SSC CHSL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Using the algebraic identity for the given expression: a² + b² + c² - ab - bc - ca = ½ [ (a - b)² + (b - c)² + (c - a)² ]

Substituting the given values of a = 2022, b = 2021, and c = 2020: a - b = 2022 - 2021 = 1 b - c = 2021 - 2020 = 1 c - a = 2020 - 2022 = -2

Substituting these differences back into the identity: ½ [ 1² + 1² + (-2)² ] = ½ [ 1 + 1 + 4 ] = ½ [ 6 ] = 3

Explanation

The difference between compound interest (CI) and simple interest (SI) for a period of 3 years can be calculated using the direct formula:

Difference = P × (R/100)² × (300 + R)/100

Substituting the given values into the formula: Principal (P) = ₹5,000 Rate (R) = 8%

Difference = 5000 × (8/100)² × (300 + 8)/100 Difference = 5000 × (64/10000) × (308/100) Difference = 5 × 6.4 × 3.08 Difference = 32 × 3.08 Difference = 98.56

Rounding off 98.56 to the nearest rupee gives ₹99.

Explanation

The standard trigonometric values are: tan 45° = 1 cos 60° = ½ sin 30° = ½

Substituting these values into the given expression: 3 tan² 45° + 2 cos 60° - 2 sin 30° = 3 × 1² + 2 × ½ - 2 × ½ = 3 + 1 - 1 = 3

Explanation

When the price of an item increases by r%, the percentage reduction in consumption required to keep the expenditure constant is given by the formula:

Percentage reduction = (r / (100 + r)) × 100%

Here, the price of rice is raised by 20%, so r = 20. Substituting this value into the formula:

Percentage reduction = (20 / (100 + 20)) × 100% Percentage reduction = (20 / 120) × 100% Percentage reduction = (1 / 6) × 100% Percentage reduction = 16.66...%

Rounding to one decimal place, the required percentage reduction in consumption is 16.7%.

Explanation

The marked price of the mobile phone is ₹36,000. When purchasing with a credit card, the discount rate increases to 15%.

The selling price can be calculated directly using the formula: Selling Price = Marked Price × (1 - Discount Percentage / 100)

Substituting the given values: Selling Price = 36000 × (1 - 15/100) Selling Price = 36000 × 0.85 Selling Price = 30600

Pooja pays ₹30,600 for the mobile phone.

Explanation

The given expression is in the algebraic form (a³ + b³ + c³ - 3abc) / (a + b + c), where a = 59, b = 54, and c = 57.

Using the standard algebraic identity: a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)

Dividing both sides by (a + b + c) gives: (a³ + b³ + c³ - 3abc) / (a + b + c) = a² + b² + c² - ab - bc - ca

This can be rewritten in terms of squared differences as: ½ [(a - b)² + (b - c)² + (c - a)²]

Substituting the values of a, b, and c into this expression: a - b = 59 - 54 = 5 b - c = 54 - 57 = -3 c - a = 57 - 59 = -2

Plugging these differences into the formula: ½ [5² + (-3)² + (-2)²] = ½ [25 + 9 + 4] = ½ [38] = 19

Explanation

The surface area of a sphere is given by the formula: Surface Area = 4πr²

where r is the radius of the sphere.

Given that the diameter of the sphere is 16.8 cm, the radius r is: r = 16.8 / 2 = 8.4 cm

Using π = 22/7, we substitute the values into the formula: Surface Area = 4 × (22/7) × 8.4 × 8.4 Surface Area = 4 × 22 × 1.2 × 8.4 Surface Area = 88 × 10.08 Surface Area = 887.04 cm²

Explanation

Average speed is calculated by dividing the total distance travelled by the total time taken.

First, determine the total distance: Total Distance = 124 + 96 + 105 = 325 km

Next, calculate the time taken for each segment of the journey using the formula Time = Distance / Speed: Time taken by bus = 124 / 31 = 4 hours Time taken by the first car journey = 96 / 16 = 6 hours Time taken by the second car journey = 105 / 7 = 15 hours

Now, find the total time taken for the entire journey: Total Time = 4 + 6 + 15 = 25 hours

Using these values, compute the average speed: Average Speed = Total Distance / Total Time = 325 / 25 = 13 km/h

Explanation

A number is divisible by 33 if and only if it is divisible by both 3 and 11.

