SSC CHSL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

To find the mean age of the children, use the formula for the weighted mean:

Mean = ∑(f_i × x_i) ÷ ∑f_i

where x_i represents the ages and f_i represents the number of children (frequency).

Calculate the product of each age and its frequency: 6 × 8 = 48 7 × 3 = 21 8 × 7 = 56 9 × 2 = 18 10 × 20 = 200

Sum of these products: ∑(f_i × x_i) = 48 + 21 + 56 + 18 + 200 = 343

Total number of children: ∑f_i = 8 + 3 + 7 + 2 + 20 = 40

Calculate the mean: Mean = 343 ÷ 40 = 8.575 years

Explanation

The average of the first n natural numbers is given by the formula:

Average = (n + 1)/2

Here, the natural numbers are from 1 to 69, so n = 69. Substituting this value into the formula:

Average = (69 + 1)/2 = 70/2 = 35

Explanation

The total surface area of a rectangular box with dimensions a, b, and c is given by the formula: TSA = 2(ab + bc + ca)

Let the edges of the box be 3x, 5x, and 8x.

Substituting these values into the surface area equation: 2((3x × 5x) + (5x × 8x) + (8x × 3x)) = 632 2(15x² + 40x² + 24x²) = 632 2 × 79x² = 632 158x² = 632 x² = 4 x = 2

The volume of a rectangular box is given by the formula: Volume = abc

Substituting the dimensions in terms of x: Volume = 3x × 5x × 8x = 120x³

Using x = 2: Volume = 120 × 2³ = 120 × 8 = 960 cm³

Explanation

The total weight of the box and the gift articles is 25 kg. Converting this weight into grams: Total weight = 25 × 1000 = 25000 grams

The ratio of the weight of the box to the weight of the gift articles is given as 1:9. The sum of the ratio terms is: Total parts = 1 + 9 = 10 parts

The weight of the gift articles represents 9 parts out of the total 10 parts. Using the ratio formula, the weight of the articles is: Weight of articles = (9/10) × 25000 = 9 × 2500 = 22500 grams

Explanation

When equal distances are covered at two different speeds, say v₁ and v₂, the average speed for the entire journey is given by the harmonic mean formula:

Average Speed = 2v₁v₂ / (v₁ + v₂)

Substituting the given speeds, v₁ = 40 km/h and v₂ = 20 km/h:

Average Speed = (2 × 40 × 20) / (40 + 20) Average Speed = 1600 / 60 Average Speed = 80 / 3 Average Speed = 26 ⅔ km/h

Explanation

Let the total number of enrolled voters be N.

The fraction of voters who cast their votes is 83%, or 0.83. The fraction of valid votes is 100% - 4% = 96%, or 0.96 of the cast votes. Party A received 83%, or 0.83, of the valid votes, which equals 156611 votes.

This relation is expressed by the formula: N × 0.83 × 0.96 × 0.83 = 156611

Simplify the product of the decimals: 0.83 × 0.96 × 0.83 = 0.661344

Now, solve for N: N = 156611 / 0.661344 N ≈ 236807.03

Taking the integer part yields 236807.

Explanation

By the law of reflection, the angle of incidence is equal to the angle of reflection. This geometric relationship forms two similar right-angled triangles between the lamp-post and the mirror, and the man and the mirror.

Using the property of similar triangles: height of lamp-post / distance of lamp-post from mirror = height of man / distance of man from mirror

Let h be the height of the lamp-post. Substituting the given values into the relation: h / 6.6 = 1.25 / 2

Solving for h: h = (1.25 × 6.6) ÷ 2 h = 8.25 ÷ 2 h = 4.125 ≈ 4.13 m

Explanation

The principal amount for the installment period is the difference between the cash price and the cash down payment: Principal (P) = £550 - £250 = £300

The installment paid after one month is £310. The interest (I) paid on the principal is: I = £310 - £300 = £10

The time period (T) is 1 month, which is 1/12 of a year.

Using the simple interest formula: I = (P × R × T) / 100

Substituting the values to find the annual rate of interest (R): 10 = (300 × R × 1) / (12 × 100) 10 = R / 4 R = 40%

Explanation

Area of the quadrilateral TEKI is the sum of the areas of two congruent right-angled triangles TEK and TIK, which are right-angled at E and I respectively.

