SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

Correct Answer: A

Let F and S be the present ages of the father and the son respectively.

The sum of their present ages is: F + S = 50

In 5 years, their ages will be F + 5 and S + 5. According to the given condition: F + 5 = 5(S + 5) F + 5 = 5S + 25 F - 5S = 20

Subtracting the second equation from the first gives: 6S = 30 S = 5

Substituting S = 5 into the first equation: F + 5 = 50 F = 45

Thus, the father's present age is 45 years. Five years ago, the father's age was: 45 - 5 = 40 years.

Explanation

The relationship between Cost Price (CP), Marked Price (MP), Profit percentage (P), and Discount percentage (D) is given by the standard formula:

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

Substituting the given values (CP = 4400, P = 24, and D = 12) into the formula:

MP / 4400 = (100 + 24) / (100 - 12)

MP / 4400 = 124 / 88

MP = 4400 × 124 / 88

MP = 50 × 124

MP = 6200

Explanation

The net percentage change for two successive percentage increases of a% and b% is given by the formula:

Net Increase = a + b + (a × b)/100

Given: First year increase, a = 10% Second year increase, b = 20%

Substituting these values into the formula:

Net Increase = 10 + 20 + (10 × 20)/100 Net Increase = 30 + 200/100 Net Increase = 30 + 2 Net Increase = 32%

Alternatively, if the initial price is assumed to be 100: Price after the first year = 100 × (1 + 10/100) = 110 Price after the second year = 110 × (1 + 20/100) = 110 × 1.2 = 132 Overall increase = 132 - 100 = 32%

Explanation

The total surface area of a cuboid is calculated using the formula:

Total Surface Area = 2 × (lb + bh + hl)

where l is the length, b is the breadth, and h is the height.

Given values: l = 19 cm b = 9 cm h = 4.5 cm

Substituting these values into the formula:

Total Surface Area = 2 × (19 × 9 + 9 × 4.5 + 4.5 × 19) Total Surface Area = 2 × (171 + 40.5 + 85.5) Total Surface Area = 2 × 297 Total Surface Area = 594 cm²

Explanation

For two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Since ΔABC is similar to ΔEDF, the corresponding side to DF is BC. The relationship between their areas and these sides is given by:

area(ΔABC) / area(ΔEDF) = (BC / DF)²

Substituting the given values:

1 / 4 = (8 / DF)²

Taking the square root on both sides:

1 / 2 = 8 / DF

DF = 8 × 2 = 16 cm

Explanation

Using the standard trigonometric values: sin 60° = √3/2 cos 60° = ½

Substituting these values into the given expression: (sin 60° - cos 60°) / (sin 60° + cos 60°) = (√3/2 - ½) / (√3/2 + ½) = (√3 - 1) / (√3 + 1)

To simplify, rationalize the denominator by multiplying both the numerator and the denominator by (√3 - 1): ((√3 - 1)(√3 - 1)) / ((√3 + 1)(√3 - 1)) = (√3 - 1)² / ((√3)² - 1²) = (3 - 2√3 + 1) / (3 - 1) = (4 - 2√3) / 2 = 2 - √3

Explanation

The total population of State B is 7 crore, which is equal to 700 lakhs.

The percentage of the literate population in State B is 72%, which means the percentage of the illiterate population is 100% - 72% = 28%.

The total number of illiterate people in State B is calculated as: Illiterate population = 28% of 700 lakhs = 196 lakhs.

The ratio of illiterate males to illiterate females in State B is given as 3 : 2.

The number of illiterate females in State B is: Illiterate females = 2/(3 + 2) × 196 lakhs = 2/5 × 196 = 78.4 lakhs.

Explanation

The weighted average formula is given by:

Weighted Average = (n₁ × A₁ + n₂ × A₂) / (n₁ + n₂)

Where: n₁ = 50, A₁ = 40 n₂ = 150, A₂ = 60

Substituting these values into the formula:

Weighted Average = (50 × 40 + 150 × 60) / (50 + 150) Weighted Average = (2000 + 9000) / 200 Weighted Average = 11000 / 200 Weighted Average = 55

Explanation

The average pass percentage is calculated by dividing the total number of passed candidates by the total number of candidates who appeared, multiplied by 100.

