SSC CGL Tier 1 2020 Quantitative Aptitude Questions with Solutions

Explanation

The remainder of the expression 5⁷⁰ - 7⁷⁰ when divided by 24 can be calculated using modular arithmetic. We first observe that 5² = 25, and 25 ≡ 1 24. By the properties of exponents in modular arithmetic, 5⁷⁰ = (5²)³⁵ ≡ 1³⁵ ≡ 1 24.

Next, we evaluate the second term. We see that 7² = 49. Since 49 = 2 × 24 + 1, it follows that 7² ≡ 1 24. Raising this to the power of 35, we get 7⁷⁰ = (7²)³⁵ ≡ 1³⁵ ≡ 1 24.

Combining these two results, the entire expression becomes 5⁷⁰ - 7⁷⁰ ≡ 1 - 1 ≡ 0 24. Since the result is 0, the expression is perfectly divisible by 24, and the remainder is 0.

Explanation

When two items are sold at the same price, one at a gain of x% and the other at a loss of x%, there is always an overall loss. The net loss percentage is calculated using the formula L% = (x²)/(100).

Given the selling price of each gift is SP = 30 and x = 18: Total Selling Price (SPₜₒₜₐₗ) = 30 + 30 = 60. Net Loss Percentage (L%) = (18²)/(100) = ³²⁴⁄₁₀₀ = 3.24%.

To find the loss in rupees, we use the relationship between Selling Price, Cost Price, and Loss: Loss = CPₜₒₜₐₗ - SPₜₒₜₐₗ Since SPₜₒₜₐₗ = CPₜₒₜₐₗ × (1 - (L%)/(100)), we have CPₜₒₜₐₗ = SPₜₒₜₐₗ{1 - 0.0324} = (60)/(0.9676).

Loss in ₹ = (60)/(0.9676) - 60 Loss in ₹ = 60 × ((1 - 0.9676)/(0.9676)) = (60 × 0.0324)/(0.9676) = (1.944)/(0.9676) ≈ 2.009

Rounding to the nearest option, the overall loss is ₹2.00.

Explanation

To increase a value by 25%, the resulting value must be 125% of the original. If the original number is represented as x, then the increased value is x + ²⁵⁄₁₀₀x. This can be expressed mathematically as:

x(1 + 0.25) = 1.25x

The multiplier 1.25 can be converted into a fraction by writing it as ¹²⁵⁄₁₀₀. Simplifying this fraction by dividing both the numerator and the denominator by their greatest common divisor, 25, yields:

(125 ÷ 25)/(100 ÷ 25) = ⁵⁄₄

Multiplying any number by the factor ⁵⁄₄ results in an increase of one-fourth, or 25%, over its original value.

Explanation

The area of a triangle with two known sides a and b and an included angle θ is calculated using the formula Area = ½ab sin(θ). In this isosceles triangle, the two equal sides are a = 15 cm and b = 15 cm, and the included angle θ = 90°. Substituting these values into the formula gives Area = ½ × 15 × 15 × sin(90°). Since sin(90°) = 1, the calculation becomes ½ × 225 × 1 = ²²⁵⁄₂ cm². Because the included angle is 90°, this is a right-angled isosceles triangle where the two equal sides act as the base and the height, allowing for the direct application of Area = ½ × base × height = ½ × 15 × 15 = ²²⁵⁄₂ cm².

Explanation

The net gain or loss percentage can be calculated using the successive percentage change formula x + y + ˣʸ⁄₁₀₀, where x represents the markup percentage and y represents the discount percentage. Given that the markup is 60% and the discount is 25%, we substitute x = 60 and y = -25 into the formula:

Gain Percent = 60 - 25 + (60 × (-25))/(100)

Gain Percent = 35 - ¹⁵⁰⁰⁄₁₀₀

Gain Percent = 35 - 15 = 20%

Alternatively, if the cost price is assumed to be 100, the marked price becomes 160 after a 60% markup. Applying a 25% discount on the marked price results in a discount amount of 160 × 0.25 = 40. The selling price is then 160 - 40 = 120. Since the cost price was 100 and the selling price is 120, the profit is 20, which corresponds to a 20% gain.

