SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Correct Answer: D

The square of the binomial expression (7 - 4x) is calculated using the standard algebraic identity:

(a - b)² = a² - 2ab + b²

Substituting a = 7 and b = 4x into the formula:

(7 - 4x)² = 7² - 2 × 7 × 4x + (4x)²

Calculating each term:

7² = 49

2 × 7 × 4x = 56x

(4x)² = 16x²

Combining these terms gives:

49 - 56x + 16x²

Rearranging the terms in descending order of powers of x yields:

16x² - 56x + 49

Explanation

Simple interest is computed on the principal only and remains constant for each year. For a period of two years, the total simple interest is Rs 3000, which means the simple interest for each individual year is: 3000 ÷ 2 = Rs 1500

Compound interest for the first year is equal to the simple interest, which is Rs 1500.

The total compound interest for two years is Rs 3225. The difference between the compound interest and simple interest for two years represents the interest earned in the second year on the first year's interest. This difference is: 3225 - 3000 = Rs 225

Thus, Rs 225 is the interest earned on the first year's interest of Rs 1500 over one year. The rate of interest R is calculated as: R = (225 × 100) ÷ 1500 R = 225 ÷ 15 = 15 percent

Explanation

Let M be the marked price of the toaster.

A discount of 10% means the selling price is 90% of the marked price: 0.90 × M = 18000 M = 18000 / 0.90 = 20000

When the discount is 37.5%, the selling price is 100% - 37.5% = 62.5% of the marked price: Selling price = 20000 × 0.625 = 12500

Explanation

Passing marks = 61 + 29 = 90

Since passing marks represent 24% of the maximum marks, we can write:

24% of Maximum Marks = 90

Let M be the maximum marks.

0.24 × M = 90

M = 90 / 0.24

M = 9000 / 24

M = 375

Explanation

To find the values x can take, we solve the two given inequalities.

First inequality: 7 + 3x ≥ 5 - x/2

Multiplying the entire inequality by 2: 14 + 6x ≥ 10 - x

Adding x to both sides: 14 + 7x ≥ 10

Subtracting 14 from both sides: 7x ≥ -4 x ≥ -4/7 ≈ -0.57

Second inequality: 2x + 3 ≤ 5 - 2x

Adding 2x to both sides: 4x + 3 ≤ 5

Subtracting 3 from both sides: 4x ≤ 2 x ≤ ½ = 0.5

Combining both results, we find the range for x: -4/7 ≤ x ≤ ½

The value 0 lies within this interval.

Explanation

For two numbers a and b, their third proportional c is defined by the relation a : b = b : c, which yields the formula:

c = b² / a

Substituting the given values a = 9 and b = 15:

c = 15² / 9

c = 225 / 9

c = 25

Explanation

For a quadratic equation of the form x² - Sx + P = 0, the sum of the roots is S and the product of the roots is P.

Comparing this with the given equation x² - 25x + 156 = 0: Sum of roots (S) = 25 Product of roots (P) = 156

We can find the roots by splitting the middle term -25x into two parts whose sum is -25 and whose product is 156. These two numbers are -12 and -13.

x² - 12x - 13x + 156 = 0 x(x - 12) - 13(x - 12) = 0 (x - 12)(x - 13) = 0

This yields the roots: x = 12 x = 13

Explanation

When a cone is melted and reshaped into a cylinder of the same radius, the volume remains conserved.

Volume of cylinder = Volume of cone

π × r² × h_cylinder = ⅓ × π × r² × h_cone

Since the radius (r) is the same for both the cylinder and the cone, we can cancel π × r² from both sides:

h_cylinder = ⅓ × h_cone

Substituting the given height of the cone (h_cone = 9.6 cm):

h_cylinder = ⅓ × 9.6 = 3.2 cm

Explanation

Expanding the terms in this equation: 4x(x + 3) + 7 = 4x(x - 11) + 9 4x² + 12x + 7 = 4x² - 44x + 9

Subtracting 4x² from both sides simplifies the equation to: 12x + 7 = -44x + 9 56x - 2 = 0

A quadratic equation must be of the form ax² + bx + c = 0, where a ≠ 0. In this simplified equation, the x² terms cancel out completely, leaving a first-degree linear equation. Since the coefficient of the x² term is 0, this is not a quadratic equation.

