SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

By using the algebraic identity: (x - y)² = x² + y² - 2xy

We can rewrite the equation to solve for x² + y²: x² + y² = (x - y)² + 2xy

Substituting the given values, x - y = 7 and xy = 32, into the equation: x² + y² = 7² + 2(32) x² + y² = 49 + 64 x² + y² = 113

Explanation

Let the cost price of the watch for the wholesaler be CP.

According to the given transaction details, the wholesaler sells the watch to the retailer at a profit of 33%, and the retailer sells it to the customer at a loss of 20%.

Using the formula for successive percentage changes: Final Price = CP × (1 + Profit%/100) × (1 - Loss%/100)

Substituting the given values: 2181.2 = CP × (1 + 33/100) × (1 - 20/100) 2181.2 = CP × 1.33 × 0.80 2181.2 = CP × 1.064

Solving for CP: CP = 2181.2 / 1.064 CP = 2050

Therefore, the cost to the wholesaler was Rs 2050.

Explanation

Let the tens digit of the two-digit number be x and the units digit be y. The original number is 10x + y.

The product of the digits is given as xy = 12.

When 36 is added to the number, the digits are reversed to form the number 10y + x: (10x + y) + 36 = 10y + x 9y - 9x = 36 y - x = 4 y = x + 4

Substituting y = x + 4 into the product equation: x(x + 4) = 12 x² + 4x - 12 = 0 (x + 6)(x - 2) = 0

Since the digits of a number must be positive integers, x = 2.

Using y = x + 4, the units digit y is 6.

This gives the original number as 26.

Explanation

The trigonometric values of the given angles are:

csc π/4 = √2 sin π/3 = √3/2

Substituting these values into the given equation:

x = csc π/4 - sin π/3 x = √2 - √3/2

Taking the common denominator of 2:

x = (2√2 - √3)/2

This directly yields the value of x.

Explanation

Using the fundamental trigonometric identity: sec²A - 1 = tan²A

Taking the square root: √(sec²A - 1) = tanA

Substitute this into the given expression: x = tanA / secA

Since tanA = sinA / cosA and secA = 1 / cosA, we have: x = (sinA / cosA) / (1 / cosA) x = sinA

Explanation

The volume of water required to fill the cylindrical vessel completely is equal to the difference between the volume of the cylindrical vessel and the volume of the solid right circular cone.

The volume of a cylinder is given by the formula: Volume of cylinder = π × r₁² × h₁

Using the radius of the cylinder r₁ = 5 cm and height h₁ = 8 cm: Volume of cylinder = π × 5² × 8 = 200π cubic cm

The volume of a cone is given by the formula: Volume of cone = 1/3 × π × r₂² × h₂

Using the radius of the cone r₂ = 4 cm and height h₂ = 7 cm: Volume of cone = 1/3 × π × 4² × 7 = 112π / 3 cubic cm

The volume of water required is: Volume of water = 200π - 112π / 3 = 488π / 3 cubic cm

Using the value of π = 22/7: Volume of water = 488/3 × 22/7 = 10736 / 21 ≈ 511.24 cubic cm

Explanation

In compound interest, the amount at the end of any year acts as the principal for the succeeding year. The relationship between the amounts at the end of the consecutive second and third years is:

A₃ = A₂ × (1 + R/100)

Here, the amount at the end of the second year A₂ = 24200 and the amount at the end of the third year A₃ = 26620. Substituting these values into the formula:

26620 = 24200 × (1 + R/100)

1 + R/100 = 26620 / 24200

1 + R/100 = 11 / 10

R/100 = 11/10 - 1

R/100 = 1/10

R = 10%

Explanation

Let the third number be 100.

Since the first number is 50% lesser than the third number, the first number is 100 - 50 = 50.

Since the second number is 80% lesser than the third number, the second number is 100 - 80 = 20.

To make the second number equal to the first number, it must be increased by 50 - 20 = 30.

The required percentage increase is (30 / 20) × 100% = 150%.

Explanation

Let Manoj's efficiency be 1 unit of work per hour. Since Manoj completes the work in 42 hours, the total work is 42 units.

Jayashree is 100% more efficient than Manoj, which means her efficiency is twice that of Manoj. Therefore, Jayashree's efficiency is 2 units of work per hour.

When working together, their combined efficiency is the sum of their individual efficiencies, which is 1 + 2 = 3 units of work per hour.

