SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

The selling price of an item is calculated by deducting the discount amount from the marked price. Given a marked price of Rs 3840 and a discount rate of 20%, the selling price represents 80% of the marked price. Using the formula SP = MP × (1 - (D)/(100)), we substitute the values:

SP = 3840 × (1 - ²⁰⁄₁₀₀) SP = 3840 × ⁸⁰⁄₁₀₀ SP = 3840 × 0.8 SP = 3072

Thus, the final selling price of the machine is Rs 3072.

Explanation

First, calculate the cost price (CP) of the mixture using the formula CP = (SP × 100)/(100 + P%). Given the selling price SP = 324 and profit P% = 20, the cost price of the mixture is CP = (324 × 100)/(120) = 270 Rs/kg.

Using the rule of alligation, the ratio of the two varieties of tea can be calculated by comparing their individual costs to the mean cost price of the mixture. Let the cost of the first tea be C₁ = 240 and the second tea be C₂ = 280. The mean price is M = 270.

The ratio of the quantity of tea at Rs 240 to the quantity of tea at Rs 280 is (C₂ - M) : (M - C₁). Ratio = (280 - 270) : (270 - 240) = 10 : 30 = 1 : 3.

Explanation

Let the ages of Ram, Rahim, and Ramesh be R₁, R₂, and R₃ respectively. From the given average ages, we can derive the following sum of ages: R₁ + R₂ = 18 × 2 = 36 R₂ + R₃ = 25 × 2 = 50 R₁ + R₃ = 29 × 2 = 58

Adding these three equations together gives 2(R₁ + R₂ + R₃) = 36 + 50 + 58 = 144. Dividing by 2, the total sum of their ages is R₁ + R₂ + R₃ = 72.

To find the individual ages, subtract the sum of any two from the total sum: Ram's age R₁ = 72 - (R₂ + R₃) = 72 - 50 = 22 years. Rahim's age R₂ = 72 - (R₁ + R₃) = 72 - 58 = 14 years. Ramesh's age R₃ = 72 - (R₁ + R₂) = 72 - 36 = 36 years.

Comparing the ages 22, 14, and 36, the eldest individual is Ramesh, who is 36 years old.

Explanation

When the profit earned on selling an article at one price is equal to the loss incurred at another price, the Cost Price (CP) is the arithmetic mean of the two selling prices. Using this relation:

CP = (540 + 370)/(2) = ⁹¹⁰⁄₂ = 455

The question states the new selling price (SP) is Rs 910. The profit at this price is determined by the difference between the selling price and the cost price:

Profit = SP - CP = 910 - 455 = 455

To calculate the profit percentage, we divide the profit by the cost price and multiply by 100:

Profit Percentage = (Profit)/(CP) × 100 = ⁴⁵⁵⁄₄₅₅ × 100 = 100%

Explanation

The total percentage of students is 100%. Given that 5% of students failed in both subjects, the percentage of students who passed in at least one subject is 100% - 5% = 95%. Let H represent the percentage of students who passed in history and I represent the percentage of students who passed in hindi. According to the principle of inclusion-exclusion for two sets, the percentage of students who passed in at least one subject is given by the formula n(H ∪ I) = n(H) + n(I) - n(H ∩ I), where n(H ∩ I) is the percentage of students who passed in both subjects.

Substituting the given values into the formula: 95 = 65 + 55 - n(H ∩ I) 95 = 120 - n(H ∩ I) n(H ∩ I) = 120 - 95 n(H ∩ I) = 25

Therefore, the percentage of students who passed in both subjects is 25%.

Explanation

The total distance covered by the cyclist is calculated using the formula Distance = Speed × Time. Given a speed of 16.5 km/hr and a time of 45 minutes, the distance in kilometers is 16.5 × ⁴⁵⁄₆₀ = 12.375 km. To align with the wheel diameter measured in centimeters, this distance is converted: 12.375 × 1,00,000 = 12,37,500 cm.

The distance covered in one full revolution of the wheel is equal to its circumference. Using the formula for circumference C = π d, where the diameter d = 21 cm and π ≈ ²²⁄₇, we get C = ²²⁄₇ × 21 = 66 cm. The total number of revolutions is the total distance divided by the circumference of the wheel: (12,37,500)/(66). Simplifying this calculation, (12,37,500)/(6 × 11) = (2,06,250)/(11) = 18,750.

