SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

To find the year with the highest cumulative manufacturing of aerated drinks B and C, calculate the sum of the production values of B and C for each year:

For 2019: 200 + 140 = 340 thousand For 2020: 230 + 170 = 400 thousand For 2021: 210 + 195 = 405 thousand For 2022: 175 + 190 = 365 thousand

Comparing these totals, the maximum production is 405 thousand, which occurred in the year 2021.

Explanation

Using the standard trigonometric identities 1 + tan² A = sec² A and 1 + cot² A = cosec² A, the given expression simplifies as follows:

(1 + tan² A)(1 + cot² A) = sec² A × cosec² A

Since sec² A = 1/cos² A and cosec² A = 1/sin² A, the expression becomes:

sec² A × cosec² A = 1 / (sin² A cos² A)

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

1 / (sin² A (1 - sin² A))

Explanation

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides. This relation is expressed as:

Area(ABC) / Area(PQR) = (AB / PQ)²

Substituting the given values into the formula:

32 / 50 = (AB / 10)²

Simplifying the fraction on the left side:

16 / 25 = (AB / 10)²

Taking the square root of both sides:

4 / 5 = AB / 10

Solving for AB:

AB = (4 / 5) × 10 AB = 8

Explanation

If four numbers a, b, c, and d are in proportion, then the ratio of the first two is equal to the ratio of the last two, which is represented by the formula: a/b = c/d

Here, d is the fourth proportion of a, b, and c.

Given the numbers a = 8, b = 20, and c = 15, let the fourth proportion be x.

Substituting these values into the formula: 8/20 = 15/x

Solving for x: x = (20 × 15) / 8 x = 300 / 8 x = 37.5

Explanation

Profit is defined as a percentage of the Cost Price (CP). Given that the profit is 20% and the profit amount is ₹800: 20% of CP = 800 CP = 800 / 0.20 = ₹4000

The Selling Price (SP) is the sum of the Cost Price and the profit: SP = CP + Profit SP = 4000 + 800 = ₹4800

A discount of 20% is given on the Marked Price (MP) to sell the cycle. Therefore, the Selling Price is 80% of the Marked Price: SP = 80% of MP 4800 = 0.80 × MP MP = 4800 / 0.80 = ₹6000

Explanation

The remaining fraction of wine in the cask after successive replacement operations is given by the formula: Final Wine / Total Capacity = (1 - x₁/V) × (1 - x₂/V) × (1 - x₃/V)

In this formula, V represents the total capacity of the cask (600 litres), while x₁, x₂, and x₃ are the volumes of mixture replaced with water in each step. Here: V = 600 x₁ = y x₂ = 120 x₃ = 120

The final ratio of wine to water in the cask is 12 : 13. Therefore, the fraction of wine remaining in the cask is: 12 / (12 + 13) = 12/25

Substituting these values into the formula: 12/25 = (1 - y/600) × (1 - 120/600) × (1 - 120/600)

Simplifying the term 1 - 120/600: 1 - 120/600 = 1 - 1/5 = 4/5

Substitute this back into the equation: 12/25 = (1 - y/600) × 4/5 × 4/5 12/25 = (1 - y/600) × 16/25

Dividing both sides of the equation by 16/25: 12/16 = 1 - y/600 3/4 = 1 - y/600

Rearranging the equation to solve for y: y/600 = 1 - 3/4 y/600 = 1/4 y = 600 / 4 y = 150

Explanation

The simple interest SI is 888 (where the leading 1 is a typographical artifact from question numbering).

Using the standard simple interest formula:

SI = (P × R × T) / 100

where: P is the principal sum = 7400 T is the time period in years = 36 months = 3 years R is the rate of interest per annum

Substituting the values into the formula to solve for R:

888 = (7400 × R × 3) / 100

888 = 74 × R × 3

888 = 222 × R

R = 888 / 222

R = 4

The rate of interest is 4% per annum.

Explanation

The initial cost price of the first article is 48,000.

Selling price after a 12% loss is calculated as: 48,000 × (1 - 12/100) = 48,000 × 0.88 = 42,240

This amount is used as the cost price for the second item.

Selling price of the second item after a 20% gain is: 42,240 × (1 + 20/100) = 42,240 × 1.20 = 50,688

The overall gain is the difference between the final selling price and the initial cost price: Overall gain = 50,688 - 48,000 = 2,688

Explanation

The algebraic identity for the sum of cubes is x³ + y³ = (x + y)³ - 3xy(x + y).