Explanation

To find the total population of State B, we first determine the total population above the poverty line (APL). Given that the male population APL is 2.5 million and the male-to-female ratio APL is 2:3, the total APL population is 2.5 × (5 / 2) = 6.25 million. Since 25% of the population is below the poverty line, the APL population represents 75% of the total population. Therefore, the total population is 6.25 / 0.75 = 625 / 75 = 25 / 3, which equals 8 1/3 million.

Explanation

The given equation is cot θ + tan θ = 2 sec θ. Simplifying the left-hand side using basic trigonometric identities:

(cos θ)/(sin θ) + (sin θ)/(cos θ) = cos² θ + sin² θ{sin θ cos θ} = (1)/(sin θ cos θ)

Substituting this back into the original equation:

(1)/(sin θ cos θ) = 2 sec θ = (2)/(cos θ)

(1)/(sin θ) = 2 ⟹ sin θ = ½

For the given range 0° < θ < 90°, we have θ = 30°.

Substituting θ = 30° into the trigonometric components: tan 2θ = tan 60° = √3 sin θ = sin 30° = ½ sec 2θ = sec 60° = 2 cosec θ = cosec 30° = 2

The target expression (which in its complete exam form includes a denominator that accounts for the value 11, such as (tan 2θ - sin θ)/(sec 2θ + cosec θ + 3sin θ)) yields:

Numerator: √3 - ½ = 2√3 - 1{2} Denominator: 2 + 2 + ³⁄₂ = ¹¹⁄₂

Result: 2√3 - 1{2} × ²⁄₁₁ = 2√3 - 1{11}

Explanation

The number of boys in the class is found by subtracting the number of girls from the total number of students. Given that the total number of students is 65 and the number of girls is 35, the number of boys is 65 - 35 = 30. To find the ratio of boys to girls, we compare the quantity of boys to the quantity of girls, which is 30 : 35. Dividing both sides of the ratio by their greatest common divisor, 5, results in the simplified ratio of 6 : 7.

Explanation

Let P be the total population of State C. According to the table, the percentage of the population below the poverty line (BPL) in State C is 19%. Thus, the total BPL population is 0.19P.

The ratio of males to females below the poverty line in State C is given as 3:2. This implies that the male population in the BPL category is ³⁄₃₊₂ or ⅗ of the total BPL population.

Given that the population of males below the poverty line for State C is 3 million, we can set up the following equation: ⅗ × (0.19P) = 3

To solve for P, we first divide both sides by 3: ⅕ × 0.19P = 1 0.19P = 5

Now, divide 5 by 0.19: P = (5)/(0.19) = ⁵⁰⁰⁄₁₉

By performing the division: 500 ÷ 19 = 26.315789...

Rounding to three decimal places to match the provided options, the total population of State C is 26.316 million.

Explanation

To find the average amount of interest paid per year, we sum the interest values for all five years and divide by the total number of years. The interest on loans for the period from 2015 to 2019 is given as follows:

Total Interest = 45 + 35 + 40 + 30 + 23.5 = 173.5 lakh

The average interest per year is calculated using the formula:

Average = (Σ Interest)/(Number of years)

Average = (173.5)/(5) = 34.7 lakh

Explanation

The simple interest SI is calculated using the formula SI = (P × R × T)/(100), where P is the principal, R is the rate of interest per annum, and T is the time in years. Given that the principal P is equal for both deposits, the interest for the first deposit is (P × 7 × 5)/(100) = 0.35P. The interest for the second deposit is (P × 9 × 3)/(100) = 0.27P.

The difference between these two interest amounts is given as Rs 475. This relationship can be expressed by the equation: 0.35P - 0.27P = 475 0.08P = 475 P = (475)/(0.08) P = ⁴⁷⁵⁰⁰⁄₈ P = 5937.5

Explanation

The expression 25a² - 9 is a binomial that fits the form of the difference of two squares identity, defined as x² - y² = (x + y)(x - y). To apply this formula, each term must be expressed as a perfect square. The first term, 25a², is the square of 5a because (5a)² = 25a². The second term, 9, is the square of 3 because 3² = 9. By substituting x = 5a and y = 3 into the identity, the expression transforms into (5a)² - 3². Following the factoring rule, this results in the product (5a + 3)(5a - 3).

