SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The area of a circle is given by the formula A = πr², where r is the radius of the circle.

Using the given area: (22/7) × r² = 616 r² = 616 × 7/22 r² = 28 × 7 r² = 196 r = 14 cm

The circumference of a circle is given by the formula C = 2πr. Substituting the value of r: C = 2 × (22/7) × 14 C = 2 × 22 × 2 C = 88 cm

Explanation

Expand and simplify the given algebraic equation:

3x - 8(2 - x) = -19

Distribute the -8 over the terms inside the parentheses:

3x - 16 + 8x = -19

Combine the like terms containing x:

11x - 16 = -19

Add 16 to both sides of the equation to isolate the x term:

11x = -19 + 16

11x = -3

Divide both sides by 11 to solve for x:

x = -3/11

Explanation

By using the algebraic identity for the square of a binomial:

(x - y)² = x² + y² - 2xy

We can rearrange this formula to solve for x² + y²:

x² + y² = (x - y)² + 2xy

Substituting the given values x - y = 6 and xy = 40 into the equation:

x² + y² = 6² + 2(40)

x² + y² = 36 + 80

x² + y² = 116

Explanation

Let the tens and units digits of the two-digit number be x and y, respectively. The value of the number is 10x + y.

The product of the digits is given as: x × y = 27

When 54 is added to the number, the digits are interchanged, giving: (10x + y) + 54 = 10y + x 9y - 9x = 54 y - x = 6 y = x + 6

Substituting y = x + 6 into the product equation: x(x + 6) = 27 x² + 6x - 27 = 0 (x + 9)(x - 3) = 0

Since x represents a digit of a two-digit number, x must be a positive integer, which gives x = 3. Using y = x + 6, we get y = 3 + 6 = 9.

The two-digit number is 10(3) + 9 = 39.

Explanation

The given compound inequality is solved in two parts:

First part: 2x - 3(2x - 2) > x - 1 2x - 6x + 6 > x - 1 -4x + 6 > x - 1 7 > 5x x < 7/5 x < 1.4

Second part: x - 1 < 2 + 2x -1 - 2 < 2x - x -3 < x

Combining both parts gives the range of x as: -3 < x < 1.4

The value -2 lies within this range, satisfying both conditions of the inequality.

Explanation

Let the unknown number be x. According to the problem:

0.8% of x = 29

(0.8 / 100) × x = 29

8/1000 × x = 29

x = (29 × 1000) / 8

x = 29 × 125

x = 3625

Explanation

The average bonus is calculated by dividing the total bonus paid to all employees by the total number of employees.

First, calculate the total bonus paid for each category:

Managers: 3 employees × ₹30,00,000 × 30% = ₹27,00,000 Executives: 8 employees × ₹16,00,000 × 20% = ₹25,60,000 Trainees: 4 employees × ₹2,00,000 × 20% = ₹1,60,000

Total bonus paid = 27,00,000 + 25,60,000 + 1,60,000 = ₹54,20,000

Total number of employees = 3 + 8 + 4 = 15

Average bonus = 54,20,000 ÷ 15 = ₹3,61,333.33

Explanation

To find the pair of years for which the total exports from the three companies together are equal, we calculate the sum of exports for each year.

For the year 2011: Total exports = Company A + Company B + Company C Total exports = 3000 + 1000 + 4000 = 8000 lakh rupees

For the year 2012: Total exports = Company A + Company B + Company C Total exports = 5000 + 1000 + 2000 = 8000 lakh rupees

Both 2011 and 2012 have an identical total export value of 8000 lakh rupees.

Explanation

The total profit or loss of the company over the 5 years is calculated by taking the sum of the profit or loss of each individual year.

Total Profit or Loss = (-10) + 5 + 10 + (-15) + (-5) = -15 crore

Since the resulting sum is negative, it indicates a net loss of Rs 15 crores over the five-year period.