By letting x = sin² θ and y = cos² θ, we can write: sin⁶ θ + cos⁶ θ = (sin² θ)³ + (cos² θ)³ sin⁶ θ + cos⁶ θ = (sin² θ + cos² θ)³ - 3 sin² θ cos² θ (sin² θ + cos² θ)

Using the fundamental trigonometric identity sin² θ + cos² θ = 1, the expression simplifies to: sin⁶ θ + cos⁶ θ = 1³ - 3 (sin θ cos θ)² (1) sin⁶ θ + cos⁶ θ = 1 - 3 (sin θ cos θ)²

Given that sin θ cos θ = √2/3, we substitute this value into the simplified expression: sin⁶ θ + cos⁶ θ = 1 - 3 (√2/3)² sin⁶ θ + cos⁶ θ = 1 - 3 (2/9) sin⁶ θ + cos⁶ θ = 1 - 2/3 sin⁶ θ + cos⁶ θ = 1/3

Explanation

Using the algebraic identity for the square of a trinomial:

(d + e + f)² = d² + e² + f² + 2(de + ef + fd)

Substitute the given values (d + e + f) = 14 and (d² + e² + f²) = 96 into the formula:

14² = 96 + 2(de + ef + fd)

196 = 96 + 2(de + ef + fd)

Subtract 96 from both sides:

100 = 2(de + ef + fd)

Divide by 2:

de + ef + fd = 50

Explanation

The relation between the Selling Price, Marked Price, and Discount Percentage is: Selling Price = Marked Price × (1 - Discount Percentage / 100)

Substituting the given values: 2325 = Marked Price × (1 - 22.5 / 100) 2325 = Marked Price × 0.775 Marked Price = 2325 / 0.775 Marked Price = 3000

The discount offered is the difference between the Marked Price and the Selling Price: Discount = Marked Price - Selling Price Discount = 3000 - 2325 = 675

Explanation

In the right-angled triangle CAB, with ∠CAB = 90°, the hypotenuse BC is calculated using the Pythagorean theorem:

BC = √(AB² + AC²) BC = √(10² + 24²) = √(100 + 576) = √676 = 26 cm

The altitude AD to the hypotenuse BC in a right-angled triangle is given by the formula:

AD = (AB × AC) / BC

Substituting the given values:

AD = (10 × 24) / 26 = 240 / 26 ≈ 9.23 cm

Explanation

Using the trigonometric identity: sec² A - tan² A = 1

For an acute angle A, the trigonometric ratio tan A is positive, which gives: tan A = √(sec² A - 1)

Given: 8 sec A = 17 sec A = 17/8

Substituting the value of sec A into the formula: tan A = √((17/8)² - 1) tan A = √(289/64 - 1) tan A = √(225/64) tan A = 15/8

Explanation

Given the expression g = 2 - √3, find the reciprocal 1/g by rationalizing the denominator: 1/g = 1 / (2 - √3) = (2 + √3) / ((2 - √3)(2 + √3)) = (2 + √3) / (4 - 3) = 2 + √3

Now, calculate the difference between g and 1/g: g - 1/g = (2 - √3) - (2 + √3) = -2√3

The algebraic identity for the difference of cubes is: g³ - 1/g³ = (g - 1/g)³ + 3(g - 1/g)

Substituting the value of g - 1/g into this formula: g³ - 1/g³ = (-2√3)³ + 3(-2√3) g³ - 1/g³ = -24√3 - 6√3 g³ - 1/g³ = -30√3

Explanation

Let the cost price of the item be CP.

The marked price is set 50% above the cost price: Marked Price = CP × (1 + 0.50) = 1.5 × CP

The selling price after a 40% discount on the marked price is: Selling Price = Marked Price × (1 - 0.40) = 1.5 × CP × 0.60 = 0.90 × CP

Given that the selling price is ₹1,305: 0.90 × CP = 1305 CP = 1305 / 0.90 = 1450

Explanation

Curved Surface Area (CSA) of a cylinder is given by the formula: CSA = 2πrh

First, convert the dimensions to meters: Diameter = 84 cm = 0.84 m Radius r = 0.84 / 2 = 0.42 m Height h = 5 m

Substitute these values into the formula: CSA = 2 × (22/7) × 0.42 × 5 CSA = 2 × 22 × 0.06 × 5 CSA = 13.2 m²

The cost of painting the curved surface is the product of the area and the rate: Cost = CSA × Rate Cost = 13.2 × 15 = 198

Explanation

Using the standard trigonometric identity: cosec²θ - cot²θ = 1

Subtracting cosec²θ and 1 from both sides of the equation gives: cot²θ - cosec²θ = -1

Explanation

The number of votes received by the winning candidate is calculated by sequentially applying the given percentages to the total number of eligible voters.

