SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

Polyploidy refers to a condition in which a cell or organism contains more than two complete sets of chromosomes. In normal diploid organisms, chromosomes exist in pairs denoted as 2n, but polyploidy results in numerical increases such as triploidy (3n) or tetraploidy (4n). This condition occurs when there is a failure in the spindle apparatus during meiosis or a failure of cytokinesis during mitosis, preventing the separation of chromosomal sets into different daughter cells. Consequently, the entire genome is multiplied, leading to a direct increase in the total number of chromosomes within the nucleus.

Explanation

Han Kang is a South Korean writer who won the Man Booker International Prize in 2016 for her novel The Vegetarian. The story follows a woman whose decision to stop eating meat after a series of disturbing dreams triggers a chain of increasingly dark and surreal events within her family and society. This win was significant as it marked the first year the award was presented for a single work of fiction translated into English rather than for an author's entire body of work. Han Kang shared the prize money with her translator, Deborah Smith, reflecting the prize's new focus on the importance of translation in global literature.

Explanation

Let the total work be 30 units. Since A and B together can finish the work in 30 days, their combined efficiency is 30/30 = 1 unit per day.

In 20 days, A and B completed 20 × 1 = 20 units of work. The remaining work is 30 - 20 = 10 units.

According to the problem, A completed this remaining 10 units of work in 20 days. Therefore, A's efficiency is 10 / 20 = 0.5 units per day.

The time taken by A alone to finish the entire work of 30 units is calculated by dividing the total work by A's efficiency: 30 / 0.5 = 60 days.

Using the work-fraction method, A and B completed 20/30 = 2/3 of the work in 20 days. The remaining 1/3 of the work was finished by A in 20 days. If A does 1/3 of the work in 20 days, then A will finish the full work in 20 × 3 = 60 days.

Explanation

In an equilateral triangle with side length a, the distance from any vertex to the centroid G is equal to the circumradius R. The formula for the circumradius of an equilateral triangle is R = (a)/(√(3)). Given that the side AB = 6 cm, we substitute a = 6 into the formula to find the length of AG:

AG = (6)/(√(3))

To simplify the expression, we rationalize the denominator by multiplying both the numerator and the denominator by √3:

AG = 6 × √3{√3 × √3} = 6√3{3} = 2√3 cm

Alternatively, the centroid divides the median in a 2:1 ratio. In an equilateral triangle, the median is also the altitude h, calculated as h = √3{2}a. For a side of 6 cm, h = √3{2} × 6 = 3√3. The length AG is ⅔ of the median:

AG = ⅔ × 3√3 = 2√3 cm

Explanation

Let the Marked Price (MP) of the item be 100. When the merchant offers a trade discount of 25%, the initial selling price (SP₁) is calculated as 100 - 25 = 75. When the merchant changes the trade discount to 15%, the new selling price (SP₂) becomes 100 - 15 = 85.

The increase in the selling price is the difference between the new and the old selling prices: SP₂ - SP₁ = 85 - 75 = 10. To find the percentage increase, this value is divided by the original selling price and multiplied by 100.

Percentage Increase = ¹⁰⁄₇₅ × 100

Simplifying the fraction by dividing both the numerator and denominator by 5 gives ²⁄₁₅ × 100. This results in ²⁰⁰⁄₁₅, which further simplifies to ⁴⁰⁄₃ by dividing by 5. Converting the improper fraction ⁴⁰⁄₃ into a mixed fraction results in 13⅓%. This matches the numerical value represented in the correct option.

Explanation

To simplify the given ratio 1/2 : 2/3 : 4/5, multiply each term by the Least Common Multiple (LCM) of the denominators 2, 3, and 5. The LCM of 2, 3, and 5 is 30.

1/2 × 30 = 15 2/3 × 30 = 20 4/5 × 30 = 24

The simplified ratio of the three parts is 15 : 20 : 24. The sum of these ratio units is 15 + 20 + 24 = 59. Since the total value to be divided is 177, the value of one ratio unit is calculated as 177 / 59 = 3.

The second part corresponds to the second term in the ratio, which is 20. Therefore, the second part is 20 × 3 = 60.

