SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

We can simplify the expression by finding a common denominator for the two fractions:

1/(1 + cosA) + 1/(1 - cosA) = [(1 - cosA) + (1 + cosA)] / [(1 + cosA)(1 - cosA)]

Simplifying the numerator and the denominator, we get:

Numerator = 1 - cosA + 1 + cosA = 2 Denominator = (1 + cosA)(1 - cosA) = 1 - cos²A

Using the fundamental trigonometric identity sin²A + cos²A = 1, we can write the denominator as sin²A:

1/(1 + cosA) + 1/(1 - cosA) = 2/sin²A

Since cosecA = 1/sinA, the expression simplifies to:

2cosec²A

This is equal to the given value of x.

Explanation

Correct Answer: D

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

Total Boys = 40 + 35 + 15 + 35 + 40 + 20 = 185

Total Girls = 10 + 25 + 10 + 35 + 30 + 20 = 130

The ratio of boys to girls is 185 : 130. Dividing both terms of the ratio by their common factor of 5 yields the simplified ratio:

185/5 : 130/5 = 37 : 26

Explanation

To find the number of students who scored 20 or more but less than 40 marks, subtract the number of students who scored 40 and above from the number of students who scored 20 and above.

From the given cumulative data table: Number of students scoring 20 and above = 61 Number of students scoring 40 and above = 25

Therefore, the number of students scoring 20 or more but less than 40 is: 61 - 25 = 36

Explanation

The beginning of the year 2015 corresponds to the end of the year 2014. Since the GDP at the end of 2013 (which is the beginning of 2014) was 3 trillion, and the GDP growth rate for the year 2014 was 4%, the GDP at the beginning of 2015 can be calculated as:

GDP at the beginning of 2015 = GDP at the end of 2013 × (1 + GDP growth rate for 2014 / 100)

Substituting the given values:

GDP at the beginning of 2015 = 3 × (1 + 4 / 100) = 3 × 1.04 = 3.12 trillion

Explanation

The average marks scored by the student across the five subjects is calculated by dividing the sum of the marks by the total number of subjects.

Sum of marks = 30 + 65 + 80 + 75 + 70 = 320 Total number of subjects = 5 Average marks = 320 / 5 = 64

After deducting 5 points for poor attendance, the student's net average marks scored is: Net average marks = 64 - 5 = 59

Explanation

To factorise the quadratic expression x² + 3x - 18, we look for two numbers whose product is the constant term -18 and whose sum is the coefficient of the middle term 3.

These two numbers are 6 and -3, since: 6 × (-3) = -18 6 + (-3) = 3

Splitting the middle term using these numbers: x² + 3x - 18 = x² + 6x - 3x - 18

Grouping the terms to factorise: x(x + 6) - 3(x + 6) = (x + 6)(x - 3)

Explanation

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

Relative speed = 17 + 22 = 39 km/hr

The time they travel before meeting is 44 minutes. Converting this time into hours:

Time = 44/60 hours = 11/15 hours

Using the distance formula:

Distance = Relative speed × Time

Distance = 39 × 11/15 = 13 × 11/5 = 143/5 = 28.6 km

Explanation

A quadratic equation with a given sum of roots S and product of roots P is expressed by the formula: x² - Sx + P = 0

Given that the sum of the roots S = -7 and the product of the roots P = 12, substituting these values into the relation yields: x² - (-7)x + 12 = 0 x² + 7x + 12 = 0

Explanation

Using the formula for successive percentage discounts: Effective Discount = d₁ + d₂ - (d₁ × d₂) / 100

Here, the first discount d₁ is 20% and the second discount d₂ (received as cash back) is 10%.

Substituting these values into the formula: Effective Discount = 20 + 10 - (20 × 10) / 100 Effective Discount = 30 - 2 Effective Discount = 28%

Alternatively, if the marked price of the item is ₹100, a 20% discount reduces the price to ₹80. Applying a 10% cash back on this amount saves an additional ₹8, making the final effective cost ₹72. This represents a total saving of ₹28 on the original ₹100, which corresponds to an effective discount of 28%.

