SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The average speed of an object is calculated using the formula:

Average Speed = Distance / Time

Substituting the given values:

Distance = 189 km Time = 3.5 hours

Average Speed = 189 / 3.5 = 54 km/h

To convert the speed from kilometers per hour (km/h) to meters per second (m/s), multiply by 5/18:

Average Speed = 54 × 5/18 = 15 m/s

Explanation

To simplify the expression, find a common denominator for the two terms:

sec A / (sec A - 1) + sec A / (sec A + 1)

= [sec A (sec A + 1) + sec A (sec A - 1)] / [(sec A - 1)(sec A + 1)]

Multiply the terms in the numerator and the denominator:

= (sec²A + sec A + sec²A - sec A) / (sec²A - 1)

= 2sec²A / (sec²A - 1)

Using the trigonometric identity sec²A - 1 = tan²A, substitute this into the denominator:

= 2sec²A / tan²A

Express sec A and tan A in terms of sin A and cos A:

= 2(1/cos²A) / (sin²A/cos²A)

= 2 / sin²A

= 2cosec²A

Explanation

The slope m₁ of the line passing through the points (x₁, y₁) = (-6, 3) and (x₂, y₂) = (2, 7) is calculated using the formula:

m₁ = (y₂ - y₁) / (x₂ - x₁)

Substituting the given values:

m₁ = (7 - 3) / (2 - (-6)) = 4 / 8 = ½

The slope m₂ of a line perpendicular to another line with slope m₁ satisfies the relation:

m₂ × m₁ = -1

Substituting m₁ = ½ into the equation:

m₂ × ½ = -1 m₂ = -2

Explanation

Total number of employees = 3 + 8 + 3 = 14

The annual salary is given in lakhs of rupees, where 1 lakh = 100,000 rupees.

The total bonus for each category of employees is calculated as follows:

Total bonus for Managers = 3 × 60,00,000 × 50% = 90,00,000 rupees

Total bonus for Executives = 8 × 20,00,000 × 30% = 48,00,000 rupees

Total bonus for Trainees = 3 × 4,00,000 × 10% = 1,20,000 rupees

The total bonus for all 14 employees combined is: 90,00,000 + 48,00,000 + 1,20,000 = 1,39,20,000 rupees

The average bonus per employee is: 1,39,20,000 ÷ 14 = 994285.71 ≈ 994286 rupees

Explanation

The total exports from all three companies together for the year 2014 is: 1000 + 4000 + 3000 = 8000 lakh rupees

The total exports from all three companies together for the year 2015 is: 3000 + 3000 + 2000 = 8000 lakh rupees

Comparing these sums, the total exports for both 2014 and 2015 are equal to each other.

Explanation

To find the total profit or loss of the company over the last 5 years, we sum the annual profit and loss values:

Total = (-20) + 10 + 5 + (-15) + 15 = -5 crores

A negative result represents a net loss of ₹5 crores.

Explanation

To find the percentage of Steel exports out of the total exports, first calculate the sum of all exports from the given table:

Total exports = 650 + 725 + 750 + 825 + 650 = 3600 million USThe export value of Steel is 825 million US.

Using the percentage formula:

Percentage of Steel = (Steel exports / Total exports) × 100

Percentage of Steel = (825 / 3600) × 100 = 825 / 36 = 22.916... %

Rounding to two decimal places, this is equal to 22.92%.

Explanation

If two individuals can complete a project in x and y days respectively, their combined work rate per day is 1/x + 1/y. Here, both P and Q can complete the project in 12 days, so x = 12 and y = 12.

Their combined rate of work per day is: 1/12 + 1/12 = 2/12 = 1/6 of the project per day.

The number of days required to complete the entire project (100%) working together is 6 days.

To find the number of days required to complete 50% of the project, we multiply the total time by 0.5: 6 × 0.5 = 3 days.

Explanation

We can solve the compound inequality by splitting it into two parts.

