these two sums: 555 - 504 = 51 years.

Alternatively, this can be solved using the shortcut formula for the age of a new entrant: Age of new entrant = Old Average + (New Total Number × Increase in Average) Substituting the given values: Teacher's age = 14 + (37 × 1) = 51 years.

Explanation

The dealer sells 875 grams while charging for 1000 grams, making a profit of 125 grams on every 875 grams sold. The gain percentage is calculated as (Error / Actual Weight) × 100, which is (125 / 875) × 100 = 1/7 × 100 = 14 2/7%.

Explanation

To find the percentage by which B's salary is less than A's, we use the standard relation for percentage comparisons. If one value is r% more than another, the second value is less than the first by [(r)/(100 + r) × 100]%.

Given that A's salary is r = 50% more than B's, we substitute this into the formula: Percentage less = [(50)/(100 + 50) × 100]% Percentage less = ⁵⁰⁄₁₅₀ × 100 = ⅓ × 100 = 33 ⅓%

Alternatively, assuming B's salary is 100 units, A's salary is 100 + 50 = 150 units. The difference is 50 units. To find the percentage less relative to A, we calculate (⁵⁰⁄₁₅₀) × 100, which simplifies to 33 ⅓%.

Explanation

The speed of the current is calculated using the formula v = (D - U)/(2), where D is the speed of the boat downstream (along the current) and U is the speed of the boat upstream (against the current). Given D = 14 km/hr and U = 8 km/hr, substituting these values into the formula gives v = (14 - 8)/(2) = ⁶⁄₂ = 3 km/hr.

Explanation

The distance covered by both trains at the point of overtaking is identical. The passenger train travels at a speed of 80 km/hr for 4 hours, so the total distance covered is D = 80 km/hr × 4 hr = 320 km. The goods train started 6 hours before the passenger train and continued to travel during the 4 hours it took for the passenger train to catch up, resulting in a total travel time of 6 + 4 = 10 hours. If v is the speed of the goods train, the distance it covers is v × 10. Equating the distances gives 10v = 320, which simplifies to v = 32 km/hr.

Explanation

The rate of interest can be calculated using the simple interest formula SI = (P × R × T)/(100). Given that the principal P is Rs. 1 (which is 100 paise), the simple interest SI is 1 paisa, and the time T is 1 month (which is ¹⁄₁₂ year).

Substituting these values into the formula: 1 = (100 × R × 1)/(12){100}

Simplifying the equation, the 100 in the numerator and denominator cancel out, leaving: 1 = (R)/(12) R = 12%

Alternatively, since 1 paisa interest is earned on 100 paise in one month, the monthly interest rate is 1%. To find the annual rate (rate percent per annum), we multiply the monthly rate by 12 months, resulting in 12% per annum.

Explanation

The given equation is (a)/(b) + ᵇ⁄ₐ = 1. By taking the least common multiple on the left side, we obtain (a² + b²)/(ab) = 1. Multiplying both sides of the equation by ab gives a² + b² = ab, which can be rearranged as a² - ab + b² = 0.

The algebraic identity for the sum of cubes is a³ + b³ = (a + b)(a² - ab + b²). By substituting the derived value a² - ab + b² = 0 into this identity, the expression becomes a³ + b³ = (a + b) × 0. Consequently, the value of a³ + b³ is 0.

Explanation

The expression p(p² + 3p + 3) can be expanded as p³ + 3p² + 3p. This expansion closely resembles the binomial formula (a + b)³ = a³ + 3a²b + 3ab² + b³. Specifically, when b = 1, the formula becomes (p + 1)³ = p³ + 3p² + 3p + 1.

By subtracting 1 from both sides, we get p³ + 3p² + 3p = (p + 1)³ - 1. Substituting the given value p = 99 into this equation yields (99 + 1)³ - 1. This simplifies to 100³ - 1, and since 100³ = 1,000,000, the calculation results in 1,000,000 - 1 = 999,999.