Explanation

Let the selling price of one watch be S and the cost price of one watch be C.

According to the given condition, the profit on selling 9 watches is equal to the selling price of 4 watches: 9S - 9C = 4S 5S = 9C

The profit percentage is given by the formula: Profit Percentage = (Profit / Total Cost Price) × 100

Substituting the values: Profit Percentage = (4S / 9C) × 100 Profit Percentage = (4S / 5S) × 100 = (4/5) × 100 = 80%

Explanation

We are given the system of linear equations: 2x + 3y = 0 (Equation 1) 3x - 4y = 34 (Equation 2)

From Equation 1, we can express x in terms of y: 2x = -3y x = -3/2 y

Substituting this value of x into Equation 2: 3(-3/2 y) - 4y = 34 -9/2 y - 4y = 34 -17/2 y = 34 y = -4

Now, substituting y = -4 back to find x: x = -3/2 × (-4) = 6

Using the values of x and y, we calculate the required expression: x - y = 6 - (-4) = 6 + 4 = 10

Explanation

Let the side of the square be a. The diagonal of a square with side length a is given by the formula d = a√2.

Given that the diagonal is 14√2 cm, we set up the equation: a√2 = 14√2 a = 14 cm

For a circle inscribed inside a square, the diameter of the circle is equal to the side length of the square. Therefore, the radius r of the circle is half of the side length of the square: r = a/2 = 14/2 = 7 cm

The area of the circle is given by the formula πr². Using the value of π = 22/7: Area = (22/7) × 7² = (22/7) × 49 = 22 × 7 = 154 sq cm

Explanation

Since ST is parallel to QR in triangle PQR, the corresponding angles ∠PST and ∠PQR are equal, and ∠PTS and ∠PRQ are equal. Therefore, by Angle-Angle similarity, triangle PST is similar to triangle PQR.

The point S divides the side PQ internally in the ratio 4:1, which means PS:SQ = 4:1. The total length PQ is represented by PS + SQ. The ratio of the side PS of the smaller triangle to the side PQ of the larger triangle is:

PS/PQ = PS / (PS + SQ) = 4 / (4 + 1) = 4/5

For similar triangles, the ratio of corresponding sides is equal:

ST/QR = PS/PQ

Substituting the given value of QR = 15 cm into the equation:

ST/15 = 4/5

ST = (4/5) × 15 = 12 cm

Explanation

To find the value of x, we can simplify the given algebraic expression by taking a common denominator:

x = 1/(1 + cosA) + 1/(1 - cosA)

x = ((1 - cosA) + (1 + cosA)) / ((1 + cosA)(1 - cosA))

By simplifying the numerator and multiplying the terms in the denominator using the identity (a - b)(a + b) = a² - b², we get:

x = 2 / (1 - cos²A)

Using the fundamental trigonometric identity sin²A + cos²A = 1, we can substitute 1 - cos²A with sin²A:

x = 2 / sin²A

Since cosecA = 1/sinA, the expression becomes:

x = 2cosec²A

Explanation

Let the three numbers be a, b, and c.

The average of the three numbers is 64, which means their sum is: a + b + c = 3 × 64 = 192

The first number, a, is 1/3 times the sum of the other two numbers, b and c: a = (b + c) / 3 b + c = 3a

Substituting the value of b + c into the sum equation: a + 3a = 192 4a = 192 a = 192 / 4 = 48

Explanation

The standard trigonometric values are: cos 45° = 1/√2 sec 30° = 2/√3

Substituting these values into the given equation: x = cos 45° - sec 30° x = 1/√2 - 2/√3

Taking the common denominator as √2 × √3 = √6: x = (1 × √3 - 2 × √2) / √6 x = (√3 - 2√2)/√6

Explanation

Efficiency ratio of Ajit and Badrinath is 2 : 1. Since efficiency is inversely proportional to the time taken to complete a job, the ratio of time taken by Ajit and Badrinath is 1 : 2.

