SSC CHSL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

Let the cost price of the goods be CP = 100.

The shopkeeper marks the goods at 80% above the cost price, so the marked price is: MP = 100 × (1 + 80/100) = 180

A discount of 48% is allowed on this marked price, which gives the selling price: SP = 180 × (1 - 48/100) SP = 180 × 52/100 = 93.6

The loss percentage is calculated as: Loss% = (CP - SP)/CP × 100 Loss% = (100 - 93.6)/100 × 100 = 6.4%

Converting the decimal to a mixed fraction: 6.4% = 6 + 4/10% = 6 2/5 %

Explanation

According to the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be strictly greater than the length of the remaining side. For a triangle with sides of length a, b, and S, this relationship can be expressed as:

|a - b| < S < a + b

Substituting the given side lengths 9 and 24 into the inequality:

|24 - 9| < S < 24 + 9

15 < S < 33

This inequality defines the mathematically permissible range of lengths for the third side S of the triangle.

Explanation

Correct Answer: D

Given the equation: 2x + 2/x = 5

Dividing the entire equation by 2 gives: x + 1/x = 5/2

Using the algebraic identity for the sum of cubes: x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x)

Substituting the value of x + 1/x = 5/2 into this identity: x³ + 1/x³ = (5/2)³ - 3(5/2) x³ + 1/x³ = 125/8 - 15/2 x³ + 1/x³ = 125/8 - 60/8 x³ + 1/x³ = 65/8

Now, to find the value of x³ + 1/x³ + 2, add 2 to the obtained value: x³ + 1/x³ + 2 = 65/8 + 2 x³ + 1/x³ + 2 = (65 + 16)/8 x³ + 1/x³ + 2 = 81/8

Explanation

The new monthly savings can be calculated using the relation:

Savings = Income - Expenditure

Original monthly income of Bhushan = ₹45,000 Since his income increases by 22%, the new monthly income is: New Income = 45,000 × (1 + 22/100) = 45,000 × 1.22 = ₹54,900

Original monthly expenditure of Bhushan = ₹33,000 Since his expenditure increases by 11%, the new monthly expenditure is: New Expenditure = 33,000 × (1 + 11/100) = 33,000 × 1.11 = ₹36,630

Subtracting the new expenditure from the new income gives the new monthly savings: New Savings = 54,900 - 36,630 = ₹18,270

Explanation

Using the algebraic identity (a + b)² = a² + 2ab + b², we can square both sides of the given equation:

(x + 1/x)² = 7²

Expanding the left side:

x² + 2(x)(1/x) + 1/x² = 49

x² + 2 + 1/x² = 49

Subtracting 2 from both sides yields:

x² + 1/x² = 49 - 2

x² + 1/x² = 47

Explanation

In concentric circles, the radius of the smaller circle drawn to the point of contact is perpendicular to the tangent chord of the larger circle and bisects it. This forms a right-angled triangle where the hypotenuse is the radius of the larger circle (R = 13 cm), one side is the radius of the smaller circle (r = 12 cm), and the other side is half of the length of the chord (L/2).

Using Pythagoras' theorem: (L/2)² = R² - r² (L/2)² = 13² - 12² (L/2)² = 169 - 144 = 25 L/2 = √25 = 5 cm

The total length of the chord L is: L = 2 × 5 = 10 cm

Explanation

From the given equation: sin A + sin² A = 1 sin A = 1 - sin² A

Using the fundamental trigonometric identity sin² A + cos² A = 1, we can write: sin A = cos² A

Squaring both sides of this equation yields: sin² A = cos⁴ A

To find the value of the expression cos² A + cos⁴ A, we substitute these relations into the expression: cos² A + cos⁴ A = sin A + sin² A

Since it is given that sin A + sin² A = 1, we have: cos² A + cos⁴ A = 1

Explanation

The sum of the current ages of all 6 family members is calculated by multiplying the number of members by their average age: 6 × 25 = 150 years.