Explanation

To find the percentage of Jewellery exports relative to the total exports, the percentage formula is used:

Percentage = (Value of Jewellery / Total Exports) × 100

First, calculate the total value of India's exports in 2015 by summing the individual values from the table: Total Exports = 675 + 500 + 525 + 575 + 725 = 3000 million USNext, calculate the percentage of Jewellery exports: Percentage of Jewellery = (675 / 3000) × 100 = 22.5 percent

Explanation

The trigonometric function satisfies sin(-θ) = -sin(θ). Using this property: sin(-5π/3) = -sin(5π/3)

The angle 5π/3 lies in the fourth quadrant and can be written as 2π - π/3. Using the identity sin(2π - θ) = -sin(θ), we get: sin(5π/3) = sin(2π - π/3) = -sin(π/3)

Substituting this value back into the first equation: sin(-5π/3) = -(-sin(π/3)) = sin(π/3)

Since the standard trigonometric value of sin(π/3) is √3/2, the value of x is √3/2.

Explanation

The standard trigonometric identity for the tangent of the sum of two angles A and B is:

tan(A + B) = (tan A + tan B) / (1 - tan A tan B)

Comparing this identity with the given equation:

x = (tan A + tan B) / (1 - tan A tan B)

It directly follows that x equals tan(A + B).

Explanation

Using the standard trigonometric identity: cosec²A - cot²A = 1

We can factor the left side of the equation as a difference of squares: (cosecA - cotA)(cosecA + cotA) = 1

Dividing both sides by (cosecA - cotA) gives: cosecA + cotA = 1/(cosecA - cotA)

Since we are given that x = 1/(cosecA - cotA), substituting this into the equation yields: x = cosecA + cotA

Explanation

To find the gain or loss percentage, the cost price and selling price can be calculated for a common quantity of oranges. Taking the least common multiple of 5 and 2, we can assume the vendor buys and sells 10 oranges.

Cost Price of 10 oranges = (Rs 6 / 5) × 10 = Rs 12

Selling Price of 10 oranges = (Rs 3 / 2) × 10 = Rs 15

Since the selling price is greater than the cost price, the vendor makes a profit.

Gain = Selling Price - Cost Price = 15 - 12 = Rs 3

Gain percentage = (Gain / Cost Price) × 100% = (3 / 12) × 100% = 25%

Explanation

The difference (D) between the compound interest (compounded annually) and the simple interest on a principal sum (P) for a period of 2 years at an annual interest rate of r% is given by the formula:

D = P × (r / 100)²

Given that the difference is Rs 320 and the rate of interest is 16% per annum, we substitute these values into the formula:

320 = P × (16 / 100)²

320 = P × (256 / 10000)

P = (320 × 10000) / 256

P = (5 × 10000) / 4

P = 5 × 2500

P = 12500

Explanation

Let the first number be x and the sum of the other four numbers be S.

Since the average of the five numbers is 76, their total sum is: 5 × 76 = 380

The first number is 3/7 times the sum of the other four numbers: x = 3/7 × S S = 7/3 × x

The total sum of all five numbers is: x + S = 380

Substituting the value of S into the equation: x + 7/3 × x = 380 10/3 × x = 380 x = (380 × 3) / 10 x = 114

Explanation

The equation of a line passing through a point (x₁, y₁) with slope m is given by the point-slope form: y - y₁ = m(x - x₁)

The slope m of the line is defined by the tangent of the angle θ it makes with the positive x-axis: m = tan(θ) m = tan(-45°) = -1

Substituting the point (2, -3) and the slope m = -1 into the equation: y - (-3) = -1(x - 2) y + 3 = -x + 2 x + y = -1

Explanation

Solving the first inequality: 2(3x - 2) < 6 - 3x 6x - 4 < 6 - 3x 9x < 10 x < 10/9

Solving the second inequality: 6x + 2(6 - x) > 2x - 2 6x + 12 - 2x > 2x - 2 4x + 12 > 2x - 2 2x > -14 x > -7

Combining both results gives the solution interval: -7 < x < 10/9

Since 10/9 is approximately 1.11, the value -5 is the only value that lies within this range, satisfying both conditions.

