SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The company has issued a total of 6,00,000 shares.

According to the data table, Ejaz initially holds a 50% share of the company.

Initial number of shares held by Ejaz = 50% of 6,00,000 = 3,00,000 shares

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

New number of shares held by Ejaz = 3,00,000 + 15,000 = 3,15,000 shares

Explanation

In a general parallelogram, the diagonals bisect each other at their point of intersection, but they do not cross at right angles. The angle of intersection between the diagonals depends on the side lengths and internal angles of the parallelogram. The diagonals are perpendicular only when all four sides of the parallelogram are equal in length, which is not true for a standard, general parallelogram. Therefore, the property of perpendicular diagonals does not hold for a general parallelogram.

Explanation

Let x be the measure of the supplementary angle.

Since the sum of supplementary angles is 180°, the given angle and its supplementary angle must add up to 180°.

The measure of the given angle is (2/3)x.

Setting up the equation: x + (2/3)x = 180°

Multiplying the entire equation by 3 to clear the fraction: 3x + 2x = 540° 5x = 540° x = 108°

The measure of the given angle is: (2/3) × 108° = 72°

Explanation

The relationship between Selling Price, Marked Price, and Discount is given by the formula:

Selling Price = Marked Price × (1 - Discount Percentage / 100)

Given: Selling Price = Rs 855 Discount Percentage = 5%

Substituting these values into the formula:

855 = Marked Price × (1 - 5/100) 855 = Marked Price × 95/100 855 = Marked Price × 19/20

Marked Price = (855 × 20) / 19 Marked Price = 45 × 20 Marked Price = Rs 900

Explanation

The simple interest is calculated using the formula: Simple Interest = (Principal × Rate × Time) / 100.

Let the common rate of interest per annum be R%.

For Jairam: Principal = Rs. 8000 Time = 17 years Interest = (8000 × R × 17) / 100 = 1360R

For Karnik: Principal = Rs. 4700 Time = 16 years Interest = (4700 × R × 16) / 100 = 752R

The total interest received from both is Rs. 21120.

1360R + 752R = 21120 2112R = 21120 R = 10

Thus, the rate of interest per annum is 10%.

Explanation

The given expression is cos(2π/3).

Using the trigonometric reduction formula cos(π - θ) = -cos(θ) where θ = π/3:

cos(2π/3) = cos(π - π/3) = -cos(π/3)

Since the cosine of π/3 is ½, we substitute this value to get:

-cos(π/3) = -½

Thus, the value of x is -½.

Explanation

The area of a trapezium is calculated using the formula: Area = ½ × (a + b) × h

where a and b are the lengths of the parallel sides, and h is the distance between them.

Substituting the given values into the formula: 21 = ½ × (8.3 + b) × 3

Multiplying both sides by 2: 42 = (8.3 + b) × 3

Dividing both sides by 3: 14 = 8.3 + b

Solving for b: b = 14 - 8.3 b = 5.7 cm

Explanation

Using the standard trigonometric definitions and half-angle identities:

cosec A = 1/sin A cot A = cos A/sin A

Adding these two expressions gives:

cosec A + cot A = (1 + cos A)/sin A

Applying the half-angle formulas, where 1 + cos A = 2 cos²(A/2) and sin A = 2 sin(A/2) cos(A/2):

cosec A + cot A = 2 cos²(A/2) / (2 sin(A/2) cos(A/2))

Simplifying the fraction by canceling common terms:

cosec A + cot A = cos(A/2) / sin(A/2) = cot(A/2)

Since cot(A/2) = x, it follows that x = cosec A + cot A.

Explanation

The condition for three points A(x₁, y₁), B(x₂, y₂), and C(x₃, y₃) to be collinear is that the area of the triangle formed by them is zero, which is given by the relation: x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂) = 0

Substituting the coordinates of A(x, 2), B(-2, 1), and C(6, -3) into this relation: x(1 - (-3)) + (-2)(-3 - 2) + 6(2 - 1) = 0 x(4) - 2(-5) + 6(1) = 0 4x + 10 + 6 = 0 4x + 16 = 0 4x = -16 x = -4

Explanation

To find the gain or loss percentage, the quantity of bananas bought and sold can be equalized. Taking the least common multiple of 2 and 10 gives 10 bananas.