To find the sum of their ages 4 years ago, which is just before the birth of the youngest member, we subtract 4 years for each of the 6 members: 150 - (6 × 4) = 150 - 24 = 126 years.

Just before the birth of the youngest member, the family consisted of only 5 members. Therefore, the average age of the family at that time is the sum of their ages 4 years ago divided by 5: 126 / 5 = 25.2 years.

Explanation

Using the fundamental trigonometric identity: sin² θ + cos² θ = 1

For cos θ = -9/17, which satisfies the condition cos θ < 0: sin² θ = 1 - (-9/17)² sin² θ = 1 - 81/289 sin² θ = (289 - 81) / 289 sin² θ = 208/289

Explanation

From the given table, the number of students studying Commerce in College A is 520, in College B is 550, in College C is 515, and in College D is 690. Comparing these values, 515 is the smallest number, which corresponds to College C.

Explanation

The volume V of a right circular cone is given by the formula: V = 1/3 × π × r² × h

Using the base diameter of 42 cm, the radius r of the base is: r = 42 / 2 = 21 cm

The height h of the cone is 24 cm.

Substituting these values into the volume formula: V = 1/3 × 22/7 × 21 × 21 × 24 V = 22 × 21 × 24 V = 11088 cm³

Explanation

For a dishonest dealer, the gain percentage is calculated using the formula:

Gain % = (True Weight - Faulty Weight) / Faulty Weight × 100%

Given: True Weight = 1000 gm Faulty Weight = 920 gm

Substituting these values into the formula:

Gain % = (1000 - 920) / 920 × 100% Gain % = 80 / 920 × 100% Gain % = 200 / 23 % Gain % ≈ 8.70%

Explanation

Let the total work be the least common multiple of 15 and 25, which is 75 units.

Efficiency of A = 75 ÷ 15 = 5 units per day Efficiency of B = 75 ÷ 25 = 3 units per day

Combined efficiency of A and B = 5 + 3 = 8 units per day Work completed by A and B in 5 days = 8 × 5 = 40 units Remaining work = 75 - 40 = 35 units

This remaining work is completed by A and C in 4 days. Combined efficiency of A and C = 35 / 4 = 8.75 units per day Efficiency of C = 8.75 - 5 = 3.75 units per day (or 15/4 units per day)

Time taken by C alone to complete the total work: Time = 75 ÷ (15/4) = (75 × 4) ÷ 15 = 20 days

Explanation

Correct Answer: D

To find the time when both runners meet at the starting point for the first time, we calculate the Least Common Multiple (LCM) of the individual times they take to complete one full round of the circular track.

First, calculate the time taken by Zeenat to complete one round: T₁ = Track length / Speed of Zeenat = 1.2 km / 18 km/h = 1/15 hours Converting to minutes: T₁ = (1/15) × 60 = 4 minutes

Next, calculate the time taken by Nasreen to complete one round: T₂ = Track length / Speed of Nasreen = 1.2 km / 27 km/h = 2/45 hours Converting to minutes: T₂ = (2/45) × 60 = 8/3 minutes

The time when they meet at the starting point is the LCM of T₁ and T₂: LCM(4, 8/3) = LCM(4/1, 8/3)

Using the formula for LCM of fractions, LCM(a/b, c/d) = LCM(a, c) / HCF(b, d): LCM(4, 8) / HCF(1, 3) = 8 / 1 = 8 minutes

Explanation

The divisibility rule for 11 states that a number is divisible by 11 if the difference between the sum of its digits at odd positions and the sum of its digits at even positions is either 0 or a multiple of 11.

For the number 752563: Sum of digits at odd positions (first, third, and fifth digits) = 7 + 2 + 6 = 15 Sum of digits at even positions (second, fourth, and sixth digits) = 5 + 5 + 3 = 13 The difference between these sums is 15 - 13 = 2

Since this difference is not 0 or a multiple of 11, the number is not divisible by 11.