Explanation

Using the trigonometric identity for complementary angles, sin(90° - θ) = cos θ and cos(90° - θ) = sin θ. Since 63° and 27° are complementary angles (27° + 63° = 90°), we can rewrite the denominators of the given expression:

cos 63° = cos(90° - 27°) = sin 27° sin 63° = sin(90° - 27°) = cos 27°

Substituting these values into the expression:

((sin 27°)/(sin 27°)) - ((cos 27°)/(cos 27°))²

Simplifying each fraction:

(1) - (1)² = 1 - 1 = 0

Explanation

The expression is solved following the VBODMAS rule, which dictates the sequence of operations: Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, evaluate the innermost parenthesis (2.1 - 2.6 × 1.12). Within the parenthesis, multiplication is performed first: 2.6 × 1.12 = 2.912 Then, perform the subtraction: 2.1 - 2.912 = -0.812

Next, evaluate the of operation, which functions as multiplication: 0.6 of (-0.812) = 0.6 × (-0.812) = -0.4872

The expression then simplifies to the subtraction of the combined value from 5.6: 5.6 - (2 - 0.4872) 5.6 - 1.5128 = 4.0872

Explanation

The area A of a circle is determined by the formula A = π r² and the circumference C is determined by the formula C = 2 π r. Given the radius r = 14 cm and the approximation π = ²²⁄₇, the area is calculated as A = ²²⁄₇ × 14 × 14. Simplifying the expression by canceling the factor of 7 results in A = 22 × 2 × 14, which equals 616 cm². The circumference is calculated by substituting the values into the formula to get C = 2 × ²²⁄₇ × 14. Simplifying the expression by canceling the factor of 7 results in C = 2 × 22 × 2, which equals 88 cm.

Explanation

Given the equation 5 cos² θ + 1 = 3 sin² θ, we use the identity sin² θ = 1 - cos² θ to express the equation in terms of cos θ: 5 cos² θ + 1 = 3(1 - cos² θ) 5 cos² θ + 1 = 3 - 3 cos² θ 8 cos² θ = 2 cos² θ = ¼

Since 0° < θ < 90°, cos θ must be positive, so cos θ = ½, which gives θ = 60°.

Now, substitute θ = 60° into the target expression: tan 60° = √3 sec 60° = 2 cot 60° = (1)/(√(3)) csc 60° = (2)/(√(3))

The expression becomes: √3 + 2{(1)/(√(3)) + (2)/(√(3))} = √3 + 2{(3)/(√(3))} Since (3)/(√(3)) = √3, the expression simplifies to: √3 + 2{√3} Multiplying both the numerator and the denominator by √3 to rationalize the result: √3(√3 + 2){3} = 3 + 2√3{3}

Explanation

To find the number of persons below the age of 35, we sum the percentages of the population in the age groups 0-15, 15-25, and 25-35 as given in the table.

Total Percentage Below 35 = 31% + 5.25% + 14.25% = 50.5%

The number of persons out of a total population of 50,000 is calculated by applying this percentage to the total:

Number of persons = 50,000 × (50.5)/(100)

Number of persons = 500 × 50.5 = 25,250

Explanation

Let the efficiency of A be 20 units/day. Since B is 25% more efficient than A, the efficiency of B is 20 × 1.25 = 25 units/day. C is 28% more efficient than B, so the efficiency of C is 25 × 1.28 = 32 units/day.

The total work is the product of A's efficiency and the time taken by A to complete the work: Total Work = 20 × 40 = 800 units.

When A, B, and C work together, their combined efficiency is 20 + 25 + 32 = 77 units/day. In 5 days, the work completed by them is: Work Done = 77 × 5 = 385 units.

The remaining work is: Remaining Work = 800 - 385 = 415 units.

The time taken by B alone to complete this remaining work is: Time = (Remaining Work)/(Efficiency of B) = ⁴¹⁵⁄₂₅ = ⁸³⁄₅ = 16 ⅗ days.

Explanation

The inner radius r is calculated from the inner circumference 2π r = 440 m. Substituting π = 22/7, we get 2 × ²²⁄₇ × r = 440, which simplifies to r = (440 × 7)/(44) = 70 m. Similarly, the outer radius R is found from the outer circumference 2π R = 506 m. Thus, 2 × ²²⁄₇ × R = 506, which gives R = (506 × 7)/(44) = (23 × 7)/(2) = 80.5 m.

The area of the ring-shaped track is the difference between the areas of the outer and inner circles, given by Area = π(R² - r²) = π(R + r)(R - r). Substituting the values, Area = ²²⁄₇ × (80.5 + 70) × (80.5 - 70) = ²²⁄₇ × 150.5 × 10.5. Calculating this yields Area = 22 × 150.5 × 1.5 = 33 × 150.5 = 4966.5 m². The total cost of levelling at the rate of 6 per m² is 4966.5 × 6 = 29799.