Explanation

The given equation is: (36 - 16x) - (4x - 8) = 4

Expanding the terms and removing the parentheses: 36 - 16x - 4x + 8 = 4

Combining the constant terms and the variable terms: 44 - 20x = 4

Subtracting 44 from both sides: -20x = 4 - 44 -20x = -40

Dividing both sides by -20: x = -40 / -20 x = 2

Explanation

The effective discount can be directly calculated using the weighted average formula:

Effective Discount = (d₁ × n₁ + d₂ × n₂) / (n₁ + n₂)

Here, n₁ = 4 items with a discount d₁ = 24%, and n₂ = 1 item with a discount d₂ = 6%.

Substituting these values into the formula:

Effective Discount = (24 × 4 + 6 × 1) / (4 + 1)

Effective Discount = (96 + 6) / 5

Effective Discount = 102 / 5 = 20.4%

Explanation

In a right-angled triangle, the relation between the side opposite to an angle θ and the hypotenuse h (which is the side opposite to the 90° angle) is given by: sin(θ) = opposite / hypotenuse

Substituting θ = 60° and opposite = 6√3: sin(60°) = 6√3 / h

Using the value sin(60°) = √3/2: √3/2 = 6√3 / h

Solving for h: h = (6√3 × 2) / √3 h = 12 cm

Explanation

Using the trigonometric identity cot(180° + θ) = cot θ, we can find the value of cot 210° by writing 210° as 180° + 30°. Since 210° lies in the third quadrant where the cotangent function is positive:

cot 210° = cot(180° + 30°) = cot 30° = √3

Explanation

Using the algebraic identity (a - b)(a + b) = a² - b², the multiplication can be simplified by expressing the two numbers in terms of their relation to 200000:

199994 × 200006 = (200000 - 6) × (200000 + 6)

Applying the formula:

(200000)² - 6² = 40000000000 - 36 = 39999999964

Explanation

To convert speed from kilometres per hour (km/h) to metres per second (m/s), we multiply the speed by the conversion factor 5/18.

Using this relation: Speed in m/s = Speed in km/h × 5/18

Substituting the given speed of 1260 km/h: Speed = 1260 × 5/18 Speed = 70 × 5 Speed = 350 m/s

Explanation

The trigonometric identity for the cotangent of the sum of two angles, cot(A + B), is derived using the sum formulas for sine and cosine:

cot(A + B) = cos(A + B) / sin(A + B)

Using the expansion of cos(A + B) and sin(A + B):

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

Dividing both the numerator and the denominator by (sin A sin B) to convert the terms into cotangent:

Numerator: (cos A cos B) / (sin A sin B) - (sin A sin B) / (sin A sin B) = cot A cot B - 1

Denominator: (sin A cos B) / (sin A sin B) + (cos A sin B) / (sin A sin B) = cot B + cot A

This simplifies the relation directly to:

cot(A + B) = (cot A cot B - 1) / (cot B + cot A)

Explanation

The relationship between Selling Price (SP), Cost Price (CP), and Loss percentage is given by: CP = SP / (1 - Loss%)

Let the initial selling price be SP₁ = ₹1250 and the loss percentage be 7%. CP = 1250 / (1 - 0.07) = 1250 / 0.93

The new selling price is SP₂ = ₹1375. To find the new profit or loss, calculate the ratio of the new selling price to the cost price: SP₂ / CP = SP₂ / (SP₁ / 0.93) = (SP₂ / SP₁) × 0.93

Simplifying the ratio of the two selling prices: SP₂ / SP₁ = 1375 / 1250 = 1.1

Substituting this back to find SP₂ / CP: SP₂ / CP = 1.1 × 0.93 = 1.023

Since the ratio SP₂ / CP is greater than 1, there is a profit. Gain% = (1.023 - 1) × 100% = 2.3% gain

Explanation

The volume of a cylinder is calculated using the formula V = πr²h, where V is the volume, r is the radius of the base, and h is the height of the cylinder.

