SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Let the monthly incomes of the two persons be 4x and 5x.

Since both persons save ₹50 per month, their expenditures are 4x - 50 and 5x - 50 respectively.

The ratio of their expenditures is given as 7:9. Setting up the equation:

(4x - 50) / (5x - 50) = 7/9

Cross-multiplying the terms:

9 × (4x - 50) = 7 × (5x - 50)

36x - 450 = 35x - 350

Subtracting 35x from both sides and adding 450 to both sides:

x = 100

Substituting the value of x back into the income terms:

First person's income = 4 × 100 = 400 Second person's income = 5 × 100 = 500

Explanation

To find the total number of scooters of all four types produced by the company during the first six months of 1992, we sum the total production values given for each of the six months:

Total Production = January + February + March + April + May + June

Substituting the monthly totals from the table:

Total Production = 100 + 105 + 108 + 110 + 110 + 90 Total Production = 623

Explanation

Let the population of the country in 2000 be P₀.

The population declines at a rate of 1% (r = 0.01) every decade. The time duration from 2000 to 2030 is 30 years, which corresponds to exactly 3 decades (n = 3).

The population after n decades is given by the compounding decline formula: P_n = P₀ × (1 - r)ⁿ

Substituting the given values: 1940598 = P₀ × (1 - 0.01)³ 1940598 = P₀ × (0.99)³

Calculating (0.99)³: (0.99)³ = 0.970299

Solving for P₀: P₀ = 1940598 ÷ 0.970299 P₀ = 2000000

Explanation

Total Cost Price (CP) of the machine includes both the purchasing cost and the repair expenses: CP = ₹2000 + ₹500 = ₹2500

The Selling Price (SP) is: SP = ₹4000

The profit made on the sale is: Profit = SP - CP = ₹4000 - ₹2500 = ₹1500

The profit percentage is calculated using the formula: Profit Percentage = (Profit / CP) × 100 Profit Percentage = (1500 / 2500) × 100 = 60%

Explanation

The length of the diagonal of a cuboid with length l, breadth b, and height h is given by the formula:

Diagonal = √(l² + b² + h²)

Substituting the given dimensions of the cuboid, l = 20 m, b = 5 m, and h = 4 m:

Diagonal = √(20² + 5² + 4²) Diagonal = √(400 + 25 + 16) Diagonal = √441 Diagonal = 21 m

Explanation

Volume of a rectangular carton is given by the formula: Volume = length × width × height

First, convert the dimensions of the carton from inches to feet: Length = 48/12 = 4 feet Width = 27/12 = 2.25 feet

Using the volume formula: 22.5 = 4 × 2.25 × height 22.5 = 9 × height Height = 22.5/9 = 2.5 feet

Convert the height back to inches: Height = 2.5 × 12 = 30 inches

Explanation

By the weighted average formula, the total cost of the mixture is the sum of the costs of its individual components: (W₁ × P₁) + (W₂ × P₂) = (W₁ + W₂) × P_m

where W₁ and W₂ are the weights of the two types of wheat, P₁ and P₂ are their respective prices per kg, and P_m is the price per kg of the final mixture.

Given: W₁ = 5 kg, P₁ = ₹18 W₂ = 2 kg P_m = ₹20

Let P₂ be the price per kg of the costlier wheat: (5 × 18) + (2 × P₂) = (5 + 2) × 20 90 + 2P₂ = 7 × 20 90 + 2P₂ = 140 2P₂ = 50 P₂ = 25

Explanation

Using the complementary angle identity, cot θ = tan(90° - θ).

For θ = 15°: cot 15° = tan(90° - 15°) = tan 75°

Since tan 75° = 1 / cot 75°, substitute the given value: cot 15° = 1 / (2 - √3)

To simplify, rationalize the denominator by multiplying the numerator and the denominator by the conjugate 2 + √3: cot 15° = (2 + √3) / ((2 - √3)(2 + √3)) cot 15° = (2 + √3) / (2² - (√3)²) cot 15° = (2 + √3) / (4 - 3) cot 15° = 2 + √3

Explanation

Let the amount of the electricity bill be x.