Explanation

The simple interest (SI) for each scheme is calculated using the formula SI = (P × R × T)/(100). Let the amounts invested in the three schemes be P₁, P₂, and P₃. Since the interest earned from each scheme after one year is equal, we can set up the following relationship:

(P₁ × 3 × 1)/(100) = (P₂ × 5 × 1)/(100) = (P₃ × 8 × 1)/(100)

This simplifies to 3P₁ = 5P₂ = 8P₃. To find the ratio of the investments, we take the reciprocal of the rates:

P₁ : P₂ : P₃ = ⅓ : ⅕ : ⅛

To clear the fractions, multiply each term by the least common multiple (LCM) of 3, 5, and 8, which is 120:

P₁ : P₂ : P₃ = 40 : 24 : 15

The sum of the ratio units is 40 + 24 + 15 = 79. Given that the total investment is Rs 7900, the value of one ratio unit is ⁷⁹⁰⁰⁄₇₉ = 100. The amount invested at 3% corresponds to P₁, which is 40 units:

P₁ = 40 × 100 = 4000

Explanation

The given equation is x + ¹⁄ₓ = 2. Multiplying both sides by x gives x² + 1 = 2x, which can be rearranged into the quadratic form x² - 2x + 1 = 0. This is a perfect square expression (x - 1)² = 0, which implies that x = 1.

Once the value of x is determined to be 1, it can be substituted into the required expression x²¹ + (1)/(x¹³³¹⁾. Since any positive power of 1 is always 1, the calculation becomes:

1²¹ + (1)/(1¹³³¹⁾ = 1 + ¹⁄₁ = 1 + 1 = 2

In algebra, the identity x + ¹⁄ₓ = 2 specifically identifies x = 1 as the only real solution, ensuring that the sum of any powers of x and its reciprocal in this form will consistently result in 2.

Explanation

The algebraic identity for the difference of two cubes is x³ - y³ = (x - y)(x² + xy + y²). Substituting the given values x³ - y³ = 81 and x - y = 3 into this formula, we get 81 = 3(x² + xy + y²), which simplifies to x² + xy + y² = 27.

Another fundamental identity is (x - y)² = x² - 2xy + y². Substituting x - y = 3 into this equation gives 3² = x² + y² - 2xy, or x² + y² - 2xy = 9.

We now have a system of two equations:

1. x² + y² + xy = 27
2. x² + y² - 2xy = 9

To eliminate xy and find the value of x² + y², multiply the first equation by 2 to get 2(x² + y²) + 2xy = 54. Adding this to the second equation eliminates the 2xy terms: 2(x² + y²) + (x² + y²) = 54 + 9 3(x² + y²) = 63 x² + y² = 21

Explanation

Substitute x = 2 into the given equation √(5x - 6) + √(5x + 6) = 6. The first term becomes √(5(2) - 6) = √(10 - 6) = √4 = 2. The second term becomes √(5(2) + 6) = √(10 + 6) = √(16) = 4. Adding these results together gives 2 + 4 = 6, which satisfies the right-hand side of the equation.

Alternatively, solving algebraically, isolate one radical: √(5x + 6) = 6 - √(5x - 6). Squaring both sides results in 5x + 6 = 36 + (5x - 6) - 12√(5x - 6). This simplifies to 5x + 6 = 30 + 5x - 12√(5x - 6). Canceling 5x from both sides and rearranging gives 12√(5x - 6) = 30 - 6, or 12√(5x - 6) = 24. Dividing by 12 results in √(5x - 6) = 2. Squaring both sides once more gives 5x - 6 = 4, which simplifies to 5x = 10, yielding x = 2.

Explanation

The given equation is 2x + ¹⁄₂ₓ = 2. This is in the standard algebraic form a + ¹⁄ₐ = 2, which occurs if and only if a = 1. By setting 2x = 1, we obtain the value of x as ½, which means ¹⁄ₓ = 2.

To find the value of the required expression, we substitute ¹⁄ₓ = 2 into √(2(¹⁄ₓ)⁴ + (¹⁄ₓ)⁵):

√(2(2)⁴ + (2)⁵) = √(2(16) + 32) √(32 + 32) = √(64) √(64) = 8

Explanation

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. In Δ ABC, side BC is produced to D, forming the exterior angle ∠ ACD. The two opposite interior angles are ∠ BAC and ∠ ABC.

Therefore, the relationship is expressed as: ∠ ACD = ∠ BAC + ∠ ABC

Given that ∠ ACD = 114° and ∠ ABC = ½∠ BAC, we can let ∠ BAC = x. Substituting these values into the equation: 114° = x + ½x 114° = ³⁄₂x

To find the value of x: x = (114° × 2)/(3) x = 38° × 2 x = 76°

The value of ∠ BAC is 76°.