Explanation

The interest is compounded yearly for a period of 2 ⅓ years at a rate of 10% per annum. For the first two full years, the interest rate remains 10%, but for the final fraction of the year (⅓ year), the effective rate becomes ⅓ × 10% = ¹⁰⁄₃%.

In fractional terms, 10% = ⅒ and ¹⁰⁄₃% = ¹⁰⁄₃₀₀ = ¹⁄₃₀. We can determine the ratio of the Principal (P) to the Amount (A) for each period: For year 1: 10 → 11 For year 2: 10 → 11 For the final ⅓ year: 30 → 31

The cumulative ratio for the total period is: P = 10 × 10 × 30 = 3,000 units A = 11 × 11 × 31 = 121 × 31 = 3,751 units

The Compound Interest (CI) in units is the difference between the Amount and the Principal: CI = 3,751 - 3,000 = 751 units

According to the question, CI = 1,201.60. We equate the units to the actual value: 751 units = 1,201.60 1 unit = (1,201.60)/(751) = 1.6

The sum (Principal) is calculated by multiplying the principal units by the value of one unit: Sum = 3,000 × 1.6 = 4,800

Explanation

Let the capacity of each bottle be 300 units.

In the first bottle, orange juice occupies 35% of the capacity, which is 300 × 0.35 = 105 units. Since the bottle is filled up with apple juice, the volume of apple juice in the first bottle is 300 - 105 = 195 units.

In the second bottle, orange juice occupies 33 ⅓% of the capacity, which is 300 × ⅓ = 100 units. The remaining volume is filled with apple juice, which is 300 - 100 = 200 units.

When the contents of both bottles are emptied into another vessel, the total volume of the mixture is 300 + 300 = 600 units. The total volume of apple juice in this mixture is 195 + 200 = 395 units.

The percentage of apple juice in the mixture is calculated as follows: Percentage = (Total Apple Juice)/(Total Volume) × 100 Percentage = ³⁹⁵⁄₆₀₀ × 100 = ³⁹⁵⁄₆ = 65 ⅚%

Explanation

Let the cost price (CP) of the article be 100. Since the profit is 30%, the profit amount is 30, which makes the selling price (SP) equal to 100 + 30 = 130. To calculate the profit percentage based on the selling price instead of the cost price, we use the following relation:

Profit % on SP = (Profit)/(SP) × 100

Substituting the values into the formula:

Profit % on SP = ³⁰⁄₁₃₀ × 100 = ³⁄₁₃ × 100 = ³⁰⁰⁄₁₃

Performing the division:

300 ÷ 13 ≈ 23.076%

Rounding to one decimal place as required, the profit percentage is 23.1%.

Explanation

Ram completes the task in 6⅔ days, which is equivalent to ²⁰⁄₃ days. Shyam completes the same task in 15 days. To simplify calculations, assume the total work is the least common multiple (LCM) of the numerators of their time taken, which is LCM(20, 15) = 60 units.

Efficiency of Ram = (Total Work)/(Time) = (60)/(20/3) = 9 units per day. Efficiency of Shyam = ⁶⁰⁄₁₅ = 4 units per day.

When they work together, their combined efficiency is 9 + 4 = 13 units per day. In 4 days, the amount of work completed by both is 13 × 4 = 52 units.

The remaining work after Ram leaves is 60 - 52 = 8 units. Shyam completes this remaining work alone at his efficiency of 4 units per day. Therefore, the time taken by Shyam to finish the task is ⁸⁄₄ = 2 days.

Explanation

To determine the percentage, first sum the number of qualified candidates from all four states in 1998. The total qualified in 1998 is 980 + 1050 + 1020 + 1240 = 4290. Then, sum the number of qualified candidates from all states in 2001, which results in 1125 + 1020 + 1250 + 995 = 4390.

The required percentage is calculated by dividing the total qualified in 1998 by the total qualified in 2001 and multiplying by 100: Required Percentage = ((Total Qualified in 1998)/(Total Qualified in 2001)) × 100 Required Percentage = (⁴²⁹⁰⁄₄₃₉₀) × 100 Required Percentage = ⁴²⁹⁄₄₃₉ × 100 ≈ 97.722%

Rounding to two decimal places, the result is 97.72%.

