SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The given expression uses the algebraic identity where if a + b + c = 0, then a³ + b³ + c³ = 3abc. In the numerator, let a = x - y, b = y - z, and c = z - x. Calculating their sum gives:

a + b + c = (x - y) + (y - z) + (z - x) = 0

Since the sum of the terms is zero, the numerator (x - y)³ + (y - z)³ + (z - x)³ simplifies to 3(x - y)(y - z)(z - x). Substituting this back into the original expression:

(3(x - y)(y - z)(z - x))/(6(x - y)(y - z)(z - x))

As x ≠ y ≠ z, the factors (x - y), (y - z), and (z - x) are non-zero and can be cancelled from the numerator and the denominator. This leaves:

³⁄₆ = ½

Explanation

To find what percent the number of toys manufactured by factory F2 is of the number of toys manufactured by factory F4, use the standard percentage formula:

Percentage = (Value of F2)/(Value of F4) × 100

According to the table, the number of toys manufactured by factory F2 is 450 and the number of toys manufactured by factory F4 is 150. Substituting these values into the formula gives:

Percentage = ⁴⁵⁰⁄₁₅₀ × 100

Simplifying the fraction ⁴⁵⁰⁄₁₅₀ yields 3. Therefore:

Percentage = 3 × 100 = 300 percent

Thus, factory F2 manufactured 300 percent of the toys manufactured by factory F4.

Explanation

The total marked price for five products at ₹500 each is 5 × 500 = ₹2,500. The store first applies a discount of 5%, so the price after the initial discount is 2,500 × (1 - 0.05) = 2,500 × 0.95 = ₹2,375. Because this intermediate amount of ₹2,375 exceeds the ₹2,000 threshold, the customer qualifies for the additional 10% discount. This second discount is calculated on the already discounted price, making the final payable amount 2,375 × (1 - 0.10) = 2,375 × 0.90 = ₹2,137.5.

Explanation

The circumference C of a circle is calculated using the formula C = 2π r, where r represents the radius. Given that the circumference is 220 cm, and taking the value of π as ²²⁄₇, the relationship is expressed as:

220 = 2 × ²²⁄₇ × r

220 = ⁴⁴⁄₇ × r

To solve for r, multiply both sides by 7 and divide by 44:

r = (220 × 7)/(44)

r = 5 × 7

r = 35 cm

Explanation

The perimeter of a square is defined by the formula P = 4s, where s is the length of one side. Given that the perimeter is 124 metres, we can determine the side length by dividing the perimeter by 4: s = ¹²⁴⁄₄ = 31 metres.

The area A of a square is determined by the formula A = s². By substituting the calculated side length of 31 metres into the area formula, we get: A = 31² = 961 m².

Explanation

In triangle XYZ, the incentre P is the point where the internal angle bisectors of the triangle intersect. A standard geometric property states that the angle formed at the incentre by two vertices of a triangle is equal to 90° plus half of the third vertex angle.

The formula to calculate ∠ XPZ is: ∠ XPZ = 90° + (∠ XYZ)/(2)

Given that ∠ XYZ = 50°, we substitute this value into the formula: ∠ XPZ = 90° + (50°)/(2) ∠ XPZ = 90° + 25° ∠ XPZ = 115°

This relationship is derived from the fact that the sum of angles in △ XPZ is 180°, where the angles at X and Z are half of the original angles of △ XYZ. Substituting the angle sum property of the main triangle into the sub-triangle calculation leads directly to this result.

Explanation

The total shift in the percentage value from a 13½% loss to an 11% gain is the numerical sum of the two percentages. This is because a loss represents a value below the cost price (CP) and a gain represents a value above it.

Total percentage difference = 13.5% + 11% = 24.5%

This 24.5% difference in the percentage of the cost price corresponds to the actual price increase of ₹1,274. Let the cost price be CP.

24.5% of CP = 1274 (24.5)/(100) × CP = 1274 CP = (1274 × 100)/(24.5) CP = (1274 × 1000)/(245)

To simplify the calculation, divide both numbers by their common factors: CP = (182 × 1000)/(35) (dividing 1274 and 245 by 7) CP = (26 × 1000)/(5) (dividing 182 and 35 by 7) CP = 26 × 200 = 5200

The cost price of the article is ₹5,200.