In the right-angled triangle TEK, the radius TE is perpendicular to the tangent KE. Applying the Pythagorean theorem: TK² = TE² + KE²

Substituting the given values of TK = 17 cm and TE = 8 cm: 17² = 8² + KE² 289 = 64 + KE² KE² = 225 KE = 15 cm

The area of the right-angled triangle TEK is: Area = ½ × KE × TE = ½ × 15 × 8 = 60 sq.cm

Since triangle TIK is congruent to triangle TEK, its area is also 60 sq.cm.

The total area of quadrilateral TEKI is: Area = Area(TEK) + Area(TIK) = 60 + 60 = 120 sq.cm

Explanation

When solid metal cubes are melted to form a new cube, the total volume remains conserved. The volume of a cube with side s is given by the formula V = s³.

Calculating the volumes of the three given cubes: V₁ = 6³ = 216 cm³ V₂ = 8³ = 512 cm³ V₃ = 10³ = 1000 cm³

The total volume of the new cube is the sum of these volumes: V = 216 + 512 + 1000 = 1728 cm³

Let S be the side length of the new cube: S³ = 1728 S = 12 cm

The total surface area of a cube with side S is given by the formula A = 6S². Substituting the side length of the new cube: A = 6 × 12² = 6 × 144 = 864 cm²

Explanation

Using the periodic property of the tangent function, tan(n × 180° + θ) = tan θ for any integer n.

We can express 570° as: 570° = 3 × 180° + 30°

Applying the identity: tan 570° = tan(3 × 180° + 30°) = tan 30°

Since the standard trigonometric value of tan 30° is 1/√3, the value of tan 570° is 1/√3.

Explanation

The total surface area (TSA) of a cylinder is given by the formula: TSA = CSA + 2πr²

where CSA is the curved surface area and r is the radius of the base.

Given that the diameter of the base is 14 cm, the radius r is 14/2 = 7 cm. The curved surface area CSA is 352 cm².

Using the value of π ≈ 22/7, we calculate the area of the two circular bases: 2πr² = 2 × (22/7) × 7 × 7 = 2 × 22 × 7 = 308 cm²

Now, substitute these values into the TSA formula: TSA = 352 + 308 = 660 cm²

Explanation

In a right-angled triangle ABC, the hypotenuse AC can be calculated using the Pythagorean theorem: AC = √(AB² + BC²)

Substituting the given values AB = 15 cm and BC = 20 cm: AC = √(15² + 20²) AC = √(225 + 400) AC = √625 = 25 cm

The area of triangle ABC can be expressed in two ways using different base-height pairs: Area = ½ × AB × BC Area = ½ × AC × BD

Equating the two expressions for the area gives the relation for the altitude BD to the hypotenuse: BD = (AB × BC) / AC

Substituting the values: BD = (15 × 20) / 25 BD = 300 / 25 = 12 cm

Explanation

To find the time taken when A, B, and C work together, assume the total work is the Least Common Multiple (LCM) of the given days: 15, 30, and 45.

Total Work = LCM(15, 30, 45) = 90 units

Now, determine the daily efficiency of each pair: Efficiency of A and B = 90 ÷ 15 = 6 units/day Efficiency of B and C = 90 ÷ 30 = 3 units/day Efficiency of C and A = 90 ÷ 45 = 2 units/day

Adding these efficiencies together gives twice the combined efficiency of A, B, and C: 2 × (A + B + C) = 6 + 3 + 2 = 11 units/day Combined efficiency of A, B, and C = 11/2 units/day

The number of days they will take to complete the work together is: Total Work ÷ Combined Efficiency = 90 ÷ (11/2) = 180/11 = 16 4/11 days

Explanation

The difference between the ages of two people remains constant over time. Therefore, Monu is always 24 years older than his sister.

Let the sister's age in the year 2000 be S. Monu's age in the year 2000 is 3S.

Using the constant age difference: 3S - S = 24 2S = 24 S = 12

Thus, in the year 2000, Monu's age was: 3 × 12 = 36 years

Monu's age in 2010 is his age in 2000 plus 10 years: 36 + 10 = 46 years

Explanation

Using the successive percentage change formula: Net percentage change = x + y + xy/100

Here, x represents the markup percentage, which is +50, and y represents the discount percentage, which is -30.