The reduction obtained is 5% of the bill amount, which is given as ₹20. Using the percentage relation:

5% of x = 20

(5 / 100) × x = 20

x = (20 × 100) / 5

x = 400

Explanation

The relationship between the cost price (CP) and the printed price (MP) when the discount percentage (D) and profit percentage (P) are known is given by the formula:

CP / MP = (100 - D) / (100 + P)

Substituting the given values of D = 10 and P = 12 into the formula:

CP / MP = (100 - 10) / (100 + 12)

CP / MP = 90 / 112

Simplifying the fraction by dividing both the numerator and the denominator by 2 yields 45/56, which represents the ratio 45:56.

Explanation

Since the policeman and the thief run for the exact same duration of time until the chase ends, the distance covered by each is directly proportional to their respective speeds.

Ratio of distance covered by the policeman to the thief = Ratio of their speeds = 13 : 11

Let the distance covered by the policeman be 13x and the distance covered by the thief be 11x. The difference between these distances is the initial gap of 225 metres: 13x - 11x = 225 2x = 225 x = 112.5 metres

The distance run by the thief before being caught is: 11x = 11 × 112.5 = 1237.5 metres

Explanation

According to the divisibility rule of 3, a number is completely divisible by 3 if the sum of its individual digits is divisible by 3.

For the number 7345932, the sum of the digits is: 7 + 3 + 4 + 5 + 9 + 3 + 2 = 33

Since 33 is a multiple of 3 (3 × 11 = 33), the number 7345932 is divisible by 3.

Explanation

The amount A after n years at a compound interest rate of r% per annum on a principal sum P is given by the formula:

A = P(1 + r/100)ⁿ

Using the given values for 2 years and 3 years:

10,890 = P(1 + r/100)²

11,979 = P(1 + r/100)³

Dividing the equation for 3 years by the equation for 2 years gives the multiplier for one year:

1 + r/100 = 11,979 / 10,890 1 + r/100 = 1.1

Substituting this value back into the equation for 2 years:

P × (1.1)² = 10,890 P × 1.21 = 10,890 P = 10,890 / 1.21 P = 9,000

Explanation

The algebraic identity for the given expression is: x³ + y³ + z³ - 3xyz = ½(x + y + z)[(x - y)² + (y - z)² + (z - x)²]

Substituting the given values x = 32, y = 33, and z = 35 into this formula:

First, calculate the sum of the variables: x + y + z = 32 + 33 + 35 = 100

Next, calculate the squared differences: (x - y)² = (32 - 33)² = (-1)² = 1 (y - z)² = (33 - 35)² = (-2)² = 4 (z - x)² = (35 - 32)² = (3)² = 9

Sum of the squared differences: 1 + 4 + 9 = 14

Substitute these values back into the identity: ½ × 100 × 14 = 50 × 14 = 700

Explanation

The weight of the teacher can be calculated directly using the average deviation formula:

Weight of the teacher = Old average + (New total number of people × Increase in average)

Here, the old average is 51 kg. When the teacher is included, the total number of people becomes 39 + 1 = 40, and the average increases by 51.2 - 51 = 0.2 kg.

Substituting these values:

Weight of the teacher = 51 + (40 × 0.2) = 51 + 8 = 59 kg.

Explanation

The formula for the amount accrued under annual compound interest is: A = P(1 + R/100)ⁿ

Here, the amount A is ₹ 3364, the rate of interest R is 16%, and the time period n is 2 years. Substituting these values into the formula:

3364 = P(1 + 16/100)² 3364 = P(1 + 4/25)² 3364 = P(29/25)² 3364 = P × 841/625 P = (3364 × 625) / 841 P = 4 × 625 P = 2500

Explanation

The surface area of a sphere is given by the formula 4πr², where r represents the radius of the sphere.

Substituting the given surface area of 1386 cm² and using the value of π as 22/7:

4 × (22/7) × r² = 1386

Rearranging the terms to solve for r²:

r² = (1386 × 7) / (4 × 22)

r² = (63 × 7) / 4

r² = 441 / 4

Taking the square root on both sides:

r = √(441 / 4)

r = 21/2

r = 10.5 cm

Explanation

Using the algebraic identity (a + b)² - 4ab = (a - b)², we can simplify the expression by substituting a = 957 and b = 932.

This gives: (957 - 932)² = 25² = 625

This yields the simplified value directly.

Explanation

Using the rule of alligation, the ratio of the quantity of cheaper rice to the quantity of dearer rice is calculated as:

Ratio = (Price of dearer rice - Mean price) / (Mean price - Price of cheaper rice)

Substituting the given values:

Ratio = (80 - 75) / (75 - 60) = 5 / 15 = 1 / 3

This simplifies to 1:3.

