SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The total time duration between 10:00 PM and 6:00 AM is exactly 8 hours.

The values given in the table represent percentages of the total duration, which sum up to 100% (20 + 20 + 30 + 25 + 5 = 100).

The percentage of time spent in Light sleep is 30% and in Extremely light sleep is 25%.

The combined percentage for Light sleep or Extremely light sleep is: 30% + 25% = 55%

To find the actual time in hours, calculate 55% of the total 8 hours: 8 × 0.55 = 4.4 hours

Explanation

Using the trigonometric identity sin(-θ) = -sin(θ), the given expression can be written as:

sin(-4π/3) = -sin(4π/3)

Expressing 4π/3 in terms of π and its fractional part gives:

-sin(4π/3) = -sin(π + π/3)

Using the quadrant relation sin(π + θ) = -sin(θ), this simplifies to:

-sin(π + π/3) = -(-sin(π/3)) = sin(π/3)

Since the standard trigonometric value of sin(π/3) is √3/2, we get:

x = √3/2

Explanation

Let the number be x. Increasing a number to 107% of itself means the net increase is 7% of the original number. This relation is expressed as:

7% of x = 28

(7 / 100) × x = 28

x = (28 × 100) / 7

x = 4 × 100

x = 400

Explanation

Using the standard trigonometric expansion formulas:

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

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

Subtracting the second equation from the first yields:

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

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

Explanation

When a journey covers two equal distances at 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 values, where v₁ = 24 km/h and v₂ = 36 km/h:

Average speed = (2 × 24 × 36) / (24 + 36)

Average speed = 1728 / 60 = 28.8 km/h

Explanation

Using the standard trigonometric identity: sec²A - tan²A = 1

Applying the algebraic identity for the difference of squares, a² - b² = (a - b)(a + b), we can express this as: (secA - tanA)(secA + tanA) = 1

Since it is given that secA - tanA = x, substituting this into the identity yields: x(secA + tanA) = 1

Rearranging the terms to solve for x gives: x = 1/(secA + tanA)

Explanation

To find the percentage of students who chose Engineering who are girls, first calculate the total number of students in the Engineering stream.

Total Engineering students = Boys in Engineering + Girls in Engineering Total Engineering students = 100 + 25 = 125

The number of girls who chose Engineering is 25.

Percentage of Engineering students who are girls = (Number of girls in Engineering / Total Engineering students) × 100 Percentage of Engineering students who are girls = (25 / 125) × 100 = 20%

Explanation

The cumulative production values represent the total production from the beginning of January up to the end of each respective month.

To find the number of cars manufactured in the months of April and May combined, we subtract the cumulative production up to the end of March from the cumulative production up to the end of May:

Cumulative production up to May = 2320 Cumulative production up to March = 1450

Cars manufactured in April and May = Cumulative production up to May - Cumulative production up to March Cars manufactured in April and May = 2320 - 1450 = 870

Alternatively, we can calculate the individual production for each month and then add them together: Production in April = Cumulative production in April - Cumulative production in March = 1950 - 1450 = 500 Production in May = Cumulative production in May - Cumulative production in April = 2320 - 1950 = 370

Total production in April and May = 500 + 370 = 870

Explanation

First, find the total distance jogged during the week by summing the distance for each day: Total distance = 3.5 + 2 + 3 + 0.5 + 3.5 + 2.5 + 3.5 = 18.5 km

Next, determine the number of calories burned per kilometer: Calorie burn rate = 400 calories / 5 km = 80 calories/km

Multiply the total distance jogged by the calorie burn rate to find the total calories burned: Total calories = 18.5 km × 80 calories/km = 1480 calories

Explanation

The trigonometric function sin(330°) can be evaluated using the quadrant reduction formula. Since 330° lies in the fourth quadrant, the sine value is negative.