Total eligible voters = 10000

Number of votes polled = 10000 × 85% = 10000 × 0.85 = 8500

Since 6% of the polled votes were invalid, the percentage of valid votes is 94%. Number of valid votes = 8500 × 94% = 8500 × 0.94 = 7990

The winning candidate secured 60% of these valid votes. Votes received by the winning candidate = 7990 × 60% = 7990 × 0.60 = 4794

Explanation

The table provides the cumulative distribution of marks for 100 students.

To find the number of students who obtained marks less than 10 in Mathematics, we use the values from the Mathematics row: The total number of students who obtained 0 and above is 100. The number of students who obtained 10 and above is 92.

The number of students who scored less than 10 is the difference between the total number of students (0 and above) and those who scored 10 and above:

Students with marks less than 10 = 100 - 92 = 8

Explanation

Percentage of girls in B. Ag. is calculated using the formula:

Percentage of girls = (Number of girls in B. Ag. ÷ Total students in B. Ag.) × 100

From the table, the number of girls in B. Ag. is 48, and the number of boys is 58.

Total students in B. Ag. = 58 + 48 = 106

Percentage of girls = (48 ÷ 106) × 100 = 45.28%

Explanation

The difference between compound interest and simple interest on a principal P for a period of two years at an annual interest rate R is given by the formula:

D = P × (R/100)²

Substituting the given values, where difference D = ₹56 and rate R = 8%:

56 = P × (8/100)²

56 = P × (2/25)²

56 = P × 4/625

P = (56 × 625) / 4

P = 14 × 625 = 8750

Explanation

The given expression is (2x + 13 - y)(2x - 13 - y).

By regrouping the terms, the expression can be rewritten in the form of the difference of squares identity, (a + b)(a - b) = a² - b².

Let a = 2x - y and b = 13. This gives: ((2x - y) + 13)((2x - y) - 13)

Applying the identity: (2x - y)² - 13²

Expanding the square term using (p - q)² = p² - 2pq + q²: (2x)² - 2(2x)(y) + y² - 169 4x² - 4xy + y² - 169

Rearranging the terms yields: 4x² + y² - 4xy - 169

Explanation

Let the measure of each of the two equal angles in the isosceles triangle be x.

The sum of the equal angles is x + x = 2x.

According to the given condition, the unequal angle is 4 times this sum, which is 4 × 2x = 8x.

Since the sum of all interior angles of a triangle is always 180°: x + x + 8x = 180° 10x = 180° x = 18°

Thus, each equal angle of the triangle measures 18°.

Explanation

The six-digit number is N = 4a6b9c. For N to be divisible by 99, it must be divisible by both 9 and 11.

The sum of the digits of N is: S = 4 + a + 6 + b + 9 + c = 19 + a + b + c

For N to be divisible by 9, this sum of the digits S must be a multiple of 9.

For N to be divisible by 11, the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions must be a multiple of 11: (4 + 6 + 9) - (a + b + c) = 19 - (a + b + c)

Let 19 - (a + b + c) = 11k, where k is an integer. Since a, b, and c are single-digit integers from 0 to 9, their sum must lie in the range 0 to 27. The possible values for a + b + c are: 19 - (a + b + c) = 11, which gives a + b + c = 8 19 - (a + b + c) = 0, which gives a + b + c = 19

Now we check which of these values makes the total sum of digits S divisible by 9: If a + b + c = 8, then S = 19 + 8 = 27, which is divisible by 9. If a + b + c = 19, then S = 19 + 19 = 38, which is not divisible by 9.

Thus, the only valid sum of the digits of N is 27.

Explanation

Since x varies directly as y, the relationship is represented by the formula x = ky, where k is the constant of variation.