Let the time taken by Ajit to complete the job be x days. This means Badrinath takes 2x days.

According to the question, Ajit finishes the job in 30 days less than Badrinath: 2x - x = 30 x = 30

So, Ajit takes 30 days and Badrinath takes 60 days to complete the job individually.

Working together, the time taken is: (Time taken by Ajit × Time taken by Badrinath) / (Time taken by Ajit + Time taken by Badrinath) = (30 × 60) / (30 + 60) = 1800 / 90 = 20 days

Explanation

The trigonometric expression cosecA - cotA can be simplified by writing the functions in terms of sine and cosine:

cosecA - cotA = 1/sinA - cosA/sinA = (1 - cos A)/sin A

Taking the reciprocal of this expression gives:

1/(cosecA - cotA) = sin A / (1 - cos A)

Using the half-angle trigonometric identities, we know that:

sin A = 2 sin(A/2) cos(A/2) 1 - cos A = 2 sin²(A/2)

Substituting these identities into the expression:

1/(cosecA - cotA) = [2 sin(A/2) cos(A/2)] / [2 sin²(A/2)]

Simplifying the fraction by canceling the common terms 2 and sin(A/2) from the numerator and the denominator:

1/(cosecA - cotA) = cos(A/2) / sin(A/2) = cot(A/2)

Since cot(A/2) = x, it follows that x = 1/(cosecA - cotA).

Explanation

The area of a square in terms of its diagonal d is given by the formula:

Area = ½ × d²

Substituting the given diagonal length d = 10 cm into the formula:

Area = ½ × 10² Area = ½ × 100 Area = 50 sq cm

Explanation

When a body covers equal distances at two different speeds, say v₁ and v₂, the average speed for the entire journey is given by the harmonic mean of the two speeds:

Average Speed = 2v₁v₂ / (v₁ + v₂)

Given: Going speed (v₁) = 32 km/h Return speed (v₂) = 68 km/h

Substituting these values into the formula:

Average Speed = (2 × 32 × 68) / (32 + 68) Average Speed = 4352 / 100 Average Speed = 43.52 km/hr

Explanation

The given equation of the line is 3x - 6y = 4.

To find the slope, we convert this equation into the standard slope-intercept form, y = mx + c, where m represents the slope of the line:

6y = 3x - 4

Dividing the entire equation by 6:

y = (3/6)x - 4/6

y = ½x - ⅔

Comparing this with y = mx + c, the slope m of the given line is ½.

Since parallel lines have identical slopes, any line parallel to the line 3x - 6y = 4 will also have a slope of ½.

Explanation

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

When the second train passes the station, the goods train has already been traveling for 5 hours. The second train then takes 9 hours to overtake the goods train. This means the goods train travels for a total of 14 hours (5 + 9) from the station until the point where it is overtaken.

Since both trains cover the exact same distance from the station to the point of overtaking, their distances can be equated using the relation Distance = Speed × Time.

The distance covered by the second train in 9 hours at 70 km/hr is: 70 × 9 = 630 km

The goods train covers this same distance of 630 km in 14 hours: 14 × v = 630 v = 630 / 14 v = 45 km/hr

Explanation

The weight-to-height ratio is determined by dividing each individual's weight by their height.

For Aarohi, the weight-to-height ratio is 55 ÷ 1.61 ≈ 34.16.

For Aarti, the ratio is 68 ÷ 1.65 ≈ 41.21.

For Aarushi, the ratio is 71 ÷ 1.68 ≈ 42.26.

For Aasha, the ratio is 59 ÷ 1.56 ≈ 37.82.

Comparing these ratios, Aarohi has the smallest value of 34.16, which is the least weight-to-height ratio.

Explanation

The least percentage profit of the company is 10%, which occurred in the year 2013.