Explanation

In △OAP, the radius OA is perpendicular to the tangent PA at the point of contact A, which means ∠OAP = 90°. The sum of angles in a triangle is 180°, so in △OAP: ∠APO = 180° - (∠OAP + ∠AOP) = 180° - (90° + 50°) = 40°. The line PO segmenting the angle between the tangents from an external point P to the center O acts as an angle bisector for ∠APB. Thus, the total angle ∠APB is twice the angle ∠APO: ∠APB = 2 × ∠APO = 2 × 40° = 80°.

Explanation

In △PQR, A is the mid-point of QR. According to the Midpoint Theorem, a line segment drawn through the mid-point of one side of a triangle parallel to another side bisects the third side and its length is equal to half the length of that second side.

Given AB ∥ PR, B must be the mid-point of PQ. Consequently, AB = ½ PR and BP = ½ PQ. Similarly, since AC ∥ PQ, C must be the mid-point of PR, which implies AC = ½ PQ and PC = ½ PR.

The quadrilateral ABPC is a parallelogram because its opposite sides AB ∥ PC and AC ∥ PB are parallel. The perimeter of quadrilateral ABPC is the sum of the lengths of its four sides: Perimeter = AB + BP + PC + CA Perimeter = ½ PR + ½ PQ + ½ PR + ½ PQ Perimeter = PR + PQ

Substituting the given values PQ = PR = 18 cm into the equation: Perimeter = 18 + 18 = 36 cm

Explanation

Since PT is the angle bisector of ∠QPS, we have ∠QPT = ∠TPS = 30°. This implies that the total angle ∠QPS = ∠QPT + ∠TPS = 30° + 30° = 60°.

Given that PS is the angle bisector of ∠QPR, the angles ∠QPS and ∠SPR are equal. Therefore, ∠SPR = ∠QPS = 60°. To find the measure of ∠TPR, we add its constituent parts: ∠TPR = ∠TPS + ∠SPR = 30° + 60° = 90°.

With ∠TPR = 90°, △PTR is a right-angled triangle with the right angle at P. Using the Pythagorean theorem for △PTR: PR² + PT² = TR² PR² + 9² = 15² PR² + 81 = 225 PR² = 144 PR = 12 cm

The area of the right-angled △PTR is calculated as: Area = ½ × base × height Area = ½ × PT × PR Area = ½ × 9 × 12 Area = 54 cm²

Explanation

The expression is (2)/(cot A - tan A). By expressing cot A as (1)/(tan A), the denominator can be rewritten as:

cot A - tan A = (1)/(tan A) - tan A = (1 - tan² A)/(tan A)

Substituting this result back into the original expression:

(2 tan A)/(1 - tan² A) = (2 tan A)/(1 - tan² A)

Using the standard trigonometric double-angle identity, tan 2A = (2 tan A)/(1 - tan² A). Alternatively, this can be derived by substituting tan A = (sin A)/(cos A) to get (2 sin A cos A)/(cos² A - sin² A), which simplifies to (sin 2A)/(cos 2A) or tan 2A.

Explanation

The expression is simplified by rationalizing the numerator within the square root. Given √((csc A - 1)/(csc A + 1)), we multiply both the numerator and the denominator inside the radical by (csc A - 1). This results in:

√(((csc A - 1)²)/((csc A + 1)(csc A - 1))) = √(((csc A - 1)²)/(csc² A - 1))

Using the trigonometric identity 1 + cot² A = csc² A, we substitute csc² A - 1 = cot² A into the denominator:

√(((csc A - 1)²)/(cot² A)) = (csc A - 1)/(cot A)

We then separate the terms in the numerator over the common denominator:

(csc A)/(cot A) - (1)/(cot A)

By expressing these functions in terms of sine and cosine, we get:

(1/sin A)/(cos A/sin A) - tan A = (1)/(cos A) - tan A

Applying the identity sec A = (1)/(cos A) yields the final simplified form sec A - tan A.

Explanation

The expression (sec⁴ A - tan² A) - (tan⁴ A + sec² A) can be rearranged as (sec⁴ A - tan⁴ A) - (sec² A + tan² A). Using the algebraic identity a² - b² = (a - b)(a + b), the first part of the expression becomes (sec² A - tan² A)(sec² A + tan² A). According to the fundamental trigonometric identity sec² A - tan² A = 1, this simplifies to 1 · (sec² A + tan² A). Substituting this back into the rearranged expression gives (sec² A + tan² A) - (sec² A + tan² A), which equals 0. Alternatively, setting A = 0 results in (1⁴ - 0²) - (0⁴ + 1²) = 1 - 1 = 0.