Explanation

The value J1 represents the sum of the expenditure of company L over the five years P, Q, R, S, and T. Adding these values gives J1 = 200 + 150 + 300 + 100 + 400 = 1150. The value J2 represents the sum of the expenditure of company M over the same five years. Adding these values gives J2 = 500 + 350 + 250 + 150 + 100 = 1350.

To find the ratio J1:J2, we use the calculated sums: J1:J2 = 1150:1350

Dividing both terms by 10 simplifies the ratio to 115:135. Further dividing both terms by their common factor 5 gives: 115 ÷ 5 = 23 135 ÷ 5 = 27

The simplified ratio J1:J2 is 23:27.

Explanation

To find the weight of the new person, use the formula: Weight of new person = Weight of replaced person + (Number of persons × Increase in average).

The problem states that there are 12 persons and the average weight increases by 3.5 kg. The total increase in the weight of the entire group is calculated as 12 × 3.5 = 42 kg.

This total increase occurs because the new person is heavier than the person who was replaced. By adding this total increase to the weight of the person who left (58 kg), the weight of the new person is determined: 58 kg + 42 kg = 100 kg.

Since the calculated weight of the new person is 100 kg, statement I is correct.

Explanation

The given equation is K + (1)/(K) + 2 = 0. Multiplying the entire equation by K results in the quadratic equation K² + 2K + 1 = 0. This is a perfect square trinomial that can be factored as (K + 1)² = 0. Solving for K gives the unique value K = -1.

To find the value of the expression K¹⁰ + K¹¹, substitute K = -1: (-1)¹⁰ + (-1)¹¹

A negative number raised to an even power results in a positive value, so (-1)¹⁰ = 1. A negative number raised to an odd power results in a negative value, so (-1)¹¹ = -1. Therefore, the expression evaluates to 1 + (-1) = 0.

Explanation

To find the value of cos 45° sin 15°, we use the product-to-sum trigonometric identity cos A sin B = ½ [sin(A+B) - sin(A-B)]. Substituting A = 45° and B = 15° into the formula gives:

cos 45° sin 15° = ½ [sin(45° + 15°) - sin(45° - 15°)]

cos 45° sin 15° = ½ [sin 60° - sin 30°]

Substituting the standard trigonometric values sin 60° = √3{2} and sin 30° = ½:

cos 45° sin 15° = ½ [√3{2} - ½] = √3 - 1{4}

Alternatively, using the direct value of cos 45° = (1)/(√(2)) and calculating sin 15° = sin(45° - 30°) = √3-1{2√2}, the product is calculated as (1)/(√(2)) × √3 - 1{2√2} = √3 - 1{4}.

Explanation

For any real number X, if X + (1)/(X) = 2, the only possible value for X is 1. This can be shown by rearranging the equation into the quadratic form X² - 2X + 1 = 0, which factors as (X - 1)² = 0, leading to X = 1. Substituting X = 1 into the required expression, we get 1³ + (1)/(1³⁾ = 1 + 1 = 2.

Alternatively, using the algebraic identity for a³ + b³, we have X³ + (1)/(X³⁾ = (X + (1)/(X))³ - 3(X)((1)/(X))(X + (1)/(X)). By substituting the given value X + (1)/(X) = 2, the expression becomes 2³ - 3(1)(2) = 8 - 6 = 2.

Explanation

The data in the table displays a direct positive correlation between height and weight. By examining the values in ascending order, it is evident that as height increases from 135 cm to 156 cm, the corresponding weight values also consistently rise from 45 to 69. Specifically, the pairs (135, 45), (139, 55), (141, 58), (142, 60), (145, 61), (155, 68), and (156, 69) show that for every increment in the height column, there is a subsequent increment in the weight column. This monotonic relationship confirms that weight increases as height increases.

Explanation

In a system of concentric circles, a chord of the larger circle that is tangent to the smaller circle is bisected by the point of tangency. The radius of the smaller circle r is perpendicular to this chord at the point of contact. This configuration forms a right-angled triangle where the radius of the larger circle R is the hypotenuse, the radius of the smaller circle r is one leg, and half the length of the chord x is the other leg.