To test 185592 for divisibility by 3, calculate the sum of its digits: 1 + 8 + 5 + 5 + 9 + 2 = 30 Since 30 is a multiple of 3, the number is divisible by 3.

To test 185592 for divisibility by 11, calculate the difference between the sum of the digits at odd places and the sum of the digits at even places: Sum of digits at odd positions: 1 + 5 + 9 = 15 Sum of digits at even positions: 8 + 5 + 2 = 15 Difference: 15 - 15 = 0 Since the difference is 0, the number is divisible by 11.

Explanation

The present value P of a loan repaid in two equal yearly instalments is given by the formula:

P = I / (1 + r) + I / (1 + r)²

where I is the instalment amount and r is the annual interest rate.

Given: I = 6,897 r = 10% = 0.1

Substituting the values into the formula:

P = 6,897 / 1.1 + 6,897 / 1.1² P = 6,270 + 5,700 P = 11,970

The total amount paid over two years is:

Total Paid = 2 × 6,897 = 13,794

The total interest charged is the difference between the total amount paid and the principal loan amount:

Total Interest = Total Paid - P Total Interest = 13,794 - 11,970 = 1,824

Explanation

Using the trigonometric sum-to-product identities:

sin A + sin B = 2 sin((A + B)/2) cos((A - B)/2) cos A + cos B = 2 cos((A + B)/2) cos((A - B)/2)

For the given expression, let A = 4θ and B = 2θ.

The numerator simplifies to: sin 4θ + sin 2θ = 2 sin 3θ cos θ

The denominator simplifies to: cos 4θ + cos 2θ = 2 cos 3θ cos θ

Dividing the numerator by the denominator: (2 sin 3θ cos θ) / (2 cos 3θ cos θ) = sin 3θ / cos 3θ = tan 3θ

Explanation

The area of a quadrilateral with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄) in cyclic order is given by the coordinate geometry area formula:

Area = ½ | (x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) - (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁) |

Substituting the given vertices (-3, 2), (5, 4), (7, -6), and (-5, -4) into the formula:

First term sum = (-3 × 4) + (5 × -6) + (7 × -4) + (-5 × 2) = -12 - 30 - 28 - 10 = -80

Second term sum = (2 × 5) + (4 × 7) + (-6 × -5) + (-4 × -3) = 10 + 28 + 30 + 12 = 80

Area = ½ | -80 - 80 | = ½ | -160 | = 80 square units

Explanation

Let the cost price of 1000g of goods be ₹100.

The shopkeeper normally sells this quantity at a 20% profit, making the normal selling price ₹120.

Due to the faulty weighing machine, he actually delivers only 900g of goods. The cost price of this delivered quantity to the shopkeeper is: (900 / 1000) × 100 = ₹90

He charges 10% less than his normal selling price. Thus, his actual selling price is: 120 - (10% of 120) = ₹108

The actual profit is: Actual Selling Price - Actual Cost Price = 108 - 90 = ₹18

The actual profit percentage is: (Profit / Actual Cost Price) × 100 = (18 / 90) × 100 = 20%

Explanation

The number of admissions grows exponentially every year. Using the compound growth formula:

A = P × (1 + R/100)ⁿ

Here, the initial number of admissions P is 2400, the annual growth rate R is 5%, and the time period n is 3 years.

Substituting these values into the formula:

A = 2400 × (1 + 5/100)³ A = 2400 × (1.05)³ A = 2400 × 1.157625 A = 2778.3

Rounding 2778.3 to the nearest integer yields 2778.

Explanation

Let the fourth proportional to 5, 8, and 30 be x.