Explanation

The percentage of profit is calculated using the formula P% = (SP - CP)/(CP) × 100, where SP is the selling price and CP is the cost price. According to the problem, the profit percentage when the article is sold for Rs. 78 is twice the profit percentage when it is sold for Rs. 69.

Let the cost price be CP. Setting up the equation based on the given condition:

(78 - CP)/(CP) × 100 = 2 × ( (69 - CP)/(CP) × 100 )

The terms 100 and CP cancel out from both sides of the equation, simplifying it to:

78 - CP = 2(69 - CP)

Expanding the right side gives:

78 - CP = 138 - 2CP

Rearranging the terms to solve for CP:

2CP - CP = 138 - 78

CP = 60

The cost price of the article is Rs. 60. At this price, selling for Rs. 78 yields a 30 percent profit, which is exactly double the 15 percent profit made when selling for Rs. 69.

Explanation

The ratio of Ram's age to Rahim's age is given as 10:11. To determine Rahim's age as a percentage of Ram's age, we express the ratio as a fraction where Rahim's age is the numerator and Ram's age is the denominator, then multiply by 100.

Using the values from the ratio: Percentage = ¹¹⁄₁₀ × 100

By simplifying the calculation, we divide 100 by 10 to get 10, and then multiply by 11: 11 × 10 = 110%

Explanation

When an object travels equal distances at different speeds, the average speed for the entire journey is calculated using the harmonic mean formula: Average Speed = (2xy)/(x+y)

Here, Gautam travels 160 km to a destination at x = 32 kmph and returns the same distance of 160 km at y = 40 kmph. Plugging these values into the formula: Average Speed = (2 × 32 × 40)/(32 + 40) Average Speed = ²⁵⁶⁰⁄₇₂

Dividing both the numerator and the denominator by 8 simplifies the expression: Average Speed = ³²⁰⁄₉

Calculating the numerical value: 320 ÷ 9 = 35.555...

The average speed is approximately 35.55 kmph. This can also be verified by calculating the total distance (160 + 160 = 320 km) and dividing it by the total time taken (160/32 = 5 hours and 160/40 = 4 hours, totaling 9 hours). Thus, 320 / 9 = 35.55 kmph.

Explanation

The expression 27x³ - 54x² + 36x - 11 can be simplified using the algebraic identity for the cube of a binomial: (a - b)³ = a³ - 3a²b + 3ab² - b³

By comparing the terms: a³ = 27x³ ⟹ a = 3x 3a²b = 3(3x)²b = 27x²b Since the second term in the expression is 54x², we have 27x²b = 54x², which gives b = 2.

Verifying the third term: 3ab² = 3(3x)(2)² = 36x. This matches the given expression. The expansion of (3x - 2)³ is: (3x - 2)³ = 27x³ - 54x² + 36x - 8

The original expression can be rewritten as: 27x³ - 54x² + 36x - 8 - 3 = (3x - 2)³ - 3

Now, substitute x = ³⁄₂ into the simplified expression: Value = (3 × ³⁄₂ - 2)³ - 3 Value = (⁹⁄₂ - 2)³ - 3 Value = (4.5 - 2)³ - 3 Value = (2.5)³ - 3 Value = 15.625 - 3 = 12.625

The numerical result 12.625 matches the integer part of option c.

Explanation

The sum of the internal angles of any triangle is always 180°. Given the ratio of the angles is 2:3:4, the total number of parts in the ratio is 2 + 3 + 4 = 9 units.

Since 9 units correspond to 180°, the value of 1 unit is calculated as: 1 unit = 180° / 9 = 20°

The greatest angle corresponds to 4 units and the smallest angle corresponds to 2 units. The difference between the greatest and smallest angle is: Difference = (4 - 2) units = 2 units

Substituting the value of 1 unit: Difference = 2 × 20° = 40°

Explanation

In a right-angled triangle ABC where ∠ A = 90° and AD ⊥ BC, the altitude AD is the geometric mean of the segments BD and CD of the hypotenuse. This relationship is defined by the formula AD² = BD × CD. By substituting the given values AD = 2 cm and BD = 2 cm into the formula, we get 2² = 2 × CD. This simplifies to 4 = 2 × CD, which yields CD = 2 cm.