Using the Pythagorean theorem: R² = r² + x²

Substituting the given values R = 20 cm and r = 16 cm: 20² = 16² + x² 400 = 256 + x² x² = 400 - 256 x² = 144 x = √(144) = 12 cm

Since x represents half the length of the chord, the total length of the chord is 2 × 12 = 24 cm.

Explanation

The circumference of a circle is defined by the formula C = 2π r. To find the difference between the radii of two circles, we can use the linear relationship between the circumference and the radius:

C₂ - C₁ = 2π r₂ - 2π r₁ = 2π(r₂ - r₁)

Rearranging the formula to solve for the difference in radii (Δ r):

Δ r = (C₂ - C₁)/(2π)

Substituting the given circumferences C₂ = 440 cm and C₁ = 308 cm, and using π = ²²⁄₇:

Δ r = (440 - 308)/(2 × 22){7}

Δ r = (132)/(44){7}

Δ r = (132 × 7)/(44)

Since 132 = 44 × 3, the expression simplifies to:

Δ r = 3 × 7 = 21 cm

Explanation

The algebraic identity for the cube of a binomial difference is (x - y)³ = x³ - y³ - 3xy(x - y). By substituting x = a and y = ¹⁄ₐ into this identity, we get:

(a - ¹⁄ₐ)³ = a³ - (¹⁄ₐ)³ - 3(a)(¹⁄ₐ)(a - ¹⁄ₐ) (a - ¹⁄ₐ)³ = a³ - (1)/(a³) - 3(a - ¹⁄ₐ)

Rearranging the terms to match the given expression: a³ - (1)/(a³) = (a - ¹⁄ₐ)³ + 3(a - ¹⁄ₐ)

The left-hand side of this rearranged identity is a³ - (1)/(a³), which is equivalent to the expression 3[a - ¹⁄ₐ] + [a - ¹⁄ₐ]³. Thus, the expression simplifies to a³ - (1)/(a³).

Explanation

A number is divisible by both 7 and 11 if and only if it is divisible by their Least Common Multiple (LCM). Since 7 and 11 are both prime numbers, their LCM(7, 11) = 7 × 11 = 77. To find the total count of numbers from 1 to 430 that are divisible by 77, we divide the upper limit by the LCM and consider the integer portion of the quotient. Applying the formula n = ⌊ 430 / 77 ⌋ gives 5.58..., which means there are 5 such numbers. These numbers are 77 × 1 = 77, 77 × 2 = 154, 77 × 3 = 231, 77 × 4 = 308, and 77 × 5 = 385.

Explanation

Total work can be assumed as the Least Common Multiple (LCM) of the individual completion times, 12 and 8, which is 24 units. The efficiency of Vijay is 24 / 12 = 2 units per week, and the efficiency of Praveen is 24 / 8 = 3 units per week. They work in alternate weeks starting with Praveen, so in one cycle of 2 weeks, they complete 3 + 2 = 5 units of work.

In 4 full cycles, which represent 8 weeks, the total work completed is 4 × 5 = 20 units. The remaining work is 24 - 20 = 4 units. In the 9th week, it is Praveen's turn, and he completes 3 units of work. After the 9th week, the remaining work is 4 - 3 = 1 unit. In the 10th week, it is Vijay's turn. Since Vijay's efficiency is 2 units per week, the time he takes to complete the remaining 1 unit is 1 / 2 = 0.5 weeks. Therefore, the total time to complete the assignment is 9 + 0.5 = 9.5 weeks.

Explanation

A proportion a : b :: c : d is defined by the relation (a)/(b) = (c)/(d). Substituting the given values 14 : 30 :: 7 : x into this formula gives:

¹⁴⁄₃₀ = ⁷⁄ₓ

To solve for x, we apply the property that the product of the extremes equals the product of the means (ad = bc):

14 × x = 30 × 7

14x = 210

x = ²¹⁰⁄₁₄

x = 15

This can also be solved by observing the scaling factor between the antecedents. Since 14 is divided by 2 to obtain 7, the same operation must be applied to the consequent 30 to maintain the ratio. Therefore, x = ³⁰⁄₂ = 15.