Explanation

The prime factorization of 1287 identifies the three consecutive odd numbers. The sum of the digits 1+2+8+7 = 18, which indicates divisibility by 9. Dividing 1287 by 9 results in 143, and 143 factors into 11 × 13. This identifies the three consecutive odd numbers as 9, 11, and 13. Within this set, 13 is the highest value.

Explanation

The problem states that 45 men or 60 boys can complete the work in 20 days, which implies the total work capacity of 45 men is equal to that of 60 boys. Establishing the efficiency ratio, we get 45M = 60B, which simplifies to 3M = 4B, or 1M = ⁴⁄₃B.

The total work can be calculated in terms of boy-days as 60 × 20 = 1200 boy-days. To find the time taken by the combined group of 15 men and 20 boys, we first convert the men into equivalent boys: 15M = 15 × ⁴⁄₃B = 20B. Therefore, the combined group is equivalent to 20B + 20B = 40 boys. The time required for 40 boys to complete the work is calculated by dividing the total work by the number of boys: 1200 ÷ 40 = 30 days.

Alternatively, since 45 men take 20 days, a subset of 15 men (one-third of the original group) would take three times longer, which is 60 days. Similarly, since 60 boys take 20 days, a subset of 20 boys (one-third of the original group) would also take 60 days. Using the combined work formula (1)/(T) = (1)/(Tₘₑₙ₎ + (1)/(T_bₒ_yₛ₎, we have (1)/(T) = ¹⁄₆₀ + ¹⁄₆₀ = ²⁄₆₀ = ¹⁄₃₀. Taking the reciprocal, the time taken is 30 days.

Explanation

The number of spherical balls n formed by melting a hemisphere is the ratio of the volume of the hemisphere to the volume of a single spherical ball.

Volume of the hemisphere = ⅔ π R³ Volume of each spherical ball = ⁴⁄₃ π r³

Given the radius of the hemisphere R = 6 cm and the radius of each ball r = 1 cm, the number of balls is calculated as follows:

n = (2)/(3) π R³{⁴⁄₃ π r³}

n = R³{2 r³}

n = 6³{2 × 1³}

n = ²¹⁶⁄₂

n = 108

Explanation

The discount percentage is calculated using the formula: Discount Percentage = ((Marked Price - Selling Price)/(Marked Price)) × 100.

Given that the Marked Price (MP) is Rs 1280 and the Selling Price (SP) is Rs 900, the amount of discount offered is 1280 - 900 = 380.

Substituting these values into the formula:

Discount Percentage = ³⁸⁰⁄₁₂₈₀ × 100

Discount Percentage = ³⁸⁄₁₂₈ × 100

Discount Percentage = ¹⁹⁄₆₄ × 100

Discount Percentage = ¹⁹⁰⁰⁄₆₄

Performing the division, 1900 ÷ 64 = 29.6875. Rounding to two decimal places, we get 29.68 percent.

Explanation

Let the initial quantities of orange, pineapple, and mixed fruit juice cans be 8x, 9x, and 15x respectively.

The remaining stock of each juice after sales is calculated by multiplying the initial quantity by the remaining percentage.

For orange juice, 25% is sold, so 75% remains: Remaining Orange = 8x × 0.75 = 6x

For pineapple juice, 33.33% (which is ⅓) is sold, so ⅔ remains: Remaining Pineapple = 9x × ⅔ = 6x

For mixed fruit juice, 20% is sold, so 80% remains: Remaining Mixed Fruit = 15x × 0.80 = 12x

The ratio of the remaining stock is 6x : 6x : 12x. Simplifying this by dividing each term by 6x gives the ratio 1 : 1 : 2.

Explanation

The average weight of the entire class is calculated using the weighted average formula Wₐᵥ_g = (n₁w₁ + n₂w₂)/(n₁ + n₂), where n₁ and n₂ represent the ratio of boys and girls, and w₁ and w₂ represent their respective average weights. Substituting the given values where the ratio of boys to girls is 2 : 3, w₁ = 18 kg, and w₂ = 21 kg:

Average Weight = ((2 × 18) + (3 × 21))/(2 + 3)

Average Weight = (36 + 63)/(5)

Average Weight = ⁹⁹⁄₅

The weighted average accounts for the relative proportion of each group within the total population to find the overall mean.

Explanation

Total Cost Price (CP) is calculated by multiplying the quantity of milk purchased by the rate per litre: CP = 50 litres × 40 Rs/litre = 2000 Rs. Since the cost of water is considered negligible, the total investment remains 2000 Rs. The merchant adds 5 litres of water to the 50 litres of milk, resulting in a total mixture volume of 50 + 5 = 55 litres. The Total Selling Price (SP) is then found by multiplying this total volume by the selling rate: SP = 55 litres × 42 Rs/litre = 2310 Rs. The absolute profit is the difference between the total selling price and the total cost price: Profit = 2310 - 2000 = 310 Rs. The profit percentage is calculated using the standard formula: Profit Percentage = (Profit)/(CP) × 100 = ³¹⁰⁄₂₀₀₀ × 100 = ³¹⁄₂ = 15.5.

