SSC CGL Tier 1 2020 Quantitative Aptitude Questions with Solutions

Explanation

For equations of the form x cos A - y sin A = p and x sin A + y cos A = q, the relation x² + y² = p² + q² holds because squaring and adding the equations eliminates the trigonometric terms. Applying the identity (x cos A - y sin A)² + (x sin A + y cos A)² = x²(cos² A + sin² A) + y²(sin² A + cos² A) and using the fundamental identity sin² A + cos² A = 1:

x² + y² = 1² + 4² x² + y² = 1 + 16 = 17

The question requires the value of 17x² + 17y². By multiplying the calculated sum by 17, we get:

17(x² + y²) = 17 × 17 = 289

Explanation

The area of the circular disc is given by the formula A = π r². Substituting the given value 0.64π = π r², we find the radius r² = 0.64, which gives r = 0.8 m. The distance covered by the disc in a single revolution is equal to its circumference, C = 2π r. Using π = 22/7, the circumference is C = 2 × ²²⁄₇ × 0.8 = (35.2)/(7) m.

The total distance traveled by the disc is 1.408 km, which is 1408 m. The number of revolutions n is the total distance divided by the circumference:

n = (1408)/(35.2 / 7) = (1408 × 7)/(35.2)

n = (14080 × 7)/(352)

Since 1408 divided by 352 is exactly 4, the calculation becomes:

n = 40 × 7 = 280

Explanation

Let the cost price (CP) of the article be 100%. The difference between the two profit percentages represents the change in the selling price. Initially, the article was sold at a gain of 18%, and in the second scenario, it was sold at a gain of 25%.

The difference in the gain percentages is: 25% - 18% = 7%

This 7% increase in profit is caused by the ₹ 49 increase in the selling price. Therefore, we can equate the percentage difference to the monetary difference: 7% of CP = 49

To find the cost price (100%), we calculate: 1% of CP = ⁴⁹⁄₇ = 7 100% of CP = 7 × 100 = 700

The cost price of the article is ₹ 700.

Explanation

To solve the given equation 3ᵃ = 27ᵇ = 81ᶜ, we first express all terms with the same base: 3ᵃ = (3³)ᵇ = (3⁴)ᶜ 3ᵃ = 3³ᵇ = 3⁴ᶜ

Comparing the exponents, we get the relationship: a = 3b = 4c

We are given that abc = 144. Expressing b and c in terms of a: b = ᵃ⁄₃ c = ᵃ⁄₄

Substituting these into the product equation: a · ᵃ⁄₃ · ᵃ⁄₄ = 144 a³{12} = 144 a³ = 144 × 12 = 1728 a = ³√(1728) = 12

Now, determine the values of b and c: b = ¹²⁄₃ = 4 c = ¹²⁄₄ = 3

Substitute a, b, and c into the expression 12(¹⁄ₐ + (1)/(2b) + (1)/(5c)): 12(¹⁄₁₂ + ¹⁄₂₍₄₎ + ¹⁄₅₍₃₎) = ¹²⁄₁₂ + ¹²⁄₈ + ¹²⁄₁₅ = 1 + 1.5 + 0.8 = 3.3 = ³³⁄₁₀

Explanation

The algebraic identity for the given expression is a³ + b³ + c³ - 3abc = (a + b + c)[(a + b + c)² - 3(ab + bc + ca)]. Substituting the given values a + b + c = 9 and ab + bc + ca = -22 into this formula yields:

a³ + b³ + c³ - 3abc = 9 × [9² - 3(-22)] a³ + b³ + c³ - 3abc = 9 × [81 + 66] a³ + b³ + c³ - 3abc = 9 × 147

By performing the multiplication: 9 × 147 = 1323

This formula allows for a direct calculation without needing to solve for individual variables a, b, or c.

Explanation

The relationship between Cost Price (CP) and Marked Price (MP) is established by the standard ratio: (CP)/(MP) = (100 - Discount%)/(100 + Gain%)

Substituting the given values where MP = 888, Discount = 22%, and Gain = 11%: CP = 888 × (100 - 22)/(100 + 11) CP = 888 × ⁷⁸⁄₁₁₁

Since 888 divided by 111 equals 8, the calculation simplifies to: CP = 8 × 78 CP = 624

Explanation

The percentage growth of maize in the year 2019 over the year 2015 is calculated using the formula: Percentage Growth = (Production in 2019 - Production in 2015)/(Production in 2015) × 100

From the data provided in the table, the production of maize in 2015 was 490 tonnes, and in 2019, it was 840 tonnes. Substituting these values into the formula: Percentage Growth = (840 - 490)/(490) × 100 Percentage Growth = ³⁵⁰⁄₄₉₀ × 100

Simplifying the fraction ³⁵⁰⁄₄₉₀ by dividing both the numerator and denominator by 70 gives ⁵⁄₇. Percentage Growth = ⁵⁄₇ × 100 ≈ 71.4285%

Rounding this value to two decimal places yields 71.43%.

