SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The expression is simplified by using standard trigonometric identities:

Numerator: 1 + tan²A = sec²A = 1/cos²A

Denominator: cosec²A × tanA = (1/sin²A) × (sinA/cosA) = 1/(sinA cosA)

Substituting these values into the expression (1 + tan²A) / (cosec²A × tanA) gives:

(1/cos²A) / (1/(sinA cosA)) = (1/cos²A) × (sinA cosA) = sinA/cosA = tanA

Explanation

Using the successive percentage change formula, the net profit or loss percentage is given by:

Net change % = x + y + (xy / 100)

Here, the markup percentage x is +50% and the discount percentage y is -40%.

Substituting these values into the formula:

Net change % = 50 - 40 + (50 × (-40)) / 100 Net change % = 10 - 20 Net change % = -10%

The negative sign indicates a net decrease, which represents a loss of 10%.

Explanation

Correct Answer: B

To find the value of a³ - 1/a³, we first determine the value of a - 1/a using the algebraic identity: (a - 1/a)² = a² + 1/a² - 2

Given that a² + 1/a² = 7/3, substitute this value into the equation: (a - 1/a)² = 7/3 - 2 (a - 1/a)² = 1/3

Taking the square root on both sides: a - 1/a = 1/√3

Now, use the algebraic identity for the difference of cubes: a³ - 1/a³ = (a - 1/a)³ + 3(a - 1/a)

Substitute a - 1/a = 1/√3 into this identity: a³ - 1/a³ = (1/√3)³ + 3(1/√3) a³ - 1/a³ = 1/(3√3) + 3/√3

To add these fractions, express them with a common denominator of 3√3: 3/1 = 9/3 3/√3 = 9/(3√3)

Now, add the terms: a³ - 1/a³ = 1/(3√3) + 9/(3√3) = 10/(3√3)

Explanation

Using the rule of alligation:

Quantity of coffee / Quantity of chicory = (Cost of chicory - Mean price) / (Mean price - Cost of coffee)

Substituting the given values:

Quantity of coffee / Quantity of chicory = (100 - 60) / (60 - 45) Quantity of coffee / Quantity of chicory = 40 / 15 Quantity of coffee / Quantity of chicory = 8 / 3

Given that the quantity of coffee is 40 kg:

40 / Quantity of chicory = 8 / 3 Quantity of chicory = 40 × 3 / 8 Quantity of chicory = 5 × 3 Quantity of chicory = 15 kg

Explanation

By the Exterior Angle Theorem, the exterior angle of a triangle is equal to the sum of its two opposite interior angles. In triangle PQR, when QR is produced to S, the exterior angle at vertex R is ∠PRS, and the two opposite interior angles are ∠RPQ and ∠PQR.

Therefore, we can set up the equation: ∠PRS = ∠RPQ + ∠PQR

Substituting the given values into the equation: 9x - 15 = 2x + 4x + 15

Simplifying the right side of the equation: 9x - 15 = 6x + 15

Subtracting 6x from both sides: 3x - 15 = 15

Adding 15 to both sides: 3x = 30

Dividing by 3: x = 10

Explanation

Using the algebraic identity: (a + b)² = a² + b² + 2ab

Substituting the given values into the equation: 10² = 58 + 2ab 100 = 58 + 2ab 2ab = 100 - 58 2ab = 42 ab = 21

Explanation

The relationship between Marked Price (MP), Cost Price (CP), Gain Percentage, and Discount Percentage is given by the formula:

MP / CP = (100 + Gain%) / (100 - Discount%)

Substituting the given values into the formula:

MP / 153 = (100 + 20) / (100 - 15)

MP / 153 = 120 / 85

MP = 153 × 120 / 85

Since 153 and 85 are both divisible by 17:

MP = 9 × 120 / 5

MP = 9 × 24

MP = 216

Explanation

Let the cost of a cup be c and the cost of a plate be p.