Explanation

To reduce the fraction 2530/1430 to its lowest terms, first simplify it by dividing both the numerator and the denominator by their common factor of 10. This gives 253/143.

Next, find the greatest common divisor of 253 and 143. Using the divisibility rule for 11, we find that both numbers are divisible by 11: 253 ÷ 11 = 23 143 ÷ 11 = 13

Dividing both the numerator and the denominator of 253/143 by 11 yields 23/13. Since both 23 and 13 are prime numbers, the fraction is in its simplest form.

Explanation

To find the range of x, solve the two inequalities separately.

For the first inequality: 5x - 3 ≥ 3 + x/2 Multiply the entire inequality by 2: 10x - 6 ≥ 6 + x Subtract x and add 6 to both sides: 9x ≥ 12 x ≥ 4/3

For the second inequality: 4x - 2 ≤ 6 + x Subtract x and add 2 to both sides: 3x ≤ 8 x ≤ 8/3

Combining both results gives the range: 4/3 ≤ x ≤ 8/3

Since 4/3 is approximately 1.33 and 8/3 is approximately 2.67, the value 2 satisfies this inequality range.

Explanation

The sum of an arithmetic progression when the first and last terms are known is given by the formula:

S = (n / 2) × (a + l)

In this formula, S represents the sum of the series, n represents the number of terms, a is the first term, and l is the last term.

Substituting the given values:

-88 = (n / 2) × (32 - 43)

-88 = (n / 2) × (-11)

-176 = -11n

n = 16

Explanation

The difference between compound interest and simple interest on a principal sum P for 2 years at an annual interest rate R is given by the formula:

Difference = P × (R / 100)²

Substituting the given values, where Difference = ₹81 and R = 18%:

81 = P × (18 / 100)² 81 = P × (324 / 10000)

Rearranging the equation to solve for P:

P = (81 × 10000) / 324 P = 10000 / 4 P = 2500

Explanation

The line segment joining the points (x₁, y₁) and (x₂, y₂) is divided by the Y-axis in the ratio -x₁ : x₂.

For the given points (12, -1) and (-3, 4), the x-coordinates are x₁ = 12 and x₂ = -3.

Substituting these values into the formula gives: -12 : -3 = 4 : 1

This positive ratio indicates that the Y-axis divides the segment internally in the ratio 4:1.

Explanation

The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula m = (y₂ - y₁) / (x₂ - x₁).

Since parallel lines have equal slopes, the slope m₁ of the line passing through (4,3) and (y,0) must equal the slope m₂ of the line passing through (-1,-2) and (3,0).

First, compute the slope m₂: m₂ = (0 - (-2)) / (3 - (-1)) = 2/4 = ½

Next, express the slope m₁: m₁ = (0 - 3) / (y - 4) = -3 / (y - 4)

Since m₁ = m₂: -3 / (y - 4) = ½

Cross-multiplying to solve for y: y - 4 = -6 y = -2

Explanation

Let the number of 1-rupee, 50-paise, and 10-paise coins be 2x, 3x, and 5x respectively.

Converting the coin values into rupees: Value of 1-rupee coin = ₹1.00 Value of 50-paise coin = ₹0.50 Value of 10-paise coin = ₹0.10

The total value of all coins in the bag is ₹20. We can express the sum of values as: (2x × 1.00) + (3x × 0.50) + (5x × 0.10) = 20 2x + 1.5x + 0.5x = 20 4x = 20 x = 5

The number of 50-paise coins is 3x: 3 × 5 = 15

Explanation

The total weight of the class is calculated by multiplying the total number of students by the average weight of the full class: 60 × 66.5 = 3990 kg

The total weight of the girls in the class is: 33 × 62 = 2046 kg

The number of boys in the class is: 60 - 33 = 27 boys

The total weight of the boys is the difference between the total weight of the class and the total weight of the girls: 3990 - 2046 = 1944 kg

The average weight of the boys is: 1944 ÷ 27 = 72 kg

Explanation

Let the number be x. Since increasing the number by 32 makes it 116% of itself, the increase of 32 corresponds directly to the 16% increment over the original 100%.