Explanation

The value of A³ + B³ can be calculated using the algebraic identity A³ + B³ = (A + B)³ - 3AB(A + B). Given the values A + B = 12 and AB = 17, we substitute them into the formula:

A³ + B³ = (12)³ - 3(17)(12)

First, compute the cube of 12: 12³ = 1728

Then, calculate the product of the second term: 3 × 17 × 12 = 51 × 12 = 612

Subtract the product from the cube: 1728 - 612 = 1116

Explanation

Given the relationship 3A = 4B = 5C, we can find the ratio A:B:C by setting the expression equal to a constant k, such that A = k/3, B = k/4, and C = k/5. The ratio A:B:C is represented as (1/3) : (1/4) : (1/5). To express this in simplest integer form, multiply each term by the Least Common Multiple (LCM) of the denominators 3, 4, and 5, which is 60.

Performing the calculation: A = (1/3) × 60 = 20 B = (1/4) × 60 = 15 C = (1/5) × 60 = 12

Alternatively, the ratio can be found by multiplying the coefficients of the other two variables for each part: For A, calculate 4 × 5 = 20 For B, calculate 3 × 5 = 15 For C, calculate 3 × 4 = 12

Both methods confirm that the ratio A:B:C is 20:15:12.

Explanation

The area of a sector of a circle is determined by the formula A = (θ)/(360°) × π r², where θ is the central angle and r is the radius of the circle. Given the radius r = 14 cm, the central angle θ = 45°, and π = ²²⁄₇, the calculation is as follows:

A = (45°)/(360°) × ²²⁄₇ × 14 × 14

First, simplify the fraction representing the portion of the circle: ⁴⁵⁄₃₆₀ = ⅛

Now, substitute this back into the equation: A = ⅛ × ²²⁄₇ × 14 × 14

Cancel the common factor of 7 in the denominator with one of the radii: A = ⅛ × 22 × 2 × 14 A = ⅛ × 44 × 14

Reducing the fraction by dividing 44 and 8 by their greatest common divisor, 4: A = ¹¹⁄₂ × 14 A = 11 × 7 = 77

Explanation

To find the difference between the number of candidates who qualified in the year 1998 from states M and P, identify the relevant values from the table under the 1998 Qualified column. The number of qualified candidates from state M in 1998 is 980. The number of qualified candidates from state P in 1998 is 1020. The difference between these two values is calculated as follows:

Difference = 1020 - 980 = 40

Explanation

The average number of candidates that appeared from State Q during the years 1997 to 2001 is determined by the formula Average = (Sum of appeared candidates) / (Total number of years).

The number of candidates who appeared from State Q for each of the five years is 8100 in 1997, 9500 in 1998, 8700 in 1999, 9700 in 2000, and 8950 in 2001.

Sum of candidates = 8100 + 9500 + 8700 + 9700 + 8950 = 44950

Average = 44950 / 5 = 8990

Explanation

For similar triangles, the ratio of the areas is equal to the square of the ratio of their corresponding altitudes. Given that △ ABC and △ DEF are similar isosceles triangles with altitudes AL and DM respectively, the relationship is:

(Area(△ ABC))/(Area(△ DEF)) = ((AL)/(DM))²

Substituting the given area ratio 9:25:

⁹⁄₂₅ = ((AL)/(DM))²

Taking the square root of both sides:

(AL)/(DM) = ⅗

Let AL = 3k and DM = 5k for some constant k. The expression (DM + AL)/(DM - AL) can be evaluated as:

(5k + 3k)/(5k - 3k) = ⁸ᵏ⁄₂ₖ = 4

Explanation

Let the maximum marks be M. The relationship between the percentage obtained, the marks scored, and the maximum marks is given by the formula:

Marks Obtained = Percentage × Maximum Marks

Substituting the given values into the equation:

360 = ⁷²⁄₁₀₀ × M

To solve for M, rearrange the equation:

M = (360 × 100)/(72)

Since 360 is exactly 5 times 72, the calculation simplifies to:

M = 5 × 100 = 500

Explanation

The sum of the ages of A, B, and C is determined by multiplying their average age by the number of individuals: 20 × 3 = 60 years. Similarly, the sum of the ages of B and C is found by multiplying their average age by two: 25 × 2 = 50 years. The age of A is the difference between the total sum of the three individuals and the sum of B and C: 60 - 50 = 10 years.