The time taken by both of them to finish the work together is calculated by dividing the total work by their combined efficiency, which gives 42 ÷ 3 = 14 hours.

Explanation

The sum of the interior angles of a triangle is always 180°.

Let the measures of the three angles be x, 2x, and 3x based on the ratio 1:2:3.

Using the angle sum property of a triangle: x + 2x + 3x = 180° 6x = 180° x = 30°

The measure of the biggest angle is 3x: 3 × 30° = 90°

Explanation

The total revenue for all 13 years is obtained by multiplying the number of years by their average: 13 × 70 = 910 lakhs

The sum of the revenues for the first 7 years is: 7 × 65 = 455 lakhs

The sum of the revenues for the last 7 years is: 7 × 77 = 539 lakhs

In this calculation, the 7th year is included in both the first 7 years and the last 7 years. Therefore, adding these two sums counts the 7th year twice. The revenue of the 7th year is calculated by subtracting the total 13-year revenue from the sum of the first 7 and last 7 years: (455 + 539) - 910 = 994 - 910 = 84 lakhs

Explanation

To find Chintan's share of the profit, use the ratio division formula:

Share = (Individual ratio term / Sum of all ratio terms) × Total profit

The ratio of profit sharing among Akram, Bipin, and Chintan is 3:2:7.

Sum of the ratio terms = 3 + 2 + 7 = 12

The total profit is ₹144000.

Chintan's share = (7 / 12) × 144000

Chintan's share = 7 × 12000 = 84000

Explanation

The area of a kite with diagonals of lengths d₁ and d₂ is calculated using the formula:

Area = ½ × d₁ × d₂

Since the diagonals of a kite intersect at right angles (90°), the kite can be split into two triangles by one of its diagonals. The area of the kite is the sum of the areas of these two triangles, which simplifies directly to half the product of the diagonals.

Explanation

Total cost of daily tickets for 30 days = 30 × 50 = 1500

Cost of the Monthly Pass = 1125

Total savings = 1500 - 1125 = 375

Percentage savings = (Total savings / Total cost of daily tickets) × 100 = (375 / 1500) × 100 = 25%

Explanation

The weight to height ratio is calculated by dividing the weight in kg by the height in meters.

Calculating the ratio for each person:

Apeksha: 59 / 1.8 = 32.78 kg/m

Apsara: 51 / 1.72 = 29.65 kg/m

Aradhana: 62 / 1.57 = 39.49 kg/m

Archana: 74 / 1.68 = 44.05 kg/m

Apsara has the least weight to height ratio of 29.65 kg/m.

Explanation

The year with the least percentage profit is 2012, where the profit is 5%.

The formula relating profit percentage, revenue, and expenditure is: Profit % = (Revenue - Expenditure) / Expenditure × 100

Rearranging the formula to solve for Revenue gives: Revenue = Expenditure × (1 + Profit % / 100)

Substituting the given expenditure of Rs 525 crore and the profit of 5% into the equation: Revenue = 525 × (1 + 5 / 100) Revenue = 525 × 1.05 Revenue = 551.25 crore

Explanation

The average number of children per house is calculated using the formula for the weighted mean:

Average = ∑(x × f) / ∑ f

where x represents the number of children and f represents the number of houses.

First, calculate the total number of houses: ∑ f = 7 + 7 + 13 + 3 = 30

Next, calculate the total number of children: ∑(x × f) = (0 × 7) + (1 × 7) + (2 × 13) + (3 × 3) = 0 + 7 + 26 + 9 = 42

Dividing the total number of children by the total number of houses gives: Average = 42 ÷ 30 = 1.4

Explanation

Dhiraj's initial shareholding is 20% of the total issued shares. Since the company has issued 6,00,000 shares, Dhiraj's initial number of shares is calculated as:

Initial shares = 20% of 6,00,000 = 0.20 × 6,00,000 = 1,20,000 shares

When Chinmay sells 15,000 of his shares to Dhiraj, Dhiraj's total shares increase by 15,000:

Final shares = 1,20,000 + 15,000 = 1,35,000 shares

Explanation

The centroid of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is calculated using the formula:

(x, y) = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3)

Substituting the coordinates of the given vertices A(-2,5), B(3,-2), and C(5,3) into the formula:

x-coordinate = (-2 + 3 + 5)/3 = 6/3 = 2

y-coordinate = (5 + (-2) + 3)/3 = 6/3 = 2

Explanation

When two objects move towards each other, their relative speed is the sum of their individual speeds.