Cost price of 10 bananas = (10 / 2) × 1 = Rs 5 Selling price of 10 bananas = Rs 7

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

Gain percentage = ((Selling Price - Cost Price) / Cost Price) × 100 Gain percentage = ((7 - 5) / 5) × 100 = (2 / 5) × 100 = 40%

Explanation

Solving the first inequality: x - 5 ≤ 2x - 3 x ≥ -2

Solving the second inequality: 2x - ½ ≥ 5x + 2 -½ - 2 ≥ 3x -5/2 ≥ 3x x ≤ -5/6

Combining both inequalities gives the range: -2 ≤ x ≤ -5/6

Since -5/6 is approximately -0.83, the solution set is the interval [-2, -0.83]. The value -1 lies within this interval, satisfying both inequalities.

Explanation

To find the combined mean of the students in the three divisions, we use the weighted mean formula:

Weighted Mean = (n₁X₁ + n₂X₂ + n₃X₃) / (n₁ + n₂ + n₃)

Here, the number of students and their respective mean marks are: For division A: n₁ = 45, X₁ = 56 For division B: n₂ = 40, X₂ = 52 For division C: n₃ = 40, X₃ = 50

First, calculate the total sum of marks: Total sum = (45 × 56) + (40 × 52) + (40 × 50) Total sum = 2520 + 2080 + 2000 = 6600

Next, find the total number of students: Total students = 45 + 40 + 40 = 125

Now, divide the total sum of marks by the total number of students to find the combined mean: Combined Mean = 6600 / 125 = 52.8

Explanation

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

199992 × 200008 = (200000 - 8) × (200000 + 8)

Applying the identity with a = 200000 and b = 8:

(200000)² - 8² = 40000000000 - 64

40000000000 - 64 = 39999999936

Explanation

For a quadratic equation ax² + bx + c = 0 to have real roots, its discriminant (D) must be greater than or equal to zero (D ≥ 0), where D is calculated using the formula:

D = b² - 4ac

For the equation 2x² - 7x + 6 = 0, we have the coefficients: a = 2, b = -7, c = 6

Substituting these values into the discriminant formula: D = (-7)² - 4 × 2 × 6 D = 49 - 48 D = 1

Since D = 1, which is greater than zero, the equation has real and distinct roots.

Explanation

To find the solution set of the system of linear equations: 4x - 3y = 10 3x + y = 14

We can use the substitution method. From the second equation, we express y in terms of x: y = 14 - 3x

Substituting this expression for y into the first equation: 4x - 3(14 - 3x) = 10 4x - 42 + 9x = 10 13x - 42 = 10 13x = 52 x = 4

Substituting x = 4 back into the equation for y: y = 14 - 3(4) y = 14 - 12 y = 2

Thus, the solution set is {(4, 2)}.

Explanation

If 16% of the electricity bill is deducted, the remaining percentage of the bill to be paid is 100% - 16% = 84%.

Let the total bill amount be B. The relationship is:

84% of B = 1806

B × (84/100) = 1806

B = (1806 × 100) / 84

B = 2150

Explanation

Let the total work be 60 units, which is the least common multiple of 20 and 12.

The efficiency of P is 60 ÷ 20 = 3 units per day. The efficiency of Q is 60 ÷ 12 = 5 units per day. Their combined efficiency is 3 + 5 = 8 units per day.

The target is to complete 40% of the project, which equals 40% of 60 = 24 units.

The time required to complete this work when working together is: Time = Work ÷ Combined Efficiency = 24 ÷ 8 = 3 days.

Explanation

The bonus paid to each category of employees is calculated as a percentage of their annual salary.