This relation can be written as: 16% of x = 32

Substituting this into the standard percentage formula: (16 / 100) × x = 32 x = (32 × 100) / 16 x = 2 × 100 x = 200

Explanation

The sum of the interior angles of an n-sided polygon is given by the formula: Sum = (n - 2) × 180°

Substituting the given sum of 1800° into the formula: (n - 2) × 180° = 1800°

Dividing both sides by 180°: n - 2 = 10

Adding 2 to both sides yields: n = 12

Explanation

To find the time taken by P and Q working together to complete the entire project, we use the work formula T = (a × b) / (a + b), where a and b represent the individual times taken by P and Q.

Substituting the given values: T = (60 × 30) / (60 + 30) = 1800 / 90 = 20 days

Since they only need to complete 90% of the project, we calculate 90% of the total combined time: Required time = 90% of 20 = 0.90 × 20 = 18 days

Explanation

The total surface area (TSA) of a hemisphere with radius r is given by the formula: TSA = 3πr²

The curved surface area (CSA) of a hemisphere is given by the formula: CSA = 2πr²

By dividing the formula for curved surface area by the formula for total surface area, we obtain the direct relation: CSA = (2/3) × TSA

Substituting the given value of the total surface area: CSA = (2/3) × 166.32 CSA = 2 × 55.44 CSA = 110.88 sq cm

Explanation

Let the cost price of the pass be CP and the marked price be MP.

When a discount of 20% is given, the selling price is 80% of the marked price: SP = 0.80 × MP

Since the profit earned is 44%, the selling price is also 144% of the cost price: SP = 1.44 × CP

Equating the two expressions for SP: 0.80 × MP = 1.44 × CP MP = (1.44 / 0.80) × CP = 1.8 × CP

When a 29% discount is offered, the new selling price is 71% of the marked price: New SP = 0.71 × MP New SP = 0.71 × 1.8 × CP = 1.278 × CP

This represents a profit of: Profit % = (1.278 - 1) × 100% = 27.8%

Explanation

Using the trigonometric identity tan(-θ) = -tan(θ), the expression can be simplified as: tan(-5π/3) = -tan(5π/3)

Expressing 5π/3 as a difference from 2π: 5π/3 = 2π - π/3

Using the identity tan(2π - θ) = -tan(θ), we get: -tan(2π - π/3) = -(-tan(π/3)) = tan(π/3)

Since the exact value of tan(π/3) is √3, we find that x = √3.

Explanation

According to the half-angle trigonometric identity, the relationship between cos(A) and sin(A/2) is derived from the double-angle formula for cosine:

cos(A) = 1 - 2 sin²(A/2)

By rearranging the terms to solve for sin²(A/2):

2 sin²(A/2) = 1 - cos(A)

Dividing both sides by 2:

sin²(A/2) = (1 - cos(A))/2

Taking the square root of both sides gives:

sin(A/2) = √[(1 - cos(A))/2]

Since sin(A/2) = x, we obtain x = √[(1 - cosA)/2].

Explanation

The given expression is x = cos A cot A / (1 - sin A).