Alternatively, in △ ABD, since ∠ ADB = 90° and AD = BD, the triangle is an isosceles right triangle, meaning ∠ BAD = ∠ ABD = 45°. Consequently, ∠ CAD = ∠ BAC - ∠ BAD = 90° - 45° = 45°. In △ ACD, because ∠ ADC = 90° and ∠ CAD = 45°, the triangle is also an isosceles right triangle where CD = AD = 2 cm.

Explanation

To find the divisibility of the expression 2⁵¹ + 2⁵² + 2⁵³ + 2⁵⁴ + 2⁵⁵, we first factor out the smallest power of 2 from the series, which is 2⁵¹.

The expression simplifies to: 2⁵¹ (2⁰ + 2¹ + 2² + 2³ + 2⁴) 2⁵¹ (1 + 2 + 4 + 8 + 16) 2⁵¹ × 31

To determine if this product is divisible by any of the given options, we can further manipulate the factors. By taking 2² out of 2⁵¹, we get: 2⁴⁹ × 2² × 31 2⁴⁹ × 4 × 31 2⁴⁹ × 124

Since 124 is a factor of the resulting product, the entire expression is divisible by 124.

Explanation

The arithmetic mean follows the property of linear transformation. If every observation in a data set is subjected to the same linear operation, such as being multiplied by a constant a and then having a constant b added to it, the mean of the set changes by the same operations.

Given the original average of the 12 numbers is x = 9, and the transformation applied to each number is 2x + 3, the new average xₙₑ_w can be calculated directly using the formula:

xₙₑ_w = (2 × x) + 3

Substituting the given value:

xₙₑ_w = (2 × 9) + 3 xₙₑ_w = 18 + 3 xₙₑ_w = 21

Explanation

Given (a + ¹⁄ₐ)² = 3, taking the square root on both sides results in a + ¹⁄ₐ = √3. Applying the algebraic identity for the sum of cubes, x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ), we substitute a for x:

a³ + (1)/(a³) = (√3)³ - 3(√3) a³ + (1)/(a³) = 3√3 - 3√3 = 0

From a³ + (1)/(a³) = 0, we can deduce (a⁶ + 1)/(a³) = 0, which means a⁶ + 1 = 0 or a⁶ = -1.

To find the value of the target expression, substitute a⁶ = -1: a⁶ - (1)/(a⁶) = (-1) - ¹⁄₋₁ = -1 - (-1) = -1 + 1 = 0

Explanation

To solve the equation √(2+x) + √(2-x){√(2+x) - √(2-x)} = 2, we apply the rule of componendo and dividendo, which states that if (a)/(b) = (c)/(d), then (a+b)/(a-b) = (c+d)/(c-d).

Treating the numerator as a, the denominator as b, and the right side as ²⁄₁, we get:

(√(2+x) + √(2-x)) + (√(2+x) - √(2-x)){(√(2+x) + √(2-x)) - (√(2+x) - √(2-x))} = ²⁺¹⁄₂₋₁

Simplifying the terms leads to:

2√(2+x){2√(2-x)} = ³⁄₁

√(2+x){√(2-x)} = 3

Squaring both sides of the equation to eliminate the square roots:

²⁺ˣ⁄₂₋ₓ = 9

2 + x = 9(2 - x)

2 + x = 18 - 9x

10x = 16

x = ¹⁶⁄₁₀ = ⁸⁄₅

Explanation

For any two similar triangles, the ratio of their corresponding sides is equal to the ratio of their perimeters. Given that △ ABC ∼ △ PQR, the relationship between the perimeters and the sides is expressed as:

(Perimeter(△ ABC))/(Perimeter(△ PQR)) = (AB)/(PQ)

Substituting the provided values into the formula:

36/24 = (AB)/(10)

Simplifying the ratio of the perimeters 36/24 to its lowest terms gives 3/2. The equation becomes:

3/2 = (AB)/(10)

Solving for AB through cross-multiplication:

2 × AB = 30

AB = 30/2 = 15 cm

Explanation

In right-angled triangle ABC, since ∠ B = 90°, the side AC is the hypotenuse. According to the Pythagorean theorem, AB² + BC² = AC². Substituting the given values, 8² + BC² = 10², which simplifies to 64 + BC² = 100. This results in BC² = 36, giving BC = 6 cm. This triangle follows the standard Pythagorean triplet (6, 8, 10). The area of a right-angled triangle is calculated using the formula ½ × base × height. Using BC as the base and AB as the height, the area is ½ × 6 × 8 = 24 sq.cm.

