SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

To find the value of tan 22.5° using the formula tan(x/2) = (1 - cos x) / sin x, we can substitute x = 45° so that x/2 = 22.5°.

The trigonometric values for 45° are: cos 45° = 1/√2 sin 45° = 1/√2

Substituting these values into the given formula: tan 22.5° = (1 - 1/√2) / (1/√2)

To simplify, multiply both the numerator and the denominator by √2: tan 22.5° = (√2 - 1) / 1 tan 22.5° = √2 - 1

Explanation

Following the BODMAS rule, the operations must be performed in the order of Division, Multiplication, Addition, and then Subtraction.

The given equation is: 42 ÷ 2 + a × 3 - 22 = 8

First, perform the division operation: 42 ÷ 2 = 21

Substitute this value back into the equation: 21 + a × 3 - 22 = 8

Next, perform the multiplication: a × 3 = 3a

Substitute this term back: 21 + 3a - 22 = 8

Combine the constant terms: 3a - 1 = 8

Solve for a by adding 1 to both sides and dividing by 3: 3a = 8 + 1 3a = 9 a = 9 ÷ 3 a = 3

Explanation

The weighted average formula is used to find the average marks of all the children:

Average = (n₁a₁ + n₂a₂ + n₃a₃ + n₄a₄) / (n₁ + n₂ + n₃ + n₄)

Here, the number of children in each group and their respective average marks are: Group 1: n₁ = 7, a₁ = 20 Group 2: n₂ = 3, a₂ = 45 Group 3: n₃ = 4, a₃ = 15 Group 4: n₄ = 2, a₄ = 0

Substitute these values to find the total marks: Total marks = (7 × 20) + (3 × 45) + (4 × 15) + (2 × 0) = 140 + 135 + 60 + 0 = 335

Total number of children = 7 + 3 + 4 + 2 = 16

Average marks of all 16 children = 335 / 16 = 20.9375

Rounding to one decimal place gives 20.9.

Explanation

The final price for a customer paying in cash is calculated using successive discounts on the marked price. The formula for the final selling price after successive discounts is:

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

Here, the marked price is ₹54,000, the first discount d₁ is 10%, and the second cash discount d₂ is 5%. Substituting these values:

Final Price = 54,000 × (1 - 10/100) × (1 - 5/100) Final Price = 54,000 × 0.9 × 0.95 Final Price = 54,000 × 0.855 Final Price = ₹46,170

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 = lb

Substituting the given values of length (l) = 7 m, breadth (b) = 4 m, and height (h) = 9 m:

Area of the four walls = 2 × 9 × (7 + 4) = 18 × 11 = 198 m² Area of the ceiling = 7 × 4 = 28 m²

Total area to be whitewashed = 198 + 28 = 226 m²

The rate of whitewashing is 7.50 per m².

Total cost of whitewashing = Total area × Rate Total cost of whitewashing = 226 × 7.50 = 1695

Explanation

The selling price of an article after allowing successive discounts can be calculated directly using the formula:

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

Here, the Marked Price is £80, and the two successive discounts, d₁ and d₂, are both 5%.

Selling Price = 80 × (1 - 5/100) × (1 - 5/100)

Selling Price = 80 × 0.95 × 0.95

Selling Price = 80 × 0.9025

Selling Price = 72.20

Explanation

A number is divisible by 15 if it is divisible by both 3 and 5.

For a number to be divisible by 5, its last digit must be either 0 or 5. The number 2995 ends with 5, so it is divisible by 5.

For a number to be divisible by 3, the sum of its digits must be divisible by 3. The sum of the digits of 2995 is 2 + 9 + 9 + 5 = 25. Since 25 is not divisible by 3, 2995 is not divisible by 3.

Consequently, 2995 is not divisible by 15.

Explanation

Given that sin θ = √3/2 and θ is an acute angle, the value of θ is 60°.