By the definition of proportion, if four quantities a, b, c, and x are in proportion, then: a/b = c/x

Substituting the given values a = 5, b = 8, and c = 30 into the formula: 5/8 = 30/x

Solving for x by cross-multiplication: x = (8 × 30) / 5 x = 8 × 6 x = 48

Explanation

Using the algebraic identity for the square of a binomial:

(x - 2y)² = x² - 4xy + 4y²

This can be rearranged as:

(x - 2y)² = (x² + 4y²) - 4(xy)

Substitute the given values x² + 4y² = 40 and xy = 6 into the equation:

(x - 2y)² = 40 - 4 × 6 (x - 2y)² = 40 - 24 (x - 2y)² = 16

Taking the square root on both sides gives:

x - 2y = ±4

Since it is given that x > 2y, the difference x - 2y must be positive. Thus, the positive value is selected:

x - 2y = 4

Explanation

The total number of students in each city is given in the last column of the table. Comparing these total values:

Ahmedabad = 6679 Bengaluru = 7156 Bhopal = 6873 Chennai = 7375 New Delhi = 7098 Hyderabad = 7688 Kolkata = 8155

Comparing the numbers, 8155 is the largest value. Therefore, Kolkata has the maximum number of students.

Explanation

The efficiency (or speed) of the man and the boy are in the ratio 4:3. Let the daily work done by the man be 4 units and by the boy be 3 units.

The combined daily work of both is 4 + 3 = 7 units.

Since they complete the entire work together in 24 days, the total work is: Total work = Combined efficiency × Number of days = 7 × 24 = 168 units

The number of days the boy alone will take to complete this work is: Time taken = Total work / Boy's efficiency = 168 / 3 = 56 days

Explanation

To find the value of the algebraic expression x² + y² for the given values x = 1 and y = 2, substitute the values of x and y directly into the expression:

x² + y² = 1² + 2²

Calculating the squares of the numbers: 1² = 1 2² = 4

Adding these values together: 1 + 4 = 5

Explanation

Applying the BODMAS rule, the expression is evaluated step-by-step from the innermost brackets to the outermost brackets. The given expression 9 ÷ [5 + 7 ÷ (9 + 9 ÷ (9 + 9 ÷ 4))] is simplified as follows:

First, evaluate the innermost division: 9 ÷ 4 = 9/4

Next, solve the innermost addition: 9 + 9/4 = 45/4

Then, perform the next division and addition: 9 + 9 ÷ (45/4) = 9 + 9 × 4/45 = 9 + 4/5 = 49/5

Now, simplify the terms inside the square bracket: 5 + 7 ÷ (49/5) = 5 + 7 × 5/49 = 5 + 5/7 = 40/7

Finally, evaluate the outermost division: 9 ÷ (40/7) = 9 × 7/40 = 63/40

Explanation

For an equilateral triangle, the circumradius R and the inradius r are related by the standard geometric formula R = 2r.

Given that the circumradius R = 18 cm, we can find the inradius r by substituting this value into the relation: r = R / 2 r = 18 / 2 r = 9 cm

Explanation

Correct Answer: D

Using the sum of weighted averages formula: Total marks of class = Total marks of girls + Total marks of boys

Given: Total students = 45 Number of girls = 20 Number of boys = 45 - 20 = 25 Class average = 36 Girls' average = 42

Let the average marks of the boys be A_b.

Substituting the values into the equation: 45 × 36 = 20 × 42 + 25 × A_b 1620 = 840 + 25 × A_b 25 × A_b = 1620 - 840 25 × A_b = 780 A_b = 780 / 25 = 31.20

Explanation

The net percentage change when a value is successively changed by x% and y% is given by the formula:

Net change = x + y + xy/100 %

Here, the number is first increased by 10% (x = 10) and then decreased by 20% (y = -20).

Substituting these values into the formula:

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

Net change = -10 - 200/100

Net change = -10 - 2 = -12%

The negative sign indicates a net decrease of 12%.

Explanation

Let the total number of votes polled (voters who cast their vote) be V.

The distribution of the polled votes among the three candidates is: Votes polled for A = 45% of V = 0.45V Votes polled for B = 35% of V = 0.35V Votes polled for C = 20% of V = 0.20V

After accounting for invalid votes, the valid votes received by each candidate are: Valid votes for A = 70% of 0.45V = 0.70 × 0.45V = 0.315V Valid votes for B = 80% of 0.35V = 0.80 × 0.35V = 0.28V Valid votes for C = 100% of 0.20V = 0.20V

The difference between the valid votes of A and B is given as 882: 0.315V - 0.28V = 882 0.035V = 882 V = 882 / 0.035 = 25200

The number of valid votes C received is: 0.20V = 0.20 × 25200 = 5040

Explanation

Using the trigonometric co-function identity, sin θ = cos(90° - θ).