First, solve the left inequality: 4(x + 5) - 3 > 6 - 4x 4x + 20 - 3 > 6 - 4x 4x + 17 > 6 - 4x 8x > -11 x > -11/8 x > -1.375

Second, solve the right inequality: 6 - 4x ≥ x - 5 11 ≥ 5x x ≤ 11/5 x ≤ 2.2

Combining both parts gives the range for x: -1.375 < x ≤ 2.2

Among the given choices, only the value -1 lies within this interval. Substituting x = -1 back into the expression yields 13 > 10 ≥ -6, which confirms the inequality holds true.

Explanation

The reflection of a point (x, y) in the origin changes the signs of both coordinates to (-x, -y). Applying this to the point R(a, b) gives the coordinates of R1 as (-a, -b).

The reflection of a point (x, y) in the x-axis changes only the sign of the y-coordinate to (x, -y). Applying this transformation to R1(-a, -b) results in the coordinates (-a, b).

We are given that this final point is (-4, 2). Equating the coordinates: -a = -4, which gives a = 4 b = 2

Thus, the coordinates of the original point R(a, b) are (4, 2).

Explanation

The slope of the line QR containing Q(2, -1) and R(0, 5) is calculated as: m = (5 - (-1)) / (0 - 2) = 6 / -2 = -3

Since the altitude PS is perpendicular to the side QR, the slope of PS is the negative reciprocal of the slope of QR: Slope of PS = -1 / -3 = 1/3

Using the point-slope form of a linear equation, the equation of the line PS passing through P(1, 2) with a slope of 1/3 is: y - 2 = (1/3)(x - 1)

Multiplying the entire equation by 3 gives: 3y - 6 = x - 1

Rearranging the terms to standard form yields: x - 3y = -5

Explanation

The given expression is sec 300° = x.

The angle 300° lies in the fourth quadrant, where the secant and cosine functions are positive.

Using the trigonometric identity: sec(360° - θ) = sec θ

We can express 300° as: sec 300° = sec(360° - 60°) = sec 60°

Since secant is the reciprocal of cosine: sec 60° = 1 / cos 60°

Given that cos 60° = ½, we substitute this value: sec 60° = 1 / (½) = 2

Thus, x = 2.

Explanation

Average speed is calculated using the formula:

Average Speed = Distance / Time

Substituting the given values into the formula:

Average Speed = 225 km / 2.5 hours = 90 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 = 90 × 5/18 = 25 m/s

Explanation

Let the required angle be x. The complementary angle of x is 90° - x.

According to the problem, the angle is greater than its complementary angle by 30°: x = (90° - x) + 30° 2x = 120° x = 60°

Explanation

Total cost price of the rice is ₹3,600. To earn a profit of 15%, the total selling price must be:

Total Selling Price = Cost Price × (1 + Profit Percentage/100) = 3600 × 1.15 = ₹4,140

The initial quantity of rice bought is 8 quintals. Since 10% of the rice is lost in transportation, the quantity available for sale is:

Remaining Rice = 8 × (1 - 0.10) = 7.2 quintals

To find the required selling rate per quintal:

Selling Rate = Total Selling Price / Remaining Rice = 4140 / 7.2 = ₹575 per quintal

Explanation

To find the sum of 3/5 and 7/9, first determine the least common multiple of the denominators 5 and 9, which is 45.

Convert each fraction to have this common denominator: 3/5 = (3 × 9) / (5 × 9) = 27/45 7/9 = (7 × 5) / (9 × 5) = 35/45

Now, add the numerators together over the common denominator: 27/45 + 35/45 = (27 + 35) / 45 = 62/45

Explanation

Total marked price of 3 items = 3 × ₹200 = ₹600

The purchase of 3 items is treated as a combination of a group of 2 items and 1 single item to apply the given discount rates.

Discount on the group of 2 items = 14% of (2 × ₹200) = 14% of ₹400 = ₹56

Discount on the remaining 1 item = 5% of ₹200 = ₹10

Total discount obtained = ₹56 + ₹10 = ₹66

Effective discount percentage = (Total Discount / Total Marked Price) × 100 = (66 / 600) × 100 = 11%

Explanation

The area of a circle is given by the formula A = πr², which means the area is directly proportional to the square of the radius. When a quantity increases by a certain percentage twice (since r² = r × r), the overall percentage increase can be determined using the successive percentage change formula:

Net percentage change = x + y + (xy / 100)

Since the radius increases by 16%, we substitute x = 16 and y = 16 into the formula:

Net percentage increase in area = 16 + 16 + (16 × 16 / 100) Net percentage increase in area = 32 + (256 / 100) Net percentage increase in area = 32 + 2.56 Net percentage increase in area = 34.56%

Explanation

To expand the expression, use the algebraic identities (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b².