From the given information, we can set up the following system of linear equations:

3c + 5p = 1080 2c + 4p = 840

Dividing the second equation by 2, we get: c + 2p = 420 c = 420 - 2p

Substituting this value of c into the first equation: 3(420 - 2p) + 5p = 1080 1260 - 6p + 5p = 1080 1260 - p = 1080 p = 180

Substituting p = 180 back to find c: c = 420 - 2(180) c = 420 - 360 c = 60

The ratio of the cost of a cup to the cost of a plate is: c:p = 60:180 = 1:3

Explanation

The length of the direct common tangent to two circles touching each other externally is given by the formula:

Length of tangent = 2√(Rr)

where R and r are the radii of the two circles.

Given: Radius of the bigger circle, R = 12 cm Radius of the smaller circle, r = 8 cm

Substituting these values into the formula:

Length of tangent AB = 2√(12 × 8) Length of tangent AB = 2√96 Length of tangent AB = 2 × 4√6 Length of tangent AB = 8√6 cm

Explanation

Using the algebraic identity a² - b² = (a - b)(a + b), we can rewrite the expression as:

cos⁴ A - sin⁴ A = (cos² A - sin² A)(cos² A + sin² A)

Using the fundamental trigonometric identity cos² A + sin² A = 1, the expression simplifies to:

cos² A - sin² A

By substituting the identity cos² A = 1 - sin² A, we get:

(1 - sin² A) - sin² A = 1 - 2sin² A

Explanation

The formula for compound interest is CI = P × [(1 + R/100)ⁿ - 1], where P is the principal, R is the rate of interest per annum, and n is the time in years.

Substituting the given values, P = 62500, R = 8, and n = 2:

CI = 62500 × [(1 + 8/100)² - 1] CI = 62500 × [(1.08)² - 1] CI = 62500 × [1.1664 - 1] CI = 62500 × 0.1664 CI = 10400

Alternatively, the effective compound interest rate for 2 years at 8% per annum is given by the formula x + y + xy/100:

Effective rate = 8 + 8 + (8 × 8)/100 = 16.64%

Compound Interest = 16.64% of 62500 = (16.64 / 100) × 62500 = 16.64 × 625 = 10400

Explanation

By the Angle-Angle-Angle (AAA) similarity criterion, two triangles are similar if their corresponding angles are equal.

Comparing the given angles of ΔPQR and ΔDEF: ∠P = 52° and ∠F = 52°, which means vertex P corresponds to vertex F. ∠Q = 74° and ∠E = 74°, which means vertex Q corresponds to vertex E. ∠R = 54° and ∠D = 54°, which means vertex R corresponds to vertex D.

Matching the corresponding vertices in the correct sequence yields: P → F Q → E R → D

Thus, ΔPQR is similar to ΔFED, represented as ΔPQR ~ ΔFED.

Explanation

If a, b, and c are three numbers, their fourth proportion x is given by the formula x = (b × c) / a.

Substituting the given values a = 15, b = 55, and c = 81 into the formula:

x = (55 × 81) / 15

Simplifying this by dividing both 55 and 15 by 5:

x = (11 × 81) / 3

Now dividing 81 by 3:

x = 11 × 27

x = 297

Explanation

The total area to be whitewashed includes the area of the four walls and the area of the ceiling.

Area of the four walls = 2h(l + b) Area of the ceiling = l × b

Given dimensions: Length (l) = 7 m Breadth (b) = 5 m Height (h) = 3 m

Total area to be whitewashed = 2h(l + b) + l × b Substituting the given values: Total area = 2 × 3 × (7 + 5) + 7 × 5 Total area = 6 × 12 + 35 Total area = 72 + 35 = 107 m²

The rate of whitewashing is ₹16 per m².

Total cost of whitewashing = Total area × Rate Total cost = 107 × 16 = ₹1,712

Explanation

Let the standard length of the measurement tape be 100 cm and the cost price of 1 cm of cloth be ₹1.