Substituting θ = 60° into the expression cos 3θ gives: cos 3θ = cos(3 × 60°) = cos 180°

Since cos 180° = -1, the value of cos 3θ is -1.

Explanation

The expression on the left-hand side can be simplified using the laws of exponents:

(5 × 5 × 5 × 5 × 5)⁵ × (5 × 5 × 5)⁵ ÷ 5 = (5⁵)⁵ × (5³)⁵ ÷ 5¹ = 5²⁵ × 5¹⁵ ÷ 5¹ = 5²⁵⁺¹⁵⁻¹ = 5³⁹

To express 5³⁹ as a power of 125, we write 125 as 5³:

5³⁹ = (5³)¹³ = (125)¹³

Comparing this to the right-hand side, the required exponent value is 13.

Explanation

The volume of a cuboid is calculated using the formula:

Volume = length × breadth × height

Given the dimensions of the cuboid: Length = 7.5 m Breadth = 22 m Height = 13 m

Substituting these values into the formula:

Volume = 7.5 m × 22 m × 13 m Volume = 165 × 13 Volume = 2145 m³

Explanation

The average of first n multiples of a number k is given by the formula k × (n + 1) / 2. Here, the number is k = 6 and the number of terms is n = 10. Substituting these values into the formula gives 6 × (10 + 1) / 2 = 3 × 11 = 33. Alternatively, because the multiples form an arithmetic progression, the average is simply the mean of the first term 6 and the last term 60, which is (6 + 60) / 2 = 33.

Explanation

Using the formula for percentage reduction: New Price = Original Price × (1 - r/100)

Here, the original price of the bike is ₹75,500 and the reduction rate (r) is 17%.

New Price = 75,500 × (1 - 0.17) New Price = 75,500 × 0.83 New Price = 62,665

Explanation

The compound interest for the first year is ₹350. The compound interest for the second year is ₹420.

In compound interest, the interest for any year is the interest on the preceding year's amount. Therefore, the difference between the compound interest of the second year and the first year represents the interest earned on the first year's interest.

Difference in interest = ₹420 - ₹350 = ₹70

This difference of ₹70 is the interest earned on the first year's interest of ₹350 over a period of one year.

Rate of interest (R) = (Difference in interest / Interest of first year) × 100%

R = (70 / 350) × 100%

R = (1 / 5) × 100% = 20%

Explanation

The area of a quadrilateral with diagonal d and perpendicular offsets h₁ and h₂ from the opposite vertices is given by the formula:

Area = ½ × d × (h₁ + h₂)

Substituting the given values of Area = 336, h₁ = 16, and h₂ = 12:

336 = ½ × d × (16 + 12)

336 = ½ × d × 28

336 = 14 × d

d = 336 / 14

d = 24

Therefore, the length of the diagonal is 24 units.

Explanation

The numerator of the given expression matches the standard algebraic expansion of (a - b)³: a³ - 3a²b + 3ab² - b³ = (a - b)³

By substituting a = 62 and b = 22, the numerator simplifies to: (62 - 22)³ = 40³

The denominator is: 8 × 8 × 8 = 8³

Dividing the simplified numerator by the denominator: 40³ / 8³ = (40 / 8)³ = 5³ = 125

Explanation

The area of a parallelogram is given by the formula: Area = base × height

Let the height of the parallelogram be h cm. Since the base is twice as long as its corresponding height, the base is 2h cm.

Substituting these values into the formula: 2h × h = 144 2h² = 144 h² = 72 h = √72 h = 6√2 cm

Explanation

The mean proportional between two numbers a and b is given by the formula √(ab).

Given the two numbers: a = 64 b = 4096

Substituting these values into the formula: Mean proportional = √(64 × 4096)

Using the property of square roots, this can be simplified as: Mean proportional = √64 × √4096

Since √64 = 8 and √4096 = 64, we compute: Mean proportional = 8 × 64 = 512

Explanation

Let the total work be 120 units, which is the least common multiple of 40 and 60.