Using the given values x = 45 and y = 9, we can find the value of k: 45 = k × 9 k = 45 ÷ 9 = 5

Now, using the constant k = 5, we calculate the value of x when y = 50: x = 5 × 50 = 250

Explanation

Using the relative speed formula for two objects moving in opposite directions, the relative speed is the sum of their individual speeds:

Relative Speed = Speed of Train 1 + Speed of Train 2 Relative Speed = 40 km/h + 32 km/h = 72 km/h

Converting the relative speed into metres per second: Relative Speed = 72 × 5/18 m/s = 20 m/s

The total distance to be covered for the trains to completely clear each other is the sum of their lengths: Total Distance = Length of Train 1 + Length of Train 2 Total Distance = 125 m + 99 m = 224 m

The time taken from the moment they meet to completely clear each other is calculated as: Time = Total Distance / Relative Speed Time = 224 m / 20 m/s = 11.2 seconds

Explanation

The time taken by Ramu alone to complete the work can be calculated using the direct rate formula:

1/Ramu = 1/Together - 1/Raju

Substituting the given values into the formula:

1/Ramu = 1/8 - 1/12

To subtract these fractions, find the least common multiple of 8 and 12, which is 24:

1/Ramu = 3/24 - 2/24 = 1/24

Thus, Ramu alone can complete the work in 24 days.

Explanation

Correct Answer: D

The numerator of the given expression is of the form a² + b² + c² + 2ab + 2bc + 2ca, where a = 0.09, b = 0.04, and c = 0.16.

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

Calculating the sum: a + b + c = 0.09 + 0.04 + 0.16 = 0.29

Therefore, the numerator simplifies to: (0.29)²

The denominator of the expression is: 0.3 × 0.3 + 0.2 × 0.2 + 0.4 × 0.4 = 0.09 + 0.04 + 0.16 = 0.29

Dividing the simplified numerator by the denominator: (0.29)² / 0.29 = 0.29

Explanation

Using the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b), the terms in the numerator can be factored.

For the first term, x² - 1 = (x - 1)(x + 1). Dividing this by (x - 1) simplifies to x + 1.

For the second term, x² - 9 = (x - 3)(x + 3). Dividing this by (x - 3) simplifies to x + 3.

Subtracting the second simplified term from the first gives: (x + 1) - (x + 3) = x + 1 - x - 3 = -2

Explanation

Let the cost price of the bag be x.

The difference in profit percentages is: 27% - 12% = 15%

This 15% difference in profit is equal to the increase in selling price, which is ₹900.

15% of x = 900 (15/100) × x = 900 x = (900 × 100) / 15 x = 60 × 100 x = 6000

The cost price of the bag is ₹6,000.

Explanation

Applying the difference of squares formula, (a - b)(a + b) = a² - b², to the first two terms: (3x - 2y)(3x + 2y) = (3x)² - (2y)² = 9x² - 4y²

Now, multiplying this product by the third term of the expression: (9x² - 4y²)(9x² + 4y²)

Applying the difference of squares formula once more with a = 9x² and b = 4y²: (9x² - 4y²)(9x² + 4y²) = (9x²)² - (4y²)² = 81x⁴ - 16y⁴

Explanation

When two objects move in opposite directions along a circular track, their relative speed is the sum of their individual speeds.

Relative speed = Speed of A + Speed of B Relative speed = 33 + 47 = 80 m/sec

To meet for the first time, the combined distance covered by both runners must be equal to the total length of the circular track.

Time taken for the first meeting = Track length / Relative speed Time taken for the first meeting = 1400 / 80 = 17.5 seconds

Since the runners continue moving at constant speeds, the time interval between any two consecutive meetings remains the same. Therefore, the time taken to meet for the second time is exactly twice the time taken for the first meeting.

Time for the second meeting = 2 × 17.5 = 35 seconds

Explanation

To find the combined average weight of both classes, we use the weighted average formula: Weighted Average = (n₁A₁ + n₂A₂) / (n₁ + n₂)

Instead of using the actual number of students, we can simplify the ratio of students to make the calculation faster: n₁ : n₂ = 64 : 32 = 2 : 1

Now, substitute the simplified ratio and the respective averages into the formula: Combined Average = (2 × 48 + 1 × 24) / (2 + 1) Combined Average = (96 + 24) / 3 Combined Average = 120 / 3 = 40 kg

Explanation

Using the successive percentage change formula, the present population is calculated as:

Present Population = P₀ × (1 + r₁/100) × (1 - r₂/100)

Here, the initial population P₀ is 60,000, the first-year growth rate r₁ is 5%, and the second-year decline rate r₂ is 10%.

Substituting these values into the formula:

Present Population = 60,000 × (1 + 5/100) × (1 - 10/100) Present Population = 60,000 × 1.05 × 0.90 Present Population = 60,000 × 0.945 Present Population = 56,700

Explanation

The length of an arc of a circle is calculated using the formula L = (θ / 360°) × 2πr, where θ is the angle subtended at the centre and r is the radius.