While buying, the tailor uses a tape of length 105 cm. Thus, he gets 105 cm of cloth but pays for only 100 cm. Actual Cost Price of 105 cm of cloth = ₹100

While selling, the tailor uses a tape of length 95 cm. Thus, he delivers 95 cm of cloth but charges the price of 100 cm. Selling Price of 95 cm of cloth = ₹100 Selling Price of 1 cm of cloth = ₹100/95 = ₹20/19 Selling Price of 105 cm of cloth = 105 × 20/19 = ₹2100/19

Total Gain = Selling Price of 105 cm - Cost Price of 105 cm Total Gain = 2100/19 - 100 = 200/19 = 10 10/19

Since the actual Cost Price was ₹100, the gain percent is calculated as: Gain % = (Gain / Cost Price) × 100 = (200/19 / 100) × 100 = 200/19 % = 10 10/19 %

Explanation

In the Accounts department, the percentage of female employees is 10%, which means the percentage of male employees is 90%. Since there are 198 male employees in the Accounts department, the number of female employees in this department is 198 × 10/90 = 22.

In the R | D department, the percentage of female employees is 60%, which means the percentage of male employees is 40%. Since there are 100 male employees in the R | D department, the number of female employees in this department is 100 × 60/40 = 150.

The total number of female employees in these two departments combined is 22 + 150 = 172.

Explanation

In the Accounts department, the percentage of female employees is 10%, which means the percentage of male employees is 90%. Since there are 198 male employees in the Accounts department, the number of female employees in Accounts is: 198 × 10/90 = 22

In the R | D department, the percentage of female employees is 60%, which means the percentage of male employees is 40%. Since there are 100 male employees in the R | D department, the number of female employees in R | D is: 100 × 60/40 = 150

Combining the female employees from both departments gives: 22 + 150 = 172

Explanation

By the property of circles, angles subtended by the same arc in the same segment are equal. Since both ∠BAC and ∠BDC are subtended by the minor arc BC at the circumference in the major sector of the circle, we have:

∠BAC = ∠BDC = 73°

In triangle ABC, the sum of all interior angles is 180°:

∠BAC + ∠ABC + ∠ACB = 180°

Substituting ∠BAC = 73° into the equation:

73° + ∠ABC + ∠ACB = 180°

∠ABC + ∠ACB = 180° - 73° = 107°

Explanation

For a number to be divisible by 88, it must be divisible by both 8 and 11.

Divisibility by 8: The number formed by the last three digits, y24, must be divisible by 8. Testing digit values for y from 0 to 9, the possible values that make y24 divisible by 8 are y ∈ {0, 2, 4, 6, 8}.

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 a multiple of 11 (including 0).

Sum of odd-positioned digits = 7 + 0 + 5 + 3 + 2 = 17 Sum of even-positioned digits = 8 + x + 3 + y + 4 = 15 + x + y

Difference = (15 + x + y) - 17 = x + y - 2

For the number to be divisible by 11, this difference must be a multiple of 11: x + y - 2 = 0 or x + y - 2 = 11

This gives: x + y = 2 or x + y = 13

The least value of x + y is 2, which is achievable when y = 0 and x = 2, or when y = 2 and x = 0. Both cases yield valid single-digit integers for x and y.

Explanation

The compound ratio of two ratios a:b and c:d is defined as (a × c) : (b × d).