Number of employees who passed in the first year = 84% of 75 = (84 / 100) × 75 = 63

Number of employees who passed in the second year = 52% of 50 = (52 / 100) × 50 = 26

Total number of employees who passed over both years = 63 + 26 = 89

Total number of employees who appeared over both years = 75 + 50 = 125

Average rate of pass percentage = (Total passed / Total appeared) × 100 Average pass percentage = (89 / 125) × 100 = 89 × 4/5 = 356 / 5 = 71.2%

Converting 71.2% into a mixed fraction: 71.2% = 71 2/10 % = 71 1/5 %

Explanation

The mean proportion of two numbers a and b is given by the formula √(a × b).

Given the two numbers: a = 0.72 b = 2.85

Substitute these values into the formula: Mean proportion = √(0.72 × 2.85) Mean proportion = √2.052

Calculating the square root: √2.052 ≈ 1.432

Rounding 1.432 to one decimal place yields 1.4.

Explanation

The length of the direct common tangent (L) between two circles with radii r₁ and r₂ and distance between their centers d is given by the formula:

L = √(d² - (r₁ - r₂)²)

Substituting the given values, where L = √35, d = 6, r₁ = r, and r₂ = 3:

√35 = √(6² - (r - 3)²)

Squaring both sides of the equation:

35 = 36 - (r - 3)²

Rearranging the terms:

(r - 3)² = 36 - 35 (r - 3)² = 1

Taking the square root on both sides:

r - 3 = ±1

This gives two possible values for r:

r = 3 + 1 = 4 r = 3 - 1 = 2

Explanation

Percentage decrease in income is calculated using the formula: Percentage Decrease = (Original Income - New Income) / Original Income × 100

Substituting the given values: Original Income = ₹42,800 New Income = ₹37,236

Decrease in income = 42,800 - 37,236 = 5,564

Percentage Decrease = (5,564 / 42,800) × 100 Percentage Decrease = 5,564 / 428 Percentage Decrease = 13%

Explanation

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

Substituting the given value of the surface area into the formula: 4 × (22/7) × r² = 1386 88/7 × r² = 1386 r² = (1386 × 7) / 88 r² = 110.25 r = 10.5 cm (or 21/2 cm)

The volume of a sphere is given by the formula V = 4/3 πr³.

Substituting the calculated radius into the volume formula: V = 4/3 × (22/7) × (21/2)³ V = 4/3 × (22/7) × (9261/8) V = 11 × 21 × 21 V = 4851 cm³

Explanation

The formula for the upstream speed of a boat is: Upstream Speed = Speed of boat in still water - Speed of the stream

Substituting the given values into the formula: 1 km/h = 3 km/h - Speed of the stream Speed of the stream = 3 km/h - 1 km/h = 2 km/h

Explanation

According to the properties of arithmetic mean, if each observation in a dataset is multiplied by a constant k, the mean of the new dataset is also multiplied by the same constant k.

Using this relation: New Average = Old Average × k

Given that the old average is 23 and the constant multiplier k is 13: New Average = 23 × 13 = 299

Explanation

According to the Basic Proportionality Theorem (Thales's Theorem), if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. Since DE is parallel to BC, we have:

AD / DB = AE / EC

Substituting the given algebraic expressions into the relation:

x / (x - 3) = (x + 3) / (x - 2)

Cross-multiplying the terms:

x(x - 2) = (x - 3)(x + 3)

Expanding both sides using the algebraic identity (a - b)(a + b) = a² - b²:

x² - 2x = x² - 9

Subtracting x² from both sides:

-2x = -9

Dividing by -2:

x = 4.5

Explanation

Let the present age of the person be x. Since the father is 25 years older, the father's present age is x + 25.

Five years ago: The person's age was x - 5. The father's age was x + 25 - 5 = x + 20.

The sum of their ages five years ago was 39: (x - 5) + (x + 20) = 39 2x + 15 = 39 2x = 39 - 15 2x = 24 x = 12

Explanation

The area of a sector of a circle can be calculated directly using the formula relating the radius and the arc length:

Area = ½ × r × l

where r is the radius of the circle and l is the length of the arc.

Substituting the given values of r = 24 units and l = 7.5 units:

Area = ½ × 24 × 7.5 Area = 12 × 7.5 Area = 90 square units

Explanation

In any triangle, a median divides the triangle into two smaller triangles of equal area. Since L is the mid-point of side BC, AL is the median of ΔABC.