Explanation

In any triangle ABC, the sum of the interior angles is A + B + C = 180°. This relationship can be rewritten as A + B = 180° - C. Dividing both sides by 2 gives the identity (A+B)/(2) = 90° - (C)/(2).

Substituting this identity into the given expression sin((A+B)/(2)) = √3{2} yields sin(90° - (C)/(2)) = √3{2}. Applying the complementary angle formula sin(90° - θ) = cos θ, the equation simplifies to cos((C)/(2)) = √3{2}.

Since cos 30° = √3{2}, we find that (C)/(2) = 30°. To find the required value, we calculate sin((C)/(2)) = sin 30°. Using standard trigonometric values, sin 30° = ½.

Explanation

The amount A after compound interest is calculated using the formula A = P(1 + (R)/(100))ⁿ. Given the principal P = 64,000, rate R = 7.5%, and time n = 3 years, the rate can be expressed in fractional form as 7.5/100 = 3/40.

Substituting the values into the formula: A = 64,000(1 + ³⁄₄₀)³ A = 64,000(⁴³⁄₄₀)³ A = 64,000 × (43 × 43 × 43)/(40 × 40 × 40) A = 64,000 × (79,507)/(64,000) A = 79,507

The compound interest is the difference between the total amount and the principal: CI = A - P CI = 79,507 - 64,000 CI = 15,507

Explanation

Let H be the height of the temple and x be the horizontal distance between the building and the temple. According to the problem, the height of the building is 30 m.

From the foot of the building, the angle of elevation to the top of the temple is 60°: tan 60° = (H)/(x) ⟹ √3 = (H)/(x) ⟹ x = (H)/(√3)

From the top of the building, the angle of elevation to the top of the temple is 30°. The vertical distance from the top of the building to the top of the temple is H - 30: tan 30° = (H - 30)/(x) ⟹ (1)/(√3) = (H - 30)/(x) ⟹ x = (H - 30)√3

Equating the two expressions for x: (H)/(√3) = (H - 30)√3 H = 3(H - 30) H = 3H - 90 2H = 90 H = 45 m

Explanation

The rate of the filling tap is 1/50 tank per minute, and the rate of the emptying tap is 1/75 tank per minute. When both taps are open simultaneously, the net filling rate is the difference between these two rates: Net rate = 1/50 - 1/75 To subtract these fractions, we use the least common multiple of 50 and 75, which is 150: Net rate = 3/150 - 2/150 = 1/150 tank per minute. The tank is already half-filled, meaning there is 1 - 1/2 = 1/2 of the tank's capacity left to be filled. The time required to fill this remaining volume is calculated by dividing the volume by the net filling rate: Time = (1/2) / (1/150) = 150 / 2 = 75 minutes.

Explanation

The volume of a solid sphere is calculated using the formula V = ⁴⁄₃ π r³, where r is the radius. When the three spheres are melted to form a single larger sphere, the total volume of the material remains conserved. Therefore, the volume of the new sphere with radius R is the sum of the volumes of the three smaller spheres.

Setting up the equation based on the conservation of volume: ⁴⁄₃ π R³ = ⁴⁄₃ π r₁³ + ⁴⁄₃ π r₂³ + ⁴⁄₃ π r₃³

The constant factor ⁴⁄₃ π can be divided out from both sides of the equation, simplifying it to: R³ = r₁³ + r₂³ + r₃³

To find the radius R, take the cube root of both sides: R = (r₁³ + r₂³ + r₃³)^⅓

Explanation

To find the effective discount for successive discounts d₁ and d₂, we use the formula D = d₁ + d₂ - d₁ × d₂{100}.

For the first shopkeeper, the effective discount is 30 + 6 - (30 × 6)/(100) = 36 - 1.8 = 34.2%. For the second shopkeeper, the effective discount is 20 + 16 - (20 × 16)/(100) = 36 - 3.2 = 32.8%.

Since the base price of Rs. 700 is the same for both, the difference in their selling prices is equal to the difference in their total discount percentages. Difference in discount percentages = 34.2% - 32.8% = 1.4%.