Explanation

The area of an equilateral triangle is determined by the formula Area = √3{4} a², where a represents the length of one side. By substituting the given area into this formula, we have the equation 36√3 = √3{4} a². We can eliminate √3 from both sides of the equation, resulting in 36 = a²{4}. Multiplying both sides by 4 gives a² = 144. Taking the positive square root of both sides, we find that the side length a is 12 cm. Since an equilateral triangle has three equal sides, its perimeter is calculated using the formula Perimeter = 3a. Substituting the side length into this formula, the perimeter is 3 × 12 = 36 cm.

Explanation

Since the sum of squares of real numbers is zero only when each square term is individually equal to zero, we can set up the following equations: a + b - 2 = 0 ⟹ a + b = 2 b + c - 5 = 0 ⟹ b + c = 5 c + a - 5 = 0 ⟹ c + a = 5

Adding these three equations together yields 2(a + b + c) = 2 + 5 + 5 = 12, which simplifies to a + b + c = 6. By subtracting the individual equations from this sum, we find the values of the variables: a = (a + b + c) - (b + c) = 6 - 5 = 1 b = (a + b + c) - (c + a) = 6 - 5 = 1 c = (a + b + c) - (a + b) = 6 - 2 = 4

The expression to be evaluated is √((b + c)ᵃ + (c + a)ᵇ - 1). Substituting the derived values a = 1, b = 1, b + c = 5, and c + a = 5 into the expression gives: √(5¹ + 5¹ - 1) = √(5 + 5 - 1) = √9 = 3

Explanation

The given equation is 2 sin A + csc A = 2 √2. Since csc A = (1)/(sin A), we can rewrite the equation as 2 sin A + (1)/(sin A) = 2 √2. Multiplying the entire equation by sin A results in the quadratic equation 2 sin² A - 2 √2 sin A + 1 = 0.

This quadratic expression is a perfect square of the form (√2 sin A - 1)² = 0. Solving for sin A, we get sin A = (1)/(√(2)). For the given range 0° < A < 90°, this corresponds to A = 45°. At this angle, the value of cos A is also (1)/(√(2)).

To find the value of 2(sin⁴ A + cos⁴ A), we substitute the values of sin A and cos A: 2(((1)/(√(2)))⁴ + ((1)/(√(2)))⁴) = 2(¼ + ¼) = 2(½) = 1

Explanation

To determine the average production of wheat over the specified five-year period from 2015 to 2019, the values provided in the wheat column are summed and then divided by the total number of years.

The formula for the average is: Average = (Total Production)/(Number of Years)

Summing the production values for wheat from 2015 to 2019: Total = 2500 + 4218 + 3482 + 4500 + 3300 = 18000 tonnes

The number of years in the data set is 5. Applying the formula: Average = ¹⁸⁰⁰⁰⁄₅ = 3600 tonnes

Explanation

To determine the remaining work, first find the total work by calculating the Least Common Multiple (LCM) of the individual time taken by A, B, and C. LCM(24, 20, 18) = 360 units.

Next, calculate the daily efficiency of each person: Efficiency of A = 360 / 24 = 15 units/day. Efficiency of B = 360 / 20 = 18 units/day. Efficiency of C = 360 / 18 = 20 units/day.

B and C work together for 6 days. Their combined efficiency is 18 + 20 = 38 units/day. Work completed by B and C in 6 days = 38 × 6 = 228 units.

The remaining work to be completed by A is: Remaining work = 360 - 228 = 132 units.

The number of days required by A alone to finish this remaining task is calculated by dividing the remaining work by A's efficiency: Time = 132 / 15 = 44 / 5 = 8 ⅘ days.

Explanation

The difference between compound interest paid yearly and half-yearly for a period of one year is equivalent to the interest earned on the interest of the first half-year. When interest is compounded yearly for one year, the interest is identical to the simple interest for that year. When compounded half-yearly, the year is treated as two compounding periods of six months each.

Annual rate R = 8%, so the half-yearly rate r = (8%)/(2) = 4%. The difference D for two compounding periods over one year is calculated using the formula: D = P × (ʳ⁄₁₀₀)²

Substituting the given values: P = 7800 r = 4 D = 7800 × (⁴⁄₁₀₀)² D = 7800 × ¹⁶⁄₁₀₀₀₀ D = 78 × 0.16 D = 12.48

This calculation isolates the extra interest generated because the interest from the first six months is added to the principal for the calculation of interest in the second six months.