Area(ΔABC) = 2 × Area(ΔABL)

Given that Area(ΔABL) = 15 cm², substituting this value gives:

Area(ΔABC) = 2 × 15 = 30 cm²

Explanation

To find the value, we simplify the expression step-by-step:

First, evaluate the term inside the parenthesis: 5/15 of 3/12 = 1/3 × 1/4 = 1/12

Next, add 1/15 to this result: 1/12 + 1/15 = 5/60 + 4/60 = 9/60 = 3/20

Now, multiply by 1 7/12, which is equal to 19/12: 3/20 × 19/12 = 19/80

Subtracting this from 15/8 gives: 15/8 - 19/80 = 150/80 - 19/80 = 131/80

This is in the form of a/b, where a = 131 and b = 80. Since 131 is a prime number, a and b are relatively prime to each other.

We calculate the value of 4b - a: 4b - a = 4 × 80 - 131 = 320 - 131 = 189

Explanation

We use the algebraic identity: x² - y² = (x + y)(x - y)

Given the equations: x = tan A + sin A y = tan A - sin A

Adding the two equations: x + y = 2 tan A

Subtracting the second equation from the first: x - y = 2 sin A

Substituting these into the identity: x² - y² = (2 tan A)(2 sin A) = 4 tan A sin A

Now, we calculate the product xy: xy = (tan A + sin A)(tan A - sin A) = tan² A - sin² A

Expressing tan A in terms of sin A and cos A: xy = sin² A / cos² A - sin² A xy = sin² A (1 / cos² A - 1) xy = sin² A ((1 - cos² A) / cos² A) xy = sin² A (sin² A / cos² A) xy = sin² A tan² A

Taking the square root on both sides: √(xy) = sin A tan A

Substituting sin A tan A = √(xy) back into the expression for x² - y²: x² - y² = 4√(xy)

Explanation

Let the total number of valid votes be V.

Since there are only two candidates, the losing candidate received 46% of the valid votes (100% - 54% = 46%).

The difference in the percentage of valid votes between the winning and losing candidates is 54% - 46% = 8%.

We are given that the winning candidate won by 88 votes. Therefore, we can set up the equation:

8% of V = 88

0.08 × V = 88

V = 88 ÷ 0.08 = 1100

The total number of votes polled is the sum of the valid votes and the invalid votes:

Total votes polled = Valid votes + Invalid votes

Total votes polled = 1100 + 58 = 1158

Explanation

To find the number of students who secured a raw score of 135 in Maths, the required percentage is calculated using the maximum marks for the subject, which is 150.

Percentage of marks = (Raw score ÷ Maximum marks) × 100 Percentage of marks = (135 ÷ 150) × 100 = 90%

Looking at the Maths column in the given table, the percentages obtained by the students are: Ayush: 90% Aman: 100% Sajal: 90% Rohit: 80% Muskan: 80% Tanvi: 70% Tarun: 65%

Two students, Ayush and Sajal, obtained exactly 90% in Maths, meaning 2 students secured a raw score of 135.

Explanation

The discount percentage is calculated using the formula:

Discount Percentage = ((Marked Price - Selling Price) / Marked Price) × 100

Substituting the given values:

Marked Price = 2165 Selling Price = 1710.35

Discount = 2165 - 1710.35 = 454.65

Discount Percentage = (454.65 / 2165) × 100 = 45465 / 2165 = 21%

Explanation

To find the percentage increase required to restore a reduced value back to its original level, we use the percentage change formula:

Percentage increase = [r / (100 - r)] × 100

where r is the percentage reduction. Given that the salary was reduced by r = 20%:

Percentage increase = [20 / (100 - 20)] × 100 Percentage increase = (20 / 80) × 100 Percentage increase = ¼ × 100 = 25%

Alternatively, let the original salary of the employees be ₹100. After a reduction of 20%, the new salary becomes ₹80. To restore this salary back to the original ₹100, it needs to be increased by ₹20. The percentage increase required on the reduced salary of ₹80 is:

(20 / 80) × 100 = 25%

Explanation

The percentage reduction in consumption required to keep the expenditure constant when the price of a commodity increases by r% is given by the formula:

Reduction % = (r ÷ (100 + r)) × 100

Substituting the given price increase of r = 36 into the formula:

Reduction % = (36 ÷ (100 + 36)) × 100 Reduction % = (36 ÷ 136) × 100 Reduction % = (9 ÷ 34) × 100 Reduction % = 900 ÷ 34 Reduction % ≈ 26.47%

Rounding off to one decimal place, the consumption should be decreased by 26.5%.