Using the identity sin(360° - θ) = -sin(θ) and setting θ = 30°:

sin(330°) = sin(360° - 30°) = -sin(30°)

Since sin(30°) = 1/2, we get:

sin(330°) = -1/2

Explanation

For any consecutive years under compound interest, the amount at the end of the later year is the sum of the amount at the end of the preceding year and the interest accrued on it during that year.

Therefore, the interest earned in the 3rd year is the difference between the amounts at the end of the 3rd year and the 2nd year: Interest = 23820.32 - 22472 = 1348.32

The rate of interest (r) can be calculated using the formula: r = (Interest / Amount at the end of 2nd year) × 100 r = (1348.32 / 22472) × 100 r = 134832 / 22472 r = 6%

Explanation

The reflection of any point (x, y) with respect to the origin is given by the formula (-x, -y). By substituting the coordinates of the point (2, -4) into this formula, the reflected point becomes (-2, -(-4)), which simplifies to (-2, 4).

Explanation

The sum of the interior angles of a triangle is always 180°. Let the measures of the three angles in the ratio 3:4:5 be 3x, 4x, and 5x.

Sum of angles = 3x + 4x + 5x = 180° 12x = 180° x = 15°

The largest angle corresponds to the largest part of the ratio, which is 5.

Largest angle = 5 × 15° = 75°

Explanation

If the third proportional of two numbers a and b is c, then the relation between them is given by the formula a/b = b/c, which simplifies to c = b²/a.

Substituting the given values, a = 24 and b = 36:

c = 36² / 24 c = (36 × 36) / 24 c = 1296 / 24 c = 54

Explanation

Cost price (CP) of 21 apples = ₹7 CP of 1 apple = 7/21 = ₹1/3

Selling price (SP) of 5 apples = ₹2 SP of 1 apple = ₹2/5

Since SP is greater than CP, there is a gain. Gain = SP - CP = 2/5 - 1/3 = 1/15

Gain percent = (Gain / CP) × 100 Gain percent = (1/15) / (1/3) × 100 = (1/15) × 3 × 100 = (1/5) × 100 = 20%

Explanation

Using the fundamental trigonometric identity: 1 + cot²A = cosec²A

Taking the square root on both sides gives: √(1 + cot²A) = cosecA

Substituting this into the given equation: x = 1/cosecA

Since sinA and cosecA are reciprocal trigonometric ratios, we have: 1/cosecA = sinA

Thus, the expression simplifies directly to: x = sinA

Explanation

The given compound inequality consists of two separate inequalities that must be satisfied simultaneously.

First, solve 3x - 5(x - 1) > 2x - 1: 3x - 5x + 5 > 2x - 1 -2x + 5 > 2x - 1 6 > 4x x < 1.5

Second, solve 2x - 1 < 2 + 4x: -1 - 2 < 4x - 2x -3 < 2x x > -1.5

Combining these two results gives the range of x as: -1.5 < x < 1.5

The value -1 satisfies this inequality, as -1.5 < -1 < 1.5.

Explanation

The total yearly expenses are calculated by summing the costs of all items: Total expenses = 12 + 6 + 6 + 4 + 4 = 32 lakh

The combined expense of Labour and Interest is: Labour + Interest = 6 + 4 = 10 lakh

To find the percentage of these combined expenses relative to the total, use the percentage formula: Percentage = (Combined expense / Total expenses) × 100% Percentage = (10 / 32) × 100% = 31.25%

Explanation

Using the algebraic identity for the square of a binomial: (x - y)² = x² - 2xy + y²

We can rearrange this formula to solve for the required term: x² + y² = (x - y)² + 2xy

Substitute the given values x - y = -8 and xy = 45 into this equation: x² + y² = (-8)² + 2(45) x² + y² = 64 + 90 x² + y² = 154

Explanation

A square is a regular quadrilateral with all four sides of equal length. If the length of each side of the square is represented by s, the formula for calculating its area is the product of its length and width. Since all sides are equal, the area is s × s, which is written as s². This mathematical relation shows that the area of a square is equal to the square of its side length.