Substituting these values into the formula: Net percentage change = 50 - 30 + (50 × -30)/100 Net percentage change = 20 - 1500/100 Net percentage change = 20 - 15 Net percentage change = 5%

Alternatively, assuming the cost price is ₹100, the marked price is ₹150 after a 50% markup. Offering a 30% discount on ₹150 reduces the price by ₹45, resulting in a selling price of ₹105. This represents a 5% gain.

Explanation

To find the highest average sale per company for the products, calculate the total sales for each product across all three companies and divide by 3: Product A total sale = 2750 + 2500 + 3500 = 8750 Product B total sale = 3250 + 3000 + 3150 = 9400 Product C total sale = 2250 + 1750 + 2400 = 6400

Product B has the highest total sales, yielding the highest average sale of 9400 ÷ 3 ≈ 3133.33.

To find the lowest average sale per product for the companies, calculate the total sales for each company across all three products and divide by 3: Company X total sale = 2750 + 3250 + 2250 = 8250 Company Y total sale = 2500 + 3000 + 1750 = 7250 Company Z total sale = 3500 + 3150 + 2400 = 9050

Company Y has the lowest total sales, yielding the lowest average sale of 7250 ÷ 3 ≈ 2416.67.

Explanation

Correct Answer: B

Let the speed of Arun be u m/s and the speed of Bhaskar be v m/s. The length of the race is 3 km, which is 3000 m.

In the first case, Arun gives Bhaskar a start of 400 m and beats him by 30 seconds. This means Bhaskar runs 2600 m in 30 seconds more than the time Arun takes to run 3000 m. 2600/v = 3000/u + 30

In the second case, Arun gives Bhaskar a time lead of 2.5 minutes (150 seconds) and gets beaten by 500 m. This means Bhaskar runs 3000 m in 150 seconds more than the time Arun takes to run 2500 m. 3000/v = 2500/u + 150

Let 1/u = x and 1/v = y. Simplifying the two equations: 260y - 300x = 3 6y - 5x = 0.3

From the second equation, 300x = 360y - 18. Substituting this into the first equation: 260y - (360y - 18) = 3 -100y + 18 = 3 100y = 15 y = 0.15

Using y = 0.15 in the simplified second equation: 6(0.15) - 5x = 0.3 0.9 - 5x = 0.3 5x = 0.6 x = 0.12

The time taken by Arun to complete the 3 km race is 3000x = 3000 × 0.12 = 360 seconds, which is 6 minutes. The time taken by Bhaskar to complete the 3 km race is 3000y = 3000 × 0.15 = 450 seconds, which is 7.5 minutes.

The difference between their times is 7.5 - 6 = 1.5 minutes.

Explanation

By the Midpoint Theorem of triangles, the line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half of its length. Since P and Q are the midpoints of AB and AC respectively, the relationship between PQ and BC is BC = 2 PQ.

The segment PQ is divided by point R in the ratio PR : RQ = 1 : 5. Given that PR = 6 cm, the length of RQ is calculated as 5 × PR = 5 × 6 = 30 cm.

The total length of PQ is the sum of PR and RQ: PQ = 6 + 30 = 36 cm

Using the midpoint relation, the length of BC is: BC = 2 × 36 = 72 cm

Explanation

By the rate-of-work formula, when individuals work together, their combined rate is the sum of their individual rates:

1/A + 1/B + 1/C = 1/Total

Substitute the given values where A completes the work in 20 days, B in 10 days, and the combined group in 5 days:

1/20 + 1/10 + 1/C = 1/5

Solve for 1/C by subtracting the individual rates of A and B from the combined rate:

1/C = 1/5 - 1/20 - 1/10

Using a common denominator of 20:

1/C = 4/20 - 1/20 - 2/20

1/C = 1/20

C = 20 days

Explanation

Let the measure of the smallest angle of the triangle be x.

According to the problem, the smallest angle is 42° less than the greatest angle. This means the greatest angle is x + 42°.

The remaining angle is 24° more than the smallest angle, which gives its measure as x + 24°.