Relative speed = 6 km/hr + 11 km/hr = 17 km/hr

The time given is 42 minutes. Converting this into hours: Time = 42 / 60 hours = 0.7 hours

The initial distance between them is the product of their relative speed and the time taken to meet: Distance = Relative speed × Time = 17 km/hr × 0.7 hours = 11.9 km

Explanation

The value of cot 135° can be calculated using the trigonometric identity cot(180° - θ) = -cot θ.

Substituting θ = 45°: cot 135° = cot(180° - 45°) = -cot 45°

Since cot 45° = 1: cot 135° = -1

Thus, the value of x is -1.

Explanation

The angle θ between two lines with slopes m₁ and m₂ is given by the formula:

tan θ = |(m₂ - m₁) / (1 + m₁ m₂)|

Given the slopes of the two lines: m₁ = 1/√3 m₂ = √3

Substituting these values into the formula:

tan θ = |(√3 - 1/√3) / (1 + √3 × 1/ √3)|

The numerator simplifies to: √3 - 1/√3 = (3 - 1) / √3 = 2/ √3

The denominator simplifies to: 1 + √3 × 1/√3 = 1 + 1 = 2

Now, compute tan θ: tan θ = (2/√3) / 2 = 1/ √3

Since tan 30° = 1/ √3, the angle between the two lines is 30°.

Explanation

Using the fundamental trigonometric identity, sin²A + cos²A = 1, we can rewrite the term inside the square root as:

sin²A = 1 - cos²A

Taking the square root on both sides gives:

√(1 - cos²A) = sin A

Substituting this back into the original expression for x:

x = sin A / cos A

Using the quotient identity of trigonometry, the ratio of sin A to cos A is equal to tan A:

x = tan A

Explanation

The given expression consists of two inequalities that must be solved simultaneously.

First, solve the inequality: x + 2(3-5x) > -1 - 2x

Expanding and simplifying both sides: x + 6 - 10x > -1 - 2x -9x + 6 > -1 - 2x -7x > -7

Dividing by -7 and reversing the inequality sign: x < 1

Second, solve the inequality: -1 - 2x < 5 - x/3

Multiplying the entire inequality by 3 to clear the fraction: -3 - 6x < 15 - x -5x < 18

Dividing by -5 and reversing the inequality sign: x > -18/5 x > -3.6

Combining both results, x must satisfy the range: -3.6 < x < 1

Among the given options, only -3 lies within this range.

Explanation

Let the measure of the angle be x.

Two angles are complementary if their sum is 90°. Therefore, the complementary angle of x is 90° - x.

Based on the given condition: x = 4(90° - x) x = 360° - 4x 5x = 360° x = 72°

Explanation

A kite is a quadrilateral with two pairs of equal-length adjacent sides. The diagonal connecting the vertices where the unequal sides meet acts as a line of symmetry. This axis of symmetry perpendicularly bisects the other diagonal, which means that one diagonal of a kite always bisects the other.

Explanation

If a person A is n times as efficient as person B and completes a work in d days less than B, the time taken by both of them working together is given by the formula: Time together = n × d / (n² - 1)

Given that Akhilesh is n = 2 times as efficient as Bansi and finishes the job in d = 39 days less than Bansi, the time taken working together is: Time together = 2 × 39 / (2² - 1) = 78 / 3 = 26 days

Explanation

To find the value of x, solve the given linear equation: 4x - 3 = 3x + 8

Subtracting 3x from both sides: x - 3 = 8

Adding 3 to both sides: x = 11

Substitute the value of x into the expression (x + 1)³: (x + 1)³ = (11 + 1)³ = 12³

Calculating the cube of 12: 12³ = 12 × 12 × 12 = 1728

Explanation

The prime numbers between 58 and 62 are 59 and 61. The sum of these prime numbers is 59 + 61 = 120.

Explanation

Let the marked price of the item be 100 units.