The relationship between revenue, expenditure, and profit percentage is given by the formula:

Revenue = Expenditure × (1 + % Profit / 100)

Given that the expenditure in the year with the least percentage profit is Rs 275 crore and the percentage profit is 10%:

Revenue = 275 × (1 + 10 / 100) Revenue = 275 × 1.1 Revenue = 302.5 crore

Explanation

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

Average = ∑(f_i × x_i) / ∑(f_i)

where x_i represents the number of children and f_i represents the number of houses (frequency).

First, calculate the total number of children in the society: Total children = (0 × 4) + (1 × 13) + (2 × 10) + (3 × 5) = 0 + 13 + 20 + 15 = 48

Next, calculate the total number of houses: Total houses = 4 + 13 + 10 + 5 = 32

Divide the total number of children by the total number of houses to find the average: Average = 48 / 32 = 1.5

Explanation

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

According to the data table, Ejaz holds a 35% share in the company.

The initial number of shares held by Ejaz is calculated as: 35% of 6,00,000 = (35/100) × 6,00,000 = 2,10,000 shares

When Basu sells 15,000 of his shares to Ejaz, these shares are added directly to Ejaz's existing holding.

The new total number of shares held by Ejaz is: 2,10,000 + 15,000 = 2,25,000 shares

Explanation

Since the marked price of each item is the same, the effective discount can be calculated directly using the weighted average of the individual discounts.

Rabia buys 5 items, which are grouped as 4 items at a 33% discount and 1 item at a 22% discount.

Effective discount percentage = (Discount rate on 4 items × 4 + Discount rate on 1 item × 1) ÷ Total quantity of items

Substituting the values:

Effective discount percentage = (33 × 4 + 22 × 1) ÷ (4 + 1) = (132 + 22) ÷ 5 = 154 ÷ 5 = 30.8%

Explanation

Let the two numbers be a and b.

The mean proportion of a and b is given by the formula: √(ab) = 8

Squaring both sides: ab = 64 a = 64/b

The third proportion of a and b is given by the formula: b²/a = 64

Substituting the value of a into the third proportion formula: b² / (64/b) = 64 b³ / 64 = 64 b³ = 4096 b = 16

Substituting b = 16 back into the equation for a: a = 64 / 16 a = 4

Thus, the two numbers are 4 and 16.

Explanation

Let the time taken by Baldev to complete the work be x days. Since Akash is 3 times as good a workman as Baldev, Akash takes x/3 days to complete the same work.

Given that Akash completes the work in 40 days less than Baldev: x - x/3 = 40 2x/3 = 40 x = 60

Thus, Baldev takes 60 days to complete the work, and Akash takes 20 days.

To find the time taken when they work together, we use the formula for combined work: Combined time = (Time by Akash × Time by Baldev) / (Time by Akash + Time by Baldev) Combined time = (20 × 60) / (20 + 60) = 1200 / 80 = 15 days.

Explanation

The compound interest formula is: A = P(1 + R/100)ⁿ

Here, the Amount (A) is Rs 23328, the Principal (P) is Rs 20000, and the time (n) is 2 years. Substituting these values into the formula: 23328 = 20000(1 + R/100)²

Divide both sides by 20000 to simplify the ratio: 23328 / 20000 = (1 + R/100)² 11664 / 10000 = (1 + R/100)²

Taking the square root on both sides: √(11664 / 10000) = 1 + R/100 108 / 100 = 1 + R/100 1.08 = 1 + R/100 R / 100 = 0.08 R = 8

Explanation

According to the property of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. This relation is expressed as:

Area(ΔABC) / Area(ΔPQR) = (AB / PQ)²

Substituting the given values into the formula:

324 / Area(ΔPQR) = (18 / 12)²

Simplifying the ratio of the corresponding sides 18/12 to 3/2:

324 / Area(ΔPQR) = (3 / 2)²

324 / Area(ΔPQR) = 9 / 4

Area(ΔPQR) = (324 × 4) / 9

Area(ΔPQR) = 36 × 4 = 144 sq cm

Explanation

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

Substituting the given points (8, -2) and (3, -1) into the formula: m = (-1 - (-2)) / (3 - 8) = 1 / -5 = -1/5

The product of the slopes of two perpendicular lines is -1. If mₚ represents the slope of the perpendicular line, then: m × mₚ = -1 (-1/5) × mₚ = -1 mₚ = 5

Explanation

For any n-gonal prism, the number of edges is 3n and the number of vertices is 2n, where n represents the number of sides of the base polygon. Since a pentagonal prism has a five-sided pentagon as its base, n = 5. The number of edges is 3 × 5 = 15, and the number of vertices is 2 × 5 = 10.