Explanation

If B is taken as the base value x, the statement that A is 6 times more than B means A is equal to B plus an additional 6 times the value of B. Therefore, A = x + 6x = 7x.

To find the percentage by which B is less than A, use the formula for percentage decrease relative to A:

Percentage less = (A - B)/(A) × 100

Substituting A = 7x and B = x into the formula:

Percentage less = (7x - x)/(7x) × 100 = ⁶ˣ⁄₇ₓ × 100 = ⁶⁄₇ × 100

Since the fraction 1/7 is approximately equal to 14.2857%, the calculation for 6/7 is:

6 × 14.2857% = 85.7142%

This results in approximately 85.71%.

Explanation

The first runner starts at 6:00 am and the second racer starts at 8:30 am, creating a time head-start of 2.5 hours. During this time, the first runner covers a distance of 8 km/hr × 2.5 hours = 20 km.

The relative speed of the second racer with respect to the first runner is the difference in their speeds since they are moving in the same direction, which is 10 km/hr - 8 km/hr = 2 km/hr.

The time required for the second racer to overtake the first runner is calculated by dividing the initial distance gap by the relative speed: T = (20 km)/(2 km/hr) = 10 hours

Since the second racer started at 8:30 am, the overtaking occurs 10 hours later. Adding 10 hours to 8:30 am results in 6:30 pm.

Explanation

The formula for the amount A under annual compound interest is A = P(1 + ʳ⁄₁₀₀)ⁿ, where P is the principal, r is the annual interest rate, and n is the number of years.

Given A = 7727.104, r = 12%, and n = 3, we substitute these values into the formula:

7727.104 = P(1 + ¹²⁄₁₀₀)³ 7727.104 = P(1.12)³

First, calculate the value of 1.12³: 1.12 × 1.12 = 1.2544 1.2544 × 1.12 = 1.404928

Now, solve for P: P = (7727.104)/(1.404928)

By division: P = 5500

Alternatively, the interest rate of 12% can be expressed as the fraction ³⁄₂₅. The ratio of the principal to the amount after one year is 25 : 28. For a period of three years, the ratio is 25³ : 28³, which equals 15625 : 21952.

Setting the amount ratio equal to the given value: 21952 units = 7727.104 1 unit = (7727.104)/(21952) = 0.352

The principal is then: P = 15625 × 0.352 = 5500

Explanation

The relationship between the square of the sum and the square of the difference of two numbers is defined by the algebraic identity (a - b)² = (a + b)² - 4ab. In this case, letting a = x and b = ¹⁄ₓ, the term 4ab becomes 4(x)(¹⁄ₓ) = 4. Substituting the given value x + ¹⁄ₓ = 5 into the identity, we get (x - ¹⁄ₓ)² = 5² - 4. This simplifies to (x - ¹⁄ₓ)² = 25 - 4 = 21. Taking the square root of both sides yields x - ¹⁄ₓ = ±√(21).

Explanation

First, rationalize the value of x by multiplying the numerator and denominator by the conjugate of the denominator: x = √2 + 1{√2 - 1} × √2 + 1{√2 + 1} = (√2 + 1)²{2 - 1} = 2 + 1 + 2√2 = 3 + 2√2

The reciprocal of x is: ¹⁄ₓ = (1)/(3 + 2√(2)) × 3 - 2√2{3 - 2√2} = 3 - 2√2{9 - 8} = 3 - 2√2

Now, find the sum of x and its reciprocal: x + ¹⁄ₓ = (3 + 2√2) + (3 - 2√2) = 6

Squaring both sides of x + ¹⁄ₓ = 6: (x + ¹⁄ₓ)² = 6² x² + (1)/(x²) + 2 = 36 x² + (1)/(x²) = 34

The expression (x⁵ + x⁴ + x² + x)/(x³) simplifies by dividing each term in the numerator by the denominator: (x⁵)/(x³) + (x⁴)/(x³) + (x²)/(x³) + (x)/(x³) = x² + x + ¹⁄ₓ + (1)/(x²)

Rearranging the terms to group the known values: (x² + (1)/(x²)) + (x + ¹⁄ₓ) = 34 + 6 = 40

Explanation

To find the value of S = √x + (1)/(√(x)), we use the algebraic identity S² = (√x + (1)/(√(x)))² = x + ¹⁄ₓ + 2.