After applying the initial 25% discount, the price of the item becomes: 100 - 25% of 100 = 75 units

The customer then gets 22% cash back on this discounted price of 75 units. Cash back amount = 22% of 75 = (22 / 100) × 75 = 16.5 units

The final price paid by the customer is: 75 - 16.5 = 58.5 units

The total savings from the initial price of 100 units is: 100 - 58.5 = 41.5 units

This represents an effective discount of: (41.5 / 100) × 100% = 41.5%

Explanation

To find the number, use the standard percentage formula:

Value = (Percentage / 100) × Base

Let the unknown base number be x. Substituting the given values into the formula:

39 = (5.2 / 100) × x

Rearranging the terms to solve for x:

x = (39 × 100) / 5.2

x = 3900 / 5.2

Multiplying both the numerator and the denominator by 10 to eliminate the decimal:

x = 39000 / 52

Simplifying the fraction by dividing both numbers by 13:

x = 3000 / 4

x = 750

Explanation

The simple interest is calculated using the standard formula SI = (P × R × T) / 100, where P represents the principal sum, R represents the annual rate of interest, and T represents the time period in years.

Substituting the given values into this formula: 8100 = (P × 6 × 9) / 100

8100 = 54P / 100

P = (8100 × 100) / 54

P = 150 × 100

P = 15000

Explanation

The sum of an arithmetic progression is given by the formula S = (n/2) × (a + l), where S is the sum of the series, n is the number of terms, a is the first term, and l is the last term.

Substituting the given values, first term a = 29, last term l = -49, and sum S = -140:

-140 = (n/2) × (29 - 49) -140 = (n/2) × (-20) -140 = n × (-10) n = 14

Explanation

Sum of the weights of Sudesh, Jyothi, and Ritu = 89 × 3 = 267 kg

Sum of the weights of Sudesh and Jyothi = 98 × 2 = 196 kg

Sum of the weights of Jyothi and Ritu = 83 × 2 = 166 kg

Adding the two-person sum values together: (Sudesh + Jyothi) + (Jyothi + Ritu) = 196 + 166 = 362 kg Sudesh + 2 × Jyothi + Ritu = 362 kg

Subtracting the sum of all three individuals from this equation isolates the weight of Jyothi: Jyothi = (Sudesh + 2 × Jyothi + Ritu) - (Sudesh + Jyothi + Ritu) Jyothi = 362 - 267 = 95 kg

Explanation

Total collection is the product of the ticket price and the number of visitors.

Let the initial price of the entry ticket be 7p and the increased price be 17p. Let the initial number of visitors be 9v and the decreased number of visitors be 5v.

Initial collection = 7p × 9v = 63pv New collection = 17p × 5v = 85pv

Ratio of initial collection to new collection = 63pv : 85pv = 63 : 85

Since 85 is greater than 63, the total collection has increased in the ratio 63:85.

Explanation

To simplify the expression √((1 - sin A)/(1 + sin A)), multiply both the numerator and the denominator inside the square root by (1 + sin A):

√(((1 - sin A)(1 + sin A))/((1 + sin A)(1 + sin A)))

Using the trigonometric identity 1 - sin² A = cos² A in the numerator:

√((1 - sin² A)/(1 + sin A)²) = √(cos² A/(1 + sin A)²)

Taking the square root of both the numerator and the denominator:

cos A/(1 + sin A)

Explanation

Let the cost price of the machine be CP lakhs.

When the machine is sold for Rs 43 lakhs, the loss is: Loss = CP - 43

When the machine is sold for Rs 53 lakhs, the gain is: Gain = 53 - CP

The problem states that the gain is 7 times the former loss: Gain = 7 × Loss 53 - CP = 7 × (CP - 43) 53 - CP = 7CP - 301 8CP = 354 CP = 354 / 8 = 44.25 lakhs

Explanation

To find the ratio of boys to girls, calculate the total number of boys and the total number of girls across all divisions.

Total number of boys = 30 + 30 + 25 + 30 + 40 + 20 = 175 Total number of girls = 30 + 35 + 10 + 10 + 30 + 40 = 155

The ratio of boys to girls is 175 : 155. Dividing both terms by their greatest common divisor, which is 5, yields 35 : 31.

Explanation

The number of students who have scored marks 20 or more but less than 40 is calculated by finding the difference between the number of students who scored 20 and above and those who scored 40 and above.

From the given data table: Number of students scoring 20 and above = 69 Number of students scoring 40 and above = 10

Subtracting the number of students scoring 40 and above from the number of students scoring 20 and above gives: 69 - 10 = 59

Explanation

The GDP at the beginning of 2015 is equivalent to the GDP at the end of 2014.

Using the percentage increase formula:

Value_new = Value_initial × (1 + r/100)

Here, the initial value is the GDP at the end of 2013, which is 1 trillion. The growth rate r during the year 2014 is 4%.