For the manager, the total bonus is 1 × 60,00,000 × 30% = ₹18,00,000.

For the executives, the total bonus is 8 × 12,00,000 × 30% = ₹28,80,000.

For the trainees, the total bonus is 5 × 2,00,000 × 10% = ₹1,00,000.

The total bonus paid to all employees is ₹18,00,000 + ₹28,80,000 + ₹1,00,000 = ₹47,80,000.

The total number of employees is 1 + 8 + 5 = 14.

The average bonus per employee is the total bonus divided by the total number of employees: Average bonus = 47,80,000 / 14 ≈ ₹3,41,429.

Explanation

The total exports for the three companies in the years 2011 and 2015 are calculated as follows:

Total exports in 2011 = 2000 + 5000 + 2000 = 9000 lakh rupees

Total exports in 2015 = 3000 + 4000 + 2000 = 9000 lakh rupees

Since both years have total exports of 9000 lakh rupees, the total exports from the three companies together are equal for the years 2011 and 2015.

Explanation

The total profit or loss of the company over the last five years is calculated by taking the sum of the profit or loss values for each year from 2011 to 2015.

Total = 10 + (-20) + (-15) + 10 + (-25)

Total = 10 - 20 - 15 + 10 - 25

Total = -40 crore

A negative result indicates a financial loss. Therefore, the company incurred a total loss of Rs 40 crores over the five-year period.

Explanation

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

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

First, calculate the total value of India's exports in 2015 by summing the individual values:

Total Exports = 625 + 725 + 825 + 750 + 625 = 3550 million USThe value of software exports is given as 725 million US.

Substituting these values into the formula:

Percentage of Software = (725 / 3550) × 100

Percentage of Software = 72500 / 3550 = 20.4225... %

Rounding to two decimal places, this is equal to 20.42%.

Explanation

To find the value of x, solve the system of inequalities separately.

For the first inequality: 3x + 5(4 - 3x) > 2 - 4x 3x + 20 - 15x > 2 - 4x 20 - 12x > 2 - 4x 20 - 2 > 12x - 4x 18 > 8x x < 18/8 x < 2.25

For the second inequality: 2 - 4x < 3x - x/3 Multiply all terms by 3 to clear the fraction: 6 - 12x < 9x - x 6 - 12x < 8x 6 < 20x x > 6/20 x > 0.3

Combining both conditions gives the interval: 0.3 < x < 2.25

The only value from the given options that lies within this range is 2.

Explanation

Let the marked price of the washing machine be MP. The selling price at a discount of d% is calculated as: Selling Price = MP × (100 - d) / 100

With a 9% discount, the selling price is Rs 14000: 14000 = MP × (100 - 9) / 100 14000 = MP × 91 / 100 MP = (14000 × 100) / 91

With a 22% discount, the new selling price is: New Selling Price = MP × (100 - 22) / 100 New Selling Price = MP × 78 / 100

Substituting the value of MP: New Selling Price = ((14000 × 100) / 91) × (78 / 100) New Selling Price = 14000 × 78 / 91

Simplifying by dividing both 78 and 91 by their greatest common divisor 13: New Selling Price = 14000 × 6 / 7 New Selling Price = 2000 × 6 New Selling Price = 12000

Explanation

Total Cost Price of rice = Rs 3,344 Total quantity of rice purchased = 22 quintals

Since 24% of the rice is lost in transportation, the percentage of rice remaining is 100% - 24% = 76%. Remaining quantity of rice = 22 × 0.76 = 16.72 quintals

To earn a profit of 30%, the total Selling Price must be: Total Selling Price = Cost Price × 1.30 Total Selling Price = 3,344 × 1.30 = Rs 4,347.2

The selling price per quintal of the remaining rice is calculated by dividing the total Selling Price by the remaining quantity of rice: Selling rate per quintal = 4,347.2 / 16.72 = Rs 260

Explanation

Using the algebraic identity (a - b)² = a² - 2ab + b², we can substitute a = 4x and b = 3y:

(4x - 3y)² = (4x)² - 2(4x)(3y) + (3y)²

Simplifying each term gives:

(4x - 3y)² = 16x² - 24xy + 9y²

Explanation

The reflection of a point (x, y) in the origin maps it to (-x, -y). Therefore, the reflection of P(a, b) in the origin gives the coordinates of P1 as (-a, -b).