Substituting the given values, L = 33π and θ = 132°:

33π = (132 / 360) × 2πr

Dividing both sides of the equation by π:

33 = (132 / 180) × r

Simplifying the fraction 132/180 by dividing both terms by 12:

132 ÷ 12 = 11 180 ÷ 12 = 15

This simplifies the equation to:

33 = (11 / 15) × r

Solving for r:

r = (33 × 15) / 11

r = 3 × 15

r = 45 cm

Explanation

The relationship between the marked price (MP), successive discounts, and the final selling price (SP) is given by the formula:

SP = MP × (1 - d₁/100) × (1 - d₂/100)

Here, the first discount d₁ is 15%, the second discount d₂ is 5%, and the final selling price SP is ₹3,068.5.

Substituting these values into the formula:

3068.5 = MP × (1 - 15/100) × (1 - 5/100) 3068.5 = MP × 0.85 × 0.95 3068.5 = MP × 0.8075 MP = 3068.5 / 0.8075 MP = 3800

Explanation

When interest is compounded half-yearly, the rate of interest is halved and the time period is doubled.

The effective rate of interest per half-year period is r = 10% / 2 = 5%. The number of conversion periods for 1.5 years is n = 1.5 × 2 = 3 half-years.

The formula for compound interest is: CI = P × ((1 + r/100)ⁿ - 1)

Substituting the given values into the formula: 5044 = P × ((1 + 5/100)³ - 1) 5044 = P × ((21/20)³ - 1) 5044 = P × (9261/8000 - 1) 5044 = P × (1261/8000)

Solving for P: P = (5044 × 8000) / 1261 P = 4 × 8000 P = 32000

Explanation

Given the ratio a:b = 3:7, we can directly substitute a = 3 and b = 7 into the expression (2a + 3b):(3a + 2b) because the terms are homogeneous and of the same degree.

Substituting these values: First part: 2a + 3b = 2 × 3 + 3 × 7 = 6 + 21 = 27 Second part: 3a + 2b = 3 × 3 + 2 × 7 = 9 + 14 = 23

This yields the ratio 27:23.

Explanation

Under simple interest, the interest accrued each year remains constant.

The simple interest earned in 1 year is the difference between the amounts at the end of the fourth year and the third year: Interest for 1 year = ₹825 - ₹750 = ₹75

The simple interest for 3 years is: 3 × ₹75 = ₹225

The principal sum is the amount after 3 years minus the interest accumulated over those 3 years: Principal = ₹750 - ₹225 = ₹525

Explanation

The volume of steel used in a hemispherical bowl is the difference between the volume of the outer hemisphere and the volume of the inner hemisphere.

The volume V of a hollow hemisphere is given by the formula: V = 2/3 π (R³ - r³)

where R is the outer radius and r is the inner radius.

Given: Inner radius r = 4 cm Thickness t = 1 cm Outer radius R = r + t = 4 + 1 = 5 cm

Substituting these values into the formula: V = 2/3 π (5³ - 4³) V = 2/3 π (125 - 64) V = 2/3 π × 61 V = 122/3 π V = 40 ⅔ π cm³

Explanation

According to the angle sum property of a triangle, the sum of all three interior angles is always 180°.

(x - 15)° + (x + 45)° + (x + 60)° = 180° 3x + 90 = 180 3x = 90 x = 30

Substituting the value of x back into the expressions for the angles: First angle = 30 - 15 = 15° Second angle = 30 + 45 = 75° Third angle = 30 + 60 = 90°

Since one of the angles of the triangle measures exactly 90°, it is a right-angled triangle.

Explanation

Using the trigonometric identity cosec² A - cot² A = 1, we can express it as (cosec A - cot A)(cosec A + cot A) = 1.

Let cosec A + cot A = x = a√b. It follows that cosec A - cot A = 1/x.

To find the value of cosec A and cot A, we add and subtract these two equations: 2 cosec A = x + 1/x = (x² + 1)/x 2 cot A = x - 1/x = (x² - 1)/x

Dividing 2 cot A by 2 cosec A yields: cos A = (2 cot A) / (2 cosec A) = (x² - 1)/(x² + 1)

Substituting x = a√b gives x² = a²b. Thus, the expression becomes: (a²b - 1)/(a²b + 1) = cos A

Explanation

To evaluate the given expression, we apply the BODMAS rule, which dictates the order of operations as Brackets, Orders (of), Division, Multiplication, Addition, and Subtraction.