The combined daily work rate of A and B is 120 / 40 = 3 units/day. The daily work rate of A alone is 120 / 60 = 2 units/day.

The daily work rate of B alone is the difference between their combined rate and A's rate: 3 - 2 = 1 unit/day.

Three-fourth of the total work is ¾ × 120 = 90 units.

The time taken by B alone to complete this three-fourth of the work is 90 units ÷ 1 unit/day = 90 days.

Explanation

The difference between the compound interest for 3 years and 2 years is equal to the interest earned in the 3rd year.

First, the accumulated amount at the end of 2 years (A₂) is given by the formula: A₂ = P(1 + R/100)²

Substituting the given values: A₂ = 5000(1 + 10/100)² A₂ = 5000(1.1)² A₂ = 5000 × 1.21 = 6050

The interest for the 3rd year is the interest earned on this 2nd-year amount: Interest in 3rd year = A₂ × R/100 Interest in 3rd year = 6050 × 10/100 = 605

Explanation

Given the equation: cot θ = cot 30° cot 60°

Using the standard trigonometric values: cot 30° = √3 cot 60° = 1/√3

Substituting these values into the equation: cot θ = √3 × (1/ √3) cot θ = 1

Since θ is given to be an acute angle, the value of θ is: θ = 45°

Using the value of θ, we find: 2θ = 2 × 45° = 90°

Explanation

The average cost of the mixture can be directly computed using the weighted average formula:

Average Cost = (w₁C₁ + w₂C₂ + w₃C₃) ÷ (w₁ + w₂ + w₃)

Here, the ratios of the quantities represent the weights, which are w₁ = 3, w₂ = 1, and w₃ = 2. The respective costs per kg are C₁ = ₹50, C₂ = ₹60, and C₃ = ₹75.

Substituting these values into the formula:

Average Cost = (3 × 50 + 1 × 60 + 2 × 75) ÷ (3 + 1 + 2)

Average Cost = (150 + 60 + 150) ÷ 6

Average Cost = 360 ÷ 6 = ₹60 per kg

Explanation

By using the algebraic identity: (A - B)² = (A + B)² - 4AB

Let A = x and B = 2y. This gives the relation: (x - 2y)² = (x + 2y)² - 4(x)(2y) (x - 2y)² = (x + 2y)² - 4(2xy)

Substitute the given values of x + 2y = 10 and 2xy = 9 into the equation: (x - 2y)² = 10² - 4(9) (x - 2y)² = 100 - 36 (x - 2y)² = 64

Taking the square root on both sides: x - 2y = ±√64 x - 2y = ±8

Hence, one of the possible values for x - 2y is 8.

Explanation

Using the successive discount formula, the equivalent discount of two successive discounts x% and y% is given by the formula: x + y - xy/100

First, calculate the equivalent discount for the first two discounts of 20% and 10%: 20 + 10 - (20 × 10)/100 = 30 - 2 = 28%

Next, calculate the equivalent discount for this result of 28% and the third discount of 5%: 28 + 5 - (28 × 5)/100 = 33 - 1.4 = 31.6%

Alternatively, if the initial marked price is assumed to be 100, the final selling price after three successive discounts is: 100 × (1 - 0.20) × (1 - 0.10) × (1 - 0.05) = 100 × 0.80 × 0.90 × 0.95 = 68.4

The equivalent single discount is: 100 - 68.4 = 31.6%

Explanation

Let the purchase price of P be x.