Explanation

Let the total number of eligible voters be V.

Number of votes cast = 80% of V = (4/5) × V Number of valid votes = 95% of the cast votes = (19/20) × (4/5) × V Number of votes secured by the candidate = 75% of the valid votes = (3/4) × (19/20) × (4/5) × V

We are given that this value is equal to 10545: (3/4) × (19/20) × (4/5) × V = 10545

Canceling 4 in the numerator and denominator: V × (19 × 3) / (20 × 5) = 10545 V × 57 / 100 = 10545 V = (10545 × 100) / 57 V = 185 × 100 = 18500

Explanation

Time taken by A to complete the whole work is 3 × 5 = 15 days. Time taken by B to complete the whole work is 7.5 × 4 = 30 days.

Taking the least common multiple of 15 and 30, let the total work be 30 units. Efficiency of A = 30 / 15 = 2 units per day. Efficiency of B = 30 / 30 = 1 unit per day. Combined efficiency of A and B = 2 + 1 = 3 units per day.

The required amount of work to be completed is 4/5 of the total work: 4/5 × 30 = 24 units.

The number of days required by A and B working together to complete this work: 24 / 3 = 8 days.

Explanation

Convert the mixed fractions into improper fractions: 1 ¼ = 5/4 2 ⅗ = 13/5 1 ⁷/₁₃ = 20/13

Now, substitute these fractions back into the given expression: (5/4 ÷ 13/5) ÷ 20/13

Perform the operations inside the parentheses first: 5/4 ÷ 13/5 = 5/4 × 5/13 = 25/52

Next, divide the result by the remaining fraction: 25/52 ÷ 20/13 = 25/52 × 13/20

Simplify the expression by canceling out common terms: (25 × 13) / (52 × 20) = (5 × 5 × 13) / (4 × 13 × 4 × 5) = 5 / (4 × 4) = 5/16

Explanation

Income equals the sum of expenditure and savings. Initial savings = Income - Expenditure = 12,000 - 10,000 = 2,000. The ratio of expenditure to savings is 10,000 : 2,000 = 5 : 1.

Using the weighted average formula for percentage changes: Percentage increase in Income = (Ratio of Expenditure × Percentage increase in Expenditure + Ratio of Savings × Percentage increase in Savings) / Total Ratio

Let the percentage increase in savings be x%. 12% = (5 × 5% + 1 × x%) / (5 + 1) 12% × 6 = 25% + x% 72% = 25% + x% x% = 47%

Explanation

The algebraic identity for the square of a binomial is: (a + b)² = a² + b² + 2ab

Letting a = x and b = 1/x, we get: (x + 1/x)² = x² + 1/x² + 2(x)(1/x) (x + 1/x)² = x² + 1/x² + 2

Rearranging the formula to solve for x² + 1/x²: x² + 1/x² = (x + 1/x)² - 2

Substituting the given value of x + 1/x = 8.5 into the equation: x² + 1/x² = (8.5)² - 2 x² + 1/x² = 72.25 - 2 x² + 1/x² = 70.25

Explanation

For College C, the number of participants across the five subjects are 125 in Hindi, 80 in English, 250 in Maths, 150 in Science, and 120 in GK. The range is the difference between the maximum and minimum values. The maximum number of participants is 250 and the minimum is 80. Thus, the range of participants from College C is 250 - 80 = 170.

For College E, the number of participants across the five subjects are 112 in Hindi, 105 in English, 180 in Maths, 80 in Science, and 183 in GK. The sum of these participants is 112 + 105 + 180 + 80 + 183 = 660. Since there are 5 subjects, the average number of participants per Olympiad from College E is 660 ÷ 5 = 132.

The required difference between the range of College C and the average of College E is 170 - 132 = 38.

Explanation

Total weight of the 6 students can be calculated using the average weight formula: Total weight = Average weight × Number of students Total weight = 46 × 6 = 276 kg

The sum of the weights of the five students who recall their weight is: 50 + 52 + 40 + 44 + 48 = 234 kg

The weight of the 6th student is the difference between the total weight of the group and the sum of the weights of the other five students: Weight of the 6th student = 276 - 234 = 42 kg

Explanation

For a number to be divisible by 22, it must be divisible by both 2 and 11.