For the given ratios 3:4 and 6:7, the compound ratio is: (3 × 6) : (4 × 7) = 18 : 28

We are given that this compound ratio is equal to 36:x. Equating the two ratios: 18/28 = 36/x

Solving for x: x = (36 × 28) / 18 x = 2 × 28 x = 56

Explanation

The intersection point of the internal angle bisectors of a triangle is the incenter. For any triangle ABC where the internal angle bisectors of ∠B and ∠C meet at X, the relation between the angle ∠BXC and the vertex angle ∠BAC is given by the formula:

∠BXC = 90° + ½ ∠BAC

Substituting the given value of ∠BAC = 30° into the formula:

∠BXC = 90° + ½ × 30° ∠BXC = 90° + 15° ∠BXC = 105°

Explanation

Correct Answer: C

The difference between the total surface area and the lateral surface area of a cube with side a is given by the formula:

Difference = Total Surface Area - Lateral Surface Area Difference = 6a² - 4a² = 2a²

Given the side of the cube a = 12 cm, substituting this value into the formula yields:

Difference = 2 × 12² Difference = 2 × 144 = 288 cm²

Explanation

The price after the first discount is calculated as: Price after first discount = Listed Price × (1 - d₁/100) Price after first discount = 1500 × (1 - 0.20) = 1500 × 0.80 = 1200

To find the additional discount d₂ required to reduce the price from 1200 to the final net price of 1104: Additional discount amount = 1200 - 1104 = 96

The additional discount percentage is: d₂ = (Additional discount amount / Price after first discount) × 100 d₂ = (96 / 1200) × 100 = 8%

Explanation

Disha's copying rate is 75/12 = 25/4 pages per hour.

Mohit's copying rate is 75/10 = 15/2 pages per hour.

Working together, their combined copying rate is 25/4 + 15/2 = 55/4 pages per hour.

The time required to copy a 150-page manuscript is: Time = 150 / (55/4) = 600/55 = 120/11 = 10 10/11 hours.

Explanation

The relation between the marked price M and the final selling price S after two successive discounts of d₁% and d₂% is given by the formula:

S = M × (1 - d₁/100) × (1 - d₂/100)

Given that the selling price S is ₹5,130, and the successive discounts are d₁ = 10% and d₂ = 5%, we substitute these values into the formula:

5130 = M × (1 - 10/100) × (1 - 5/100)

5130 = M × 9/10 × 19/20

5130 = M × 171/200

M = (5130 × 200) / 171

M = 30 × 200 = 6000

Explanation

Formula for simple interest: SI = (P × R × T) / 100

Let the principal be P. Since the amount returned is three times the principal, the Amount A = 3P. The simple interest earned is: SI = A - P = 3P - P = 2P

Using the formula with time T = 25 years: 2P = (P × R × 25) / 100 2 = 25R / 100 2 = R / 4 R = 8%

Explanation

The length of a direct common tangent to two circles is given by the formula:

L = √(d² - (R - r)²)

where: d is the distance between the centres of the two circles, which is 48 cm. R is the radius of the larger circle, which is 24 cm. r is the radius of the smaller circle, which is 18 cm.

Substituting the given values into the formula:

L = √(48² - (24 - 18)²) L = √(48² - 6²)

Using the algebraic identity a² - b² = (a - b)(a + b):

L = √((48 - 6) × (48 + 6)) L = √(42 × 54) L = √(6 × 7 × 6 × 9) L = √(36 × 9 × 7) L = 6 × 3 × √7 L = 18√7 cm

Explanation

In a circle, a tangent at any point is perpendicular to the radius through the point of contact. Let T be the point of tangency on the circle. This forms a right-angled triangle OTP, where the right angle is at T.

Applying the Pythagorean theorem to triangle OTP:

OP² = OT² + PT²

Substituting the given values, where the distance from the centre OP = 5 cm and the radius OT = 3 cm:

5² = 3² + PT² 25 = 9 + PT² PT² = 25 - 9 PT² = 16 PT = 4 cm

Explanation

To find the total number of vacant berths in Trains B and C, first calculate the total capacity of all classes for both trains:

Total capacity for one train = 200 + 200 + 400 + 300 + 500 = 1600 Total capacity for Trains B and C = 1600 × 2 = 3200

Next, calculate the total number of reserved berths in Train B and Train C:

Total reserved in Train B = 190 + 183 + 348 + 215 + 414 = 1350 Total reserved in Train C = 198 + 191 + 364 + 291 + 484 = 1528 Total reserved in both trains = 1350 + 1528 = 2878