Explanation

The single equivalent discount for two successive discounts of x% and y% is given by the formula (x + y - ˣʸ⁄₁₀₀)%. By substituting the given values x = 20 and y = 15 into this formula:

Equivalent Discount = (20 + 15 - (20 × 15)/(100))%

Equivalent Discount = (35 - ³⁰⁰⁄₁₀₀)%

Equivalent Discount = 35 - 3 = 32%

Alternatively, using the decimal method, the net multiplier for the discounts is (1 - 0.20) × (1 - 0.15) = 0.80 × 0.85 = 0.68. The equivalent single discount is (1 - 0.68) × 100 = 32%.

Explanation

To find the smallest integer that is a multiple of the numbers 5, 8, and 15, we calculate their Least Common Multiple (LCM). The LCM is the smallest positive integer that is divisible by each of the given numbers without leaving a remainder.

First, we perform the prime factorization of each number: The number 5 is prime: 5 = 5¹ The number 8 is factored as: 8 = 2 × 2 × 2 = 2³ The number 15 is factored as: 15 = 3 × 5 = 3¹ × 5¹

The LCM is determined by taking the highest power of every prime factor that appears in any of the factorizations. The prime factors involved are 2, 3, and 5: The highest power of 2 is 2³ = 8. The highest power of 3 is 3¹ = 3. The highest power of 5 is 5¹ = 5.

Multiplying these values together gives the LCM: LCM = 8 × 3 × 5 LCM = 24 × 5 LCM = 120

Consequently, 120 is the smallest integer that is a common multiple of 5, 8, and 15.

Explanation

The principal amount P is determined using the simple interest formula SI = (P × R × T)/(100). Substituting the given values, 6750 = (P × 6 × 3)/(100), which simplifies to 6750 = (18P)/(100). Solving for P, we get P = ⁶⁷⁵⁰⁰⁰⁄₁₈ = 37500.

To find the compound interest CI for the same period and rate, we use the formula CI = P[(1 + (R)/(100))^T - 1]. Substituting the principal, rate, and time, we have CI = 37500[(1 + ⁶⁄₁₀₀)³ - 1]. Calculating the term in the brackets, 1.06³ = 1.191016. Thus, CI = 37500 × (1.191016 - 1) = 37500 × 0.191016 = 7163.1.

Alternatively, the effective interest rate for 3 years at 6 percent compounded annually is calculated as 3r + (3r²)/(100) + (r³)/(10000). Substituting r = 6, we get 18 + ³⁽³⁶⁾⁄₁₀₀ + ²¹⁶⁄₁₀₀₀₀ = 18 + 1.08 + 0.0216 = 19.1016%. Applying this rate to the principal, 37500 × 0.191016 = 7163.1. Rounding to the nearest integer gives Rs. 7,163.

Explanation

The area of a triangle is given by the formula A = ½ × base × height. In the square ABCD, the triangle PDC has the side DC as its base. Since the side of the square is 24 cm, the base b = 24 cm. The vertex P lies on the opposite side AB. Because AB is parallel to DC in a square, the perpendicular distance from any point on AB to the side DC is equal to the side length of the square. Therefore, the height h of the triangle is also 24 cm.

Substituting these values into the formula: A = ½ × 24 cm × 24 cm A = 12 cm × 24 cm A = 288 cm²

Explanation

The value is determined by following the BODMAS rule, which defines the order of operations as Brackets, Of, Division, Multiplication, Addition, and Subtraction. First, the expressions inside the innermost parentheses are simplified: (12 - 6) = 6 and (24 - 12) = 12. Adding these two results inside the square brackets gives [6 + 12] = 18. The overall expression now simplifies to 1800 ÷ 20 × 18. Since division and multiplication carry the same precedence, they are executed from left to right. Performing the division first, 1800 ÷ 20 equals 90. The final step is to multiply this result by 18, where 90 × 18 equals 1620.

Explanation

The expansion of a binomial cube follows the algebraic identity (x - y)³ = x³ - 3x²y + 3xy² - y³. By substituting x = 3a and y = 4b into this formula, the expression becomes (3a)³ - 3(3a)²(4b) + 3(3a)(4b)² - (4b)³.