The reflection of a point (x, y) in the y-axis maps it to (-x, y). Reflecting P1(-a, -b) in the y-axis yields the coordinates (a, -b).

We are given that this final point is (5, -4). Equating the coordinates gives: (a, -b) = (5, -4)

Comparing the components: a = 5 -b = -4 → b = 4

Thus, the coordinates of the initial point P(a, b) are (5, 4).

Explanation

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

Speed in m/s = Speed in km/h × 5/18

Substituting the given speed of 1350 km/h:

Speed = 1350 × 5/18

Dividing 1350 by 18 gives 75. Then, multiplying 75 by 5 yields 375 m/s, which means the missile travels 375 metres in one second.

Explanation

The trigonometric function tangent of an angle in the fourth quadrant is negative. The angle 330° lies in the fourth quadrant and can be written as 360° - 30°.

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

Substituting θ = 30°: tan 330° = tan(360° - 30°) = -tan 30°

Since tan 30° = 1/I3, we get: tan 330° = -1/I3

Therefore, x = -1/I3.

Explanation

To add the fractions 3/4 and 8/9, find the least common multiple of the denominators 4 and 9, which is 36.

Convert each fraction to have the common denominator of 36: 3/4 = (3 × 9) / (4 × 9) = 27/36 8/9 = (8 × 4) / (9 × 4) = 32/36

Now, add the two converted fractions: 27/36 + 32/36 = (27 + 32) / 36 = 59/36

Explanation

By the fundamental trigonometric identity, sin²A + cos²A = 1, which can be rearranged to express the cosine of angle A as cosA = √(1 - sin²A).

Substituting this identity into the given equation: x = sinA / √(1 - sin²A) x = sinA / cosA

Since the ratio of the sine of an angle to its cosine is defined as the tangent of that angle, we have: x = tanA

Explanation

To simplify the expression √[(1 - cos A)/(1 + cos A)], we multiply both the numerator and the denominator inside the square root by (1 + cos A):

√[(1 - cos A)/(1 + cos A)] = √[((1 - cos A) × (1 + cos A)) / (1 + cos A)²]

Using the algebraic identity (a - b)(a + b) = a² - b², the numerator becomes:

√[(1 - cos² A) / (1 + cos A)²]

Applying the fundamental trigonometric identity 1 - cos² A = sin² A, the expression is rewritten as:

√[sin² A / (1 + cos A)²]

Taking the square root of both the numerator and the denominator yields:

sin A / (1 + cos A)

Explanation

The angle θ between two lines with slopes m₁ and m₂ is given by the formula:

tan θ = |(m₁ - m₂)/(1 + m₁m₂)|

Substituting the given slopes m₁ = 1 and m₂ = 1/√3 into the formula:

tan θ = |(1 - 1/√3) / (1 + 1 × (1/√3))|

tan θ = |(√3 - 1) / (√3 + 1)|

To simplify this expression, rationalize the denominator by multiplying the numerator and denominator by (√3 - 1):

tan θ = (√3 - 1)² / ((√3 + 1)(√3 - 1))

tan θ = (3 - 2√3 + 1) / (3 - 1)

tan θ = (4 - 2√3) / 2

tan θ = 2 - √3

Using the trigonometric identity for the tangent of a difference, tan 15° = tan(45° - 30°) = (tan 45° - tan 30°) / (1 + tan 45° tan 30°) = (1 - 1/√3) / (1 + 1/√3) = 2 - √3. Thus, θ = 15°.