Since the sum of all angles in a triangle is always 180°, we can write the equation: x + (x + 24°) + (x + 42°) = 180°

Simplifying the equation: 3x + 66° = 180° 3x = 114° x = 38°

Thus, the measure of the smallest angle of the triangle is 38°.

Explanation

Using the weighted average relation between income, expenditure, and savings:

Income change % = (Expenditure ratio × Expenditure change %) + (Savings ratio × Savings change %)

Let the original income be 100. The expenditure is 70% (ratio of 0.7) and the savings is 30% (ratio of 0.3).

Substituting the given values into the formula:

15% = (0.7 × 7.5%) + (0.3 × Savings change %)

15 = 5.25 + 0.3 × Savings change %

0.3 × Savings change % = 9.75

Savings change % = 9.75 / 0.3 = 32.5%

Explanation

Since the line segment CD is formed by joining C to M and producing it to D such that DM = CM, the point M is the midpoint of the segment CD. This gives the relation:

CD = CM + DM = 2CM

CM = CD / 2

Substituting the given value of CD = 10 cm into the equation:

CM = 10 / 2 = 5 cm

Explanation

Using the weighted average formula:

Weighted Average = (n₁ × w₁ + n₂ × w₂) / (n₁ + n₂)

Here, the number of boys is n₁ = 40 with an average weight of w₁ = 33 kg. The number of remaining students is n₂ = 72 - 40 = 32 with an average weight of w₂ = 15 kg.

To simplify the calculation, the ratio of the number of students in the two groups is 40 : 32, which reduces to 5 : 4.

Applying these ratio weights to the formula:

Average weight = (5 × 33 + 4 × 15) / (5 + 4) Average weight = (165 + 60) / 9 Average weight = 225 / 9 Average weight = 25 kg

Explanation

The area of a rhombus is calculated using the formula:

Area = ½ × d₁ × d₂

where d₁ and d₂ are the lengths of the diagonals.

Substituting the given diagonal lengths of 10 cm and 8.2 cm into the formula:

Area = ½ × 10 × 8.2 = 5 × 8.2 = 41 cm²

This yields the exact area of the rhombus.

Explanation

Using the remainder theorem relation for the sum of two numbers, if two numbers N₁ and N₂ leave remainders r₁ and r₂ when divided by a divisor d, then their sum N₁ + N₂ leaves a remainder equal to the remainder of (r₁ + r₂) when divided by d.

Here, the individual remainders are r₁ = 6 and r₂ = 7. The sum of these remainders is: 6 + 7 = 13

When the sum of the two numbers is divided by the divisor d, the remainder is 5. This implies that when the sum of the individual remainders, 13, is divided by d, it leaves a remainder of 5.

This relation can be written as: 13 = qd + 5 13 - 5 = qd 8 = qd

This means that the divisor d must be a factor of 8. The positive factors of 8 are 1, 2, 4, and 8.

Since a divisor must always be strictly greater than the remainders it produces, and the given remainders are 6 and 7, the divisor d must be greater than 7 (d > 7).

The only factor of 8 that is greater than 7 is 8. Therefore, the divisor is 8.

Explanation

Using the trigonometric co-function identity sin(90° - 2α) = cos(2α), the given expression simplifies as follows:

1/cos⁴(2α) - 1/cos²(2α)

Let θ = 2α. Substituting θ into the expression gives:

1/cos⁴θ - 1/cos²θ = sec⁴θ - sec²θ = sec²θ (sec²θ - 1)

Using the fundamental identity sec²θ - 1 = tan²θ:

= sec²θ tan²θ = sec²(2α) tan²(2α)

Explanation

The percentage increase in monthly salary is calculated using the formula:

Percentage Increase = ((This Year's Salary - Last Year's Salary) / Last Year's Salary) × 100

Substituting the given values into the formula:

Salary Increase = ₹38,640 - ₹34,500 = ₹4,140

Percentage Increase = (4,140 / 34,500) × 100

Percentage Increase = 4,140 / 345 = 12%

Explanation

The length of the transverse common tangent (TCT) is given by the formula: TCT = √(d² - (r₁ + r₂)²)

The length of the direct common tangent (DCT) is given by the formula: DCT = √(d² - (r₁ - r₂)²)