By using the trigonometric identity cot A = cos A / sin A, the expression can be rewritten as: x = (cos²A / sin A) / (1 - sin A) x = cos²A / [sin A (1 - sin A)]

Using the identity cos²A = 1 - sin²A: x = (1 - sin²A) / [sin A (1 - sin A)]

Since 1 - sin²A can be factored as (1 - sin A)(1 + sin A) using the algebraic identity a² - b² = (a - b)(a + b), we get: x = (1 - sin A)(1 + sin A) / [sin A (1 - sin A)]

Canceling the common term (1 - sin A) from both the numerator and the denominator: x = (1 + sin A) / sin A

Splitting the terms in the numerator: x = 1/sin A + sin A/sin A x = cosec A + 1 x = 1 + cosec A

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 = 15 + 35 + 30 + 10 + 20 + 30 = 140

Total number of girls = 15 + 40 + 15 + 25 + 40 + 30 = 165

Ratio of boys to girls = 140 : 165

Simplifying the ratio by dividing both terms by their common divisor 5:

140 ÷ 5 = 28 165 ÷ 5 = 33

This gives the simplified ratio of 28 : 33.

Explanation

To find the number of students who scored 20 or more but less than 40, we use the subtraction of cumulative frequencies:

Number of students in the range [20, 40) = Number of students scoring 20 and above - Number of students scoring 40 and above

From the data table: Number of students scoring 20 and above = 78 Number of students scoring 40 and above = 25

Substituting these values into the relation: 78 - 25 = 53

Explanation

The GDP of the country at the end of 2011 was 8 trillion, which serves as the starting GDP for the year 2012. The beginning of 2013 corresponds exactly to the end of 2012.

To find the GDP at the end of 2012, the growth rate of 7% for the year 2012 is applied to the GDP at the end of 2011 using the standard growth formula:

GDP at the end of 2012 = GDP at the end of 2011 × (1 + Growth rate for 2012 / 100)

Substituting the values into the formula:

GDP at the end of 2012 = 8 × (1 + 7/100) = 8 × 1.07 = 8.56 trillion

Therefore, the GDP at the beginning of 2013 was 8.56 trillion.

Explanation

The fundamental relationship between two numbers and their Highest Common Factor (HCF) and Lowest Common Multiple (LCM) is given by the formula:

Product of two numbers = HCF × LCM

Let the given number be a = 57 and the other number be b. Substituting the known values into the formula:

57 × b = 19 × 969

Solving for b:

b = (19 × 969) ÷ 57

Since 57 is divisible by 19 (19 × 3 = 57), this simplifies to:

b = 969 ÷ 3

b = 323

Explanation

The sum of the weights of Sourabh, Juana, and Rina is 90 × 3 = 270 kg.

The sum of the weights of Sourabh and Juana is 69 × 2 = 138 kg.

The sum of the weights of Juana and Rina is 94 × 2 = 188 kg.

Adding the two-person sum equations yields: (Sourabh + Juana) + (Juana + Rina) = 138 + 188 = 326 kg Sourabh + 2 × Juana + Rina = 326 kg

Subtracting the sum of all three people's weights from this combined total gives the weight of Juana: Juana = 326 - 270 = 56 kg

Explanation

To find the coefficient of x in the algebraic expression (x + 6)(7 - 3x), we expand the product:

(x + 6)(7 - 3x) = x(7 - 3x) + 6(7 - 3x)

Evaluating each term:

= 7x - 3x² + 42 - 18x

Grouping the like terms:

= -3x² + (7 - 18)x + 42 = -3x² - 11x + 42

This expansion gives a coefficient of -11 for the x term.

Explanation

Let B be the current age of the brother, and S be the current age of the sister. Since the brother is elder by 7 years, we can write: S = B - 7

Five years ago, the brother's age was B - 5, and the sister's age was: S - 5 = (B - 7) - 5 = B - 12

The product of their ages five years ago was 30: (B - 5) × (B - 12) = 30

Expanding this quadratic equation: B² - 12B - 5B + 60 = 30 B² - 17B + 30 = 0

Solving for B by factoring: (B - 15) × (B - 2) = 0

This yields B = 15 or B = 2. Since S = B - 7, choosing B = 2 would result in a negative age for the sister. Thus, the current age of the brother must be 15 years.