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) when the profit percentage (P%) and discount percentage (D%) are known is given by the formula:

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

Substituting the given values of P = 20 and D = 10:

MP/CP = (100 + 20) / (100 - 10)

MP/CP = 120 / 90

Simplifying this fraction by dividing both the numerator and the denominator by 30 gives:

MP/CP = 4/3

The ratio of the marked price to the cost price is 4:3.

Explanation

Using the algebraic identity (A - B)² = (A + B)² - 4AB, we can let A = 5x and B = 3y:

(5x - 3y)² = (5x + 3y)² - 4(5x)(3y) (5x - 3y)² = (5x + 3y)² - 60xy

From the given equations: 5x + 3y = 15 2xy = 6, which gives xy = 3

Substituting these values into the identity: (5x - 3y)² = 15² - 60(3) (5x - 3y)² = 225 - 180 (5x - 3y)² = 45

Taking the square root on both sides: 5x - 3y = √45 = √(9 × 5) = 3√5

Explanation

Correct Answer: B

To simplify the given expression, we factor out cos θ from both terms:

cos θ / (sec θ - 1) + cos θ / (sec θ + 1) = cos θ [ 1 / (sec θ - 1) + 1 / (sec θ + 1) ]

By taking the common denominator in the bracket, we get:

= cos θ [ (sec θ + 1 + sec θ - 1) / ((sec θ - 1)(sec θ + 1)) ]

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

= cos θ [ 2 sec θ / (sec² θ - 1) ]

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

= cos θ [ 2 sec θ / tan² θ ]

Since cos θ × sec θ = 1:

= 2 / tan² θ = 2 cot² θ

Explanation

Correct Answer: D

To find the principal amount, we use the formula for compound interest when the time period is in fractions of a year. The time period is 1 year and 9 months, which is 1 and 3/4 years.

The formula for the total amount A after 1 year and a fraction f of a year at rate R% compounded annually is: A = P × (1 + R/100) × (1 + f × R / 100)

Substituting the given values, where R = 10% and f = 3/4: A = P × (1 + 10/100) × (1 + 3/4 × 10 / 100) A = P × 1.1 × (1 + 7.5/100) A = P × 1.1 × 1.075 A = 1.1825P

The compound interest is the difference between the total amount and the principal: CI = A - P CI = 1.1825P - P = 0.1825P

Given that the compound interest is ₹1,460, we set up the equation: 0.1825P = 1460 P = 1460 / 0.1825 P = 8000

Explanation

Since the distance traveled upstream and downstream is the same, the ratio of the speed of the boat in still water (u) to the speed of the stream (v) is related to the upstream time (tᵤp) and downstream time (t_down) by the formula:

u/v = (tᵤp + t_down)/(tᵤp - t_down)

Substituting the given time values:

u/v = (11 + 5)/(11 - 5) = 16/6 = 8/3

v = 3/8 × u

The speed of the boat in still water is u = 2 2/9 m/s, which can be written as 20/9 m/s. Converting this speed into km/h:

u = 20/9 × 18/5 = 8 km/h

Substituting u = 8 km/h into the ratio formula:

v = 3/8 × 8 = 3 km/h

Explanation

Since A and B are positive acute angles, we can determine their values using the inverse trigonometric values for the given equations.

From the first equation: sin(A - B) = ½ A - B = 30°

From the second equation: sin(A + B) = 1 A + B = 90°

To find the values of A and B, we solve this system of linear equations. Adding the two equations gives: 2A = 120° A = 60°

Subtracting the first equation from the second gives: 2B = 60° B = 30°

Thus, A is 60° and B is 30°.

Explanation

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

Given a = 15 and b = 120, we substitute these values into the formula: c = 120² / 15 c = (120 × 120) / 15 c = 8 × 120 c = 960

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula:

Difference = P × (R / 100)²

where P is the principal sum and R is the annual rate of interest.