Using the given values r₁ = 25 cm, r₂ = 35 cm, and d = 80 cm:

TCT = √(80² - (25 + 35)²) = √(80² - 60²) = √(6400 - 3600) = √2800 = 20√7 cm

DCT = √(80² - (35 - 25)²) = √(80² - 10²) = √(6400 - 100) = √6300 = 30√7 cm

The ratio of TCT to DCT is: TCT / DCT = 20√7 / 30√7 = 2/3

Explanation

Convert the mixed fractions into improper fractions: 3 3/5 = 18/5 2 2/3 = 8/3 7 1/6 = 43/6 2 1/2 = 5/2

Now, solve the expression step-by-step using BODMAS rules:

First, simplify the innermost parentheses: 2 1/2 + 2/3 = 5/2 + 2/3 = (15 + 4)/6 = 19/6

Next, simplify the terms inside the parentheses: 7 1/6 - 19/6 = 43/6 - 19/6 = 24/6 = 4

Now, evaluate the expression inside the square brackets: 2 2/3 ÷ 4 = 8/3 × 1/4 = 2/3

Next, perform the division outside the brackets: 3 3/5 ÷ 2/3 = 18/5 × 3/2 = 27/5

Finally, perform the addition: 10 + 27/5 = (50 + 27)/5 = 77/5

Explanation

Using the successive percentage formula for selling price: Selling Price = Cost Price × (1 + Markup %) × (1 - Discount %)

Substituting the given values into the equation: 540 = Cost Price × (1 + 0.25) × (1 - 0.10) 540 = Cost Price × 1.25 × 0.90 540 = Cost Price × 1.125

Expressing the decimal 1.125 as the fraction 9/8: 540 = Cost Price × 9/8 Cost Price = 540 × 8 / 9 Cost Price = 60 × 8 = 480

Explanation

The curved surface area (CSA) of a cylinder is given by the formula:

CSA = 2πrh

Substituting the given values, radius r = 3 cm, height h = 14 cm, and π = 22/7:

CSA = 2 × (22/7) × 3 × 14 CSA = 2 × 22 × 3 × 2 CSA = 44 × 6 CSA = 264 cm²

Explanation

Equivalent discount percentage for two successive discounts d₁ and d₂ is given by the formula: d₁ + d₂ - (d₁ × d₂)/100

For Scheme (i), the equivalent discount is: 25 + 5 - (25 × 5)/100 = 30 - 1.25 = 28.75%

For Scheme (ii), the equivalent discount is: 20 + 10 - (20 × 10)/100 = 30 - 2 = 28%

The difference in discount percentages between the two schemes is: 28.75% - 28% = 0.75%

Since the marked price of the article is £1,000, the difference in the selling price under these two schemes is: 0.75% of £1,000 = (0.75/100) × 1000 = 7.5

Explanation

Let the fourth proportional to 2, 6, and 8 be x.

By the definition of proportion, if four numbers a, b, c, and d are in proportion, then a/b = c/d.

Substituting the given values into the formula: 2/6 = 8/x

Simplifying the fraction on the left side: 1/3 = 8/x

Cross-multiplying to solve for x: x = 8 × 3 x = 24

Explanation

If three numbers a, b, and c are in continued proportion, then a/b = b/c, where c is the third proportion. This gives the formula c = b² / a.

Substituting the given values, a = 36 and b = 48:

c = 48² / 36 c = (48 × 48) / 36 c = 2304 / 36 c = 64

Explanation

Since the two circles touch each other externally at point T, the distance between their centres M and N is MT + TN = 5 + 8 = 13 cm. The points lie on the line PR in the order P, M, T, N, R.

For the circle with centre M and radius MQ = 5 cm, PQ is a tangent at Q, which makes triangle PQM a right-angled triangle at Q. Applying the Pythagorean theorem: PM² = PQ² + MQ² PM² = 12² + 5² = 144 + 25 = 169 PM = 13 cm

For the circle with centre N and radius NS = 8 cm, RS is a tangent at S, making triangle NSR a right-angled triangle at S. Applying the Pythagorean theorem: NR² = RS² + NS² NR² = 15² + 8² = 225 + 64 = 289 NR = 17 cm

The total length of PR is the sum of the segments PM, MT, TN, and NR: PR = PM + MT + TN + NR = 13 + 5 + 8 + 17 = 43 cm

Explanation

Let the cost price of the article be 100.