Explanation

Correct Answer: D

Let the unknown number be x. The relationship is represented by the formula:

Percentage × Number = Value

Substituting the given values into the formula:

1.6% × x = 42

(1.6 / 100) × x = 42

x = (42 × 100) / 1.6

x = 4200 / 1.6

x = 42000 / 16

x = 2625

Explanation

Solve the first inequality: 5 - x > 1 + 5x 4 > 6x x < 2/3

Solve the second inequality: 5 - 3x ≥ 2 - 4x x ≥ -3

Combining both conditions, the value of x must satisfy: -3 ≤ x < 2/3

The value -2 lies within this interval since -3 ≤ -2 < 2/3.

Explanation

Let SP represent the selling price of one watch.

Selling price of 11 watches = 11 × SP Profit earned = 3 × SP

Using the standard relation: Cost Price = Selling Price - Profit Cost Price of 11 watches = 11 × SP - 3 × SP = 8 × SP

The formula for profit percentage is: Profit Percentage = (Profit / Cost Price) × 100 Profit Percentage = (3 × SP / 8 × SP) × 100 = (3/8) × 100 = 37.5%

Explanation

The common difference d in an arithmetic progression with given terms a_p and a_q is calculated using the formula:

d = (a_q - a_p) / (q - p)

Using the 8th term, a₈ = -17, and the 17th term, a₁₇ = -71:

d = (-71 - (-17)) / (17 - 8) d = -54 / 9 = -6

Any term aₙ can be determined from another term a_p using the relation:

aₙ = a_p + (n - p) × d

To find the 13th term using the 8th term:

a₁₃ = a₈ + (13 - 8) × d a₁₃ = -17 + 5 × (-6) a₁₃ = -17 - 30 a₁₃ = -47

Explanation

Let the speed of the goods train be G km/hr.

The goods train travels for 4 hours before the second train starts, and then both trains travel for 9 hours until the second train overtakes the goods train. Thus, the total travel time for the goods train is 4 + 9 = 13 hours.

Since the second train travels at a speed of 65 km/hr for 9 hours to cover the same distance from the station: Distance = Speed × Time 13 × G = 65 × 9 G = (65 × 9) ÷ 13 G = 5 × 9 G = 45 km/hr

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula:

D = P × (R / 100)²

where D is the difference, P is the principal sum, and R is the rate of interest per annum.

Substituting the given values into the formula:

144 = P × (4 / 100)²

144 = P × (1 / 25)²

144 = P × (1 / 625)

P = 144 × 625

P = 90000

Explanation

The Y-axis has the equation x = 0. When a line segment joining (x₁, y₁) and (x₂, y₂) is divided by the Y-axis, the ratio of division m : n is given by the formula -x₁ : x₂.

For the points (2, -4) and (-3, 2), the coordinates are x₁ = 2 and x₂ = -3.

Substituting these values into the formula: Ratio = -2 : -3 = 2 : 3.

Explanation

Let the number of 1-rupee, 50-paise, and 10-paise coins be 3x, 5x, and 2x respectively.

The value of these coins in rupees is: Value of 1-rupee coins = 3x × 1 = 3x Value of 50-paise coins = 5x × 0.5 = 2.5x Value of 10-paise coins = 2x × 0.1 = 0.2x

The total value is given as ₹34.2. Summing these values: 3x + 2.5x + 0.2x = 34.2 5.7x = 34.2 x = 6

The number of 50-paise coins is 5x = 5 × 6 = 30.

Explanation

For a point (x, y) to lie on the line 7x + 3y = 4, its coordinates must satisfy the equation. Substituting the coordinates of the point (-2, 6) into the equation:

7(-2) + 3(6) = -14 + 18 = 4

Since the left-hand side equals the right-hand side, the point satisfies the equation of the line.

Explanation

Let the total work be 1. The daily work rate of A is 1/12, and the daily work rate of B is 1/18.