The difference in selling price is 1.4% of Rs. 700: Difference = (1.4)/(100) × 700 = 1.4 × 7 = 9.8.

Explanation

The extension in a spring follows Hooke's Law, which states that the weight W hung on the spring is directly proportional to the extension x produced, formulated as W = kx. Since proportionality holds, the ratio of weight to extension remains constant: (W₁)/(x₁) = (W₂)/(x₂). Substituting the given values where W₁ = 5 kgs, x₁ = 0.4 cm, and x₂ = 5 cm, we have (5)/(0.4) = (W₂)/(5). Rearranging the equation to solve for W₂ results in W₂ = (5 × 5)/(0.4) = (25)/(0.4). This calculation simplifies to ²⁵⁰⁄₄, yielding a weight of 62.5 kgs.

Explanation

Let the cost price of one copy of the book be Rs. x. The total cost price for 12 copies is 12x. The selling price of these 12 copies is given as Rs. 1800. According to the question, the gain earned from this sale is equal to the cost price of 3 copies, which is 3x.

Using the fundamental profit and loss formula: Selling Price = Cost Price + Gain

Substituting the known values for 12 copies: 1800 = 12x + 3x 1800 = 15x

To find the cost price of a single copy, divide the total selling price by the total number of equivalent cost price units: x = ¹⁸⁰⁰⁄₁₅ x = 120

The cost price of each copy of the book is Rs. 120.

Explanation

The total percentage of people in the marriage party is 100%. The percentage of children is calculated by subtracting the percentages of women and men from the total: 100% - (32% + 54%) = 14%.

We are given that there are 196 children, so 14% of the total population equals 196. To find the value of 1% of the population, we divide the number of children by their percentage: 196 / 14 = 14. The number of men is 54% of the total population. Therefore, the total number of men is 54 × 14 = 756.

Explanation

To find the distance covered in meters, the speed must first be converted from kilometers per hour (km/hr) to meters per second (m/s). This is done using the conversion factor ⁵⁄₁₈.

The speed in m/s is calculated as: Speed = 116 × ⁵⁄₁₈ m/s

The distance formula is Distance = Speed × Time. Substituting the given values: Distance = (116 × ⁵⁄₁₈) × 18

The 18 in the denominator and the 18 in the numerator cancel each other out, simplifying the calculation to 116 × 5. This yields a total distance of 580 meters.

Explanation

The given condition is a + b = 5. We can find a set of values for a and b that satisfy this equation by looking at the constants in the target expression (a - 3)² + (b - 2)². By choosing a = 3 and b = 2, we satisfy the condition because 3 + 2 = 5. Substituting these specific values into the expression, we get:

(3 - 3)² + (2 - 2)² = 0² + 0² = 0

In competitive exams, when an expression is asked for a given constraint and a specific pair of values satisfying that constraint yields one of the options, that value is the intended answer. Since a = 3 and b = 2 satisfy the condition a + b = 5, the expression evaluates to 0.

Explanation

The given equation x² - 2x + 1 = 0 is a perfect square trinomial that can be written in the form (x - 1)² = 0. Solving for x, we take the square root of both sides to find x - 1 = 0, which gives x = 1. To find the value of the required expression, substitute x = 1 into x⁴ + (1)/(x⁴). This results in 1⁴ + (1)/(1⁴). Since 1 raised to any power remains 1, the expression simplifies to 1 + ¹⁄₁ = 1 + 1 = 2.

Explanation

In an equilateral triangle, the median is also the altitude of the triangle. The relationship between the side s and the length of the median h is given by the formula h = √3{2}s. Substituting the given value of the median, we have 12√3 = √3{2}s, which simplifies to s = 24 cm.

The area A of an equilateral triangle with side s is calculated using the formula A = √3{4}s². By substituting the value of the side, we get A = √3{4} × (24)² = √3{4} × 576 = 144√3 sq. cms. Alternatively, the area can be found directly from the median length using the formula A = (h²)/(√(3)). Plugging in h = 12√3 gives A = (12√3)²{√3} = (144 × 3)/(√(3)) = 144√3 sq. cms.