For θ = 75°: sin 75° = cos(90° - 75°) = cos 15°

We are given: cos 2A = sin 75° cos 2A = cos 15°

To find the smallest positive value of A, we equate the angles directly: 2A = 15° A = 7.5°

Explanation

The volume V of a frustum of a right circular cone is given by the formula:

V = ⅓πh(R² + r² + Rr)

where R is the radius of the base, r is the radius of the top, and h is the height of the frustum.

Given: R = 5 cm r = 3 cm h = 6 cm

Substituting these values into the formula:

V = ⅓ × π × 6 × (5² + 3² + 5 × 3)

V = 2π × (25 + 9 + 15)

V = 2π × 49

V = 98π cm³

Explanation

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

Substituting the given values, r = 3.5 cm = 7/2 cm and π = 22/7, into the formula:

Total Surface Area = 3 × (22/7) × (7/2)² Total Surface Area = 3 × (22/7) × (49/4) Total Surface Area = 3 × 11 × 7 / 2 Total Surface Area = 231 / 2 Total Surface Area = 115.5 cm²

Explanation

The number of type B geysers sold in 2010 is 189 thousand, which equals 189,000.

The number of type B geysers sold in 2012 is 145 thousand, which equals 145,000.

The difference between the sales of type B geysers in these two years is:

189,000 - 145,000 = 44,000

Explanation

Relative speed of the policeman with respect to the thief is the difference between their speeds: v = 10.5 km/h - 8 km/h = 2.5 km/h

To convert this relative speed into metres per minute: v = 2.5 × 1000 m / 60 min = 250 / 6 m/min

The relative distance covered by the policeman toward the thief in 12 minutes is: Distance = Speed × Time = (250 / 6) × 12 = 500 m

Since the initial distance between them was 650 m, the remaining distance between them after 12 minutes is: 650 m - 500 m = 150 m

Explanation

The original price of the cell phone is 20,000.

On Sunday, a flat cash discount of 1,000 is applied first: Price after Sunday discount = 20,000 - 1,000 = 19,000

An additional 5% discount is then calculated on this reduced price for using a credit card: Credit card discount = 5% of 19,000 = 0.05 × 19,000 = 950

Subtracting this additional discount gives the final amount to be paid: Final price = 19,000 - 950 = 18,050

Explanation

Correct Answer: B

The net percentage change after two successive percentage changes of x% and y% is given by the formula:

Net change % = x + y + xy/100

In this case, the price first increases by 10% (so x = 10) and then decreases by 2.5% (so y = -2.5).

Substituting these values into the formula:

Net change % = 10 + (-2.5) + (10 × -2.5)/100 Net change % = 7.5 - 25/100 Net change % = 7.5 - 0.25 Net change % = 7.25%

A positive result indicates a net increase. Thus, the final price of the tablet increases by 7.25%.

Explanation

To find the rationalising factor of √2 + √3 + √5, we can use the algebraic identity (a + b)(a - b) = a² - b².

Let a = √2 + √3 and b = √5. Multiplying the expression by √2 + √3 - √5 gives:

(√2 + √3 + √5)(√2 + √3 - √5) = (√2 + √3)² - (√5)²

Expanding the squared term:

(2 + 3 + 2√6) - 5 = 5 + 2√6 - 5 = 2√6

To make this resulting term rational, we must multiply it by √6.

Multiplying the entire conjugate term by √6 yields the complete rationalising factor:

(√2 + √3 - √5)√6

Multiplying the original expression by this factor results in:

(2√6) × √6 = 12

Since 12 is a rational number, (√2 + √3 - √5)√6 is a valid rationalising factor.

Explanation

The fourth proportional x to three numbers a, b, and c is defined by the relation a/b = c/x.

Given a = 16, b = 24, and c = 8, we substitute these values into the formula: 16/24 = 8/x

Reducing the fraction on the left side: 2/3 = 8/x

Solving for x: 2x = 24 x = 12

Explanation

The volume of a cone is given by the formula V = ⅓πr²h, where r is the radius of the base and h is the height.