Substituting the given values into the formula:

200 = P × (10 / 100)²

200 = P × (1 / 10)²

200 = P × (1 / 100)

P = 200 × 100

P = 20,000

The sum invested is 20,000.

Explanation

Let the total number of items purchased be 100 and the cost price of each item be ₹1. This makes the total cost price ₹100.

Since 15% of the items went bad, the number of items sold is 85.

To earn an overall profit of 19%, the total selling price must be ₹119.

Therefore, the selling price of each of the 85 items is: 119 ÷ 85 = 1.4

This means each of the remaining items was sold for ₹1.4, which is 40% above its cost price of ₹1.

Explanation

For a number to be divisible by 12, it must be divisible by both 3 and 4.

A number is divisible by 4 if its last two digits form a number divisible by 4. Here, the last two digits of 712x816 are 16, which is divisible by 4. This condition is satisfied for any value of x.

A number is divisible by 3 if the sum of its digits is divisible by 3.

Sum of the digits of 712x816 = 7 + 1 + 2 + x + 8 + 1 + 6 = 25 + x

For 25 + x to be divisible by 3, the value of 25 + x must be a multiple of 3. The smallest integer multiple of 3 greater than or equal to 25 is 27.

By setting 25 + x = 27, we find that the least value of x is 2.

Explanation

The total number of students who went abroad from West Bengal over the years 1994 to 1998 is calculated by summing the annual figures:

500 + 550 + 450 + 600 + 525 = 2625

This sum of 2625 is the minimum overall total among all the states and union territories listed in the table, making West Bengal the state with the least number of students going abroad over the given period.

Explanation

The volume of a hemisphere is given by the formula V = 2/3 π r³, where r is the radius of the hemisphere.

Given: Radius (r) = 2.1 m π = 22/7

Substituting these values into the volume formula: V = 2/3 × 22/7 × (2.1)³ V = 2/3 × 22/7 × 2.1 × 2.1 × 2.1 V = 2 × 22 × 0.1 × 2.1 × 2.1 V = 44 × 0.441 V = 19.404 m³

Since 1 cubic meter (m³) is equal to 1000 litres, the capacity of the hemispherical tank in litres is: 19.404 × 1000 = 19404 litres

Explanation

Correct Answer: D

The perimeter of a sector of a circle is given by the formula: Perimeter = 2r + (θ / 360°) × 2πr

Here, the radius r is 4.9 cm, the central angle θ is 144°, and π is 22/7.

Substitute these values into the formula: Perimeter = 2 × 4.9 + (144 / 360) × 2 × (22 / 7) × 4.9

Simplify the term (144 / 360) to 2/5 or 0.4: Perimeter = 9.8 + 0.4 × 2 × 22 × 0.7 Perimeter = 9.8 + 12.32 Perimeter = 22.12 cm

Explanation

Let a = 612 and b = 388.

The given expression is of the form: ((a + b)² + (a - b)²) / (a² + b²)

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

Substitute this identity into the numerator of the expression: 2(a² + b²) / (a² + b²) = 2

The term a² + b² in the numerator and denominator cancels out, simplifying the entire expression to 2.

Explanation

Using the algebraic identity for the difference of squares, we have: a² - b² = (a + b)(a - b)

Given: a = cotA + cosA b = cotA - cosA

Adding the two equations: a + b = 2 cotA

Subtracting the second equation from the first: a - b = 2 cosA

Substituting these into the difference of squares: a² - b² = (2 cotA)(2 cosA) = 4 cotA cosA

Now, we calculate the product ab: ab = (cotA + cosA)(cotA - cosA) ab = cot²A - cos²A

Expressing cotA in terms of cosA and sinA: ab = cos²A/sin²A - cos²A ab = cos²A (1/sin²A - 1) ab = cos²A ((1 - sin²A)/sin²A) ab = cos²A (cos²A/sin²A) ab = cos²A cot²A

Taking the square root of both sides: √(ab) = cosA cotA

Multiplying by 4: 4√(ab) = 4 cotA cosA

Substituting the values of a² - b² and 4√(ab) into the expression: a² - b² - 4√(ab) = 4 cotA cosA - 4 cotA cosA = 0

Explanation

Applying the algebraic identity for the difference of squares, x² - y² = (x - y)(x + y), to the numerator:

Let x = a² + b² - c² and y = a² - b² + c².