Explanation

Let Mahmood's efficiency be 1 unit per hour. Since Mahmood completes the work in 40 hours, the total work is 40 × 1 = 40 units.

Janaki is 50% more efficient than Mahmood, which means Janaki's efficiency is 1.5 units per hour.

When working together, their combined efficiency is 1 + 1.5 = 2.5 units per hour.

The time required for both of them to finish the work together is: Total work / Combined efficiency = 40 / 2.5 = 16 hours.

Explanation

Let the two-digit number be represented as 10x + y, where x is the tens digit and y is the units digit.

The product of the digits is given as 24: x × y = 24

When 45 is added to the number, the digits are interchanged: 10x + y + 45 = 10y + x 9y - 9x = 45 y - x = 5 y = x + 5

Substituting y = x + 5 into the product equation: x × (x + 5) = 24 x² + 5x - 24 = 0 (x + 8)(x - 3) = 0

Since x must be a positive single digit, we get x = 3. Using this value to find y: y = 3 + 5 = 8

Therefore, the tens digit is 3 and the units digit is 8, which gives the number 38. Adding 45 to 38 results in 83, which indeed has the digits interchanged.

Explanation

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

The formula for the volume of a cylinder is V_cylinder = π × R² × H, where R is the radius and H is the height of the cylinder.

The formula for the volume of a cone is V_cone = ⅓ × π × r² × h, where r is the radius and h is the height of the cone.

Given dimensions: Radius of the cylindrical vessel (R) = 11 cm Height of the cylindrical vessel (H) = 9 cm Radius of the cone (r) = 10 cm Height of the cone (h) = 8 cm

Substitute these values to find the volumes: Volume of the cylinder = π × 11² × 9 = 1089π cm³ Volume of the cone = ⅓ × π × 10² × 8 = 800/3 × π cm³

The volume of water required is: Volume of water = Volume of cylinder - Volume of cone Volume of water = 1089π - 800/3 × π Volume of water = (3267 - 800)/3 × π Volume of water = 2467/3 × π cm³

Using the approximation π ≈ 22/7: Volume of water = 2467/3 × 22/7 = 54274/21 ≈ 2584.48 cm³

Explanation

To simplify the expression x = sin A / (1 + cos A), we multiply both the numerator and the denominator by the conjugate of the denominator, which is 1 - cos A:

x = (sin A × (1 - cos A)) / ((1 + cos A) × (1 - cos A))

Using the algebraic identity (a + b)(a - b) = a² - b² in the denominator, we get:

x = (sin A × (1 - cos A)) / (1 - cos² A)

Since 1 - cos² A = sin² A, the expression simplifies to:

x = (sin A × (1 - cos A)) / sin² A

By canceling sin A from both the numerator and the denominator, we obtain:

x = (1 - cos A) / sin A

We can now split this into two separate fractions:

x = (1 / sin A) - (cos A / sin A)

Using the reciprocal and quotient identities, where cosec A = 1 / sin A and cot A = cos A / sin A, we arrive at:

x = cosec A - cot A

Explanation

To find the average speed in meters per second, the formula is:

Speed = Distance / Time

Given: Distance = 252 km Time = 2.5 hours

First, calculate the average speed in km/h: Speed = 252 / 2.5 = 100.8 km/h

To convert speed from km/h to m/s, multiply by 5/18: Speed = 100.8 × 5/18 = 28 m/s

Explanation

The effective discount on a combination of items can be calculated directly using the weighted average of the discount percentages, since the marked price of each individual item is the same.

Effective discount = (d₁ × q₁ + d₂ × q₂) / (q₁ + q₂)

Substituting the given values: q₁ = 2 items with discount d₁ = 24% q₂ = 5 items with discount d₂ = 38%

Effective discount = (24 × 2 + 38 × 5) / (2 + 5) Effective discount = (48 + 190) / 7 Effective discount = 238 / 7 = 34%

Explanation

To factorise the quadratic expression 24x² - 54x - 15, we first factor out the greatest common divisor of the coefficients, which is 3:

3(8x² - 18x - 5)

Next, we factor the quadratic expression inside the parentheses, 8x² - 18x - 5. We find two numbers that multiply to 8 × (-5) = -40 and add up to -18. These numbers are -20 and 2.