Given that the radius r = 13 cm and the volume V = 507π cm³, we can substitute these values into the formula to find the height:

507π = ⅓ × π × 13² × h

Dividing both sides by π:

507 = ⅓ × 169 × h

Multiplying both sides by 3:

1521 = 169h

h = 1521 / 169

h = 9 cm

Explanation

Quantity of wheat used in July is 400 kg.

Total quantity of food items used in June is calculated as: Total June quantity = 180 + 420 + 200 + 250 + 300 = 1350 kg

The required percentage is calculated as: Percentage = (Wheat in July / Total in June) × 100% Percentage = (400 / 1350) × 100% Percentage = (8 / 27) × 100% Percentage = 800 / 27%

Dividing 800 by 27 gives a quotient of 29 and a remainder of 17. This simplifies to 29 17/27%.

Explanation

The net profit percentage is determined by the combined effect of the price markup and the usage of the incorrect weight.

Let the cost price of 1 gram of the item be 1 unit.

The cost price of 1000 grams is 1000 units. Since Subir sells the items at 5% above the cost of production, the selling price for 1 kg is: 1000 × 1.05 = 1050 units

Subir delivers only 960 grams instead of 1000 grams. The actual cost price of the goods delivered to the customer is: 960 × 1 = 960 units

The actual profit earned by Subir is: Selling Price - Actual Cost Price = 1050 - 960 = 90 units

The actual profit percentage is: (Profit / Actual Cost Price) × 100 = (90 / 960) × 100 = (3 / 32) × 100 = 9.375%

Explanation

According to the Remainder Theorem, when a polynomial f(x) is divided by a linear divisor of the form ax + b, the remainder is equal to f(-b/a).

In this case, the divisor is 3x + 2. Setting this divisor to zero gives x = -2/3.

Substituting x = -2/3 into the polynomial f(x) = 15x³ - 14x² - 4x + 10:

f(-2/3) = 15(-2/3)³ - 14(-2/3)² - 4(-2/3) + 10

f(-2/3) = 15(-8/27) - 14(4/9) + 8/3 + 10

f(-2/3) = -40/9 - 56/9 + 24/9 + 90/9

f(-2/3) = (-40 - 56 + 24 + 90) / 9

f(-2/3) = 18/9 = 2

Explanation

The age of the newly joined relative can be directly calculated using the average change formula:

Age of new entrant = New average + (Original number of members × Increase in average)

Given: Original number of members = 4 Original average age = 27 years Increase in average = 3 years New average age = 27 + 3 = 30 years

Substituting these values into the formula: Age of the relative = 30 + (4 × 3) = 30 + 12 = 42 years

Alternatively, this can be solved using the sum of observations: Total age of 4 family members = 4 × 27 = 108 years Total age of 5 members after the relative joins = 5 × 30 = 150 years Age of the relative = 150 - 108 = 42 years

Explanation

To find the overall grade, we calculate the weighted average of the scores: (90 * 0.30) + (95 * 0.30) + (75 * 0.40) = 27 + 28.5 + 30 = 85.50%. Therefore, option D is the correct answer.

Explanation

Correct Answer: D

Using the algebraic identity for the difference of squares:

a² - b² = (a - b)(a + b)

Substituting a = 983 and b = 17 into the formula:

983² - 17² = (983 - 17) × (983 + 17)

Calculating the terms inside the parentheses:

983 - 17 = 966 983 + 17 = 1000

Multiplying the results:

966 × 1000 = 966000

Explanation

The remaining work can be solved using the work equivalence relation M₁ × D₁ = M₂ × D₂.

The original group of 30 men had 12 days of work remaining (24 days minus the 12 days already worked). Therefore, the remaining work is equivalent to 30 × 12 man-days.

With 10 additional men joining, the new group size is 30 + 10 = 40 men.