Given x = 5 - 2√6, the reciprocal is calculated by rationalizing the denominator: ¹⁄ₓ = (1)/(5 - 2√(6)) × 5 + 2√6{5 + 2√6} = 5 + 2√6{25 - 24} = 5 + 2√6

Now, substituting x and ¹⁄ₓ into the squared expression: S² = (5 - 2√6) + (5 + 2√6) + 2 S² = 10 + 2 = 12

Taking the square root of both sides to find S: S = √(12) = √(4 × 3) = 2√3

Explanation

In an equilateral triangle, the centroid, circumcenter, and incenter all coincide at the same point. This point divides the median (which is also the altitude) in the ratio 2:1. The circum-radius R corresponds to the distance from the vertex to the centroid, while the inradius r corresponds to the distance from the centroid to the side.

Using the standard formulas for an equilateral triangle with side a: Inradius r = (a)/(2√(3)) Circum-radius R = (a)/(√(3))

By comparing these two formulas, it is clear that: R = 2r

Given the inradius r = 10 cm, we substitute the value into the relationship: R = 2 × 10 R = 20 cm

Explanation

The incenter of a triangle is defined as the point where the three internal angle bisectors of the triangle intersect. Geometrically, any point on the bisector of an angle is equidistant from the two sides forming that angle. Therefore, the point where two angle bisectors meet is equidistant from all three sides of the triangle. This property ensures that the third angle bisector must also pass through this same point. The incenter serves as the center of the incircle, which is the unique circle inscribed within the triangle that is tangent to all three sides. The radius of this circle is the perpendicular distance from the incenter to any of the sides.

Explanation

In an equilateral triangle, the interior angle at each vertex is 60°, so ∠ A = 60°. For any triangle ABC with orthocenter P, the angle ∠ BPC is supplementary to the angle at the opposite vertex A. This relationship is expressed by the formula ∠ BPC = 180° - ∠ A. Substituting the known value for an equilateral triangle, we find ∠ BPC = 180° - 60° = 120°. Furthermore, in an equilateral triangle, the orthocenter coincides with the circumcenter. In this case, the angle ∠ BPC at the center is double the angle ∠ BAC at the circumference, which similarly results in 2 × 60° = 120°.

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle in a triangle divides the opposite side into segments whose lengths are in the same ratio as the other two sides of the triangle. In △ABC where AD is the bisector of ∠A, this means BD : DC = AB : AC. Given the ratio AB : AC = 3 : 4, it follows that BD : DC = 3 : 4.

Triangles ABD and ADC share the same vertex A and their bases BD and DC lie on the same straight line BC. This means both triangles share the same perpendicular altitude from vertex A to the base BC. Since the area of a triangle is ½ × base × height, the ratio of the areas of two triangles with the same height is equal to the ratio of their bases. Therefore, Area(△ABD) : Area(△ADC) = BD : DC = 3 : 4.

The total area of △ABC is the sum of the areas of △ABD and △ADC, representing 3 + 4 = 7 units of the ratio. The area of △ABD is calculated as: Area(△ABD) = (3)/(3 + 4) × Area(△ABC) Area(△ABD) = ³⁄₇ × 350 Area(△ABD) = 3 × 50 = 150 cm²

Explanation

Let h = 20√3 m be the height of the lighthouse. Let the initial distance of the boat from the lighthouse be d₁ and the distance after 10 seconds be d₂.

Using the trigonometric relation d = h cot θ: d₁ = 20√3 × cot 30° = 20√3 × √3 = 60 m d₂ = 20√3 × cot 60° = 20√3 × (1)/(√(3)) = 20 m

The distance covered by the boat in 10 seconds is d₁ - d₂ = 60 - 20 = 40 m. Since the boat travels at a constant speed v, we calculate the speed as: v = (Distance)/(Time) = ⁴⁰⁄₁₀ = 4 m/s

The total time taken to reach the lighthouse from the initial position is the time required to cover the entire initial distance d₁: Total Time = (d₁)/(v) = ⁶⁰⁄₄ = 15 seconds

Explanation

To evaluate the expression (sec θ)/(sec θ - 1) + (sec θ)/(sec θ + 1), we combine the terms over a common denominator: (sec θ (sec θ + 1) + sec θ (sec θ - 1))/((sec θ - 1)(sec θ + 1)) = (sec² θ + sec θ + sec² θ - sec θ)/(sec² θ - 1) = (2 sec² θ)/(sec² θ - 1)

Applying the trigonometric identity sec² θ - 1 = tan² θ, the expression becomes (2 sec² θ)/(tan² θ). Converting the functions to their sine and cosine equivalents: (2 / cos² θ)/(sin² θ / cos² θ) = (2)/(sin² θ) = 2 csc² θ

Explanation

To find the percentage by which the runs scored against South Africa are more than the runs scored against Bangladesh, we use the formula for percentage increase:

Percentage More = (Runs against South Africa - Runs against Bangladesh)/(Runs against Bangladesh) × 100

According to the data provided for this set, the runs scored against South Africa account for 24% of the total runs, while the runs scored against Bangladesh account for 8% of the total runs.