Finally, calculate the total vacant berths:

Total vacant berths = Total capacity - Total reserved berths Total vacant berths = 3200 - 2878 = 322

Explanation

The given expression is a standard algebraic identity. Expanding the product directly yields:

(a + b + c)(a² + b² + c² - ab - bc - ca) = a(a² + b² + c² - ab - bc - ca) + b(a² + b² + c² - ab - bc - ca) + c(a² + b² + c² - ab - bc - ca)

Expanding the terms gives:

a³ + ab² + ac² - a²b - abc - ca² + a²b + b³ + bc² - ab² - b²c - abc + ca² + cb² + c³ - abc - bc² - c²a

Combining like terms, the positive and negative cross-product terms cancel each other out: ab² and -ab² cancel ac² and -c²a cancel -a²b and a²b cancel bc² and -bc² cancel -b²c and cb² cancel -ca² and ca² cancel

The remaining terms are: a³ + b³ + c³ - abc - abc - abc = a³ + b³ + c³ - 3abc

This confirms the standard identity.

Explanation

Upstream speed is defined as the distance traveled upstream divided by the time taken.

Time taken = 50 minutes = 50/60 hours = 5/6 hours Distance = 10 km

Upstream speed = 10 / (5/6) = 10 × 6/5 = 12 kmph

The relation between upstream speed, speed of the boat in still water (x), and speed of the stream (y) is: Upstream speed = x - y

Given that the speed of the stream y = 3 kmph: 12 = x - 3 x = 12 + 3 = 15 kmph

Thus, the speed of the boat in still water is 15 kmph.

Explanation

The combined return percentage on multiple investments can be calculated using the weighted average formula:

Weighted Return = (w₁ × r₁ + w₂ × r₂) / (w₁ + w₂)

Here, the investment ratio is w₁:w₂ = 1:3, the return on the first investment r₁ is +10%, and the return on the second investment r₂ is -10%.

Substituting these values into the formula:

Weighted Return = (1 × 10% + 3 × (-10%)) / (1 + 3) Weighted Return = (10% - 30%) / 4 Weighted Return = -20% / 4 Weighted Return = -5%

The negative sign represents a net loss, resulting in a combined loss of 5%.

Explanation

We are given the trigonometric equations: sin(p+q) = 1 cos(p-q) = √3/2

Since sin(90°) = 1 and cos(30°) = √3/2, we can set up the following system of linear equations: p + q = 90° p - q = 30°

Adding these two equations eliminates q: (p + q) + (p - q) = 90° + 30° 2p = 120° p = 60°

Explanation

The average weight of the employees can be calculated using the weighted average formula:

Weighted Average = (m × Aw + w × Aw) / (m + w)

where m and w represent the ratio of men and women, and Aw represents their respective average weights.

Given: Ratio of men to women = 2:3 Average weight of men = 56 kg Average weight of women = 51 kg

Substituting the values into the formula:

Average Weight = (2 × 56 + 3 × 51) / (2 + 3) Average Weight = (112 + 153) / 5 Average Weight = 265 / 5 Average Weight = 53 kg

Explanation

To find the average number of appearing candidates from all four states in the year 2021, sum the number of appearing candidates (A) for each state in 2021 and divide by the total number of states.

The appearing candidates in 2021 are: Assam: 2900 Bihar: 4700 Kerala: 3400 Uttar Pradesh: 4800

Total appearing candidates in 2021 = 2900 + 4700 + 3400 + 4800 = 15800

Average number of appearing candidates = Total appearing candidates ÷ Number of states Average = 15800 ÷ 4 = 3950

Explanation

Using the income-expenditure relationship, we have:

Income = Expenses + Savings

Since the person spends 68% of his income, his savings percentage is:

Savings % = 100% - 68% = 32%

We can calculate the total expenses directly by using the ratio of spending percentage to saving percentage:

Expenses = (Spending % / Saving %) × Savings

Substituting the given values:

Expenses = (68 / 32) × 14,720

Expenses = (17 / 8) × 14,720

Expenses = 17 × 1,840

Expenses = 31,280

Thus, the total monthly expenses of the person are ₹31,280.