Splitting the middle term using these numbers gives:

8x² - 20x + 2x - 5

Grouping the terms to factor by grouping:

4x(2x - 5) + 1(2x - 5)

Factoring out the common binomial term (2x - 5) yields:

(2x - 5)(4x + 1)

Combining this with the initial common factor of 3 results in the final factored form:

3(2x - 5)(4x + 1)

Explanation

Let the third number be 100.

The first number is 40% less than 100, which gives: 100 - 40 = 60

The second number is 50% less than 100, which gives: 100 - 50 = 50

To make the second number equal to the first number, it must be increased by: 60 - 50 = 10

The percentage enhancement needed on the second number is calculated by dividing the required increase by the second number and multiplying by 100: (10 / 50) × 100% = 20%

Explanation

The slope m₁ of the line x - 6y = 8 can be found by rewriting it in the slope-intercept form, y = (1/6)x - 4/3, which gives m₁ = 1/6.

Since two perpendicular lines have slopes whose product is -1, the slope m₂ of the line passing through the points (5, a) and (4, 3) must be: m₂ = -1 / m₁ = -1 / (1/6) = -6

The formula for the slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is: m₂ = (y₂ - y₁) / (x₂ - x₁)

Substituting the coordinates (5, a) and (4, 3) into the formula: -6 = (3 - a) / (4 - 5) -6 = (3 - a) / (-1) 6 = 3 - a a = -3

Explanation

The prime numbers between 18 and 32 are 19, 23, 29, and 31.

Summing these numbers: 19 + 23 + 29 + 31 = 102

Explanation

Correct Answer: D

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

For Ashu, the ratio is 68 ÷ 1.6 = 42.5 kg/m. For Aslam, the ratio is 62 ÷ 1.75 ≈ 35.43 kg/m. For Ayush, the ratio is 63 ÷ 1.59 ≈ 39.62 kg/m. For Ayushman, the ratio is 50 ÷ 1.56 ≈ 32.05 kg/m.

Ayushman has the lowest value among all the calculated ratios.

Explanation

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

Average = (∑ (x × f)) / (∑ f)

where x represents the number of children in a house and f represents the number of houses (frequency).

First, calculate the total number of children in the society by multiplying each number of children by its corresponding frequency and summing the products: (0 × 3) + (1 × 7) + (2 × 12) + (3 × 7) = 0 + 7 + 24 + 21 = 52

Next, calculate the total number of houses in the society by summing the frequencies: 3 + 7 + 12 + 7 = 29

Dividing the total number of children by the total number of houses gives: 52 / 29 ≈ 1.793

Explanation

The company has issued a total of 6 lakh shares, which is equivalent to 6,00,000 shares.

According to the data table, Dhiraj currently holds a 5% share in the company.

Dhiraj's initial number of shares is calculated as: 5% of 6,00,000 = (5/100) × 6,00,000 = 30,000 shares

When Basu sells 15,000 of his shares to Dhiraj, Dhiraj's share count increases by 15,000.

Dhiraj's new total number of shares = 30,000 + 15,000 = 45,000 shares.

Explanation

In compound interest, the amount at the end of any year acts as the principal for the next year. Therefore, the interest earned in the third year is the difference between the amounts at the end of the third and second years.