First, calculate the sum of the two terms: x + y = (a² + b² - c²) + (a² - b² + c²) = 2a²

Next, calculate the difference of the two terms: x - y = (a² + b² - c²) - (a² - b² + c²) = 2b² - 2c² = 2(b² - c²)

Substitute these results back into the numerator: (x + y)(x - y) = 2a² × 2(b² - c²) = 4a²(b² - c²)

Dividing the simplified numerator by the denominator: 4a²(b² - c²) / (b² - c²) = 4a²

Explanation

Let the initial price of the item be 100. When successive discounts of d₁%, d₂%, and d₃% are applied, the remaining price percentage is calculated using the multiplier formula:

Remaining Price = 100 × (1 - d₁/100) × (1 - d₂/100) × (1 - d₃/100)

Substituting the given successive discounts of 10%, 20%, and 30%:

Remaining Price = 100 × (1 - 0.10) × (1 - 0.20) × (1 - 0.30) Remaining Price = 100 × 0.90 × 0.80 × 0.70 Remaining Price = 100 × 0.504 = 50.4

The single equivalent discount represents the total reduction from the original price:

Equivalent Discount = 100 - 50.4 = 49.6%

Explanation

Using the sum formula for averages, Sum = Average × Number of observations:

Total sum of ages of all 30 students = 30 × 12 = 360 years

Sum of ages of the first group of 5 students = 5 × 10 = 50 years

Sum of ages of the second group of 5 students = 5 × 14 = 70 years

Total sum of ages of these 10 students = 50 + 70 = 120 years

Remaining number of students = 30 - 10 = 20 students

Sum of ages of the remaining 20 students = 360 - 120 = 240 years

Average age of the remaining 20 students = 240 / 20 = 12 years

Explanation

The final remaining amount after successive percentage expenditures can be calculated using the successive reduction formula:

Final Remaining Amount = Monthly Income × (1 - r₁/100) × (1 - r₂/100) × (1 - r₃/100)

Here, r₁ = 30%, r₂ = 15%, and r₃ = 35%. The final remaining amount is 1,62,435. Substituting these values into the formula:

1,62,435 = Monthly Income × (1 - 30/100) × (1 - 15/100) × (1 - 35/100)

1,62,435 = Monthly Income × (70/100) × (85/100) × (65/100)

To find the Monthly Income, rearrange the terms:

Monthly Income = (1,62,435 × 100 × 100 × 100) ÷ (70 × 85 × 65)

Monthly Income = (1,62,435 × 1,00,000) ÷ (7 × 85 × 65)

By simplifying the division step-by-step with the factors in the denominator (7, 17, and 13):

1,62,435 ÷ 7 = 23,205 23,205 ÷ 17 = 1,365 (since 85 = 5 × 17) 1,365 ÷ 13 = 105 (since 65 = 5 × 13)

Now substitute these simplified values back:

Monthly Income = (105 × 1,00,000) ÷ (5 × 5)

Monthly Income = (105 × 1,00,000) ÷ 25

Monthly Income = 105 × 4,000 = 4,20,000

Explanation

The formula for the length of an arc of a circle is: L = (θ / 360°) × 2πr

Given that the radius r = 21 cm, the central angle θ = 22.5°, and π = 22/7:

L = (22.5 / 360) × 2 × (22 / 7) × 21

First, simplify the angle ratio: 22.5 / 360 = 45 / 720 = 1/16

Now substitute this back into the equation: L = (1/16) × 2 × 22 × 3 L = 132 / 16 L = 33 / 4 L = 8.25 cm