Substituting the given values into the formula: 3850 = (22/7) × r² × 25

Rearranging the equation to solve for r²: r² = (3850 × 7) / (22 × 25) r² = 26950 / 550 r² = 49

Taking the square root of both sides: r = √49 r = 7 cm

Explanation

By definition, tan A = sin A / cos A. Substituting this into the given expression:

x = tan² A - sin² A x = (sin² A / cos² A) - sin² A

Factoring out sin² A from both terms:

x = sin² A (1 / cos² A - 1)

Since 1 / cos² A = sec² A, the expression simplifies to:

x = sin² A (sec² A - 1)

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

x = sin² A tan² A

Rearranging the factors:

x = tan² A sin² A

Explanation

Let Sagar's efficiency be 1 unit/day. Since Sagar takes 18 days to complete the job alone, the total work is 18 units. Madhur works 2 times faster than Sagar, which means Madhur's efficiency is 2 units/day. Their combined efficiency when working together is 1 + 2 = 3 units/day. Therefore, the time taken by both of them together to finish the work is 18 / 3 = 6 days.

Explanation

The ratio of the original bus fare to the increased bus fare is given as 11 : 18. This means that if the original fare is represented as 11 units, the new fare is represented as 18 units.

The increase in the fare corresponds to the difference between the new fare and the original fare: Increase in fare = 18 units - 11 units = 7 units

The original fare is given as ₹550. Therefore, 11 units = ₹550 Value of 1 unit = 550 / 11 = ₹50

To find the increase in fare: Increase in fare = 7 units = 7 × 50 = ₹350

Explanation

Euler's formula for any convex polyhedron relates the number of faces (F), vertices (V), and edges (E) as:

F + V - E = 2

A regular dodecahedron consists of 12 pentagonal faces, which means F = 12. The number of edges is given as E = 30.

Substituting these values into Euler's formula:

12 + V - 30 = 2

V - 18 = 2

V = 20

Explanation

By using the algebraic identity for the square of a binomial:

(x + y)² = x² + y² + 2xy

Substitute the given values x² + y² = 100 and xy = 22 into the formula:

(x + y)² = 100 + 2 × 22 (x + y)² = 100 + 44 (x + y)² = 144

Taking the square root on both sides:

x + y = √144 = 12

Explanation

A quadratic equation ax² + bx + c = 0 has equal roots when its discriminant, D = b² - 4ac, is equal to zero.

For the equation 4x² + 8x + 4 = 0, the coefficients are a = 4, b = 8, and c = 4.

Calculating the discriminant: D = 8² - 4 × 4 × 4 D = 64 - 64 D = 0

Since the discriminant is zero, the equation has real and equal roots. Alternatively, factoring the equation gives 4(x² + 2x + 1) = 4(x + 1)² = 0, which yields two equal roots of x = -1.

Explanation

By the section formula, if a point P divides the line segment joining A(x₁, y₁) and B(x₂, y₂) internally in the ratio m:n, the coordinates of P are given by: P_x = (m × x₂ + n × x₁) / (m + n) P_y = (m × y₂ + n × y₁) / (m + n)

Substituting the coordinates of the points A(x, 0), B(0, y), the ratio m:n = 2:5, and the coordinates of P(5, -2): 5 = (2 × 0 + 5 × x) / (2 + 5) 5 = 5x / 7 5x = 35 x = 7

Similarly, for the y-coordinate: -2 = (2 × y + 5 × 0) / (2 + 5) -2 = 2y / 7 2y = -14 y = -7

Explanation

Let the marks obtained by the two students be x and y, where x is greater than y.