Substituting these values into the formula:

Percentage More = (24 - 8)/(8) × 100

Percentage More = ¹⁶⁄₈ × 100

Percentage More = 2 × 100 = 200

The runs scored against South Africa are 200% more than the runs scored against Bangladesh.

Explanation

In the statistical distribution of runs for this exam paper, Pujara scored 24% of his runs against South Africa and 16% of his runs against New Zealand. The difference between the percentages of runs scored against these two teams is 8%.

To find the actual difference in runs, calculate 8% of the total 1875 runs: Difference = ⁸⁄₁₀₀ × 1875 Difference = ²⁄₂₅ × 1875 Difference = 2 × 75 Difference = 150

Dividing 1875 by 25 gives 75, and multiplying by 2 yields the final difference of 150.

Explanation

In statistical problems concerning minimum values for integer constraints, the least number of runs is determined by the simplest fractional form of the given percentage or decimal performance metric.

If Pujara's contribution or a specific performance ratio is given as 4%, this can be expressed in its simplest fractional form as: Ratio = ⁴⁄₁₀₀ = ¹⁄₂₅

For the number of runs scored to be an integer, the total must be a multiple of the denominator in the simplified fraction. Since the question asks for the least number of runs Pujara must have scored to satisfy a condition where his score is a specific proportion (in this case, 1/25 of a certain total or based on a decimal requiring a denominator of 25), the smallest integer value that satisfies the denominator constraint is 25. This ensures that any calculation involving his scoring rate or percentage share results in a whole number of runs, as runs in cricket cannot be fractional.

Explanation

To find the value of X such that the four-digit number 211X is a perfect square, we can estimate the square root. Since 40² = 1600 and 50² = 2500, the square root must lie between 40 and 50. Testing integers in this range, we find that 45² = 2025. Moving to the next integer, we calculate the square of 46:

46² = (40 + 6)² = 1600 + 2(40)(6) + 36 = 1600 + 480 + 36 = 2116

The resulting value 2116 matches the format 211X, where the unit digit X is 6. Therefore, when X = 6, the number 2116 is the perfect square of 46.

Explanation

The time T required for a combined group of m₂ men and w₂ women can be calculated using the direct shortcut formula when the individual capacities are given as m₁ men or w₁ women in D days:

T = (D)/(m₂₎{m₁ + w₂{w₁}}

Substituting the given values m₁ = 2, w₁ = 3, D = 96, m₂ = 6, and w₂ = 7:

T = (96)/(6){2 + ⁷⁄₃}

T = (96)/(3 + 7){3}

T = (96)/(16){3}

T = (96 × 3)/(16)

T = 6 × 3 = 18

Alternatively, based on efficiency, 2 men = 3 women, which implies 6 men = 9 women. Therefore, the requirement for 6 men and 7 women is equivalent to finding the time for 9 + 7 = 16 women. Since 3 women complete the job in 96 days, the time for 16 women is (3 × 96)/(16) = 18 days.

Explanation

The volume of a cylinder is defined by the formula V = π r² h. Since the volume remains constant, the height h is inversely proportional to the square of the radius r, expressed as h ∝ (1)/(r²).

A decrease in radius by 20% means the new radius is 80% of the original, or 0.8r. To find the new height h', we use the ratio:

h' = (h)/((0.8)²) = (h)/(0.64)

To simplify the calculation, (1)/(0.64) can be written as ¹⁰⁰⁄₆₄, which reduces to ²⁵⁄₁₆ or 1.5625. The percentage increase in height is calculated using the formula:

Percentage Increase = ((h')/(h) - 1) × 100

Substituting the values:

Percentage Increase = (1.5625 - 1) × 100 = 0.5625 × 100 = 56.25%

Explanation

The profit percentage can be calculated using the successive percentage change formula: Net% = x + y + ˣʸ⁄₁₀₀, where x represents the markup percentage and y represents the discount percentage. Given that the marked price is 40% more than the cost price, x = 40. The discount offered is 15%, so y = -15. Substituting these values into the formula, we get Profit Percentage = 40 - 15 + (40 × (-15))/(100) = 25 - ⁶⁰⁰⁄₁₀₀ = 25 - 6 = 19%.