Explanation

In a circle with circumcentre O, the angle subtended by a chord at the centre is twice the angle subtended by it at any point on the major arc. The problem states that the vertex A and the chord BC lie on opposite sides of the circumcentre O. This geometric configuration implies that the circumcentre O lies within the triangle ABC, placing the vertex A on the major arc relative to the chord BC.

Given the central angle subtended by the chord BC: ∠ BOC = 150°

According to the theorem relating the central angle and the angle at the circumference for the major arc: ∠ BAC = ½ × ∠ BOC

Substituting the given value: ∠ BAC = ½ × 150° = 75°

Explanation

In any triangle ABC, the sum of the interior angles is A + B + C = 180°. The given trigonometric equations provide a system of linear equations for the angles:

From sin(B + C - A) = √3/2, we know that B + C - A can be 60° or 120°. Similarly, from tan(C + A - B) = 1, we have C + A - B = 45°.

Using B + C - A = 60° and C + A - B = 45°, we add the two equations together: (B + C - A) + (C + A - B) = 60° + 45° 2C = 105° C = 52.5°

To ensure this satisfies the triangle conditions, we find A and B. Subtracting the second equation from the first: (B + C - A) - (C + A - B) = 60° - 45° 2B - 2A = 15° ⟹ B - A = 7.5° Using the angle sum property: A + B = 180° - C = 180° - 52.5° = 127.5° Adding (B - A) and (B + A) gives 2B = 135°, so B = 67.5°. This leaves A = 127.5° - 67.5° = 60°. Since A, B, and C are all less than 90°, the triangle is acute-angled as required.

Explanation

A number is divisible by 125 if and only if the number formed by its last three digits is divisible by 125. The three-digit multiples of 125 are 125, 250, 375, 500, 625, 750, and 875. Given the available digits {2, 3, 8, 7, 5}, the only possible three-digit endings that can be formed are 375 and 875.

For the case where the number ends in 375, the digits 2 and 8 remain. These two digits can be arranged in the remaining two positions in 2! = 2 × 1 = 2 ways. This results in the five-digit numbers 28375 and 82375.

For the case where the number ends in 875, the digits 2 and 3 remain. These two digits can also be arranged in the remaining two positions in 2! = 2 × 1 = 2 ways. This results in the five-digit numbers 23875 and 32875.

Combining the possibilities from both cases, the total number of such integers is 2 + 2 = 4.

Explanation

Using the rule of alligation, the ratio of the number of technicians to the number of rest workers is calculated based on the deviation of their respective average salaries from the overall average salary.

Ratio of Technicians to Rest Workers = (12000 - 9000) : (15000 - 12000) Ratio = 3000 : 3000 = 1 : 1

Since the number of technicians is given as 7, and the ratio of technicians to rest workers is 1:1, the number of rest workers is also 7. Therefore, the total number of workers in the workshop is 7 + 7 = 14.

This can also be solved using the weighted average formula. Let n be the total number of workers: 12000n = 7 × 15000 + (n - 7) × 9000 12000n = 105000 + 9000n - 63000 3000n = 42000 n = 14

Explanation

Let h be the height of the aeroplane from the ground. For two observation points located on the same side of the object at a distance d apart, the relationship between the height and the angles of elevation α and β (where α < β) is given by the formula h = (d)/(cot α - cot β). Since the observations are made from consecutive stones, the distance d between them is 1 km. Given the angles α = 45° and β = 60°:

h = (1)/(cot 45° - cot 60°)

Substituting the trigonometric values cot 45° = 1 and cot 60° = (1)/(√(3)):

h = (1)/(1 - 1){√3} = √3{√3 - 1}

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

h = √3(√3 + 1){(√3 - 1)(√3 + 1)} = 3 + √3{3 - 1} = 3 + √3{2}

This can be written as ½ (√3 + 3) kms.