Since the last digit of 23₄5678 is 8, the number is even and thus always divisible by 2.

For divisibility by 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be 0 or a multiple of 11. Let the blank space be represented by the digit x.

Sum of digits at odd positions from the left = 2 + x + 5 + 7 = 14 + x Sum of digits at even positions from the left = 3 + 4 + 6 + 8 = 21

The difference between these sums is (21) - (14 + x) = 7 - x.

For this difference to be a multiple of 11 (in this case, 0), we set 7 - x = 0, which yields x = 7. Since 7 is a natural number, it is the correct value to fill the blank.

Explanation

Let the cost price of 1 article be CP and the selling price of 1 article be SP.

We are given that the cost price of 70 articles is equal to the selling price of 40 articles: 70 × CP = 40 × SP

Rearranging the terms gives the ratio of the selling price to the cost price: SP / CP = 70 / 40 = 7 / 4

This means if the cost price is 4 units, the selling price is 7 units.

The profit is the difference between the selling price and the cost price: Profit = SP - CP = 7 - 4 = 3 units

The profit percentage is calculated as: Profit percentage = (Profit / CP) × 100 Profit percentage = (3 / 4) × 100 = 75%

Explanation

Let the selling price of one article be SP.

The standard formula for loss is: Loss = Total Cost Price - Total Selling Price

For 80 articles, this is expressed as: Loss on 80 articles = CP of 80 articles - SP of 80 articles

We are given that the loss is equal to the selling price of 6 articles and the cost price of 80 articles is 11,180. Substituting these values into the equation: 6 × SP = 11,180 - 80 × SP 86 × SP = 11,180 SP = 11,180 / 86 SP = 130

Explanation

Using the standard algebraic identity: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Substitute the given values: 10² = a² + b² + c² + 2(30) 100 = a² + b² + c² + 60 a² + b² + c² = 100 - 60 a² + b² + c² = 40

Explanation

The initial solution of 700 grams contains 60% sugar, which is 420 grams of sugar and 280 grams of other components. To make the sugar concentration 75%, the 280 grams of other components must represent the remaining 25% of the new solution, meaning the new total weight of the solution is 1120 grams. The amount of sugar to be added is therefore 1120 grams minus the initial 700 grams, which equals 420 grams.

Explanation

The difference between two numbers x and y can be determined using the standard algebraic identity:

(x - y)² = (x + y)² - 4xy

Given that the sum of the two numbers (x + y) is 25 and their product (xy) is 144, substituting these values into the formula gives:

(x - y)² = 25² - 4 × 144 (x - y)² = 625 - 576 (x - y)² = 49

Taking the square root of both sides:

x - y = √49 = 7

The positive difference between the two numbers is 7.

Explanation

Let the total work be the Least Common Multiple (LCM) of the days taken by A and B, which is LCM(24, 80) = 240 units.

Efficiency of A = 240 ÷ 24 = 10 units per day. Efficiency of B = 240 ÷ 80 = 3 units per day.

According to the given pattern, the work is done in cycles of 3 days: Day 1: B works alone and does 3 units of work. Day 2: B works alone and does 3 units of work. Day 3: A works alone and does 10 units of work.

Total work done in one 3-day cycle = 3 + 3 + 10 = 16 units.

To complete the total work of 240 units, the number of such 3-day cycles required is: 240 ÷ 16 = 15 cycles.

Total days required to complete the work = 15 × 3 = 45 days.

Explanation

The difference between compound interest and simple interest for two conversion periods (half-years) can be calculated using the formula:

Difference = P × (r/100)²

Here, the principal P is ₹1,000. The interest rate is 20% per annum, which becomes r = 10% per half-year. The time of 1 year corresponds to 2 half-yearly compounding periods.

Substituting the values into the formula:

Difference = 1000 × (10/100)² Difference = 1000 × (1/10)² Difference = 1000 × 1/100 Difference = ₹10

Explanation

The volume of a hemisphere is given by the formula V = (2/3)πr³, where r is the radius of the hemisphere.

Substituting the given values of r = 21 cm and π = 22/7:

V = (2/3) × (22/7) × 21 × 21 × 21

Since the product of the denominators 3 × 7 equals 21, we can simplify the expression by canceling one 21 from the numerator:

V = 2 × 22 × 21 × 21

V = 44 × 441

V = 19404 cm³

Explanation

The ratio of the speeds of Rakesh and Mukesh is 4 : 3.