Since P sold the item to Q at a 20% gain, the cost price for Q is: Cost Price of Q = x × 1.20

Q then sold the item to R at a 10% loss, making the cost price for R: Cost Price of R = Cost Price of Q × 0.90 = x × 1.20 × 0.90 = 1.08x

We are given that R bought the item for ₹1,080: 1.08x = 1080 x = 1080 / 1.08 x = 1000

Explanation

The central angle for an item in a pie chart is calculated using the formula: Central Angle = (Component Value / Total Value) × 360°

Given that the total salary is ₹7,200:

For food: Central Angle = (4000 / 7200) × 360° = 200°

For education: Central Angle = (400 / 7200) × 360° = 20°

Explanation

The weighted average formula for the cost of a mixture of three items is:

Average Price = (w₁P₁ + w₂P₂ + w₃P₃) / (w₁ + w₂ + w₃)

Here, P₁ = ₹96, P₂ = ₹104, and P₃ is the unknown price of the third variety, represented as x. The mixing ratio w₁ : w₂ : w₃ is 1 : 1 : 2, and the average price of the mixture is ₹113 per kg.

Substituting these values into the formula:

113 = (1 × 96 + 1 × 104 + 2 × x) / (1 + 1 + 2)

113 = (200 + 2x) / 4

Multiplying both sides by 4:

452 = 200 + 2x

2x = 452 - 200

2x = 252

x = 126

Explanation

Correct Answer: D

The selling price after successive discounts is calculated using the formula:

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

Here, the marked price is ₹2,250, and the successive discounts are d₁ = 10% and d₂ = 20%.

Selling Price = 2250 × (1 - 10/100) × (1 - 20/100) Selling Price = 2250 × 0.9 × 0.8 Selling Price = 2250 × 0.72 Selling Price = 1620

Explanation

The surface area of a sphere is calculated using the formula:

A = 4πr²

The diameter of the sphere is given as 30 cm, which means its radius r is:

r = 30 / 2 = 15 cm

Substituting r = 15 cm and π = 3.14 into the formula:

A = 4 × 3.14 × 15²

A = 4 × 3.14 × 225

A = 900 × 3.14

A = 2826 cm²

Explanation

The effective compound interest rate for 2 years at 7% per annum can be calculated using the successive percentage change formula:

Effective Rate = R + R + (R × R)/100 Effective Rate = 7 + 7 + (7 × 7)/100 = 14.49%

The compound interest (CI) is the effective rate percentage of the principal amount:

CI = 14.49% of ₹5,50,000 CI = (14.49 / 100) × 5,50,000 CI = 14.49 × 5,500 CI = 79,695

Explanation

For a number to be divisible by 6, it must be divisible by both 2 and 3. Dividing 54321 by 6 yields a quotient of 9053 and a remainder of 3.

To find the smallest natural number to add to make it divisible by 6, we subtract the remainder from the divisor: 6 − 3 = 3

Adding 3 to 54321 gives 54324, which is an even number and has a digit sum of 18, making it divisible by both 2 and 3, and therefore divisible by 6.

Explanation

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

Let the radius of the larger spherical clay ball be R = 15 cm and the radius of each smaller spherical ball be r = 5 cm.

Since the total volume of clay remains conserved during melting, the number of smaller balls, n, that can be made is the ratio of the volume of the larger ball to the volume of a single smaller ball:

n = ((4/3) π R³) / ((4/3) π r³) n = (R / r)³

Substituting the given radii:

n = (15 / 5)³ n = 3³ n = 27

Explanation

The formula for the average monthly income of the family over the year is:

Average Monthly Income = (Total Annual Expenditure + Total Annual Savings) ÷ 12

First, calculate the total annual expenditure by multiplying the average expenditure of each period by the number of months in that period:

Expenditure for the first 4 months = 4 × 13,750 = 55,000 Expenditure for the next 3 months = 3 × 11,750 = 35,250 Expenditure for the last 5 months = 5 × 31,750 = 158,750

Total Annual Expenditure = 55,000 + 35,250 + 158,750 = 249,000

Next, determine the total annual income by adding the total annual savings to the total annual expenditure:

Total Annual Income = 249,000 + 15,550 = 264,550

Now, divide the total annual income by 12 to find the average monthly income:

Average Monthly Income = 264,550 ÷ 12 ≈ 22,045.83

Considering only the integral part, the average monthly income of the family is 22,045.