Explanation

In a triangle, the distance from a vertex to the point where the incircle touches the adjacent sides is equal to the semi-perimeter minus the length of the opposite side. Let the semi-perimeter be s.

s = (AB + BC + AC)/(2) = (16 + 20 + 24)/(2) = ⁶⁰⁄₂ = 30 cm

The point D lies on side AB, and AD is the tangent from vertex A to the incircle. According to the properties of tangents to an incircle, the length of the tangent segment from vertex A is calculated as:

AD = s - BC

Substituting the given values:

AD = 30 - 20 = 10 cm

Explanation

When two triangles are similar, denoted as △ ABC ∼ △ DEF, the ratios of their corresponding sides are equal. This property of similarity is expressed by the relation:

(AB)/(DE) = (BC)/(EF)

The problem states that 2AB = DE, which allows us to determine the ratio of the corresponding sides AB and DE as:

(AB)/(DE) = ½

Substituting this ratio and the given value of BC = 8 cm into the proportionality equation:

½ = (8)/(EF)

Solving for EF by cross-multiplication:

EF = 8 × 2 = 16 cm

Explanation

The least number exactly divisible by a given set of numbers is their Least Common Multiple (LCM). To find the LCM of 9, 24, and 36, we determine the prime factorization of each number:

9 = 3² 24 = 2³ × 3¹ 36 = 2² × 3²

The LCM is calculated by taking the highest power of every prime factor present in these factorizations. For the prime factor 2, the highest power is 2³. For the prime factor 3, the highest power is 3².

LCM = 2³ × 3² = 8 × 9 = 72

Alternatively, using the common division method, we divide the numbers by their common prime factors: Dividing 9, 24, and 36 by 3 gives 3, 8, and 12. Dividing the results by 3 again gives 1, 8, and 4. Dividing the remaining numbers by 4 gives 1, 2, and 1. The LCM is the product of the divisors and the final remainders: 3 × 3 × 4 × 2 = 72.

Explanation

Sohan is a student buying a single book, making him eligible for the standard 10% discount for purchasing one book and an additional 5% student discount. These discounts are applied successively to the marked price.

Let M be the marked price of the book. The first discount of 10% reduces the price to 90% of M, and the second discount of 5% is applied to this reduced value. The relationship between the marked price and the final selling price is determined by the formula for successive discounts: SP = M × (1 - d₁/100) × (1 - d₂/100)

Substituting the given values into the equation: 513 = M × (1 - 10/100) × (1 - 5/100) 513 = M × (90/100) × (95/100) 513 = M × (9/10) × (19/20) 513 = M × (171/200)

To find M, rearrange the equation: M = (513 × 200) / 171

Dividing 513 by 171 gives exactly 3. Therefore: M = 3 × 200 = 600

Explanation

Let the speed of the man in still water be v and the speed of the stream be s. The speed downstream is D = v + s and the speed upstream is U = v - s. Since the man rows 4 km downstream in the same time as 3 km upstream, the ratio of the speeds is D : U = 4 : 3. We can represent these speeds as D = 4k and U = 3k for some constant k.

The total time for the round trip of 48 km is 14 hours. Using the formula Time = (Distance)/(Speed), we set up the equation ⁴⁸⁄₄ₖ + ⁴⁸⁄₃ₖ = 14. Simplifying the fractions gives ¹²⁄ₖ + ¹⁶⁄ₖ = 14, which results in ²⁸⁄ₖ = 14. Solving for k, we find k = 2.

Substituting k back into the speed expressions, the downstream speed D = 4 × 2 = 8 km/h and the upstream speed U = 3 × 2 = 6 km/h. The speed of the stream is given by the formula s = (D - U)/(2). Therefore, s = (8 - 6)/(2) = ²⁄₂ = 1 km/h.

Explanation

If the initial population is x, after 20% die, 80% remain (0.8x), and after 30% of those leave, 70% of that remainder is left (0.8x * 0.7 = 0.56x). Setting 0.56x equal to 12,000 gives x = 12,000 / 0.56, which is approximately 21,428.57, making 21,428 the integral part.

Explanation

Let the initial principal be 1 unit. The sum becomes 1.25 units in 3 years, which implies the simple interest earned in 3 years is 0.25 units. The interest for 1 year is (0.25)/(3) units.