Since distance is directly proportional to speed for a given duration, the ratio of the distances covered by Rakesh and Mukesh to the meeting point is also 4 : 3.

Let the distance covered by Rakesh be 4x and by Mukesh be 3x.

The difference in their distances is equal to the start lead given to Mukesh: 4x - 3x = 60 m x = 60 m

The distance from the starting point where they meet is the total distance run by Rakesh: 4x = 4 × 60 m = 240 m

Explanation

The given equation is x² = b - ax. Rearranging the terms, we get: x² + ax = b

To find a relation involving x⁶ and x³, we cube both sides of this equation: (x² + ax)³ = b³

Using the algebraic identity (p + q)³ = p³ + q³ + 3pq(p + q), where p = x² and q = ax, we expand the left side: (x²)³ + (ax)³ + 3(x²)(ax)(x² + ax) = b³ x⁶ + a³x³ + 3ax³(x² + ax) = b³

Substitute b in place of x² + ax on the left side: x⁶ + a³x³ + 3ax³(b) = b³ x⁶ + a³x³ + 3abx³ = b³

Factoring out x³: x³(x³ + a³ + 3ab) = b³

Comparing this with the target equation x³(x³ + c) = d, we get: c = a³ + 3ab d = b³

Explanation

The formula for the depreciated value of an asset over time is A = P(1 - R/100)ⁿ, where A is the depreciated value, P is the initial purchase price, R is the annual rate of depreciation, and n is the number of years.

Substituting the given values into the formula: 36450 = 50000(1 - 10/100)ⁿ 36450 = 50000(0.9)ⁿ 36450 / 50000 = (0.9)ⁿ 3645 / 5000 = (0.9)ⁿ

Dividing both the numerator and denominator by 5 on the left-hand side: 729 / 1000 = (0.9)ⁿ (9/10)³ = (0.9)ⁿ (0.9)³ = (0.9)ⁿ

Comparing the exponents, we get: n = 3

Explanation

According to the BODMAS rule, division and multiplication are performed first.

First, perform the division: 2945 ÷ 19 = 155

Next, perform the multiplication: 11 × 5 = 55

Now, substitute these values into the given equation: 155 - 55 = 29 + a 100 = 29 + a

Solving for a: a = 100 - 29 a = 71

Explanation

By the unitary method, the cost of 1 kg of sugar is obtained by dividing the total cost of the initial quantity of sugar by that quantity:

Cost of 1 kg of sugar = 600 / 15 = 40

Using this unit price, the cost of 25 kg of sugar is calculated by multiplying the unit price by the desired quantity:

Cost of 25 kg of sugar = 25 × 40 = 1000

Explanation

The selling price of the perfume is calculated by subtracting the loss from the cost price:

Selling Price = Cost Price - Loss Selling Price = 380 - 140 = 240

Since the perfume is sold at a 20% discount on its marked price, the selling price is equal to 80% of the marked price:

Selling Price = Marked Price × (100% - Discount%) 240 = Marked Price × 80% 240 = Marked Price × 0.8

Solving for the marked price:

Marked Price = 240 / 0.8 = 300

Explanation

When a sphere passes through the eight corners of a cube, the sphere circumscribes the cube. Consequently, the diagonal of the cube is equal to the diameter of the sphere.

For a cube of side a = 14 cm, the length of its body diagonal is given by the formula: d = a√3 = 14√3 cm

The radius r of the circumscribing sphere is half of the diagonal: r = d / 2 = 7√3 cm

The volume V of a sphere is given by the formula: V = (4/3) × π × r³

Substituting the values r = 7√3 cm and π = 22/7: V = (4/3) × (22/7) × (7√3)³ V = (4/3) × (22/7) × 343 × 3√3 V = 4 × 22 × 49 × √3 V = 4312√3 cm³

Explanation

The relationship between the slant height (l), height (h), and radius (r) of a right circular cone is given by the formula:

l² = r² + h²

Substituting the given values of slant height (l = 10 cm) and height (h = 8 cm):

10² = r² + 8² 100 = r² + 64 r² = 36 r = 6 cm

The volume (V) of a cone is given by the formula:

V = (1/3) × π × r² × h

Using the value of π as 22/7, substitute the radius and height into the formula:

V = (1/3) × (22/7) × 6² × 8 V = 12 × (22/7) × 8 V = 2112 / 7 V ≈ 301.71 cm³

Explanation

When the expenditure remains constant, the price and consumption are inversely proportional to each other.