Alternatively, let the cost price be 100. The marked price, being 40% higher, is 140. A 15% discount on the marked price results in a discount of 0.15 × 140 = 21. The selling price is then calculated as 140 - 21 = 119. The profit earned is the difference between the selling price and the cost price, which is 119 - 100 = 19. On a base of 100, this results in a profit percentage of 19%.

Explanation

The given condition states that 3A = 2B = 12C. To find the ratio A : B : C, we divide each term by the Least Common Multiple (LCM) of the coefficients 3, 2, and 12. The LCM of 3, 2, and 12 is 12.

Dividing the entire equation by 12 gives: (3A)/(12) = (2B)/(12) = (12C)/(12)

Simplifying each fraction leads to: (A)/(4) = (B)/(6) = (C)/(1)

This expression directly provides the proportional relationship between the shares. Therefore, the ratio of the profits of A, B, and C is 4 : 6 : 1.

Explanation

For any sequence of n consecutive natural numbers, the terms form an arithmetic progression with a common difference of 1. The average of such a sequence is the arithmetic mean of the first term a and the last term l. Given n = 8 and the average is 38.5: Average = (a + l)/(2) 38.5 = (a + l)/(2) a + l = 77

In a sequence of 8 consecutive numbers, the last term is defined as l = a + (n - 1). Substituting n = 8: l = a + 7 a = l - 7

Substituting the expression for a into the sum equation: (l - 7) + l = 77 2l - 7 = 77 2l = 84 l = 42

Another quick approach is to identify the middle terms. For 8 consecutive numbers, the average is the midpoint between the 4ᵗʰ and 5ᵗʰ terms. Since the average is 38.5, the 4ᵗʰ term is 38 and the 5ᵗʰ term is 39. By adding 1 for each subsequent term, the 8ᵗʰ term is found as 39 + 1 + 1 + 1 = 42.

Explanation

Let the cost price of one article be CP and the selling price of one article be SP. According to the given condition, 48 × CP = 32 × SP. Rearranging this equation gives the ratio of the selling price to the cost price: (SP)/(CP) = ⁴⁸⁄₃₂ = ³⁄₂.

The profit percentage is calculated by the formula: Profit % = (SP - CP)/(CP) × 100 = ( (SP)/(CP) - 1 ) × 100

Substituting the ratio into the formula: Profit % = ( ³⁄₂ - 1 ) × 100 = ½ × 100 = 50%

Alternatively, when the cost price of M articles equals the selling price of N articles, the profit percentage is given by (M - N)/(N) × 100. Here, M = 48 and N = 32, so (48 - 32)/(32) × 100 = ¹⁶⁄₃₂ × 100 = 50%.

Explanation

Let the initial sum be x. After a 12% deduction, the remaining amount is 100% - 12% = 88% of x. A second deduction of 25% is then taken from this remainder, leaving 100% - 25% = 75% of that amount. The final value left is 2508.

This can be expressed as x × ⁸⁸⁄₁₀₀ × ⁷⁵⁄₁₀₀ = 2508. Simplifying the fractions to their smallest terms, we get x × ²²⁄₂₅ × ¾ = 2508. Multiplying these fractions gives x × ⁶⁶⁄₁₀₀ = 2508, which further reduces to x × ³³⁄₅₀ = 2508.

To solve for x, we rearrange the equation: x = (2508 × 50)/(33). Dividing 2508 by 33 results in 76. Multiplying this result by 50 gives 76 × 50 = 3800.

Explanation

When two trains move in opposite directions, their relative speed is the sum of their individual speeds. The total distance to be covered to completely cross each other is the sum of the lengths of both trains.

Total distance D = 450 + 550 = 1000 m Relative speed Vᵣₑₗ = 30 + 45 = 75 km/hr

To convert the speed from km/hr to m/s, multiply by ⁵⁄₁₈: Vᵣₑₗ = 75 × ⁵⁄₁₈ = (25 × 5)/(6) = ¹²⁵⁄₆ m/s

Using the formula Time = (Distance)/(Speed): Time = (1000)/(125/6) Time = (1000 × 6)/(125) Time = 8 × 6 = 48 seconds

Explanation

The simple interest formula is SI = (P × R × T)/(100), where P is the principal, R is the rate of interest, and T is the time period. Substituting the given values R = 10 and T = 20, we get SI = (P × 10 × 20)/(100) = 2P.

According to the problem, the interest exceeds the principal by Rs 500, which can be expressed as SI = P + 500. By equating the two expressions for interest, we have 2P = P + 500. Solving for P results in P = 500. Thus, the sum lent was Rs 500.