Let D be the number of days required for these 40 men to complete the remaining work:

30 × 12 = 40 × D 360 = 40 × D D = 360 ÷ 40 = 9 days

Explanation

Correct Answer: A

For two successive discounts of d₁% and d₂%, the equivalent single discount is given by the formula: Equivalent Discount = d₁ + d₂ - (d₁ × d₂)/100 %

Substituting the given values d₁ = 20 and d₂ = 15: Equivalent Discount = 20 + 15 - (20 × 15)/100 % = 35 - 3 % = 32%

The difference between the single discount of 35% and the equivalent successive discount of 32% is: 35% - 32% = 3%

Calculating this difference on the principal amount of ₹8,000: 3% of 8,000 = 3/100 × 8000 = 240

Explanation

The largest 6-digit number is 999999. When 999999 is divided by 52, the remainder is 39, and subtracting this remainder from 999999 gives 999960, which is the largest 6-digit number exactly divisible by 52.

Explanation

Average Speed = Total Distance / Total Time

The total distance travelled is: Total Distance = 100 km + 100 km + 100 km = 300 km

The time taken for each of the three equal sections is: t₁ = 100 / 30 = 10/3 hours t₂ = 100 / 40 = 5/2 hours t₃ = 100 / 50 = 2 hours

The total time is: Total Time = 10/3 + 5/2 + 2 = (20 + 15 + 12) / 6 = 47/6 hours

Using the average speed formula: Average Speed = 300 / (47/6) = 1800 / 47 ≈ 38.3 km/h

Explanation

The population growth over decades follows the compound interest formula:

A = P × (1 + R/100)ⁿ

Here, A represents the population in 2030, which is 1,58,18,400. P is the population in 2000. R is the decadal growth rate of 4%, and n is the number of decades between 2000 and 2030, which is 3.

Substituting these values into the formula:

1,58,18,400 = P × (1 + 4/100)³

1,58,18,400 = P × (1.04)³

1,58,18,400 = P × (26/25)³

1,58,18,400 = P × (17576 / 15625)

Rearranging the equation to solve for P:

P = 1,58,18,400 × (15625 / 17576)

Dividing 1,58,18,400 by 17576 yields exactly 900:

P = 900 × 15625

P = 1,40,62,500

Explanation

Following the BODMAS rule, the operations are performed in the order of brackets (innermost first), of, division, multiplication, addition, and subtraction.

First, solve the innermost of operator: ¼ of 7920 = 7920 ÷ 4 = 1980

Next, solve the expression inside the curly braces: -1980 + 1980 = 0

Now, evaluate the expression inside the square brackets: -77 + 0 = -77

Next, perform the division at the beginning of the expression: 1980 ÷ 9 = 220

Finally, combine the terms: 220 + (-77) = 220 - 77 = 143

Explanation

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

Relative speed = 37 km/h + 35 km/h = 72 km/h

To convert this speed into meters per second (m/s), multiply by 5/18: Relative speed = 72 × 5/18 = 20 m/s

In a circular race, the two runners will meet for the first time when their combined distance covered is equal to the length of the track.

Time taken to meet = Track length / Relative speed Time = 2500 m / 20 m/s = 125 seconds

Since 60 seconds make a minute, 125 seconds can be written as 2 minutes and 5 seconds.

Explanation

The given sides of the triangle are 5 cm, 12 cm, and 13 cm. These sides satisfy the Pythagorean theorem since 5² + 12² = 25 + 144 = 169 = 13². This confirms that the triangle is a right-angled triangle where the legs (base and height) are 5 cm and 12 cm, and the hypotenuse is 13 cm.

The area of a right-angled triangle is calculated using the formula: Area = ½ × base × height

Substituting the values: Area = ½ × 5 × 12 = 30 cm²

Explanation

The number of cakes sold by Bakery C on Monday is 245.

Since 40% of the cakes sold by Bakery C on Monday are Vanilla cakes, the remaining cakes are strawberry cakes, which represents: 100% - 40% = 60%

The number of strawberry cakes sold by Bakery C on Monday is: 245 × 60% = 245 × 0.6 = 147

Explanation

The fourth proportional x to three numbers a, b, and c is defined by the ratio a : b = c : x, which gives the formula x = (b × c) / a.

Substituting the given values a = 7, b = 13, and c = 21 into the formula:

x = (13 × 21) / 7

x = 13 × 3

x = 39