Explanation

To find the difference between the average production of barley and the average production of rice, first calculate the total production for each crop over the five-year period from 2015 to 2019.

Total production of Rice = 415 + 520 + 585 + 625 + 600 = 2745 tonnes. Average production of Rice = ²⁷⁴⁵⁄₅ = 549 tonnes.

Total production of Barley = 975 + 825 + 700 + 625 + 775 = 3900 tonnes. Average production of Barley = ³⁹⁰⁰⁄₅ = 780 tonnes.

The difference between the two averages is: Difference = Average Barley - Average Rice = 780 - 549 = 231 tonnes.

Alternatively, the average of the differences can be calculated by summing the yearly differences between barley and rice and dividing by 5: ((975-415) + (825-520) + (700-585) + (625-625) + (775-600))/(5) = (560 + 305 + 115 + 0 + 175)/(5) = ¹¹⁵⁵⁄₅ = 231 tonnes.

Explanation

The distance traveled is constant, which means speed and time are inversely proportional (S ∝ 1/T). When two quantities are inversely proportional, the product of their initial values equals the product of their final values (s₁t₁ = s₂t₂).

The student wants to reduce the time by 25%. If the initial time is t₁, the new time t₂ is: t₂ = t₁ × (1 - 0.25) = 0.75t₁ = ¾t₁

To find the new speed s₂, we substitute t₂ into the proportionality equation: s₁ × t₁ = s₂ × ¾t₁ s₂ = ⁴⁄₃s₁

The increase in speed is s₂ - s₁ = ⁴⁄₃s₁ - s₁ = ⅓s₁. To express this as a percentage increase, we calculate: Percentage Increase = (s₂ - s₁{s₁}) × 100 = ⅓ × 100 = 33.33%

Rounding to the nearest integer given in the options, the student must increase his speed by 33%.

Explanation

Given the relationship 2A = 5B = 7C, we can express the shares in terms of their ratios. By setting 2A = 5B = 7C = k, we get A = k/2, B = k/5, and C = k/7. To simplify the ratio A:B:C, we multiply each term by the least common multiple (LCM) of 2, 5, and 7, which is 70. This gives the ratio A:B:C = 35:14:10.

The total number of units in the ratio is 35 + 14 + 10 = 59. The total amount to be divided is 1,180, so the value of one unit is 1180 / 59 = 20. Since A's share is represented by 35 units, the calculation is 35 × 20 = 700.

Explanation

According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is also divisible by 9. For the number 687x29, the sum of the digits is 6 + 8 + 7 + x + 2 + 9 = 32 + x.

For the number to be divisible by 9, the expression 32 + x must be a multiple of 9. Since x is a single-digit integer, the only multiple of 9 greater than or equal to 32 that satisfies this is 36. Setting 32 + x = 36 gives x = 4. The question asks for the value of 2x, which is 2 × 4 = 8.

Explanation

Let the population two years ago be P. The population undergoes successive changes of +30% and -15%. The net percentage change can be determined using the formula x + y + ˣʸ⁄₁₀₀. Substituting x = 30 and y = -15 into the formula, the net change is 30 - 15 + (30 × (-15))/(100) = 15 - 4.5 = 10.5%. This indicates that the current population is 110.5% of the original population from two years ago. Mathematically, this is expressed as 1.105 × P = 11,050. Dividing 11,050 by 1.105 results in P = 10,000.

Explanation

Growth in production is determined by calculating the difference between the production in a given year and the production in the preceding year. For sugarcane, the annual absolute growth values are calculated as follows:

In 2016, growth = 1120 - 1000 = 120 tonnes. In 2017, growth = 1250 - 1120 = 130 tonnes. In 2018, growth = 1325 - 1250 = 75 tonnes. In 2019, growth = 1450 - 1325 = 125 tonnes.

Comparing these values, the maximum growth of 130 tonnes occurred in the year 2017.

Explanation

The volume of a sphere is calculated using the formula V = ⁴⁄₃π r³. When a larger metallic sphere is melted and recast into smaller spheres, the total volume remains constant. Thus, the number of smaller spheres n is found by dividing the volume of the large sphere by the volume of one small sphere.

The diameter of the large sphere is 40 cm, so its radius R is 20 cm. The radius of the smaller spheres r is 0.5 cm. The relationship is expressed as:

n = (4)/(3)π R³{⁴⁄₃π r³} = ((R)/(r))³

Substituting the given values into the equation:

n = ((20)/(0.5))³

n = (40)³

n = 40 × 40 × 40 = 64,000