Explanation

The perimeter of a square is given by the formula 4 × side. With a perimeter of 172 cm, the length of one side of the square is: side = 172 / 4 = 43 cm

The area of a square is calculated as side²: Area of the square = 43² = 1849 cm²

Since the area of the triangle is half the area of the square: Area of the triangle = 1849 / 2 = 924.5 cm²

Explanation

To find the annual rate of interest, first calculate the simple interest earned by subtracting the principal from the total amount: Simple Interest = Amount - Principal Simple Interest = 67,790 - 25,400 = 42,390

Using the simple interest formula: Simple Interest = (Principal × Rate × Time) / 100

Substitute the known values into the formula: 42,390 = (25,400 × Rate × 9) / 100 42,390 = 254 × Rate × 9 42,390 = 2,286 × Rate

Now, solve for the rate: Rate = 42,390 / 2,286 Rate ≈ 18.5433%

Rounding off to two decimal places, the annual rate of interest is 18.54%.

Explanation

The average number of students per college in the Commerce stream is calculated by taking the sum of Commerce students in all colleges (10100) and dividing by 4, which gives 2525. The average number of students per stream in College 2 is calculated by summing the students in all streams of College 2 (5900) and dividing by 3, which gives 1966.67. The difference between these two averages is 2525 - 1966.67 = 558.33.

Explanation

By substituting a = 2x and b = 3y, we are given a + b = 4 and a² + b² = 64. Using the algebraic identity a² + b² = (a+b)² - 2ab, we find that ab = -24. Substituting these values into the formula for a³ + b³ = (a+b)(a² + b² - ab) gives 4 * (64 - (-24)) = 352.

Explanation

The number of low-quality products sold by each company is given in the third row of the table.

Company A sold 488 low-quality products. Company B sold 550 low-quality products. Company C sold 580 low-quality products. Company D sold 220 low-quality products.

Comparing these values, 580 is the highest number, which corresponds to Company C.

Explanation

Let the two numbers be a and b.

The mean proportional of two numbers a and b is given by the formula √(ab). Here, √(ab) = 25, which simplifies to ab = 625.

The third proportional of two numbers a and b is given by the formula b²/a. Here, b²/a = 25, which simplifies to b² = 25a, or a = b²/25.

Substituting the value of a into the mean proportional equation: (b²/25) × b = 625 b³ = 15625 b = 25

Substituting b = 25 back into the equation for a: a = 25²/25 = 25

The two numbers are 25 and 25.

Explanation

The total simple interest paid is the sum of the interest accrued over three different time intervals. Let the borrowed principal be P.

The formula for total simple interest is: SI = P × (R₁T₁ + R₂T₂ + R₃T₃) / 100

The total tenure of the loan is 7 years. The breakdown of the time periods and interest rates is: For the first 3 years, the rate is 5% per annum. For the next 2 years (years 4 and 5), the rate is 8% per annum. For the remaining period beyond 5 years (7 - 5 = 2 years), the rate is 10% per annum.

Substituting these values into the formula: 12750 = P × (5 × 3 + 8 × 2 + 10 × 2) / 100 12750 = P × (15 + 16 + 20) / 100 12750 = P × 51 / 100 P = 12750 × 100 / 51 P = 250 × 100 P = 25000

Thus, the borrowed principal is ₹25,000.

Explanation

Correct Answer: D

Let M and W represent the daily work rates of a man and a woman, respectively.

Based on the given information, the total work can be represented in two ways: Total Work = 10 × (4M + 8W) = 40M + 80W Total Work = 4 × (5M + 24W) = 20M + 96W

Equating the two expressions for the same total work: 40M + 80W = 20M + 96W 20M = 16W M = 4/5 W

Now, substitute M = 4/5 W into the first expression to find the total work in terms of W: Total Work = 10 × (4 × 4/5 W + 8W) = 10 × (16/5 W + 40/5 W) = 10 × 56/5 W = 112W

Next, calculate the combined daily work rate of 1 man and 1 woman: 1M + 1W = 4/5 W + 1W = 9/5 W

The number of days required for 1 man and 1 woman to complete the job is the total work divided by their combined daily rate: Days = 112W ÷ (9/5 W) = 112 × 5/9 = 560/9 = 62.22 days

Rounding to the nearest whole number gives 62 days.