Explanation

To find the roots of the quadratic equation 21x² - 37x - 28 = 0, we split the middle term. We look for two numbers whose product is 21 × (-28) = -588 and whose sum is -37. These numbers are -49 and 12.

We can rewrite the equation as: 21x² - 49x + 12x - 28 = 0

Grouping the terms gives: 7x(3x - 7) + 4(3x - 7) = 0 (7x + 4)(3x - 7) = 0

Solving for x: 7x + 4 = 0 leads to x = -4/7 3x - 7 = 0 leads to x = 7/3

Explanation

The volume of a rectangular pool with a uniformly sloping floor is given by the formula for the volume of a prism with a trapezoidal cross-section:

Volume = ½ × (Depth at one end + Depth at the other end) × Length × Breadth

Substituting the given values:

Volume = ½ × (2 + 3) × 25 × 12

Volume = ½ × 5 × 25 × 12

Volume = 5 × 25 × 6 = 750 cubic meters

Explanation

Peevush completes 1/3 of the work in 30 days, which means he takes 30 × 3 = 90 days to complete the entire job alone.

The remaining portion of the work is 1 - 1/3 = 2/3. Sanjiv completes this remaining 2/3 of the work in 60 days, which means he takes 60 × 3/2 = 90 days to complete the entire job alone.

Using the standard work formula, if two individuals take P days and S days respectively to finish a job, the time T they take to complete the job together is:

T = (P × S) / (P + S)

Substituting the values:

T = (90 × 90) / (90 + 90) = 8100 / 180 = 45 days

Explanation

The sum Sₙ of an arithmetic progression when the first term a, the last term l, and the number of terms n are known is given by the formula:

Sₙ = (n / 2) × (a + l)

Substituting the given values, where a = 33, l = -57, and n = 16, into the formula yields:

S₁₆ = (16 / 2) × (33 + (-57))

S₁₆ = 8 × (-24)

S₁₆ = -192

Explanation

Let the required number be x. Using the percentage formula:

3.6% × x = 27

(3.6 / 100) × x = 27

x = (27 × 100) / 3.6

x = 27000 / 36

x = 750

Explanation

A kite is mathematically defined as a quadrilateral with two distinct pairs of consecutive adjacent sides that are congruent. In a kite ABCD, the adjacent sides AB and AD are equal in length, while the remaining adjacent sides CB and CD are also equal in length. This distinct pair-wise equality of adjacent sides is the fundamental defining characteristic of a kite.

Explanation

Let S, G, and R represent the weights of Sopan, Guruprasad, and Rima respectively.

Using the sum formula where Sum = Average × Number of observations:

The total weight of all three is S + G + R = 3 × 65 = 195 kg.

The total weight of Sopan and Guruprasad is S + G = 2 × 67 = 134 kg.

The total weight of Guruprasad and Rima is G + R = 2 × 57 = 114 kg.

To find the weight of Guruprasad, we use the relation:

G = (S + G) + (G + R) - (S + G + R)

Substituting the calculated sums:

G = 134 + 114 - 195 = 248 - 195 = 53 kg.

Explanation

The total collection is the product of the ticket price and the number of visitors.

Let the initial and final ticket prices be P₁ and P₂, where P₁ : P₂ = 3 : 5. Let the initial and final number of visitors be V₁ and V₂, where V₁ : V₂ = 7 : 2.

The ratio of the initial total collection (C₁) to the final total collection (C₂) is given by the product of their respective ratios: C₁ : C₂ = (P₁ × V₁) : (P₂ × V₂) C₁ : C₂ = (3 × 7) : (5 × 2) = 21 : 10

Since the value dropped from 21 to 10, the total collection has decreased in the ratio 21:10.