Expanding the first term: (x + 4)² = x² + 8x + 16

Expanding the second term: (x - 2)² = x² - 4x + 4

Adding these two expanded expressions together: (x² + 8x + 16) + (x² - 4x + 4) = 2x² + 4x + 20

Factoring out the common factor of 2: 2(x² + 2x + 10)

Explanation

The volume of the remaining cube is calculated by subtracting the volume of the hollowed-out cone from the total volume of the solid wooden cube.

Remaining Volume = Volume of the cube - Volume of the cone

The volume of a cube with side s is given by the formula s³. Since the side of the cube is 6 cm: Volume of the cube = 6³ = 216 cubic cm

The volume of a cone with radius r and height h is given by the formula ⅓ × π × r² × h. Since the diameter of the cone is equal to the side of the cube (6 cm), the radius r is 3 cm. The height h is also 6 cm. Substituting these values: Volume of the cone = ⅓ × π × 3² × 6 = 18 × π cubic cm

Using π ≈ 22/7: Volume of the cone ≈ 18 × 22/7 = 396/7 ≈ 56.57 cubic cm

Remaining Volume = 216 - 56.57 = 159.43 cubic cm, which closely matches 159.42 cubic cm.

Explanation

In any triangle ABC, D is the midpoint of the side CB, which gives CD = DB = CB/2. Since CB = 14 cm, we have CD = DB = 7 cm. The length of the median AD is also given as 7 cm, which means AD = CD = DB = 7 cm. This equality shows that the point D is equidistant from all three vertices A, B, and C. Therefore, D is the circumcenter of the triangle ABC, making CB the diameter of its circumcircle. Since any angle subtended by a diameter at the circumference of a circle is a right angle, the measure of angle CAB is 90°.

Explanation

Let the first, second, and third numbers be x, y, and z respectively.

According to the conditions: x = 2y which gives y = x/2 x = 3z which gives z = x/3

The average of the three numbers is 198: (x + y + z) / 3 = 198 x + y + z = 594

Substitute y and z in terms of x: x + x/2 + x/3 = 594

Multiply the entire equation by the common denominator 6: 6x + 3x + 2x = 594 × 6 11x = 3564 x = 324

The third number z is: z = 324 / 3 = 108

The difference between the first and the third number is: x - z = 324 - 108 = 216

Explanation

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

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

Substituting this into the given equation for x: x = sec A / tan A

Expressing the trigonometric functions in terms of sine and cosine: sec A = 1/cos A tan A = sin A/cos A

Substituting these into the expression: x = (1/cos A) / (sin A/cos A) x = 1/sin A x = cosec A

Explanation

Let G, H, and S represent the salaries of Gangadhar, Hari, and Shambhu respectively.

According to the given conditions: G = (3/11) × H S = (3/4) × H

To find the ratio of Gangadhar's salary to Shambhu's salary, we take the ratio G : S: G/S = ((3/11) × H) / ((3/4) × H) G/S = (3/11) / (3/4) G/S = (3/11) × (4/3) G/S = 4/11

This gives the required ratio of 4:11.

Explanation

The algebraic identity for the sum of cubes of two numbers is: a³ + b³ = (a + b)³ - 3ab(a + b)

Substituting the given values, a + b = 8 and ab = 15, into the identity: a³ + b³ = 8³ - 3 × 15 × 8 a³ + b³ = 512 - 360 a³ + b³ = 152

Alternatively, the two numbers can be found by observation. Since the sum of the two numbers is 8 and their product is 15, the numbers are 3 and 5. The sum of their cubes is: 3³ + 5³ = 27 + 125 = 152

Explanation

Using the algebraic identity (x + y)² + (x - y)² = 2(x² + y²), the given expression can be expanded and simplified directly.