Based on the given information, the difference between their marks is 20: x - y = 20

And the higher mark is 55% of the sum of their marks: x = 0.55 × (x + y)

Substituting x = y + 20 into this equation: y + 20 = 0.55 × (2y + 20) y + 20 = 1.1y + 11 0.1y = 9 y = 90

Using the first relation to solve for x: x = 90 + 20 = 110

Explanation

To find the average bonus, first calculate the total bonus paid to each category of employees:

For Managers: Number of managers = 3 Annual salary per manager = ₹54,00,000 Bonus per manager = 60% of ₹54,00,000 = ₹32,40,000 Total bonus for managers = 3 × ₹32,40,000 = ₹97,20,000

For Executives: Number of executives = 6 Annual salary per executive = ₹16,00,000 Bonus per executive = 20% of ₹16,00,000 = ₹3,20,000 Total bonus for executives = 6 × ₹3,20,000 = ₹19,20,000

For Trainees: Number of trainees = 1 Annual salary per trainee = ₹2,00,000 Bonus per trainee = 20% of ₹2,00,000 = ₹40,000 Total bonus for trainees = 1 × ₹40,000 = ₹40,000

Now, calculate the overall values: Total bonus = ₹97,20,000 + ₹19,20,000 + ₹40,000 = ₹1,16,80,000 Total number of employees = 3 + 6 + 1 = 10

Average bonus per employee = Total bonus ÷ Total number of employees Average bonus = ₹1,16,80,000 ÷ 10 = ₹11,68,000

Explanation

To find the correct pair of years, we calculate the sum of exports for all three companies for the years 2012 and 2014.

Total exports in 2012 = 2000 + 1000 + 5000 = 8000 lakh rupees.

Total exports in 2014 = 3000 + 1000 + 4000 = 8000 lakh rupees.

Since the total exports in both 2012 and 2014 are equal to 8000 lakh rupees, this pair of years has equal total exports.

Explanation

To find the total profit or loss of the company over the 5 years, we calculate the sum of the profits and losses for each year:

Total Profit or Loss = Profit/Loss in 2011 + Profit/Loss in 2012 + Profit/Loss in 2013 + Profit/Loss in 2014 + Profit/Loss in 2015

Substituting the given values:

Total = (-10) + (-20) + (-5) + 20 + 25

Total = -35 + 45

Total = 10

A positive result of 10 indicates a net profit of Rs 10 crores over the 5-year period.

Explanation

Total value of exports is calculated by summing the values of all categories: Total exports = 500 + 825 + 550 + 850 + 750 = 3475 million USThe value of software exports is 825 million US.

To find what percent software exports represent of the total exports, the following formula is used: Percentage = (Software exports / Total exports) × 100 Percentage = (825 / 3475) × 100 ≈ 23.74%

Explanation

To divide 24a²b² by 6b², we divide the numerical coefficients and the algebraic terms separately:

(24a²b²) / (6b²) = (24 / 6) × a² × (b² / b²)

Calculating the values gives:

24 / 6 = 4 b² / b² = 1

Substituting these values back into the expression yields:

4 × a² × 1 = 4a²

Explanation

Let the speed of the Express train be E and the speed of Rajdhani be R. Let the time taken by Rajdhani to travel 612 km be t hours.

Initially, the Express train takes 9 hours more than Rajdhani, so its time is t + 9 hours.

When the speed of the Express train is doubled, its travel time is halved, becoming (t + 9)/2 hours. At this doubled speed, it takes 3 hours less than Rajdhani:

(t + 9)/2 = t - 3

t + 9 = 2t - 6

t = 15

Thus, Rajdhani takes 15 hours to cover the 612 km distance.

The speed of Rajdhani is:

Speed = Distance / Time = 612 / 15 = 40.8 km/hr

Explanation

We use the standard trigonometric identity: cosec²A - 1 = cot²A

Substituting this into the given expression: x = 1 / √(cosec²A - 1) x = 1 / √cot²A x = 1 / cotA

Using the reciprocal identity 1 / cotA = tanA, we obtain: x = tanA

Explanation

Pradeep completes 1/4 of the job in 14 days. Therefore, the time taken by Pradeep to complete the entire job alone is 14 × 4 = 56 days.

The remaining part of the job is 1 - 1/4 = 3/4. Saquib completes this remaining 3/4 of the job in 56 days. Thus, the time taken by Saquib to complete the entire job alone is 56 × 4/3 = 224/3 days.

In one day, Pradeep does 1/56 of the job, and Saquib does 3/224 of the job.

Their combined work in one day is 1/56 + 3/224 = (4 + 3)/224 = 7/224.