The total interest for 7 years is calculated as: Interest = (0.25)/(3) × 7 = (1.75)/(3) = ¹⁷⁵⁄₃₀₀ = ⁷⁄₁₂ units.

The total amount after 7 years is the sum of the principal and the interest: Amount = 1 + ⁷⁄₁₂ = ¹⁹⁄₁₂ units.

Given that the required amount after seven years is 7,60,000: ¹⁹⁄₁₂ units = 7,60,000 1 unit = (7,60,000 × 12)/(19)

By dividing 7,60,000 by 19, we get 40,000. Thus: Principal = 40,000 × 12 = 4,80,000.

Explanation

The total consumption of coal across all six countries is calculated by summing the individual consumption values provided in the table for P, Q, R, S, T, and U.

Total Consumption = 132 + 248 + 185 + 245 + 226 + 174

By adding these values sequentially: 132 + 248 = 380 380 + 185 = 565 565 + 245 = 810 810 + 226 = 1036 1036 + 174 = 1210

Alternatively, summing by units, tens, and hundreds: Units: 2 + 8 + 5 + 5 + 6 + 4 = 30 Tens: 30 + 40 + 80 + 40 + 20 + 70 = 280 Hundreds: 100 + 200 + 100 + 200 + 200 + 100 = 900 Total: 30 + 280 + 900 = 1210

Explanation

To find the value of x³ + (1)/(x³⁾ from x⁴ + (1)/(x⁴⁾ = 194, we first reduce the power to find x + ¹⁄ₓ. Adding 2 to both sides of the given equation gives x⁴ + (1)/(x⁴⁾ + 2 = 194 + 2, which simplifies to (x² + (1)/(x²⁾)² = 196. Taking the square root, we get x² + (1)/(x²⁾ = 14.

Repeating this process, we add 2 to both sides again to get x² + (1)/(x²⁾ + 2 = 14 + 2, which simplifies to (x + ¹⁄ₓ)² = 16. Taking the square root gives x + ¹⁄ₓ = 4.

The algebraic identity for the sum of cubes is x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ). Substituting the value x + ¹⁄ₓ = 4 into this formula, we get 4³ - 3(4) = 64 - 12 = 52.

Explanation

The given expression follows the algebraic identity for the sum of cubes, which is defined as a³ + b³ = (a + b)(a² - ab + b²).

By identifying the terms in the expression (5x + 8y)(25x² + 64y² - 40xy), we can let a = 5x and b = 8y. Squaring these terms yields a² = (5x)² = 25x² and b² = (8y)² = 64y². Their product is ab = (5x)(8y) = 40xy.

Substituting these values into the identity, the expression matches the structure (a + b)(a² + b² - ab). Consequently, the result is a³ + b³. Calculating the cubes of the individual terms gives a³ = (5x)³ = 125x³ and b³ = (8y)³ = 512y³. Summing these together produces the simplified expression 125x³ + 512y³.

Explanation

The problem is solved using the work-rate formula M₁D₁{W₁} = M₂D₂{W₂}, where M represents the number of workers, D is the time in days, and W is the work completed in kilometers.

In the first phase, M₁ = 90 workers worked for D₁ = 200 days to complete W₁ = 0.5 km of the bridge. For the second phase, the remaining work is W₂ = 3 - 0.5 = 2.5 km, and the remaining time to meet the 600-day deadline is D₂ = 600 - 200 = 400 days. Let M₂ be the total number of workers required for the second phase.

Substituting the values into the formula: (90 × 200)/(0.5) = M₂ × 400{2.5}

Solving for M₂: 36000 = M₂ × 400{2.5} 36000 × 2.5 = 400 × M₂ 90000 = 400 × M₂ M₂ = ⁹⁰⁰⁰⁰⁄₄₀₀ = 225

The total number of workers needed to finish the work on time is 225. Since the contractor already has 90 workers, the number of extra workers required is 225 - 90 = 135.

Explanation

The shopkeeper's profit or loss is determined by the difference between the selling price and the cost price. Given the Cost Price CP = ₹1,200 and the Selling Price SP = ₹1,500, the profit is calculated as: Profit = SP - CP = 1,500 - 1,200 = 300

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

Substituting the values into the formula: Profit% = (300)/(1,200) × 100 = ¼ × 100 = 25%

Since the selling price is higher than the cost price, the shopkeeper earned a profit of 25 percent.