Explanation

The prime factorizations of 57 and 93 are: 57 = 3 × 19 93 = 3 × 31

The least common multiple (LCM) of two numbers is the product of the highest power of each prime factor involved in both numbers.

LCM(57, 93) = 3 × 19 × 31 = 1767

Alternatively, using the relation between LCM and the Greatest Common Divisor (GCD): LCM(a, b) = (a × b) / GCD(a, b)

The only common prime factor between 57 and 93 is 3, making GCD(57, 93) = 3. LCM(57, 93) = (57 × 93) / 3 = 19 × 93 = 1767

Explanation

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

πr² = 49π r² = 49 r = 7 cm

The diameter of the circle is twice the radius: d = 2r = 2 × 7 = 14 cm

Since the side of the square (s) is equal to the diameter of the circle: s = 14 cm

The area of the square is given by the formula s²: Area = 14² = 196 sq cm

Explanation

Using the trigonometric identity for negative angles:

sec(-θ) = sec(θ)

For θ = 5π/4, this gives:

sec(-5π/4) = sec(5π/4)

We can write the angle 5π/4 as (π + π/4). Since secant is negative in the third quadrant, we have:

sec(π + π/4) = -sec(π/4)

Using the standard trigonometric value sec(π/4) = √2, we get:

x = -√2

Explanation

The total marked price of 5 items is 5 × Rs 500 = Rs 2500.

To find the discount on purchasing 5 items, we split the purchase into a group of 2 items and a group of 3 items:

Marked price of 2 items = 2 × Rs 500 = Rs 1000 Discount on 2 items = 8% of Rs 1000 = Rs 80

Marked price of 3 items = 3 × Rs 500 = Rs 1500 Discount on 3 items = 16% of Rs 1500 = Rs 240

Total discount amount = Rs 80 + Rs 240 = Rs 320

Effective discount percentage = (Total discount / Total marked price) × 100 Effective discount percentage = (320 / 2500) × 100 = 12.8%

Explanation

Using the half-angle trigonometric identities, we can express sin A and cos A in terms of the half-angle A/2:

sin A = 2 sin(A/2) cos(A/2)

1 + cos A = 2 cos²(A/2)

Substituting these expressions into the ratio sin A / (1 + cos A) gives:

sin A / (1 + cos A) = (2 sin(A/2) cos(A/2)) / (2 cos²(A/2))

Simplifying the fraction by canceling the common factor of 2 cos(A/2) from the numerator and the denominator yields:

sin A / (1 + cos A) = sin(A/2) / cos(A/2) = tan(A/2)

Given that tan(A/2) = x, we get:

x = sin A / (1 + cos A)

Explanation

Given expression: x = 2 sec A - (1 + sin A)/cos A

Using the identity sec A = 1/cos A, we can rewrite the equation as: x = 2/cos A - (1 + sin A)/cos A

Combining the terms over the common denominator: x = (2 - (1 + sin A))/cos A x = (1 - sin A)/cos A

Now, multiply both the numerator and the denominator by (1 + sin A): x = ((1 - sin A)(1 + sin A))/(cos A(1 + sin A)) x = (1 - sin² A)/(cos A(1 + sin A))

Using the fundamental trigonometric identity 1 - sin² A = cos² A, the expression simplifies to: x = cos² A/(cos A(1 + sin A)) x = cos A/(1 + sin A)

Explanation

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

D = P × (R / 100)²

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

Given: Difference (D) = Rs 45 Rate of interest (R) = 5%

Substituting these values into the formula:

45 = P × (5 / 100)²

45 = P × (1 / 20)²

45 = P / 400

P = 45 × 400

P = 18000

Explanation

The percentage increase required to restore a price to its original value after a reduction of r% is given by the formula:

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

Substituting the given rate of reduction, r = 33:

Percentage increase = (33 / (100 - 33)) × 100 = (33 / 67) × 100 = 3300 / 67 ≈ 49.25%

Explanation

If three numbers a, b, and c are in continued proportion, then c is called the third proportional to a and b, which is defined by the relation a/b = b/c.