Since the marked price is 25% more than the cost price, the marked price is: Marked Price = 100 + 25% of 100 = 125

A discount of 40% is given on this marked price, which makes the selling price: Selling Price = 125 × (1 - 0.40) = 125 × 0.60 = 75

The loss is the difference between the cost price and the selling price: Loss = 100 - 75 = 25

The loss percentage is calculated as: Loss Percentage = (Loss / Cost Price) × 100 = (25 / 100) × 100 = 25%

Alternatively, using the successive percentage formula where the markup is +25% and the discount is -40%: Net Percentage Change = 25 - 40 + (25 × -40) / 100 = -15 - 10 = -25% The negative result indicates a loss of 25%.

Explanation

Correct Answer: D

Let the number be x. The average of x and its reciprocal 1/x is given as 2: (x + 1/x) / 2 = 2 x + 1/x = 4

We need to find the average of its cube x³ and its reciprocal 1/x³: Average = (x³ + 1/x³) / 2

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

Substitute x + 1/x = 4 into the formula: x³ + 1/x³ = 4³ - 3(4) x³ + 1/x³ = 64 - 12 = 52

Now, calculate the required average: Average = 52 / 2 = 26

Explanation

The perimeter of a semi-circle is given by the formula P = πr + 2r = r(π + 2), where r is the radius of the semi-circle.

Given that the perimeter is 56.54 cm and π is 3.14, we can substitute these values into the formula to find the radius:

56.54 = r(3.14 + 2) 56.54 = 5.14r r = 56.54 / 5.14 r = 11 cm

The diameter of the semi-circle is twice the radius:

Diameter = 2r = 2 × 11 = 22 cm

Explanation

Convert the mixed fractions into improper fractions: 3 4/7 = 25/7 16 1/2 = 33/2 2 1/2 = 5/2

Using the BODMAS rule, simplify the innermost parentheses first: 7 - 2 1/2 = 7 - 5/2 = (14 - 5)/2 = 9/2

Next, solve the terms inside the curly braces: 6² - 9/2 = 36 - 9/2 = (72 - 9)/2 = 63/2

Now, solve the terms inside the square brackets: 16 1/2 ÷ 63/2 = 33/2 ÷ 63/2 = 33/2 × 2/63 = 33/63 = 11/21

Finally, subtract this result from the first term: 25/7 - 11/21 = (25 × 3)/21 - 11/21 = 75/21 - 11/21 = 64/21

Explanation

For a constant distance, speed is inversely proportional to time. Therefore, the product of speed and time remains constant:

s₁ × t₁ = s₂ × t₂

Here, s₁ = 36 km/hr, t₁ = 85 minutes, and the target time t₂ = 51 minutes.

Substituting these values into the formula:

36 × 85 = s₂ × 51

s₂ = (36 × 85) / 51

Dividing 85 and 51 by their greatest common divisor, 17:

s₂ = (36 × 5) / 3

s₂ = 12 × 5 = 60 km/hr

Explanation

According to the Remainder Theorem, when a polynomial P(x) is divided by ax - b, the remainder is given by P(b/a).

For the polynomial P(x) = 2x³ - 3x² + 6x - 4 and divisor 2x - 3, we set 2x - 3 = 0, which gives x = 3/2.

Substituting x = 3/2 into the polynomial:

P(3/2) = 2(3/2)³ - 3(3/2)² + 6(3/2) - 4

P(3/2) = 2(27/8) - 3(9/4) + 9 - 4

P(3/2) = 27/4 - 27/4 + 9 - 4

P(3/2) = 5

The remainder when the polynomial is divided by 2x - 3 is 5.

Explanation

The total surface area of a solid hemisphere is given by the formula 3πr², where r is the radius of the hemisphere.

Given the diameter of the hemisphere is 28 cm, its radius r is 14 cm.

Substituting the value of the radius and π = 22/7 into the formula:

Total Surface Area = 3 × (22/7) × 14 × 14 Total Surface Area = 3 × 22 × 2 × 14 Total Surface Area = 66 × 28 Total Surface Area = 1848 cm²

Explanation

To convert any percentage to a decimal, the percentage value is divided by 100.