For the denominator, grouping the terms as follows yields: ((4 + 4) of 5 + 7) × 7 × 7 + 6 - 4 First, evaluate the addition inside the first bracket: 4 + 4 = 8 Next, evaluate the "of" operation: 8 of 5 = 40 Now, add 7 to the result: 40 + 7 = 47 Multiply this result by the product of the next two terms (7 × 7 = 49): 47 × 49 = 2303 Finally, perform the addition and subtraction at the end: 2303 + 6 - 4 = 2305

For the numerator, grouping the terms similarly yields: ((2 + 7) × 7 + 9 of 9 - 6) × (6 + 6) + 8 = 1664

Dividing the simplified numerator by the simplified denominator gives: 1664 / 2305

Explanation

The profit shared among partners in a partnership business is directly proportional to the product of their investment and the time period of investment.

Kapil's investment is ₹3,00,000 and he stays in the business for the entire duration of 3 years, which is 36 months. Kapil's profit equivalent = 3,00,000 × 36 = 1,08,00,000

Kunal joins after 6 months with an investment of ₹2,50,000. He stays in the business for 36 - 6 = 30 months. Kunal's profit equivalent = 2,50,000 × 30 = 75,00,000

Sitalo joins in February 2023, which is 10 months after Kapil started. Her investment is equal to the sum of Kapil's and Kunal's investments, which is ₹3,00,000 + ₹2,50,000 = ₹5,50,000. She stays in the business for 36 - 10 = 26 months. Sitalo's profit equivalent = 5,50,000 × 26 = 1,43,00,000

The ratio of their profit distribution is: Kapil : Kunal : Sitalo = 1,08,00,000 : 75,00,000 : 1,43,00,000 = 108 : 75 : 143

Explanation

Using the definition of trigonometric ratios in a right-angled triangle, if tan A = 3/4, the opposite side to angle A is 3 and the adjacent side is 4.

By Pythagoras' theorem, the hypotenuse is: √(3² + 4²) = 5

Using this, sin A is defined as the ratio of the opposite side to the hypotenuse: sin A = 3/5

Substituting sin A = 3/5 into the expression (6 sin A) / (1 - sin A): (6 × 3/5) / (1 - 3/5) = (18/5) / (2/5)

Dividing the fractions gives: 18/2 = 9

Explanation

The rate of typing for an individual is given by the formula: Rate = Pages typed / Time taken.

Raju's typing rate is 32 / 4 = 8 pages per hour. Akber's typing rate is 48 / 6 = 8 pages per hour.

When they work together on two different computers, their combined rate of typing is the sum of their individual rates: Combined Rate = 8 + 8 = 16 pages per hour.

The total time required to complete the assignment of 256 pages is: Time = Total Pages / Combined Rate = 256 / 16 = 16 hours.

Explanation

In triangle ABC, since side AB is equal to side AC, the angles opposite to these sides must also be equal. Therefore, angle ABC is equal to angle ACB.

The sum of all interior angles in a triangle is 180°. This gives the relation: angle BAC + angle ABC + angle ACB = 180°

Substituting the given value of angle BAC = 50° and setting angle ABC = angle ACB: 50° + 2 × angle ABC = 180° 2 × angle ABC = 180° - 50° 2 × angle ABC = 130° angle ABC = 65°

Explanation

Distance remains constant, so we use the inverse relationship between speed and time: S₁ × T₁ = S₂ × T₂

Substituting the given values: 40 × 7 = S₂ × 5 280 = S₂ × 5 S₂ = 280 / 5 = 56 km/h

Explanation

Let the marked price of the article be MP. The relation between the selling price (SP) and the marked price after two successive discounts of d% is given by the formula: SP = MP × (1 - d/100)²

Substituting SP = 864 and d = 25: 864 = MP × (1 - 25/100)² 864 = MP × (3/4)² 864 = MP × 9/16 MP = (864 × 16) / 9 MP = 96 × 16 = 1536

The total discount offered is the difference between the marked price and the selling price: Discount = MP - SP Discount = 1536 - 864 = 672

Explanation

Using the standard algebraic identity for the difference of squares:

(a + b)(a - b) = a² - b²

By substituting a = 5p and b = 3q into the formula:

(5p + 3q)(5p - 3q) = (5p)² - (3q)²

Simplifying the squared terms:

(5p)² = 25p² (3q)² = 9q²

Thus, the expression simplifies to:

25p² - 9q²