Let x = 1 and y = cotA.

Substituting these values into the identity yields: (1 + cotA)² + (1 - cotA)² = 2(1² + cot²A) = 2(1 + cot²A)

Using the standard trigonometric identity 1 + cot²A = cosec²A, this simplifies to: 2cosec²A

Explanation

Let the number be x. According to the problem, the sum of twice the number and thrice its reciprocal is 25/2. This can be written as:

2x + 3/x = 25/2

Substituting the value x = 6 into the equation:

2(6) + 3/6 = 12 + ½ = 12.5 = 25/2

Since the left-hand side equals the right-hand side, the number is 6.

Explanation

The standard formula for Simple Interest is SI = (P × R × T) / 100, where SI represents the Simple Interest, P is the Principal (sum), R is the rate of interest per year, and T is the time period in years.

Substituting the given values into the formula: 7728 = (P × 7 × 8) / 100

Simplifying the equation: 7728 = (P × 56) / 100

Rearranging the terms to solve for the principal P: P = (7728 × 100) / 56

P = 138 × 100

P = 13800

Explanation

To find the average bonus, first compute the total bonus paid to all employees.

The annual salary is given in lakhs of rupees, where one lakh equals 100,000 rupees.

For managers: Number of managers = 3 Annual salary per manager = 60 lakhs = ₹6,000,000 Bonus per manager = 30% of ₹6,000,000 = ₹1,800,000 Total bonus for managers = 3 × ₹1,800,000 = ₹5,400,000

For executives: Number of executives = 6 Annual salary per executive = 12 lakhs = ₹1,200,000 Bonus per executive = 30% of ₹1,200,000 = ₹360,000 Total bonus for executives = 6 × ₹360,000 = ₹2,160,000

For trainees: Number of trainees = 4 Annual salary per trainee = 4 lakhs = ₹400,000 Bonus per trainee = 20% of ₹400,000 = ₹80,000 Total bonus for trainees = 4 × ₹80,000 = ₹320,000

Summing these values: Total bonus for all employees = ₹5,400,000 + ₹2,160,000 + ₹320,000 = ₹7,880,000 Total number of employees = 3 + 6 + 4 = 13

The average bonus is the total bonus divided by the total number of employees: Average bonus = ₹7,880,000 ÷ 13 ≈ ₹606,153.85

Rounding to the nearest whole rupee yields ₹606,154.

Explanation

The total exports from the three companies together for the year 2011 is calculated as: Total exports in 2011 = 3000 + 5000 + 3000 = 11000 lakh rupees

The total exports from the three companies together for the year 2015 is calculated as: Total exports in 2015 = 2000 + 4000 + 5000 = 11000 lakh rupees

Both 2011 and 2015 have equal total exports of 11000 lakh rupees.

Explanation

The total profit or loss of the company over the five-year period is calculated by taking the algebraic sum of the profits and losses of each year:

Total Profit or Loss = 5 + 15 + (-20) + (-15) + (-5) crore

Total Profit or Loss = 20 - 40 = -20 crore

The negative sign represents a net loss, which means the company incurred a total loss of Rs 20 crores over the five years.

Explanation

To find the percentage of software exports relative to the total exports, first calculate the sum of all export values from the given data:

Total exports = 550 + 700 + 875 + 775 + 550 = 3450 million USThe export value of software is 700 million US.

Using the percentage formula:

Software percentage = (Software exports / Total exports) × 100 Software percentage = (700 / 3450) × 100 ≈ 20.29%

Explanation

The midpoint formula for a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by ((x₁ + x₂)/2, (y₁ + y₂)/2).

Let the coordinates of B be (x, y). Since P(5, -4) is the midpoint of AB with A(7, 0):

5 = (7 + x)/2 10 = 7 + x x = 3

-4 = (0 + y)/2 -8 = 0 + y y = -8

The coordinates of B are (3, -8).

Explanation

Using the algebraic identity (a - b)(a + b) = a² - b², we can simplify the multiplication by choosing a convenient base value.