Explanation

To evaluate the expression, follow the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction). The expression is:

12 - 8 ÷ 2 - (16 of -2 + 3 × 5 - 4)

First, evaluate the operations inside the parentheses.

Begin with the of operation: 16 of -2 = 16 × (-2) = -32

Next, perform the multiplication: 3 × 5 = 15

Now, perform addition and subtraction inside the parentheses: -32 + 15 - 4 = -17 - 4 = -21

The parenthetical part simplifies to -21, so the expression becomes: 12 - 8 ÷ 2 - (-21)

Next, perform the division: 8 ÷ 2 = 4

Substitute this value and simplify the subtraction of the negative number: 12 - 4 + 21

Finally, perform the addition and subtraction: 8 + 21 = 29

Explanation

First, calculate the distance covered by the first train before the second train starts. From 3 p.m. to 4 p.m., the train from station A travels for 1 hour at a speed of 83 km/h, covering a distance of 83 km.

The remaining distance between the two stations at 4 p.m. is 494 km - 83 km = 411 km.

Since the two trains are traveling towards each other, their relative speed is the sum of their individual speeds. Relative speed = 83 km/h + 54 km/h = 137 km/h.

The time taken for the trains to meet after 4 p.m. is found by dividing the remaining distance by their relative speed. Time taken = 411 km ÷ 137 km/h = 3 hours.

Adding these 3 hours to 4 p.m. gives the meeting time as 7 p.m.

Explanation

By the angle sum property of a triangle, the sum of all interior angles is 180°. Let the measure of each of the two equal angles be x.

108° + x + x = 180°

Simplifying this relation: 108° + 2x = 180° 2x = 180° - 108° 2x = 72° x = 36°

Explanation

Let √p = x. The given expression can be written as: x² + 1/9 x + k²

For this expression to be a perfect square, it must be in the form of the algebraic identity (x + a)² = x² + 2ax + a².

Comparing the coefficients of the given expression with the identity: 2a = 1/9 a = 1/18

Since the constant term is k², we have: k² = a² k² = (1/18)² k = ±1/18

Explanation

Let V be the total number of votes.

Candidate A secured 30% of V and fell short of the qualifying mark by 6 votes. Candidate B secured 40% of V and exceeded the qualifying mark by 6 votes.

The difference in percentage of votes secured by the two candidates is: 40% - 30% = 10% of V

The absolute difference in their votes is: 6 + 6 = 12 votes

We equate the percentage difference to the vote difference: 10% of V = 12 0.10 × V = 12 V = 12 / 0.10 = 120

Explanation

The sum of the first 10 observations is equal to 10 × 21 = 210.

The new average of the 11 observations is 1 less than the previous average, which is 21 - 1 = 20.

The sum of these 11 observations is equal to 11 × 20 = 220.

The value of the 11th observation is the difference between the sum of 11 observations and the sum of 10 observations, which is 220 - 210 = 10.

Explanation

To find the least 6-digit number divisible by 93, we start with the smallest 6-digit number, which is 100000.

Dividing 100000 by 93: 100000 = 93 × 1075 + 25

The remainder is 25. To find the next multiple of 93, we add the difference between the divisor and the remainder to 100000: 100000 + (93 - 25) = 100000 + 68 = 100068

Explanation

Let the two numbers be x and y. According to the given conditions: x = 3y x + y = 20

By substituting the first equation into the second: 3y + y = 20 4y = 20 y = 5

Now, substitute the value of y back into the first equation to find x: x = 3 × 5 = 15

Explanation

Let the initial income of the person be 100. Since he saves 20% of his income, his initial savings are 20 and his initial expenditure is 100 - 20 = 80.

A year later, his income increases by 25%, making his new income 125. He saves the same absolute amount as before, so his new savings are still 20. This means his new expenditure is 125 - 20 = 105.

The increase in expenditure is 105 - 80 = 25.