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3. Calculating the digit sums for the numbers in the expression: for 9435, 9 + 4 + 3 + 5 = 21; for 7593, 7 + 5 + 9 + 3 = 24; and for 2607, 2 + 6 + 0 + 7 = 15. Because 21, 24, and 15 are all multiples of 3, each individual number in the operation is divisible by 3. According to the properties of divisibility, if x, y, and z are divisible by n, then the linear combination x + y - z is also divisible by n. Therefore, the result of the calculation is divisible by 3.

Explanation

The Least Common Multiple (LCM) of two numbers is the smallest value that is divisible by both numbers. A fundamental property of the LCM states that if a number x is a multiple of another number y, then LCM(x, y) = x. Examining the given numbers, we observe that 373.2 = 186.6 × 2. Because 373.2 is an exact integer multiple of 186.6, it serves as the smallest common multiple for both values. Consequently, the LCM of 186.6 and 373.2 is 373.2.

Explanation

When a solid object is reshaped into another form, the total volume remains constant. The volume of a sphere is given by the formula Vₛ = ⁴⁄₃ π R³, where R is the radius of the sphere. Given a diameter of 18 metre, the radius R is 9 metre.

Volume of the sphere = ⁴⁄₃ π (9)³ = ⁴⁄₃ π × 729 = 4 π × 243 = 972 π cubic metre.

The sphere is drawn into a cylindrical wire. The volume of a cylinder is given by V_c = π r² h, where r is the radius and h is the length (height). Given the length h = 12 metre, we equate the volumes:

π r² × 12 = 972 π

Dividing both sides by 12 π:

r² = ⁹⁷²⁄₁₂

r² = 81

Taking the square root:

r = 9 metre

Explanation

The volume of a cuboid is given by the formula V = L × B × H. When dimensions change by percentages, the net percentage change can be calculated using the successive percentage formula a + b + ᵃᵇ⁄₁₀₀.

First, calculate the net effect of the increases in length and breadth. For a = 20% and b = 25%, the combined increase is 20 + 25 + (20 × 25)/(100) = 45 + 5 = 50%.

Next, combine this 50% increase with the 30% reduction in height. Using the same formula with a = 50% and b = -30%, the total change is 50 - 30 + (50 × (-30))/(100) = 20 - 15 = 5%.

Since the final result is positive, there is a total increase of 5% in the volume of the cuboid. Alternatively, using the multiplier method, the new volume is V × 1.20 × 1.25 × 0.70 = V × 1.05, which confirms a 5% increase.

Explanation

The fourth proportional d to three numbers a, b, and c is defined by the property that a, b, c, and d are in proportion, such that a:b = c:d. This relationship can be written as the formula d = (b × c)/(a).

Substituting the given values a = 24, b = 192, and c = 27 into the formula:

d = (192 × 27)/(24)

First, divide 192 by 24:

192 / 24 = 8

Then, multiply the result by 27:

d = 8 × 27 = 216

Explanation

The altitude h of an equilateral triangle with side length a is given by the formula: h = √3{2} a

Given the length of the altitude is 6√3 m, we substitute this into the formula: 6√3 = √3{2} a

Solving for a: a = 6√3 × 2{√3} a = 12 m

The perimeter P of an equilateral triangle is three times its side length: P = 3a P = 3 × 12 = 36 m

Explanation

The expression can be solved using the trigonometric identity for complementary angles, cos x = sin(90° - x). By setting x = 30° + θ, we find that cos(30° + θ) = sin(90° - (30° + θ)) = sin(60° - θ). Substituting this equivalent value into the given expression yields sin(60° - θ) - sin(60° - θ) = 0.

Alternatively, using the expansion formulas cos(A + B) = cos A cos B - sin A sin B and sin(A - B) = sin A cos B - cos A sin B: cos(30° + θ) = cos 30° cos θ - sin 30° sin θ = (√3)/2 cos θ - ½ sin θ sin(60° - θ) = sin 60° cos θ - cos 60° sin θ = (√3)/2 cos θ - ½ sin θ The difference between these two identical values is 0.