Let a = 300000 and b = 4.

The expression can be rewritten as: 299996 × 300004 = (300000 - 4)(300000 + 4)

Applying the identity: = 300000² - 4² = 90000000000 - 16 = 89999999984

Explanation

The weighted mean formula is used to find the combined mean of the three divisions:

Weighted mean = (n₁x̄₁ + n₂x̄₂ + n₃x̄₃) / (n₁ + n₂ + n₃)

Here, the number of students in divisions A, B, and C are n₁ = 35, n₂ = 45, and n₃ = 70, with respective means of x̄₁ = 65, x̄₂ = 61, and x̄₃ = 59.

Dividing the number of students by their greatest common divisor, 5, simplifies the ratio of students to 7 : 9 : 14.

Substituting these simplified values into the formula:

Mean = (7 × 65 + 9 × 61 + 14 × 59) / (7 + 9 + 14)

Mean = (455 + 549 + 826) / 30

Mean = 1830 / 30 = 61

Explanation

To find the value of x, we solve the compound inequality by splitting it into two parts.

First, we solve the left inequality: 5(4 - x) - 4 > 5x - 2 20 - 5x - 4 > 5x - 2 16 - 5x > 5x - 2 18 > 10x x < 1.8

Next, we solve the right inequality: 5x - 2 > 2x - 6 3x > -4 x > -4/3 x > -1.33

Combining both results gives the range for x: -1.33 < x < 1.8

Among the choices, -1 is the only value that lies within this interval, as -1.33 < -1 < 1.8.

Explanation

In a rectangle ABCD, the adjacent sides DA and AB meet at a right angle, meaning they are perpendicular to each other.

The product of the slopes of two perpendicular lines is always -1.

Let the slope of side DA be m₁ and the slope of side AB be m₂. Since DA is perpendicular to AB: m₁ × m₂ = -1

Given that the slope of side DA is 5/3: (5/3) × m₂ = -1

Solving for m₂: m₂ = -3/5

Explanation

The cosine function is even, which means cos(-225 degrees) is equal to cos(225 degrees). Since 225 degrees lies in the third quadrant where cosine is negative, cos(225 degrees) is equal to -cos(45 degrees), which simplifies to -1/√2.

Explanation

In a right-angled triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of this altitude is the geometric mean of the lengths of the two segments.

BD² = AD × DC

Substituting the given values of AD = 18 cm and DC = 50 cm:

BD² = 18 × 50 BD² = 900 BD = √900 BD = 30 cm

Explanation

Simple Interest SI = P × R × T / 100

An increase of 40% in 10 years implies that the simple interest earned is 40% of the principal P.

0.40 × P = P × R × 10 / 100 R = 4% per annum

For compound interest on ₹ 25000 for 2 years at 4% per annum, the effective rate of interest is calculated using the successive percentage formula:

Effective Rate = R + R + (R × R)/100 Effective Rate = 4 + 4 + (4 × 4)/100 = 8.16%

Compound Interest = P × Effective Rate / 100 Compound Interest = 25000 × 8.16 / 100 = 250 × 8.16 = 2040

Explanation

The given expression can be simplified using standard trigonometric identities.

We know the fundamental Pythagorean identity: 1 + cot² A = cosec² A

Substituting this into the expression inside the square root: x = √((1 + cot² A) / cot² A) x = √(cosec² A / cot² A) x = cosec A / cot A

Now, expressing cosec A and cot A in terms of sin A and cos A: cosec A = 1 / sin A cot A = cos A / sin A

Substituting these values into the simplified expression: x = (1 / sin A) / (cos A / sin A) x = 1 / cos A x = sec A

Explanation

The given expression is (cosec A - cot A) / (sec A + tan A).

Using the standard trigonometric identity cosec² A - cot² A = 1, we can factor it as: (cosec A - cot A)(cosec A + cot A) = 1 cosec A - cot A = 1 / (cosec A + cot A)

Similarly, using the standard identity sec² A - tan² A = 1, we can factor it as: (sec A - tan A)(sec A + tan A) = 1 1 / (sec A + tan A) = sec A - tan A

Substituting these expressions back into the original formula: (cosec A - cot A) / (sec A + tan A) = (cosec A - cot A) × (1 / (sec A + tan A)) = (1 / (cosec A + cot A)) × (sec A - tan A) = (sec A - tan A) / (cosec A + cot A)

Explanation

Let the maximum marks be x.