Explanation

Using the relationship Total Time = Time by Train + Time by Bus:

Total Time = (Distance by Train ÷ Speed of Train) + (Distance by Bus ÷ Speed of Bus)

Let d be the distance travelled by train in kilometers. Since the total distance is 285 km, the distance travelled by bus is 285 - d km.

Substituting the given values into the formula: d/55 + (285 - d)/40 = 6

Multiplying the entire equation by the lowest common multiple of 55 and 40, which is 440: 8d + 11(285 - d) = 6 × 440 8d + 3135 - 11d = 2640 3135 - 3d = 2640 3d = 495 d = 165

The distance Tom travelled by train is 165 km.

Explanation

Volume of the cylindrical candle is calculated using the formula V = πr²h, where r is the radius and h is the height (length). Given the diameter is 14 cm, the radius r is 7 cm, and the length h is 2 cm.

Volume of the cylinder = (22/7) × 7² × 2 = 22 × 7 × 2 = 308 cm³

Volume of each cuboidal candle is calculated using the formula V = length × width × height.

Volume of the cuboid = 7 × 11 × 2 = 154 cm³

The number of cuboidal candles that can be obtained is the volume of the cylindrical candle divided by the volume of one cuboidal candle.

Number of candles = 308 / 154 = 2

Explanation

We can express the given percentage relationship as an algebraic equation:

10% × 24% × x = 240

Converting the percentages to fractions:

(10/100) × (24/100) × x = 240

(1/10) × (24/100) × x = 240

24x/1000 = 240

To solve for x, multiply both sides by 1000 and divide by 24:

x = (240 × 1000) / 24

Simplifying this gives:

x = 10 × 1000

x = 10000

Explanation

The formula for simple interest is SI = P × R × T ÷ 100, where P is the principal sum, R is the rate of interest per annum, and T is the time in years.

Given that the simple interest is three-fifth of the sum, SI = 3/5 × P. The time T is 12 years.

Substituting these values into the formula: 3/5 × P = P × R × 12 ÷ 100

Simplifying by canceling P from both sides: 3/5 = 12 × R ÷ 100

This simplifies to: 0.6 = 0.12 × R

R = 0.6 ÷ 0.12 = 5%

Explanation

Let the total number of votes polled be T. Since the losing candidate secured 41% of the votes, the winning candidate secured the remaining 59% of the votes (100% - 41% = 59%).

The difference in their vote percentages is: 59% - 41% = 18%

This difference represents the margin of loss, which is 97,632 votes: 18% of T = 97,632 T = 97,632 ÷ 0.18 = 542,400

The number of votes obtained by the winning candidate is 59% of this total: Winning votes = 59% of 542,400 = 0.59 × 542,400 = 320,016

Explanation

Using the fundamental trigonometric identity sin²A + cos²A = 1, we can substitute cos²A = 1 - sin²A into the given equation:

12 sin²A + 2(1 - sin²A) = 7

Expanding and simplifying the equation:

12 sin²A + 2 - 2 sin²A = 7 10 sin²A = 5 sin²A = ½

Now, we find cos²A:

cos²A = 1 - sin²A = 1 - ½ = ½

Using the definition of cotangent:

cot²A = cos²A / sin²A = (½) / (½) = 1

Taking the square root of both sides gives:

cot A = ±1

Since the angle A lies in the range 0 < A < 180°, cot A is positive in the first quadrant (0 < A < 90°) and negative in the second quadrant (90° < A < 180°), which justifies both the positive and negative values.

Explanation

The earnings of workers are directly proportional to both the number of persons and the number of hours they work per day. This relationship can be expressed using the formula:

E₁ / (M₁ × H₁) = E₂ / (M₂ × H₂)

where E represents earnings, M represents the number of persons, and H represents the working hours per day.

Substituting the given values into the formula:

M₁ = 15 persons, H₁ = 12 hours, E₁ = ₹18,000 M₂ = 18 persons, H₂ = 9 hours, E₂ = unknown earnings

18,000 / (15 × 12) = E₂ / (18 × 9)

18,000 / 180 = E₂ / 162

100 = E₂ / 162

E₂ = 100 × 162 = 16,200

The 18 persons working 9 hours a day will earn ₹16,200 per week.