Explanation

To find the distance traveled by B, we first determine the lead distance A has covered during the 5-second delay. Since A travels at 34 ~m/s, the distance is Dₗₑₐ_d = 34 × 5 = 170 metres. The relative speed at which B closes this gap is the difference between their speeds: vᵣₑₗ = 44 ~m/s - 34 ~m/s = 10 ~m/s.

The time taken by B to catch A is calculated by dividing the lead distance by the relative speed: t = (170 metres)/(10 ~m/s) = 17 seconds. The total distance traveled by B during this time is D = v_B × t = 44 ~m/s × 17 s = 748 metres.

Explanation

The first 7 non-zero multiples of 17 form an arithmetic progression (AP): 17, 34, 51, 68, 85, 102, 119. For any sequence in arithmetic progression, the average is equal to the middle term if the number of terms is odd, or the mean of the first and last terms.

Using the formula for the average of an AP: Average = (a + l)/(2)

In this case, the first term a = 17 × 1 = 17 and the last (7th) term l = 17 × 7 = 119. Substituting these values: Average = (17 + 119)/(2) = ¹³⁶⁄₂ = 68

Alternatively, since there are 7 terms, the average is the 4th term of the sequence. Average = 17 × 4 = 68

Explanation

Using the standard trigonometric identities derived from (sin²θ + cos²θ) = 1, we have: sin⁴θ + cos⁴θ = (sin²θ + cos²θ)² - 2sin²θ cos²θ = 1 - 2sin²θ cos²θ sin⁶θ + cos⁶θ = (sin²θ + cos²θ)³ - 3sin²θ cos²θ(sin²θ + cos²θ) = 1 - 3sin²θ cos²θ

Substituting these identities into the given expression: 5(1 - 2sin²θ cos²θ) + 3(1 - 3sin²θ cos²θ) + 19sin²θ cos²θ Expanding the brackets: 5 - 10sin²θ cos²θ + 3 - 9sin²θ cos²θ + 19sin²θ cos²θ Grouping the constant terms and the sin²θ cos²θ terms: (5 + 3) + (-10 - 9 + 19)sin²θ cos²θ 8 + (0)sin²θ cos²θ = 8

Alternatively, since the expression is independent of θ, we can use the substitution method. Let θ = 0°, then sin 0° = 0 and cos 0° = 1: 5(0⁴ + 1⁴) + 3(0⁶ + 1⁶) + 19(0²)(1²) = 5(1) + 3(1) + 0 = 8

Explanation

The annual installment x required to discharge a debt A due in n years at simple interest rate R is given by the formula A = nx + (x · R · n(n-1))/(200). This formula accounts for the fact that each installment paid before the due date effectively earns interest until the end of the n-year period.

Substituting the given values into the formula where A = 9600, n = 5, and R = 10:

9600 = 5x + (x · 10 · 5(5 - 1))/(200)

9600 = 5x + (x · 10 · 20)/(200)

9600 = 5x + ²⁰⁰ˣ⁄₂₀₀

9600 = 5x + x

9600 = 6x

x = ⁹⁶⁰⁰⁄₆ = 1600

Alternatively, if we assume the annual installment is Rs. 100, the value of the installments at the end of 5 years would be the sum of each payment plus interest earned for the remaining time: End of Year 1: 100 + (100 · 10% · 4) = 140 End of Year 2: 100 + (100 · 10% · 3) = 130 End of Year 3: 100 + (100 · 10% · 2) = 120 End of Year 4: 100 + (100 · 10% · 1) = 110 End of Year 5: 100 + (100 · 10% · 0) = 100

Total debt discharged by a Rs. 100 installment = 140 + 130 + 120 + 110 + 100 = 600. To discharge a debt of Rs. 9,600, the installment must be 9600 · (100 / 600) = 1600.

Explanation

The quadratic expression under the radical, 36x² - 108x + 81, follows the perfect square trinomial identity (Ax - B)² = A²x² - 2ABx + B². By inspection, the first term 36x² is the square of 6x, and the constant term 81 is the square of 9. Verifying the middle term using 2ABx, we calculate 2 × 6x × 9 = 108x, which matches the middle term of the given expression. Thus, the expression under the radical can be rewritten as:

36x² - 108x + 81 = (6x - 9)²

Taking the square root of this perfect square yields:

√((6x - 9)²) = 6x - 9

Explanation

The given expression is in the form of the trigonometric identity tan(A - B) = (tan A - tan B)/(1 + tan A tan B). By comparing the terms, we identify A = 45° and B = 15°. Substituting these values into the identity, the expression simplifies to tan(45° - 15°), which is tan 30°. According to the standard trigonometric table, the value of tan 30° is 1/√3.