Explanation

The algebraic identity relating the difference of cubes to the difference of terms is a³ - b³ = (a - b)³ + 3ab(a - b). Given the values a - b = 1 and a³ - b³ = 61, we substitute them directly into the formula:

61 = (1)³ + 3ab(1) 61 = 1 + 3ab 3ab = 61 - 1 3ab = 60 ab = 20

This identity is derived from the expansion of (a - b)³ = a³ - b³ - 3ab(a - b), and substituting the known variables provides the value of the product ab efficiently.

Explanation

The given equation is (x)/(a+b) + 1 = (x)/(a-b) + (a-b)/(a+b). To solve for x, rearrange the terms to group those involving x on one side and the constant terms on the other:

1 - (a-b)/(a+b) = (x)/(a-b) - (x)/(a+b)

Simplify the left side by taking the common denominator (a+b):

((a+b) - (a-b))/(a+b) = (a+b-a+b)/(a+b) = (2b)/(a+b)

Simplify the right side by taking the common denominator (a-b)(a+b):

x ( ((a+b) - (a-b))/((a-b)(a+b)) ) = x ( (2b)/((a-b)(a+b)) )

Equating the simplified sides results in:

(2b)/(a+b) = (2bx)/((a-b)(a+b))

By dividing both sides by (2b)/(a+b), the equation reduces to:

1 = (x)/(a-b)

Multiplying both sides by (a-b) yields x = a-b.

Explanation

The centroid is the point where the three medians of a triangle intersect. A median is defined as a line segment joining a vertex of the triangle to the midpoint of the opposite side. In Euclidean geometry, the three medians of any triangle are always concurrent, meaning they meet at a single point that lies within the boundaries of the triangle. This intersection point, the centroid, also serves as the geometric center of gravity for a triangle of uniform density. A fundamental property of the centroid is that it divides each median in a ratio of 2:1, where the longer segment is the one connected to the vertex.

Explanation

In a right-angled triangle Δ ABC with ∠ B = 90°, the length of the hypotenuse AC is found using the Pythagorean theorem: AC = √(AB² + BC²). Given AB = 3 cm and BC = 4 cm, the hypotenuse AC = √(3² + 4²) = √(9 + 16) = √(25) = 5 cm. The circumcenter of a right-angled triangle lies exactly at the midpoint of the hypotenuse. If D is the circumcenter, it is equidistant from the vertices A, B, and C. Consequently, the distance BD is the circumradius of the triangle, which is equal to half the length of the hypotenuse. Thus, BD = (AC)/(2) = ⁵⁄₂ = 2.5 cm.

Explanation

In △ ABC, D and E are the midpoints of sides AB and BC. According to the Midpoint Theorem, the segment DE is parallel to AC and its length is half of AC, so DE = ½ AC. Since DE ∥ AC, △ DBE is similar to △ ABC.

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding sides. Therefore, the ratio of the area of △ DBE to the area of △ ABC is:

(Area(△ DBE))/(Area(△ ABC)) = ((DE)/(AC))² = (½)² = ¼

If the area of △ ABC is represented as 4 units, then the area of △ DBE is 1 unit. The area of the trapezium ADEC is the difference between the area of the large triangle and the small triangle:

Area(trapezium ADEC) = Area(△ ABC) - Area(△ DBE) = 4 - 1 = 3 units

The ratio of the area of △ ABC to the area of the trapezium ADEC is 4 : 3.

Explanation

In the isosceles triangle ABC, since AB = AC, the base angles ∠ B and ∠ C are equal. According to the angle sum property of a triangle, ∠ A + ∠ B + ∠ C = 180°. Substituting the given value ∠ A = 30°, we have 30° + 2∠ B = 180°, which simplifies to 2∠ B = 150°, hence ∠ B = 75°.

Since XY ∥ BC and AB acts as a transversal, the interior angles on the same side of the transversal are supplementary. Therefore, ∠ BXY + ∠ B = 180°. Substituting the value of ∠ B, we get ∠ BXY + 75° = 180°. Solving for the unknown angle, ∠ BXY = 180° - 75° = 105°.