Explanation

According to the Pythagorean theorem, three side lengths a, b, and c, where c is the longest side, form a right-angled triangle if they satisfy the relation:

a² + b² = c²

For the side lengths 12 cm, 16 cm, and 20 cm:

12² + 16² = 144 + 256 = 400 20² = 400

Since 12² + 16² = 20², these dimensions satisfy the theorem. This triplet is a multiple of the standard 3-4-5 right-angled triangle ratio scaled by a factor of 4, where 3×4 = 12, 4×4 = 16, and 5×4 = 20.

Explanation

Using the sum-average relation, the sum of ages of all workers is the product of the number of workers and their average age.

Sum of ages of the 12 original workers = 12 × 35½ = 12 × 35.5 = 426 years

When a new worker joins, the total number of workers becomes 13, and the new average age is 37 years.

Sum of ages of the 13 workers = 13 × 37 = 481 years

The age of the newly joined worker is the difference between these two sums:

Age of the new worker = 481 - 426 = 55 years

Explanation

The intersection area of two identical circles of radius r = 7 with a common chord of length 7 is formed by two symmetric circular segments.

The triangle formed by the center of one circle and the two endpoints of the common chord has all three sides of length 7, making it an equilateral triangle. Thus, the central angle θ subtended by the chord at the center of each circle is 60°, which is π/3 radians.

The area of a single circular segment is the area of the circular sector minus the area of the equilateral triangle: Area of segment = (θ/2π) × πr² - (√3/4)r² Area of segment = (1/6)πr² - (√3/4)r² = r²(π/6 - √3/4)

Since the common area of intersection consists of two such segments, the total area is: Total Area = 2 × r²(π/6 - √3/4)

Substituting r = 7: Total Area = 2 × 49(π/6 - √3/4) = 98(π/6 - √3/4)

Explanation

The volume V of a sphere is given by the formula V = (4/3) × π × r³, where r is the radius of the sphere.

Substituting V = 38.808 and π = 22/7:

38.808 = (4/3) × (22/7) × r³

38.808 = (88/21) × r³

r³ = (38.808 × 21) / 88

r³ = 0.441 × 21

r³ = 9.261

Taking the cube root on both sides:

r = 2.1 cm

The diameter of the spherical ball is twice the radius:

Diameter = 2 × r = 2 × 2.1 = 4.2 cm.

Explanation

The relationship between the number of men (M), days (D), and working hours per day (H) required to complete a constant amount of work is governed by the joint variation formula:

M₁ × D₁ × H₁ = M₂ × D₂ × H₂

Substituting the given values into the formula: M₁ = 12 men, D₁ = 24 days, H₁ = 9 hours M₂ = 8 men, H₂ = 12 hours, and D₂ is the required number of days

12 × 24 × 9 = 8 × D₂ × 12

Dividing both sides by 12: 24 × 9 = 8 × D₂

Solving for D₂: D₂ = (24 × 9) / 8 D₂ = 3 × 9 D₂ = 27 days

Explanation

The cost price of milk is 1 / 1½ = ⅔ rupees per liter. Since water is free, its cost price is 0 rupees per liter.

The price of the final mixture is 1 / 1⅞ = 8/15 rupees per liter.

By the rule of alligation, the ratio of milk to water in the mixture is: (8/15 - 0) : (⅔ - 8/15) = 8/15 : 2/15 = 4 : 1

This means that for every 4 parts of milk, 1 part of water is added.

Since the quantity of milk is 60 liters, which corresponds to 4 parts, the quantity of water corresponding to 1 part is: 60 × ¼ = 15 liters

Explanation

Simple interest (SI) is calculated using the formula: SI = (P × R × T) / 100

Let the principal amount for the first bank be K, with a rate of interest r% and time n² years. The simple interest is: SI₁ = (K × r × n²) / 100

Let the principal amount for the second bank be L, with a rate of interest r²% and time n³ years. The simple interest is: SI₂ = (L × r² × n³) / 100

Since the interest earned from both banks is the same: SI₁ = SI₂ (K × r × n²) / 100 = (L × r² × n³) / 100

Canceling the common terms from both sides: K × r × n² = L × r² × n³

To find the ratio between K and L: K / L = (r² × n³) / (r × n²) K / L = rn / 1

Thus, the ratio K : L is rn : 1.

Explanation

The marked price of 12 pairs of gloves is ₹180.