An increase of 25% in the price of sugar means the price rises by a fraction of 1/4. To maintain the same expenditure, the consumption must decrease by a fraction of: (1/4) / (1 + 1/4) = 1/5

Given that this reduction in consumption is 1.5 kg: 1/5 × Original Consumption = 1.5 kg Original Consumption = 1.5 × 5 = 7.5 kg

Therefore, the new reduced consumption is: New Consumption = 7.5 kg - 1.5 kg = 6 kg

The total money spent is £240. The increased price per kg of sugar is calculated by dividing the total expenditure by the new consumption: Increased Price = 240 / 6 = £40 per kg

Explanation

The number of Lenovo laptops sold by Store A is 50. The number of Dell laptops sold by Store D is 45.

To find the percentage by which the Lenovo laptops sold by Store A are more than the Dell laptops sold by Store D, the standard percentage increase formula is used:

Percentage More = ((Lenovo laptops sold by Store A - Dell laptops sold by Store D) / Dell laptops sold by Store D) × 100

Substituting the values from the table:

Percentage More = ((50 - 45) / 45) × 100 = (5 / 45) × 100 = (1 / 9) × 100 ≈ 11.11%

This gives an approximate value of 11.1%.

Explanation

Substitute the standard values of the trigonometric functions: cosec 30° = 2 cos 60° = ½ cot 45° = 1 sec 60° = 2

Substituting these values into the left-hand side of the equation: (x - x cosec²30°) / (1 + cosec²30°) = (x - x(2)²) / (1 + 2²) = (x - 4x) / (1 + 4) = -3x / 5

Substituting these values into the right-hand side of the equation: cos²60° + 4cot²45° - sec²60° = (½)² + 4(1)² - 2² = ¼ + 4 - 4 = ¼

Equating both sides: -3x / 5 = ¼ -3x = 5 / 4 x = -5 / 12

Explanation

Let the original price of the article be 100.

After a 25% reduction, the new price becomes: 100 - 25 = 75

To restore the price back to its original value of 100, the new price must be increased by 25.

The percentage increase required is calculated as: Percentage Increase = (Required Increase / New Price) × 100% Percentage Increase = (25 / 75) × 100% Percentage Increase = 1/3 × 100% = 33 ⅓%

Alternatively, using the standard restoration formula where a price is cut by r%: Percentage Increase = (r / (100 - r)) × 100%

Substituting r = 25: Percentage Increase = (25 / (100 - 25)) × 100% = (25 / 75) × 100% = 33 ⅓%

Explanation

To find the ratio of students playing cricket, football, and hockey, we combine the given individual ratios.

The ratio of cricket (C) to football (F) is 27 : 13. The ratio of football (F) to hockey (H) is 26 : 14, which simplifies to 13 : 7.

Since the term representing football is 13 in both cases, the combined ratio of students playing the three games is: C : F : H = 27 : 13 : 7

The sum of the ratio parts is: 27 + 13 + 7 = 47

Since each student plays exactly one game, the percentage of students playing cricket can be computed directly using their share of the total ratio parts: Percentage of cricket players = (Cricket parts / Total parts) × 100 Percentage of cricket players = (27 / 47) × 100 = 2700 / 47 % ≈ 57.446%

Rounding to two decimal places, the percentage of students playing cricket is 57.45%.

Explanation

The given data represents a cumulative frequency distribution of the less than type.

The number of students scoring less than 40 marks is 8. The number of students scoring less than 30 marks is 6.

To find the number of students who scored at least 30 marks but less than 40 marks, which corresponds to the class interval 30-40, we subtract the cumulative frequency of students scoring less than 30 from the cumulative frequency of students scoring less than 40:

Number of students = 8 - 6 = 2

Explanation

The first given equation is sec θ - cos θ = 14.

We are given that x = 14 sec θ.

Substituting the value of 14 from the first equation into the second equation: x = (sec θ - cos θ) sec θ

Multiplying the terms: x = sec²θ - sec θ cos θ

Since sec θ cos θ = 1: x = sec²θ - 1

Using the trigonometric identity sec²θ - tan²θ = 1, we get: x = tan²θ