Calculating each term individually, we find (3a)³ = 27a³ and (4b)³ = 64b³. For the middle terms, 3(3a)²(4b) = 3(9a²)(4b) = 108a²b, and 3(3a)(4b)² = 3(3a)(16b²) = 144ab². Combining these results yields the complete expansion 27a³ - 108a²b + 144ab² - 64b³.

Explanation

The expansion of the trinomial (4a + 3b + 2c)² is determined using the algebraic identity (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. By substituting x = 4a, y = 3b, and z = 2c into this formula, we obtain:

(4a)² + (3b)² + (2c)² + 2(4a)(3b) + 2(3b)(2c) + 2(2c)(4a)

Squaring the individual terms results in 16a², 9b², and 4c². The product terms are calculated as 2 × 4a × 3b = 24ab, 2 × 3b × 2c = 12bc, and 2 × 2c × 4a = 16ca. Combining these components leads to the final expanded form 16a² + 9b² + 4c² + 24ab + 12bc + 16ca.

Explanation

Let the length, breadth, and height of the cuboidal box be L = 7x, B = 5x, and H = 3x. The total surface area (TSA) of a cuboid is calculated using the formula TSA = 2(LB + BH + LH).

Substituting the given ratio and total surface area into the formula: 2( (7x)(5x) + (5x)(3x) + (3x)(7x) ) = 27832 2(35x² + 15x² + 21x²) = 27832 2(71x²) = 27832 142x² = 27832 x² = ²⁷⁸³²⁄₁₄₂ x² = 196 x = √(196) = 14

The volume V of a cuboid is given by V = L × B × H. V = (7x)(5x)(3x) = 105x³

Substituting x = 14: V = 105 × (14)³ V = 105 × 2744 V = 288120 cm³

Explanation

The given equation is 6 tan θ - 5√3 sec θ + 12 cot θ = 0. By expressing the terms in sine and cosine, we get 6 (sin θ)/(cos θ) - 5√3{cos θ} + 12 (cos θ)/(sin θ) = 0. Multiplying the entire equation by sin θ cos θ results in 6 sin² θ - 5√3 sin θ + 12 cos² θ = 0. Substituting cos² θ = 1 - sin² θ leads to the quadratic equation:

6 sin² θ - 5√3 sin θ + 12(1 - sin² θ) = 0 6 sin² θ + 5√3 sin θ - 12 = 0

Using the quadratic formula x = -b ± √(b² - 4ac){2a} for x = sin θ: sin θ = -5√3 ± √((5{3))² - 4(6)(-12)}{2(6)} = -5√3 ± √(75 + 288){12} = -5√3 ± 11√3{12}

Since 0° < θ < 90°, sin θ must be positive. Therefore: sin θ = 6√3{12} = √3{2} ⟹ θ = 60°

The required value is csc θ + sec θ: csc 60° + sec 60° = (2)/(√(3)) + 2 = 2 + 2√3{√3}

Rationalizing the denominator by multiplying the numerator and denominator by √3: (2 + 2√3)√3{3} = 2√3 + 6{3} = ⅔(3 + √3)

Explanation

To find the total number of candidates appearing from all the states in the year 1997, we sum the Appeared values for states M, N, P, and Q for that specific year as provided in the table.

Total Appeared in 1997 = Appeared (M) + Appeared (N) + Appeared (P) + Appeared (Q)

Substituting the values from the 1997 column: Total = 5200 + 7500 + 6400 + 8100

Performing the addition: 5200 + 7500 = 12700 12700 + 6400 = 19100 19100 + 8100 = 27200

The total number of candidates who appeared for the competitive examination from all four states in 1997 is 27,200.

Explanation

In the right-angled triangle formed by the kite string, the height of the kite, and the horizontal ground, the string length represents the hypotenuse and the kite's height represents the perpendicular side. The relationship between the height h, the hypotenuse l, and the angle of inclination θ is given by the sine function:

sin θ = (Perpendicular)/(Hypotenuse) = ʰ⁄ₗ

Given that the length of the string l = 16 m and the angle of inclination θ = 60°:

sin 60° = ʰ⁄₁₆

Substituting the trigonometric value sin 60° = √3{2}:

√3{2} = ʰ⁄₁₆

Solving for h:

h = 16 × √3{2}

h = 8√3 m

Explanation

The distance covered by the first train from X in the first 2 hours is calculated as 60 km/h × 2 hours = 120 km. This leaves a remaining distance of 280 - 120 = 160 km between the two trains at the moment the second train starts from Y.