At a 10% discount, the selling price for 12 pairs is: 180 × (1 - 0.10) = ₹162

The cost of a single pair of gloves is: 162 ÷ 12 = ₹13.5

The number of pairs of gloves that can be bought for ₹54 is: 54 ÷ 13.5 = 4

Explanation

Downstream speed of the boat is calculated by dividing the downstream distance by the time taken: Downstream speed = 60 / 4 = 15 km/h

The downstream speed is the sum of the speed of the boat in still water and the speed of the stream. Given that the speed of the boat in still water is 13 km/h, we can find the speed of the stream: Speed of the stream = Downstream speed - Speed of boat in still water Speed of the stream = 15 - 13 = 2 km/h

The upstream speed of the boat is the difference between the speed of the boat in still water and the speed of the stream: Upstream speed = 13 - 2 = 11 km/h

The time required to cover 30 km upstream is calculated by dividing the upstream distance by the upstream speed: Time taken = 30 / 11 = 2 8/11 hours

Explanation

The curved surface area (CSA) of a hemisphere is given by the formula: CSA = 2πr²

Given that the diameter of the hemisphere is 21 cm, the radius (r) is: r = 21/2 cm

Using the value of π = 22/7, substitute the radius into the formula: CSA = 2 × (22/7) × (21/2)² CSA = 2 × (22/7) × (21/2) × (21/2) CSA = 2 × (22/7) × (441/4) CSA = 11 × 63 CSA = 693 cm²

Explanation

Let the present ages of the father and the son be 7x and 4x respectively.

According to the given condition, the product of their ages is 1372: (7x) × (4x) = 1372 28x² = 1372 x² = 1372 / 28 x² = 49 x = 7

Using the value of x, we find their present ages: Father's present age = 7 × 7 = 49 years Son's present age = 4 × 7 = 28 years

Their ages after 8 years will be: Father's age = 49 + 8 = 57 years Son's age = 28 + 8 = 36 years

The ratio of their ages after 8 years is: 57 : 36 = 19 : 12

Explanation

To find the car types with a continuous decrease in production from 2018 to 2020, we analyze the annual production values for each type during these years.

For car Type A, the production values are 39 in 2018, 25 in 2019, and 18 in 2020. This shows a steady decline since 39 > 25 > 18.

For car Type B, the production values are 14 in 2018, 12 in 2019, and 8 in 2020. This also shows a steady decline since 14 > 12 > 8.

Both car types A and B satisfy the condition of continuous decrease in production during this period.

Explanation

Let P be the population at the beginning of the first year.

According to the given conditions, the population increases by 5% in the first year and decreases by 5% in the second year. The net relationship between the initial population and the population at the end of the second year is given by the formula:

Final Population = P × (1 + 5/100) × (1 - 5/100)

Substituting the given final population of 47,880 into the formula:

47880 = P × 1.05 × 0.95

47880 = P × 0.9975

P = 47880 / 0.9975

P = 48000

Thus, the population at the beginning of the first year was 48,000.

Explanation

The volume of a cylinder is given by the formula V = pi * r² * h. Substituting the volume of 8316 cubic centimeters and radius of 21 centimeters into the formula gives 8316 = (22/7) * 21 * 21 * h, which simplifies to 8316 = 1386 * h. Solving for h gives a depth of 6 centimeters.

Explanation

The trigonometric expansion formulas are: cos(x + y) = cos x cos y - sin x sin y sin(x - y) = sin x cos y - cos x sin y tan(2z) = 2tan z / (1 - tan²z)

Substituting these standard identities into the given expression: cos(x + y) - sin(x - y) + tan(2z) = (cos x cos y - sin x sin y) - (sin x cos y - cos x sin y) + 2tan z / (1 - tan²z)

Expanding the terms and distributing the negative sign yields: cos x cos y - sin x sin y - sin x cos y + cos x sin y + 2tan z / (1 - tan²z)

Explanation

The perimeter of a sector of a circle is given by 2r + L, where r is the radius and L is the arc length. For a radius of 4 units and an angle of 45 degrees, the arc length is (45/360) * 2 * pi * 4 = pi, making the total perimeter 2(4) + pi = 8 + pi units.

Explanation

The percentage increase in salary is calculated using the formula:

Percentage Increase = (Increase in Salary / Original Salary) × 100%

Here, the original salary is ₹2,00,000 and the new salary is ₹2,40,000.

The increase in salary is: 2,40,000 - 2,00,000 = ₹40,000

Substituting these values into the formula: Percentage Increase = (40,000 / 2,00,000) × 100% = 20%

This corresponds to the value of 20% in Option C.