Explanation

The effective rate of interest for two years at a rate of r% per annum compounded annually can be calculated using the successive percentage formula:

Effective Rate = r + r + (r × r)/100

Substituting r = 18:

Effective Rate = 18 + 18 + (18 × 18)/100 = 36 + 3.24 = 39.24%

Given that the compound interest earned is 981, this interest represents 39.24% of the present value (P):

P × 39.24% = 981

P × 39.24/100 = 981

P = (981 × 100) / 39.24

P = 98100 / 39.24 = 2500

Explanation

We are given the equations: p = cot A + cos A q = cot A - cos A

Using the algebraic identity for the difference of squares: p² - q² = (p + q)(p - q)

By adding and subtracting the two given equations: p + q = 2 cot A p - q = 2 cos A

Substituting these into the identity gives: p² - q² = (2 cot A)(2 cos A) = 4 cot A cos A

Next, we find the product pq: pq = (cot A + cos A)(cot A - cos A) = cot² A - cos² A

Expressing cot A in terms of sin A and cos A: pq = (cos² A / sin² A) - cos² A pq = cos² A (1 / sin² A - 1) pq = cos² A (cosec² A - 1) pq = cos² A cot² A

Taking the square root on both sides: √pq = cos A cot A

Substituting this relation back into the expression for p² - q²: p² - q² = 4 √pq p² - q² = √(16pq)

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) when both discount and profit percentages are known is given by the formula:

CP / MP = (100 - Discount%) / (100 + Profit%)

Given: Marked Price (MP) = 1080 Discount% = 10 Profit% = 8

Substituting these values into the formula:

CP / 1080 = (100 - 10) / (100 + 8)

CP / 1080 = 90 / 108

Solving for CP:

CP = 1080 × (90 / 108)

CP = 10 × 90

CP = 900

Explanation

The formula for the difference (D) between compound interest and simple interest on a principal sum (P) for 3 years at an annual interest rate (r) is:

D = P × (r/100)² × (3 + r/100)

Given that the difference is £76 and the rate of interest is 4%, substituting these values into the formula gives:

76 = P × (4/100)² × (3 + 4/100)

76 = P × (1/25)² × (76/25)

76 = P × (1/625) × (76/25)

Dividing both sides of the equation by 76:

1 = P / (625 × 25)

P = 625 × 25

P = 15,625

Explanation

The total simple interest accrued over successive years is calculated on the initial principal amount.

Using the simple interest relation: Total Interest = P × (r₁ + r₂) / 100

Substituting the given values: Principal (P) = ₹25,000 First year rate (r₁) = 4% Second year rate (r₂) = 5%

Total Interest = 25000 × (4 + 5) / 100 Total Interest = 25000 × 9 / 100 Total Interest = 250 × 9 = ₹2,250

The total amount payable is: Amount = Principal + Total Interest Amount = 25000 + 2250 = ₹27,250

Explanation

When two individuals complete a work in A and B hours respectively, the time taken by both of them working together is given by the formula:

Time = (A × B) / (A + B)

Given that Aditya takes A = 20 hours and Vinay takes B = 15 hours, substituting these values into the formula yields:

Time = (20 × 15) / (20 + 15) Time = 300 / 35

Simplifying the fraction by dividing the numerator and denominator by 5:

Time = 60 / 7

Converting this improper fraction into a mixed fraction:

Time = 8 4/7 hours

Explanation

A tangent to a circle is perpendicular to the radius at the point of contact. This forms a right-angled triangle where the hypotenuse is the distance from the point to the centre of the circle (d = 10 cm), one leg is the radius (r = 6 cm), and the other leg is the length of the tangent (t).