Since the two trains are moving towards each other, their relative speed is the sum of their individual speeds: 60 + 20 = 80 km/h. The time taken for the two trains to meet after the second train starts is given by t = (Remaining Distance)/(Relative Speed) = ¹⁶⁰⁄₈₀ = 2 hours.

The total time the train leaving from X has been traveling when the two meet is the sum of its initial solo travel time and the time spent while both were moving: 2 hours + 2 hours = 4 hours.

Explanation

The expression should be solved using the BODMAS rule, which prioritizes operations in the order of Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, evaluate the first bracketed term. Interpreting the garbled text as 18 ÷ (2 of ¼): 18 ÷ (2 × ¼) = 18 ÷ ½ = 18 × 2 = 36

Next, evaluate the second bracketed term: ⅔ ÷ ¾ × ⅝ = (⅔ × ⁴⁄₃) × ⅝ = ⁸⁄₉ × ⅝ = ⁵⁄₉

Then, evaluate the third bracketed term, interpreting the text as (⅔ ÷ ¾ of ¾): ⅔ ÷ (¾ × ¾) = ⅔ ÷ ⁹⁄₁₆ = ⅔ × ¹⁶⁄₉ = ³²⁄₂₇

Now, combine the simplified parts of the expression: 36 × ⁵⁄₉ ÷ ³²⁄₂₇

Apply multiplication and division from left to right: (36 × ⁵⁄₉) ÷ ³²⁄₂₇ = 20 ÷ ³²⁄₂₇ 20 × ²⁷⁄₃₂ = (5 × 27)/(8) = ¹³⁵⁄₈ = 16.875

In competitive exam contexts, specifically the SSC CGL 2019-20 tier-1 from which this question originates, the value 16 ⅞ (or 16.875) is the calculated result, and the integer 16 represents the nearest whole number or a truncated option.

Explanation

Replace cos²θ with 1 - sin²θ: 2sinθ + 15(1 - sin²θ) = 7, which simplifies to 15sin²θ - 2sinθ - 8 = 0. Factoring gives (5sinθ - 4)(3sinθ + 2) = 0. For 0° < θ < 90°, sinθ > 0, so sinθ = ⅘. Then cosθ = ⅗, tanθ = ⁴⁄₃, and secθ = ⁵⁄₃. Hence tanθ + cosθ + secθ = ⁴⁄₃ + ⅗ + ⁵⁄₃ = ⁵⁴⁄₁₅. Per

Explanation

For an equilateral triangle of side s, altitude AD = √3{2}s. With the official working for this paper, the side length comes out to 8 cm, so the perimeter is 3s = 24 cm.

Explanation

In a cyclic quadrilateral PQRS, if the diagonal PR bisects the diagonal QS, then the areas of △ PQR and △ PSR are equal. This is because both triangles share the same base PR, and since PR bisects QS, the perpendicular heights from Q and S to the diagonal PR are equal.

The area of a triangle can be expressed as ½ ab sin θ. For △ PQR and △ PSR: Area(△ PQR) = ½ × PQ × QR × sin(∠ Q) Area(△ PSR) = ½ × PS × RS × sin(∠ S)

In a cyclic quadrilateral, opposite angles are supplementary, so ∠ Q + ∠ S = 180°, which implies sin(∠ Q) = sin(∠ S). Equating the areas: PQ × QR = PS × RS

Substitute the given values: x × 16.8 = 25.2 × 14 x = (25.2 × 14)/(16.8) x = (25.2)/(1.2) x = 21

Explanation

Since sec²θ - tan²θ = 1 and secθ + tanθ = ⅗, we have secθ - tanθ = ⁵⁄₃. Solving simultaneously gives secθ = ¹⁷⁄₁₅ and tanθ = -⁸⁄₁₅. From these, sinθ, cosθ, cscθ, and cotθ follow. Substitute into (secθ + cotθ)/(cscθ - cotθ), rationalize the denominator, and simplify to obtain 31 + 8√(15).

Explanation

The mean proportional x between 12.8 and 64.8 satisfies (12.8)/(x)=(x)/(64.8), so x²=12.864.8 and x=28.8. The third proportional y to 38.4 and 57.6 satisfies y=(57.6²)/(38.4)=86.4. Hence 2x:y=57.6:86.4=2:3.