Explanation

The given expression is a quadratic function f(x) = -4x² + 20x + 5, which is in the standard form ax² + bx + c with a = -4, b = 20, and c = 5. Because the coefficient of x² is negative (a < 0), the parabola opens downwards, and the function reaches its maximum value at its vertex. The formula for the maximum value of such a quadratic expression is:

Maximum Value = (4ac - b²)/(4a)

Substituting the given constants into the formula: Maximum Value = (4(-4)(5) - (20)²)/(4(-4)) Maximum Value = (-80 - 400)/(-16) Maximum Value = ⁻⁴⁸⁰⁄₋₁₆ = 30

Alternatively, the maximum occurs at x = -b / (2a). Calculating this gives x = -20 / (2 × -4) = 2.5. Substituting x = 2.5 back into the original expression: f(2.5) = 5 + 20(2.5) - 4(2.5)² f(2.5) = 5 + 50 - 4(6.25) f(2.5) = 55 - 25 = 30

Explanation

Given the ratio 3a = 4b = 6c, we can express a, b, and c in terms of a common variable x by finding the least common multiple of 3, 4, and 6, which is 12. Dividing the entire equation by 12, we get a/4 = b/3 = c/2 = x. Thus, a = 4x, b = 3x, and c = 2x.

Substituting these into the sum equation a + b + c = 27√(29): 4x + 3x + 2x = 27√(29) 9x = 27√(29) x = 3√(29)

The required value is √(a² + b² + c²): √((4x)² + (3x)² + (2x)²) = √(16x² + 9x² + 4x²) = √(29x²) = x√(29)

Substituting the value of x: 3√(29) × √(29) = 3 × 29 = 87

Explanation

The sum of an interior angle and an exterior angle of a regular polygon is always 180°. Given that the interior angle is 170°, the exterior angle can be found as: Exterior Angle = 180° - 170° = 10°

The sum of all exterior angles for any convex polygon is 360°. In a regular polygon, all exterior angles are equal, so the number of sides n is determined by dividing the total sum of exterior angles by the measure of one exterior angle: n = (360°)/(10°) = 36

Alternatively, using the formula for the interior angle of a regular polygon: ((n - 2) × 180°)/(n) = 170° 180n - 360 = 170n 10n = 360 n = 36

Explanation

For a circle with center O and tangents PA and PB from an external point P, the relationship between the angle formed by the tangents ∠ APB and the angle ∠ OAB is defined by the formula ∠ OAB = ½ ∠ APB. This relation exists because the radii OA and OB are perpendicular to the tangents at the points of contact, making ∠ OAP = ∠ OBP = 90°. In the quadrilateral OAPB, the angle at the center ∠ AOB is 180° - ∠ APB. In the isosceles triangle OAB, the base angle ∠ OAB = (180° - ∠ AOB)/(2). Substituting the value of ∠ AOB, the expression simplifies to (180° - (180° - ∠ APB))/(2) = ½ ∠ APB. Given ∠ APB = 50°, the calculation is ∠ OAB = (50°)/(2) = 25°.

Explanation

The final number of workers after three successive annual increases is calculated using the formula for compound growth with varying rates:

A = P(1 + (R₁)/(100))(1 + (R₂)/(100))(1 + (R₃)/(100))

Substituting the initial strength P = 8000 and the growth rates R₁ = 5%, R₂ = 10%, and R₃ = 20%:

A = 8000 × (1 + ⁵⁄₁₀₀) × (1 + ¹⁰⁄₁₀₀) × (1 + ²⁰⁄₁₀₀)

A = 8000 × ¹⁰⁵⁄₁₀₀ × ¹¹⁰⁄₁₀₀ × ¹²⁰⁄₁₀₀

A = 8000 × ²¹⁄₂₀ × ¹¹⁄₁₀ × ⁶⁄₅

Simplifying the expression by dividing the initial population by the product of the denominators:

A = ⁸⁰⁰⁰⁄₁₀₀₀ × (21 × 11 × 6)

A = 8 × 21 × 11 × 6

A = 168 × 66

A = 11088

This result represents the total number of workers present at the end of the third year, which is the workforce count during the fourth year.

Explanation

For any angle X, the algebraic relationship between the sum and difference of sine and cosine is given by the identity: (sin X + cos X)² + (sin X - cos X)² = 2(sin² X + cos² X)

Substituting the given value sin X - cos X = 1 and the fundamental trigonometric identity sin² X + cos² X = 1 into the equation: (sin X + cos X)² + (1)² = 2(1) (sin X + cos X)² + 1 = 2 (sin X + cos X)² = 1

Taking the square root of both sides, we get sin X + cos X = ± 1. Since X is an angle in the first quadrant, both sin X and cos X are non-negative, meaning their sum must be positive. Thus, sin X + cos X = 1.