By applying the Pythagorean theorem: d² = r² + t²

Substituting the given values: 10² = 6² + t² 100 = 36 + t² t² = 100 - 36 t² = 64 t = √64 t = 8 cm

Explanation

According to the algebraic identity, if a + b + c = 0, then a³ + b³ + c³ = 3abc.

For the numerator, let a = u-v, b = v-w, and c = w-u. Since (u-v) + (v-w) + (w-u) = 0, the numerator simplifies to: 3(u-v)(v-w)(w-u)

For the denominator, let x = u²-v², y = v²-w², and z = w²-u². Since (u²-v²) + (v²-w²) + (w²-u²) = 0, the denominator simplifies to: 3(u²-v²)(v²-w²)(w²-u²)

Using the difference of squares identity, a² - b² = (a-b)(a+b), the denominator can be factored as: 3(u-v)(u+v)(v-w)(v+w)(w-u)(w+u)

Dividing the simplified numerator by the simplified denominator: 3(u-v)(v-w)(w-u) / [3(u-v)(u+v)(v-w)(v+w)(w-u)(w+u)]

Canceling out the common terms 3, (u-v), (v-w), and (w-u) from both the numerator and the denominator yields: 1 / [(u+v)(v+w)(w+u)]

Explanation

The single equivalent discount for two successive discounts of a% and b% is given by the formula:

Equivalent Discount = a + b - (a × b)/100

Substituting the given values of a = 15 and b = 20 into the formula:

Equivalent Discount = 15 + 20 - (15 × 20)/100 Equivalent Discount = 35 - 300/100 Equivalent Discount = 35 - 3 Equivalent Discount = 32%

Explanation

We are given the expression: sin A(1 + sin A / cos A) + cos A(1 + cos A / sin A)

Simplify the terms inside the parentheses by taking the common denominator: sin A((cos A + sin A) / cos A) + cos A((sin A + cos A) / sin A)

Factor out the common term (sin A + cos A): (sin A + cos A) (sin A / cos A + cos A / sin A)

Simplify the term in the second parentheses by finding a common denominator: sin A / cos A + cos A / sin A = (sin² A + cos² A) / (sin A cos A)

Using the fundamental trigonometric identity sin² A + cos² A = 1, this term becomes: 1 / (sin A cos A)

Now, substitute this back into the expression: (sin A + cos A) / (sin A cos A)

Split the fraction into two separate terms: sin A / (sin A cos A) + cos A / (sin A cos A) = 1 / cos A + 1 / sin A

Using the reciprocal identities 1 / cos A = sec A and 1 / sin A = cosec A, we get: sec A + cosec A

Explanation

The third proportion c of two numbers a and b is given by the formula c = b² / a.

Substituting the given values, where a = 1.8 and b = 9:

c = 9² / 1.8

c = 81 / 1.8

c = 810 / 18

c = 45

Explanation

Using the algebraic relation: (w - 1/w)² = (w + 1/w)² - 4

Substitute the given value w + 1/w = 6 into the identity: (w - 1/w)² = 6² - 4 (w - 1/w)² = 36 - 4 (w - 1/w)² = 32

Taking the square root on both sides: w - 1/w = √32 w - 1/w = √(16 × 2) w - 1/w = 4√2

Explanation

The given expression is a product of cotangent values of integer angles from 1° to 179°: cot 1° × cot 2° × ... × cot 90° × ... × cot 179°

The term cot 90° is present in this sequence. Since cot 90° = cos 90° ÷ sin 90° = 0, and all other terms are defined, finite real numbers, multiplying the entire product by zero yields a final value of 0.

Explanation

The largest five-digit number is 99999. To find the largest five-digit number exactly divisible by 88, we divide 99999 by 88 to find the remainder.

99999 ÷ 88 = 1136 with a remainder of 31

Subtracting this remainder from 99999 gives the largest five-digit number that is completely divisible by 88:

99999 - 31 = 99968