The student obtained 54 marks and failed by 36 marks. This means the minimum passing marks required are: 54 + 36 = 90

According to the problem, the passing marks correspond to 45% of the maximum marks. This relationship is represented by the formula: 45% of x = 90

Solving for x: (45 / 100) × x = 90 x = 90 × (100 / 45) x = 2 × 100 x = 200

Explanation

The relation between Cost Price (CP) and Marked Price (MP) for a given discount (d%) and profit (p%) is:

CP/MP = (100 - d) / (100 + p)

For the first case where the discount is 25% and the profit is 25%:

CP/MP = (100 - 25) / (100 + 25) = 75/125 = 3/5

For the second case where the discount is 10% and the profit is p%:

CP/MP = (100 - 10) / (100 + p) = 90 / (100 + p)

Equating both expressions for CP/MP:

3/5 = 90 / (100 + p)

Solving for p:

300 + 3p = 450

3p = 150

p = 50%

Explanation

The volume of sand remains constant when poured from the cylindrical vessel to form a right circular cone. Therefore, the volume of the cylinder equals the volume of the cone.

Volume of cylinder = π × r₁² × h₁ Volume of cone = ⅓ × π × r₂² × h₂

Equating both volumes: π × r₁² × h₁ = ⅓ × π × r₂² × h₂

Canceling π on both sides: r₁² × h₁ = ⅓ × r₂² × h₂

Substitute the given values where radius of the cylinder (r₁) is 7 cm, height of the cylinder (h₁) is 9 cm, and radius of the cone (r₂) is 10 cm: 7² × 9 = ⅓ × 10² × h₂ 49 × 9 = ⅓ × 100 × h₂ 441 = 100/3 × h₂ h₂ = (441 × 3) / 100 h₂ = 1323 / 100 h₂ = 13.23 cm

Explanation

Rate of work is given by the formula: Rate = Work / Time

Munni's baking rate = 70 / 7 = 10 cakes per hour

Combined rate of Munni and Guddi = 90 / 6 = 15 cakes per hour

Guddi's baking rate = Combined rate - Munni's rate Guddi's baking rate = 15 - 10 = 5 cakes per hour

Number of cakes Guddi can bake in 10 hours = Guddi's rate × Time Cakes = 5 × 10 = 50 cakes

Explanation

A rectangle is a special type of parallelogram in which all four interior angles are right angles. By definition, a fundamental property of all parallelograms, including rectangles, is that their diagonals bisect each other. This means that the point of intersection of the two diagonals divides each diagonal into two equal segments. This property holds true because the opposite sides of a rectangle are parallel and equal, which leads to congruent triangles being formed by the diagonals, ensuring they intersect at their mutual midpoints.

Explanation

The original fare and the increased fare are in the ratio 25:29.

Let the original fare be 25x and the new fare be 29x.

Given that the original fare is Rs. 625: 25x = 625 x = 25

The increase in the fare is the difference between the new fare and the original fare: Increase = 29x - 25x = 4x

Substituting the value of x: Increase = 4 × 25 = Rs. 100

Explanation

The algebraic identity for the difference of two squares is x² - y² = (x - y)(x + y). Substituting the given values x - y = -6 and x + y = 16 into this formula yields x² - y² = (-6) × 16 = -96.

Explanation

Let the three consecutive natural numbers be n, n + 1, and n + 2, where n represents the smallest of the three numbers.

The problem states that the square of the greatest number is greater than the product of the other two numbers by 16. This relation is expressed as:

(n + 2)² = n(n + 1) + 16

Expanding both sides of the equation yields:

n² + 4n + 4 = n² + n + 16

Subtracting n² from both sides simplifies the equation to:

4n + 4 = n + 16

Rearranging the terms to solve for n:

3n = 12

n = 4