Rearranging this formula to solve for the third proportional c gives:

c = b² / a

Substituting the given values, a = 9 and b = 24:

c = 24² / 9

c = 576 / 9

c = 64

Explanation

The total surface area of a hemisphere is given by the formula TSA = 3πr², where r represents the radius.

By substituting the given total surface area of 166.32 sq cm and using the value of π as 22/7: 166.32 = 3 × (22/7) × r² 166.32 = (66/7) × r² r² = (166.32 × 7) / 66 r² = 2.52 × 7 r² = 17.64 r = √17.64 r = 4.2 cm

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding sides. Since ΔABC is similar to ΔDEF, the ratio of the perimeter of ΔABC to the perimeter of ΔDEF is equal to AB : DE. Substituting the given side lengths of AB = 18 cm and DE = 10 cm, the ratio of the perimeters is 18 : 10, which simplifies to 9 : 5.

Explanation

The volume of a hemisphere is given by the formula:

V = (2/3) × π × r³

Substituting the radius r = 7 cm and π ≈ 22/7 into the formula:

V = (2/3) × (22/7) × 7³

V = (2/3) × 22 × 49

V = 2156 / 3

V ≈ 718.67 cubic cm

Explanation

The distance between two points when the speeds and the difference in travel times are known can be calculated using the direct relation:

Distance = (Product of Speeds / Difference of Speeds) × Difference in Time

Given: Speed of the first car = 36 km/hr Speed of the second car = 48 km/hr Difference in time = 3 hours

Substituting these values into the formula:

Distance = (36 × 48 / (48 - 36)) × 3 Distance = (36 × 48 / 12) × 3 Distance = 3 × 48 × 3 Distance = 432 km

Explanation

Let the selling price of one Titan watch be S.

The total selling price of 13 Titan watches is 13S.

The profit earned from selling these 13 watches is equal to the selling price of 3 watches, which is 3S.

Using the relation: Profit = Selling Price - Cost Price

We get: 3S = 13S - Cost Price

Cost Price = 13S - 3S = 10S

The profit percentage is calculated as: Profit Percentage = (Profit / Cost Price) × 100

Substituting the values: Profit Percentage = (3S / 10S) × 100 = 30%

Explanation

Two lines are perpendicular if the product of their slopes is -1.

The given line is 20x + 5y = 3, which can be rewritten as: 5y = -20x + 3 y = -4x + 3/5

Comparing this to y = mx + c, the slope (m₁) of this line is -4.

Since the second line is perpendicular, its slope (m₂) must satisfy: m₁ × m₂ = -1 -4 × m₂ = -1 m₂ = 1/4

The slope of the line passing through the points (-2,5) and (6,b) is calculated using the formula: m₂ = (y₂ - y₁) / (x₂ - x₁) 1/4 = (b - 5) / (6 - (-2)) 1/4 = (b - 5) / 8

Multiplying both sides by 8: 2 = b - 5 b = 7

Explanation

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

A dodecagon is a polygon with 12 sides, which means n = 12.

Substituting the value of n into the formula: Sum = (12 - 2) × 180° Sum = 10 × 180° Sum = 1800°

Explanation

A completes 80% of the work in 20 days, which means A's daily rate of work is 80% ÷ 20 = 4% per day.

The remaining 20% of the work is completed by A and B together in 4 days. Thus, their combined daily rate of work is 20% ÷ 4 = 5% per day.

B's daily rate of work is the combined rate minus A's rate: 5% - 4% = 1% per day.

To complete 100% of the work alone at a rate of 1% per day, B requires: 100% ÷ 1% = 100 days.