Explanation

In a right-angled triangle, sin θ is defined as the ratio of the perpendicular (P) to the hypotenuse (H). Given sin θ = ⅗, we can take P = 3 and H = 5. According to the Pythagorean theorem, H² = P² + B², where B represents the base of the triangle. Substituting the given values, 5² = 3² + B², which leads to 25 = 9 + B². Solving for the base, B² = 16, so B = 4. This is consistent with the standard Pythagorean triplet (3, 4, 5). Since tan θ is the ratio of the perpendicular to the base ((P)/(B)), we substitute the values to get tan θ = ¾.

Explanation

The selling price is 90 percent of the marked price, which is 90. Since the gain is 8 percent, the cost price is calculated as 90 divided by 1.08, which equals 83.33. Rounding this value to the nearest integer gives 83.

Explanation

Given the equation 4tan θ = 3, we have tan θ = ¾. Since tangent is the ratio of the perpendicular (P) to the base (B) in a right-angled triangle, we can consider P = 3 and B = 4. Using the Pythagorean triplet (3, 4, 5), the hypotenuse is H = √(3² + 4²) = 5. From this, we derive sinθ = (P)/(H) = ⅗ and cosθ = (B)/(H) = ⅘.

Now, substitute these values into the expression (4sinθ - cosθ + 1)/(4sinθ + cosθ - 1):

The numerator is calculated as: 4(⅗) - ⅘ + 1 = ¹²⁄₅ - ⅘ + ⁵⁄₅ = ¹³⁄₅

The denominator is calculated as: 4(⅗) + ⅘ - 1 = ¹²⁄₅ + ⅘ - ⁵⁄₅ = ¹¹⁄₅

Dividing the numerator by the denominator gives: (13/5)/(11/5) = ¹³⁄₁₁

Explanation

Expand the given equation (a + 1)² + (a + 2)² = 16 using the algebraic identity (x + y)² = x² + 2xy + y²: (a² + 2a + 1) + (a² + 4a + 4) = 16 2a² + 6a + 5 = 16 2a² + 6a = 11

The target expression is 4a² + 12a + 40. This can be rewritten by factoring out 2 from the terms containing the variable a: 2(2a² + 6a) + 40

By substituting the value 2a² + 6a = 11 into this expression: 2(11) + 40 = 22 + 40 = 62

Explanation

The volume V of a hemisphere is calculated using the formula V = ⅔ π r³, where r is the radius. Given the radius r = 5.6 cm and taking the value of π as ²²⁄₇, the calculation is:

V = ⅔ × ²²⁄₇ × (5.6)³

V = ⅔ × ²²⁄₇ × 5.6 × 5.6 × 5.6

Simplifying the terms by dividing 5.6 by 7:

V = ⅔ × 22 × 0.8 × 5.6 × 5.6

V = (44 × 0.8 × 31.36)/(3)

V = (35.2 × 31.36)/(3)

V = (1103.872)/(3)

V ≈ 367.957 cm³

Rounding to two decimal places, the volume of the hemisphere is 367.95 cm³.

Explanation

When two trains travel in the same direction, the relative speed is calculated as the difference between their individual speeds. In this scenario, the faster train moves at 48 km/hr and the slower train moves at 36 km/hr, resulting in a relative speed vᵣₑₗ = 48 - 36 = 12 km/hr. To perform calculations with time in seconds, we convert this speed into metres per second by multiplying by the factor ⁵⁄₁₈:

vᵣₑₗ = 12 × ⁵⁄₁₈ = ¹⁰⁄₃ m/s

The distance the faster train must cover to cross a man sitting in the slower train is exactly equal to the length of the faster train, L. Using the fundamental relation Distance = Speed × Time, we substitute the relative speed and the given time of 33 seconds:

L = ¹⁰⁄₃ × 33

L = 10 × 11 = 110 metres