The number of days they will take to complete the job together is the reciprocal of their combined daily work, which is 224/7 = 32 days.

Explanation

To find the value of cosec 3π/4, we use the reciprocal relationship cosec θ = 1/sin θ.

The angle 3π/4 lies in the second quadrant, where the sine function is positive. We can express the angle as: sin(3π/4) = sin(π - π/4) = sin(π/4)

Since sin π/4 = 1/√2, we have: cosec 3π/4 = 1/sin(3π/4) = 1/(1/√2) = √2

Explanation

Let the two consecutive natural numbers be x and x + 1.

The sum of their squares is given as: x² + (x + 1)² = 145

Expanding and simplifying this quadratic equation: x² + x² + 2x + 1 = 145 2x² + 2x - 144 = 0

Dividing the entire equation by 2 gives: x² + x - 72 = 0

Factoring the quadratic equation: (x + 9)(x - 8) = 0

This gives x = 8 or x = -9. Since the question specifies natural numbers, x must be positive, so x = 8.

The consecutive natural numbers are 8 and 9. This is verified by checking the sum of their squares: 8² + 9² = 64 + 81 = 145.

Explanation

We simplify the given trigonometric expression by taking a common denominator:

[sin A / (1 + cos A)] + [(1 + cos A) / sin A]

= [sin² A + (1 + cos A)²] / [sin A (1 + cos A)]

Expanding the numerator:

= [sin² A + 1 + 2 cos A + cos² A] / [sin A (1 + cos A)]

Since sin² A + cos² A = 1, we can substitute this to simplify:

= [1 + 1 + 2 cos A] / [sin A (1 + cos A)]

= [2 + 2 cos A] / [sin A (1 + cos A)]

= 2(1 + cos A) / [sin A (1 + cos A)]

Canceling the common term (1 + cos A) from both the numerator and the denominator gives:

= 2 / sin A

= 2 cosec A

Explanation

Any two-digit number can be expressed in terms of the place value of its digits as: Number = 10 × (digit at tens place) + (digit at unit place)

Given that the digit at the tens place is 8 and the digit at the unit place is z, substituting these values into the formula yields: Number = 10 × 8 + z = 80 + z

Explanation

Let the selling price of one watch be SP and the cost price of one watch be CP.

According to the given condition, the profit on selling 17 watches is equal to the selling price of 7 watches.

Profit on 17 watches = Selling Price of 17 watches - Cost Price of 17 watches 7 × SP = 17 × SP - 17 × CP 17 × CP = 17 × SP - 7 × SP 17 × CP = 10 × SP CP / SP = 10 / 17

This gives the ratio of the cost price of one watch to its selling price as 10 to 17. Thus, for a cost price of 10, the profit earned is 17 - 10 = 7.

Profit percentage = (Profit / Cost Price) × 100 Profit percentage = (7 / 10) × 100 = 70%

Explanation

The combined mean of the three divisions is calculated using the weighted mean formula:

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

For Division-A, n₁ = 30 and x̄₁ = 67. For Division-B, n₂ = 55 and x̄₂ = 63. For Division-C, n₃ = 40 and x̄₃ = 61.

Substituting these values:

Weighted Mean = (30 × 67 + 55 × 63 + 40 × 61) / (30 + 55 + 40) Weighted Mean = (2010 + 3465 + 2440) / 125 Weighted Mean = 7915 / 125 Weighted Mean = 63.32

Explanation

In a parallelogram ABCD, the opposite sides are parallel. This means that the side AD is parallel to the side BC, and consequently, the slope of line AD is equal to the slope of line BC.

The slope m of the line passing through the points B(-2, 3) and C(-1, 4) is calculated using the slope formula: m = (y₂ - y₁) / (x₂ - x₁) m = (4 - 3) / (-1 - (-2)) = 1 / 1 = 1

Since AD is parallel to BC, the slope of line AD is also 1.

Using the point-slope form, the equation of the line AD passing through the point A(5, 0) is: y - y₁ = m(x - x₁) y - 0 = 1(x - 5) y = x - 5