Interest earned in the 3rd year = Amount after 3 years - Amount after 2 years Interest = 33745.92 - 32448 = 1297.92

The rate of interest is calculated as: Rate = (Interest ÷ Principal for that year) × 100 Rate = (1297.92 ÷ 32448) × 100 Rate = 0.04 × 100 = 4%

Explanation

Using the trigonometric identity for the cotangent of a double angle, we have:

cot(2A) = (cot²A - 1) / 2cotA

This identity is derived from the double angle formula for tangent:

tan(2A) = 2tanA / (1 - tan²A)

Taking the reciprocal of both sides to express it in terms of cotangent:

cot(2A) = (1 - tan²A) / 2tanA

Substituting tanA = 1/cotA into the expression:

cot(2A) = (1 - 1/cot²A) / (2/cotA)

Simplifying the numerator by taking the common denominator:

cot(2A) = ((cot²A - 1) / cot²A) / (2/cotA)

Multiplying by the reciprocal of the denominator:

cot(2A) = ((cot²A - 1) / cot²A) × (cotA / 2)

cot(2A) = (cot²A - 1) / 2cotA

Explanation

Correct Answer: B

Let the side of the square be x. The area of the square is given by x² = 6√3 sq cm.

The length of the diagonal of the square, d, is related to its side by the formula d = x√2. Squaring both sides gives d² = 2x².

Substituting the value of x²: d² = 2 × 6√3 = 12√3 sq cm

Since the side of the equilateral triangle, s, equals the diagonal of the square, s = d, which means s² = d² = 12√3 sq cm.

The area of an equilateral triangle is given by the formula (√3/4)s². Substituting the value of s² into this formula: Area of the equilateral triangle = (√3/4) × 12√3 = 3 × (√3 × √3) = 3 × 3 = 9 sq cm

Explanation

The values of the trigonometric functions at 30° are: sec 30° = 2/√3 cot 30° = √3

Substituting these values into the expression: sec 30° + cot 30° = 2/√3 + √3

Taking the common denominator: (2 + (√3 × √3)) / √3 = (2 + 3) / √3 = 5/√3

Explanation

The given series 61 + 62 + 63 + ... + 110 is an arithmetic progression with the first term a = 61, the last term l = 110, and a common difference of 1.

The total number of terms n is calculated as: n = 110 - 61 + 1 = 50

The formula for the sum of an arithmetic progression is: Sₙ = (n/2) × (a + l)

Substituting the values into the formula: S₅₀ = (50/2) × (61 + 110) S₅₀ = 25 × 171 S₅₀ = 4275

Explanation

Volume of a cube is given by the formula V = a³, where a is the side length of the cube.

Given: a³ = 512 cm³ a = ∛512 = 8 cm

The total surface area of a cube is given by the formula: TSA = 6a² TSA = 6 × 8² = 6 × 64 = 384 cm²

Explanation

The cost of a daily ticket is ₹160. If daily tickets are purchased for 26 days of travel, the total cost would be: 26 × ₹160 = ₹4160

The cost of the Monthly Pass is ₹2912.

The total amount saved by purchasing the Monthly Pass instead of individual daily tickets is: ₹4160 - ₹2912 = ₹1248

The percentage of savings is calculated as: (Savings ÷ Cost of daily tickets) × 100 (1248 ÷ 4160) × 100 = 30%

Explanation

Time taken by Prahlad to complete the entire job is 30 × 3 = 90 days.

The remaining work is 1 - 1/3 = 2/3 of the job. Since Sarfaraz completes this 2/3rd of the job in 90 days, the time taken by Sarfaraz to complete the entire job alone is 90 × 3/2 = 135 days.

Using the combined work formula, the number of days they take together to complete the job is: (90 × 135) / (90 + 135) = 12150 / 225 = 54 days.

Explanation

Solving the first inequality 2x - 3 ≤ 5 + x by subtracting x and adding 3 to both sides gives x ≤ 8.

Solving the second inequality 5 - x < 1 + 5x by adding x and subtracting 1 on both sides gives 4 < 6x, which simplifies to x > 2/3.

Combining these two results, the valid range for x is 2/3 < x ≤ 8. The value 1 satisfies this condition because 2/3 < 1 ≤ 8.