Explanation

Using the fundamental trigonometric identities, the given expression can be simplified by substituting tan θ = sin θ / cos θ and sec θ = 1 / cos θ:

1 - tan²θ / sec²θ = 1 - (sin²θ / cos²θ) / (1 / cos²θ)

Simplifying the fraction:

1 - sin²θ = cos²θ

Now, simplifying the expression in the correct option:

sin²θ cot²θ = sin²θ × (cos²θ / sin²θ) = cos²θ

Both expressions simplify to cos²θ, making them equal.

Explanation

The given equation is: α sin 45° = β cos 30°

Substitute the standard trigonometric values: sin 45° = 1/√2 cos 30° = √3/2

Substitute these values into the equation: α × (1/√2) = β × (√3/2)

Rearrange the terms to find the ratio α/β: α/β = (√3/2) / (1/√2) α/β = (√3/2) × √2 α/β = √3 / √2

Now, raise both sides of the equation to the power of 4: α⁴/β⁴ = (√3 / √2)⁴ α⁴/β⁴ = ( (√3 / √2)² )² α⁴/β⁴ = (3/2)² α⁴/β⁴ = 9/4

Explanation

The selling price after successive discounts can be calculated using the formula:

Selling Price = Marked Price × (1 - d₁/100) × (1 - d₂/100)

Where: Marked Price = 800 First discount d₁ = 10% Second discount d₂ = 5%

Substituting the values into the formula:

Selling Price = 800 × (1 - 10/100) × (1 - 5/100) Selling Price = 800 × 0.90 × 0.95 Selling Price = 720 × 0.95 Selling Price = 684

Explanation

When two trains of equal length L move in the same direction, their relative speed is the difference between their individual speeds: Relative speed = 46 - 36 = 10 km/h

Converting the relative speed to meters per second: Speed = 10 × 5/18 = 25/9 m/s

The total distance covered during the overtake is the sum of the lengths of both trains: Distance = 2L

Using the formula Distance = Speed × Time: 2L = (25/9) × 36 2L = 25 × 4 2L = 100 L = 50

Explanation

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

Substitute the given values a + b + c = 15 and ab + bc + ca = 22 into the equation: 15² = a² + b² + c² + 2(22) 225 = a² + b² + c² + 44

Rearranging the terms to solve for a² + b² + c²: a² + b² + c² = 225 - 44 a² + b² + c² = 181

Explanation

In ΔABC and ΔPQR, we are given: ∠B = ∠Q ∠C = ∠R

By the Angle-Angle (AA) similarity criterion, since two angles of ΔABC are respectively equal to two angles of ΔPQR, the two triangles are similar (ΔABC ~ ΔPQR).

For two triangles to be congruent, their corresponding sides must be equal in length. Here, we are given AB = 2PQ, which means the ratio of their corresponding sides is AB/PQ = 2. Since this ratio is not equal to 1, the corresponding sides are not equal in length, and therefore the triangles are not congruent.

Explanation

The dome of the residential tower is a hemisphere. The area to be painted is its curved surface area.

Curved surface area of a hemisphere = 2πr²

The radius of the hemisphere r is given as 560 cm, which is equal to 5.6 m.

Curved surface area = 2 × (22/7) × 5.6 × 5.6 m² Curved surface area = 2 × 22 × 0.8 × 5.6 Curved surface area = 197.12 m²

The rate of painting is ₹25 per m².

Total cost of painting = Area × Rate Total cost = 197.12 × 25 Total cost = 4928

Thus, the total cost of painting the dome is ₹4,928.