Applying this formula to each given percentage:

For 20%, the decimal value is 20 ÷ 100 = 0.2.

For 0.5%, the decimal value is 0.5 ÷ 100 = 0.005.

For 0.03%, the decimal value is 0.03 ÷ 100 = 0.0003.

Thus, the corresponding decimals are 0.2, 0.005, and 0.0003.

Explanation

Given the trigonometric relation cot θ + tan θ = 2, we can substitute cot θ = 1/tan θ to form an equation in terms of tan θ. Let tan θ = x:

x + 1/x = 2 x² - 2x + 1 = 0 (x - 1)² = 0 x = 1

This gives tan θ = 1. Since θ is an acute angle, θ = 45°, which also gives cot θ = 1.

Substituting tan θ = 1 and cot θ = 1 into the expression to be evaluated:

tan¹²θ + cot¹²θ + 2tan⁵θ cot⁷θ = 1¹² + 1¹² + 2 × 1⁵ × 1⁷ = 1 + 1 + 2 = 4

Explanation

The remaining principal to be paid after the cash down payment is: Remaining Principal = ₹40,000 - ₹25,000 = ₹15,000

The total amount paid through monthly instalments is: Total Instalments = 4 × ₹4,000 = ₹16,000

The interest paid under the instalment scheme is: Interest = ₹16,000 - ₹15,000 = ₹1,000

Under the average debt method, the average principal over the instalment period of 4 months is calculated as: Average Principal = (Initial Principal + Final Principal) / 2 Average Principal = (15,000 + 0) / 2 = ₹7,500

The time period (T) is 4 months, which is 4/12 = 1/3 year.

Using the simple interest formula on the average principal: Interest = (Average Principal × Rate × Time) / 100 1,000 = (7,500 × R × 1/3) / 100 1,000 = 25 × R R = 40%

Explanation

To find the number of students who secured a raw score of at least 30 in Computer Science, we determine the minimum percentage score required. Since the maximum marks for Computer Science is 40, a score of 30 represents:

(30 ÷ 40) × 100% = 75%

Therefore, any student with a score of 75% or higher in Computer Science has obtained at least 30 marks.

Analyzing the percentage marks obtained by each student in Computer Science: Ayush obtained 80%, which corresponds to 32 marks. Muskan obtained 90%, which corresponds to 36 marks. Tarun obtained 80%, which corresponds to 32 marks.

All other students scored less than 75%. This gives a total of 3 students who secured a raw score of at least 30.

Explanation

The total surface area (TSA) of a cylinder is calculated using the formula:

TSA = 2πr(r + h)

where r is the radius and h is the height of the cylinder.

Given: Diameter = 14 cm, which gives radius r = 14/2 = 7 cm Height h = 9 cm

Substituting these values into the formula with π ≈ 22/7:

TSA = 2 × (22/7) × 7 × (7 + 9) TSA = 2 × 22 × 16 TSA = 44 × 16 TSA = 704 cm²

Explanation

Since ΔDEF and ΔGHI are similar triangles, the ratio of their corresponding sides is equal to the ratio of their perimeters.

DE / GH = Perimeter(ΔDEF) / Perimeter(ΔGHI)

Substituting the given values:

64 / 24 = Perimeter(ΔDEF) / 72

Simplifying the ratio:

8 / 3 = Perimeter(ΔDEF) / 72

Perimeter(ΔDEF) = (8 / 3) × 72 = 192 cm

The perimeter of ΔDEF is the sum of its three sides:

DE + EF + FD = 192 cm

Substituting the value of DE = 64 cm:

64 + EF + FD = 192

EF + FD = 192 - 64 = 128 cm

Explanation

The total remaining work after 2 days can be calculated in terms of man-days.

Initially, 8 workers were to complete the work in 12 days. After working for 2 days, the remaining work is equivalent to 8 workers working for 10 days: Remaining work = 8 × 10 = 80 man-days

This remaining work must be completed in the next 2 days. Let N be the total number of workers required: N × 2 = 80 N = 40 workers

Since 8 workers are already engaged, the number of additional workers required is: Additional workers = 40 - 8 = 32