Explanation

The daily rice requirement is calculated by dividing the total weight used in a week by the number of days: Daily requirement = (56 kg)/(7 days) = 8 kg/day

The month of April has 30 days and the month of May has 31 days. To find the total quantity required for both months, first determine the total number of days: Total days = 30 + 31 = 61 days

The total quantity of rice needed for April and May together is the product of the daily consumption and the total number of days: Total rice = 8 kg/day × 61 days = 488 kg

Explanation

The relationship between the Cost Price (CP) and the Printed Price or Marked Price (MP) can be directly determined using the standard formula (CP)/(MP) = (100 - Discount%)/(100 + Profit%).

Given that the discount percentage is 20% and the profit percentage is 15%, we substitute these values into the formula:

(CP)/(MP) = (100 - 20)/(100 + 15)

(CP)/(MP) = ⁸⁰⁄₁₁₅

To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 5:

80 ÷ 5 = 16 115 ÷ 5 = 23

The ratio of the cost price to the printed price is 16:23.

Explanation

The speed of an object is determined by the formula Speed = (Distance)/(Time). For the car, the speed is S_cₐᵣ = (80 km)/(2 h) = 40 km/h. For the train, the speed is Sₜᵣₐᵢₙ = (180 km)/(3 h) = 60 km/h. The ratio of the speed of the car to the speed of the train is 40 : 60. Dividing both sides by their greatest common divisor, 20, simplifies the ratio to 2 : 3.

Explanation

The cash amount is the marked price less the discount. To calculate the 2.5% discount quickly, note that 10% of 3600 is 360. Since 2.5% is one-fourth of 10%, the discount value is: Discount = ³⁶⁰⁄₄ = 90 Cash Amount = Marked Price - Discount = 3600 - 90 = 3510 Alternatively, using the formula for selling price after a discount: SP = MP × (1 - (D)/(100)) SP = 3600 × (1 - (2.5)/(100)) = 3600 × 0.975 = 3510

Explanation

When total expenditure remains constant, the quantity consumed is inversely proportional to the price. A price hike of 20% implies the ratio of the initial price to the new price is 100:120, which simplifies to 5:6. Consequently, the ratio of the initial quantity to the new quantity is the inverse, 6:5.

The reduction in quantity in terms of ratio units is 6 - 5 = 1 unit. According to the problem, this 1 unit corresponds to a reduction of 4 kg. Therefore, the initial quantity consumed was 6 × 4 = 24 kg.

To find the initial price per kg, divide the total expenditure by the initial quantity: Initial Price = 120 / 24 = 5

Alternatively, the price increase of 20% on ₹120 is 0.20 × 120 = 24. This extra amount of ₹24 is what would have been paid for the 4 kg of sugar that is no longer available. Thus, the current (increased) price per kg is 24 / 4 = 6. Since the current price is 120% of the initial price, the initial price is 6 / 1.2 = 5 per kg.

Explanation

The distance covered by the man each day forms an arithmetic progression where the first term a = 1 and the common difference d = 1. To find the total distance covered in n = 10 days, we use the formula for the sum of the first n natural numbers, which is Sₙ = (n(n + 1))/(2). Substituting n = 10 into the formula, we get S₁₀ = (10(10 + 1))/(2). Calculating this gives (10 × 11)/(2) = 5 × 11 = 55 km.

Explanation

The rate of simple interest R when a sum becomes n times itself in T years is given by the formula R = ((n - 1) × 100)/(T). Given that the sum becomes 5 times (n = 5) in 8 years (T = 8), the rate is calculated by substituting these values into the formula: R = ((5 - 1) × 100)/(8) = (4 × 100)/(8). Simplifying the expression, R = ⁴⁰⁰⁄₈ = 50%.

Explanation

The algebraic identity for the square of a trinomial is given by: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Given the values a + b + c = 15 and a² + b² + c² = 83, substitute them into the formula: 15² = 83 + 2(ab + bc + ca) 225 = 83 + 2(ab + bc + ca)

To find the value of ab + bc + ca, rearrange the equation: 2(ab + bc + ca) = 225 - 83 2(ab + bc + ca) = 142 ab + bc + ca = ¹⁴²⁄₂ ab + bc + ca = 71