Their combined daily work rate is: 1/12 + 1/18 = (3 + 2)/36 = 5/36

In 4 days of working together, the fraction of work completed is: 4 × 5/36 = 5/9

The remaining fraction of work is: 1 - 5/9 = 4/9

Explanation

Since line segment AB is parallel to line segment CD, and AD intersects BC at E, the alternate interior angles are equal, meaning angle BAE = angle CDE and angle ABE = angle DCE. Additionally, angle AEB = angle DEC as they are vertically opposite angles. This establishes that triangle AEB is similar to triangle DEC by Angle-Angle similarity.

The similarity of these triangles means the ratio of their corresponding sides is equal: AE / ED = BE / EC

Since E lies on BC, the length of BE can be written as: BE = BC - EC = 13.6 - EC

Substituting the given values into the similarity ratio: 18 / 6 = (13.6 - EC) / EC 3 = (13.6 - EC) / EC 3 × EC = 13.6 - EC 4 × EC = 13.6 EC = 3.4 cm

Explanation

In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse (the side opposite to the 90° angle).

Using the sine ratio for the 60° angle: sin 60° = opposite / hypotenuse

Let h be the length of the hypotenuse. The side opposite to the 60° angle is given as 6 cm.

sin 60° = 6 / h

Since sin 60° = (√3)/2: (√3)/2 = 6 / h

Solving for h: h = 12 / √3 h = (12 × √3) / 3 h = 4√3 cm

Explanation

The curved surface area of a cylinder is given by the formula CSA = 2πrh and the circumference of its base is given by C = 2πr, where r is the radius and h is the height of the cylinder.

Given: 2πr = 88 cm 2πrh = 1760 sq cm

Substitute 2πr = 88 into the curved surface area formula to find the height h: 88 × h = 1760 h = 1760 / 88 = 20 cm

Now, find the radius r using the circumference: 2 × (22/7) × r = 88 (44/7) × r = 88 r = 88 × (7/44) r = 14 cm

The volume of a cylinder is given by the formula V = πr²h. Substituting the values of r and h: V = (22/7) × 14 × 14 × 20 V = 22 × 2 × 14 × 20 V = 12320 cubic cm

Explanation

The relationship between the selling price (SP) and the marked price (MP) of an item is given by: SP = MP × (100 - Discount%)/100

At a 10% discount, the selling price is Rs 48000: 0.90 × MP = 48000 MP = 48000 / 0.90

To find the selling price at a 25% discount: New SP = 0.75 × MP New SP = 0.75 × (48000 / 0.90) New SP = (3/4) × (48000 × 10/9) New SP = (3/4) × (480000/9) New SP = (3/4) × (160000/3) New SP = 40000

Explanation

We can express the angle 7π/4 as 2π - π/4.

Since the angle lies in the fourth quadrant, the sine function is negative. Using the standard trigonometric identity sin(2π - θ) = -sin θ, we have:

sin(7π/4) = sin(2π - π/4) = -sin(π/4)

Substituting the standard trigonometric value sin(π/4) = 1/√2, we get:

-sin(π/4) = -1/√2

Explanation

The triple-angle identity for cosine expresses cos 3A in terms of cos A: cos 3A = 4cos³A - 3cosA

To isolate cos³A, we add 3cosA to both sides of the equation: 4cos³A = cos 3A + 3cosA

Dividing both sides of the equation by 4 yields the formula: cos³A = (cos 3A + 3cosA)/4

Explanation

Using the standard trigonometric identity: cosec²A - cot²A = 1

We can express cot²A and its relation to cosec²A as: cot²A = cosec²A - 1 cot²A + 1 = cosec²A

Now, factoring the given expression: cot⁴A + cot²A = cot²A (cot²A + 1)

Substituting the identities into this factored form gives: (cosec²A - 1) (cosec²A)

Multiplying the terms yields: cosec⁴A - cosec²A

Explanation

Aesha has the least weight-to-height ratio among all four people, with a ratio of 51 / 1.72, which is approximately 29.65. The ratios for Aditi, Advika, and Agrima are approximately 32.75, 42.11, and 32.70 respectively.