Explanation

To restore a value that has been decreased by r% to its original level, the required percentage increase is given by the formula:

Percentage increase = (r / (100 - r)) × 100%

Substituting r = 9 into the formula:

Percentage increase = (9 / (100 - 9)) × 100% Percentage increase = (9 / 91) × 100% Percentage increase = 900 / 91 % ≈ 9.89%

Explanation

To find the ratio A : B : C from the given equation 13A = 11B = 8C, we can equate them to a constant k:

A = k/13 B = k/11 C = k/8

Thus, the ratio A : B : C is: A : B : C = 1/13 : 1/11 : 1/8

To express this as a ratio of integers, we multiply each term by the product of the denominators, which is 13 × 11 × 8: A : B : C = (11 × 8) : (13 × 8) : (13 × 11) A : B : C = 88 : 104 : 143

Explanation

Let the selling price of one shoe be SP. The selling price of 21 shoes is 21 × SP.

The profit earned is equal to the selling price of 5 shoes, which is 5 × SP.

Using the relation: Cost Price (CP) = Selling Price (SP) - Profit

The cost price of 21 shoes is: CP = 21 × SP - 5 × SP = 16 × SP

The formula for profit percentage is: Profit Percentage = (Profit / Cost Price) × 100

Substituting the values: Profit Percentage = (5 × SP / 16 × SP) × 100 Profit Percentage = (5 / 16) × 100 Profit Percentage = 31.25%

Explanation

The perimeter of a sector is the sum of its arc length and the two bounding radii. The formula for the perimeter of a sector is:

Perimeter = (θ / 360°) × 2πr + 2r

Substituting the given values θ = 60°, r = 21 cm, and π = 22/7 into the formula:

Arc length = (60 / 360) × 2 × (22 / 7) × 21 Arc length = (1 / 6) × 2 × 22 × 3 Arc length = 22 cm

Using the arc length to find the total perimeter:

Perimeter = 22 + 2 × 21 Perimeter = 22 + 42 Perimeter = 64 cm

Explanation

The discriminant D of a quadratic equation ax² + bx + c = 0 determines the nature of its roots and is given by the formula: D = b² - 4ac

For a quadratic equation to have real roots, its discriminant must be greater than or equal to zero (D ≥ 0).

Applying this formula to the equation 3x² - 5x + 2 = 0, we identify the coefficients as a = 3, b = -5, and c = 2.

Substituting these values into the formula yields: D = (-5)² - 4 × 3 × 2 D = 25 - 24 D = 1

Since the discriminant is greater than zero, the equation possesses two distinct real roots.

Explanation

The area of a trapezium is calculated using the formula:

Area = ½ × (sum of parallel sides) × (distance between the parallel sides)

Substituting the given values into the formula:

Sum of parallel sides = 12.4 cm Distance between the parallel sides = 3.5 cm

Area = ½ × 12.4 × 3.5 Area = 6.2 × 3.5 Area = 21.7 sq cm

Explanation

Let the four numbers be a, b, c, and d.

The average of the four numbers is 91, which means their total sum is: a + b + c + d = 4 × 91 = 364

We are given that the first number, a, is 3/10 times the sum of the other three numbers: a = (3/10) × (b + c + d)

This relation can be rewritten to express the sum of the other three numbers in terms of a: b + c + d = 10a / 3

Substituting this expression back into the total sum equation: a + 10a / 3 = 364 13a / 3 = 364 13a = 364 × 3 13a = 1092 a = 1092 / 13 a = 84

Explanation

To find the solution for the system of linear equations, write the equations as: 4x + 3y = 11 3x + 2y = 8

Multiply the first equation by 2 and the second equation by 3 to eliminate the y terms: 8x + 6y = 22 9x + 6y = 24

Subtract the first equation from the second to solve for x: (9x - 8x) + (6y - 6y) = 24 - 22 x = 2

Substitute the value of x into the second equation: 3(2) + 2y = 8 6 + 2y = 8 2y = 2 y = 1

This yields the solution (x, y) = (2, 1).