GDP at the beginning of 2015 = 1 trillion × (1 + 4/100) = 1.04 trillion

Explanation

The average marks of the student are calculated by dividing the sum of marks in all five subjects by the total number of subjects.

Sum of marks scored = 45 + 70 + 30 + 80 + 35 = 260

Total number of subjects = 5

Average marks scored = 260 / 5 = 52

Since five points are to be deducted from this average due to poor attendance:

Net average marks scored = 52 - 5 = 47

Explanation

Let the original price of the article be 100.

After a price cut of 31%, the new price becomes: 100 - 31 = 69

To restore the price to its original value of 100, the increase required on the new price of 69 is 31.

The required percentage increase is calculated using the formula: Percentage increase = (Increase / New Price) × 100

Substituting the values: Percentage increase = (31 / 69) × 100 Percentage increase = 3100 / 69 Percentage increase ≈ 44.93%

Explanation

The trigonometric function cotangent has a period of 180°, which gives the relation cot(180° + θ) = cot θ.

For θ = 60°: cot 240° = cot(180° + 60°) = cot 60°

Since cot 60° = cos 60° / sin 60° = (1/2) / (√3/2) = 1/√3, the value of x is 1/√3.

Explanation

To find the number of cakes Tai can bake, we first calculate the individual baking rates per hour.

Akka's baking rate = 100 cakes / 20 hours = 5 cakes/hour.

Akka and Tai's combined baking rate = 75 cakes / 10 hours = 7.5 cakes/hour.

Tai's baking rate = Combined rate - Akka's rate = 7.5 cakes/hour - 5 cakes/hour = 2.5 cakes/hour.

Using Tai's rate, the number of cakes Tai can bake in 20 hours is: 2.5 cakes/hour × 20 hours = 50 cakes.

Explanation

Let the first number be x. Since the first number is twice the second and thrice the third, the second number is x/2 and the third number is x/3.

The average of the three numbers is 396, which means their sum is: x + x/2 + x/3 = 396 × 3 11x/6 = 1188 11x = 7128 x = 648

The first number is 648 and the third number is: 648/3 = 216

The difference between the first and the third number is: 648 - 216 = 432

Explanation

Reflection of a point (x, y) in the y-axis changes the sign of its x-coordinate, yielding (-x, y). Reflection of a point (x, y) in the x-axis changes the sign of its y-coordinate, yielding (x, -y).

Applying these transformations to the point Q(a, b): First, reflecting Q(a, b) in the y-axis gives Q1(-a, b). Next, reflecting Q1(-a, b) in the x-axis gives the final coordinates (-a, -b).

We are given that this final point is (-5, 3). Comparing the coordinates: -a = -5, which gives a = 5 -b = 3, which gives b = -3

Thus, the coordinates of the initial point Q are (5, -3).

Explanation

Let CP be the cost price of the machine in lakhs.

When the machine is sold for ₹43 lakhs, the loss incurred is CP - 43.

When the machine is sold for ₹52 lakhs, the gain obtained is 52 - CP.

According to the given condition, the gain is 5 times the loss: 52 - CP = 5 × (CP - 43)

Solving the equation: 52 - CP = 5CP - 215 6CP = 267 CP = 267 / 6 = 44.5

The cost price of the machine is ₹44.5 lakhs.

Explanation

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

Rearranging the terms: sec² A - 1 = tan² A

Taking the square root on both sides gives: √(sec² A - 1) = tan A

Since the given relation is √(sec² A - 1) = x, substituting the identity yields: x = tan A

Explanation

Average speed is calculated using the formula Speed = Distance / Time.

Given: Distance = 234 km Time = 1.6 hours

Average speed in km/h = 234 / 1.6 = 146.25 km/h

To convert the speed from kilometers per hour to meters per second, multiply the value by the conversion factor 5/18.

Average speed in m/s = 146.25 × 5/18 = 40.625 m/s

Explanation

Using the fundamental trigonometric identity, sin² A + cos² A = 1, we can express cos² A as 1 - sin² A.

Substituting this into the given expression: cos² A / (1 + sin A) = (1 - sin² A) / (1 + sin A)

Applying the algebraic identity a² - b² = (a - b)(a + b) to the numerator: (1 - sin A)(1 + sin A) / (1 + sin A)

Canceling the common term (1 + sin A) from the numerator and the denominator simplifies the expression to 1 - sin A.