The percentage increase in expenditure is calculated as: (Increase in expenditure / Initial expenditure) × 100% = (25 / 80) × 100% = 31.25%

Explanation

Cost Price (CP) for the retailer is calculated from the marked price after applying the 20% discount: CP = 3750 × (1 - 0.20) = 3750 × 0.80 = ₹3,000

To earn a profit of 15%, the Selling Price (SP) is calculated on this cost price: SP = CP × (1 + 0.15) = 3000 × 1.15 = ₹3,450

Explanation

The formula for average speed is the total distance divided by the total time taken.

In the first part of the journey, the distance covered is 70 miles per hour × 4 hours = 280 miles. In the second part of the journey, the distance covered is 40 miles per hour × 2 hours = 80 miles.

The total distance is 280 + 80 = 360 miles. The total time taken is 4 + 2 = 6 hours.

Average speed = 360 miles / 6 hours = 60 miles per hour.

Explanation

To find the product of (x + a) and (x + b), we use the distributive property to expand the expression:

(x + a)(x + b) = x(x) + x(b) + a(x) + a(b)

Simplifying each term gives:

x² + bx + ax + ab

By factoring out the common variable x from the middle terms, we get:

x² + (a + b)x + ab

Explanation

The formula for the total surface area of a solid right circular cylinder is: Total Surface Area = 2πr(r + h)

Given that the radius (r) is 3 cm and the total surface area is 60π cm², these values can be substituted into the formula: 60π = 2 × π × 3 × (3 + h)

Simplifying the equation: 60π = 6π(3 + h)

Dividing both sides by 6π: 10 = 3 + h

Solving for height (h): h = 7 cm

Explanation

Using the work equivalence formula: M₁ × D₁ × H₁ = M₂ × D₂ × H₂

where M represents the number of people, D represents the number of days, and H represents the number of working hours per day.

Given: M₁ = 45, D₁ = 10, H₁ = 6 M₂ = 30, H₂ = 6

Substituting these values into the formula: 45 × 10 × 6 = 30 × D₂ × 6

Since the daily hours are the same on both sides, we simplify to: 45 × 10 = 30 × D₂ 450 = 30 × D₂ D₂ = 450 / 30 D₂ = 15

Explanation

The area of an equilateral triangle with side length a is given by the formula:

Area = (√3/4) × a²

Given that the side of the equilateral triangle is a = 20 cm, we substitute this value into the formula:

Area = (√3/4) × 20² Area = (√3/4) × 400 Area = 100√3 cm²

Explanation

Discounted price of 12 pairs (1 dozen pairs) of gloves = £200 × (100 - 20)% = £200 × 0.8 = £160.

This means 12 pairs of gloves cost £160.

Number of pairs of gloves that can be purchased for £320 = 12 × (320 / 160) = 12 × 2 = 24.

Explanation

Using the trigonometric identity cos θ = -sin(θ - 90°), cos 137° can be written as: cos 137° = cos(90° + 47°) = -sin 47°

Substituting this into the original expression gives: sin 73° + cos 137° = sin 73° - sin 47°

Applying the sum-to-product formula, sin A - sin B = 2 cos((A + B)/2) sin((A - B)/2), where A = 73° and B = 47°: sin 73° - sin 47° = 2 cos((73° + 47°)/2) sin((73° - 47°)/2) = 2 cos 60° sin 13°

Substituting the known value cos 60° = ½: 2 × ½ × sin 13° = sin 13°

Explanation

The curved surface area of a cylinder is calculated using the formula: Curved Surface Area = 2πrh

Given: Diameter of the base = 12 cm Radius of the base, r = 12 / 2 = 6 cm Height of the cylinder, h = 2.45 × r = 2.45 × 6 = 14.7 cm

Substituting the given values into the formula with π = 22/7: Curved Surface Area = 2 × (22/7) × 6 × 14.7 Curved Surface Area = 2 × 22 × 6 × 2.1 Curved Surface Area = 44 × 12.6 Curved Surface Area = 554.4 cm²