Explanation

The total number of boys across all divisions and standards is calculated by summing the boys in each group: 30 + 20 + 40 + 35 + 10 + 35 = 170

The total number of girls across all divisions and standards is calculated by summing the girls in each group: 30 + 35 + 35 + 15 + 10 + 40 = 165

The ratio of total boys to total girls is: 170 : 165

Dividing both terms by their greatest common divisor, 5, yields the simplified ratio: 34 : 33

Explanation

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

Using the given data: Number of students scoring 20 and above = 67 Number of students scoring 40 and above = 21

Number of students scoring 20 or more but less than 40 = 67 - 21 = 46.

Explanation

The beginning of the year 2015 is equivalent to the end of the year 2014. Therefore, the change in GDP from the end of 2013 to the beginning of 2015 is determined solely by the GDP growth rate of the year 2014.

From the table, the GDP growth rate for 2014 is -8%.

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

Substituting the given values: GDP at the beginning of 2015 = 8 × (1 - 0.08) GDP at the beginning of 2015 = 8 × 0.92 = 7.36 trillion

Explanation

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

Total marks scored = 80 + 45 + 55 + 60 + 65 = 305

Number of subjects = 5

Average marks scored = 305 ÷ 5 = 61

Since five points are deducted from this average due to poor attendance, the calculation is:

Net average marks scored = 61 - 5 = 56

Explanation

Given the expression: x = (csc A - 1) / (csc A + 1)

Substitute csc A = 1 / sin A into the equation: x = (1 / sin A - 1) / (1 / sin A + 1) x = (1 - sin A) / (1 + sin A)

To simplify, multiply both the numerator and the denominator by (1 + sin A): x = (1 - sin A)(1 + sin A) / (1 + sin A)² x = (1 - sin² A) / (1 + sin A)²

Using the fundamental trigonometric identity sin² A + cos² A = 1, we can substitute cos² A for 1 - sin² A: x = cos² A / (1 + sin A)² x = (cos A / (1 + sin A))²

Explanation

To find the product of the algebraic terms 6ab, 3b, and 4a, we multiply their numerical coefficients and their variable parts separately.

First, multiply the numerical coefficients: 6 × 3 × 4 = 72

Next, multiply the variable components by combining the exponents of like bases: (ab) × b × a = (a × a) × (b × b) = a²b²

Combining the multiplied coefficient and variable parts yields: 72a²b²

Explanation

Euler's polyhedron formula states that for any convex polyhedron, the number of vertices (V), edges (E), and faces (F) are related by the formula V - E + F = 2.

An octahedron is a solid geometric figure with 8 faces (F = 8). Given that it has 6 vertices (V = 6), substituting these values into Euler's formula gives:

6 - E + 8 = 2

14 - E = 2

E = 12

Thus, the octahedron has 12 edges.

Explanation

Total collection is the product of the price of the entry ticket and the number of visitors.

Let the initial price of the ticket be 5x and the new price be 7x. Let the initial number of visitors be 11y and the new number of visitors be 9y.

Initial total collection = 5x × 11y = 55xy New total collection = 7x × 9y = 63xy

Comparing the initial collection to the new collection, the ratio is 55xy : 63xy, which simplifies to 55:63. Since 63 is greater than 55, the total collection has increased in the ratio of 55:63.

Explanation

The largest five-digit number is 99999. Dividing 99999 by 84 gives a quotient of 1190 and a remainder of 39.

Subtracting this remainder from the largest five-digit number yields the largest five-digit number exactly divisible by 84: 99999 - 39 = 99960

Explanation

To find the range of possible values for x, we solve both inequalities.

First inequality: 3x + 2 ≥ x - 1 2x ≥ -3 x ≥ -1.5

Second inequality: 2x - 4 ≤ 2 - x/3 Multiplying both sides by 3 to clear the fraction: 6x - 12 ≤ 6 - x 7x ≤ 18 x ≤ 18/7 ≈ 2.57

Combining both inequalities, we get the valid range for x: -1.5 ≤ x ≤ 18/7

The value -1 satisfies both conditions because it lies within this defined range as -1.5 ≤ -1 ≤ 2.57.