SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

When the expenditure on a commodity remains constant, the consumption is inversely proportional to the price. If the price increases by r%, the percentage reduction in consumption required to keep the expenditure unchanged is given by the formula:

Reduction Percentage = ( (r)/(100 + r) ) × 100

Given that the price of petrol increases by r = 28%, substitute the value into the formula:

Reduction in Consumption = ( (28)/(100 + 28) ) × 100 Reduction in Consumption = ( ²⁸⁄₁₂₈ ) × 100

Simplifying the fraction by dividing both the numerator and the denominator by 4:

⁷⁄₃₂ × 100 = ⁷⁰⁰⁄₃₂ ⁷⁰⁰⁄₃₂ = ¹⁷⁵⁄₈ = 21.875%

Rounding to two decimal places, the percentage decrease in consumption is 21.88%.

Explanation

The expression can be simplified using the trigonometric identity cot(90° - θ) = tan θ and the property that cot θ tan θ = 1.

We identify the pairs of complementary angles in the expression: cot 77° = cot(90° - 13°) = tan 13° cot 63° = cot(90° - 27°) = tan 27°

Substituting these values back into the original expression: cot 13° cot 27° cot 45° tan 27° tan 13°

By rearranging the terms, we group the reciprocal functions: (cot 13° tan 13°) · (cot 27° tan 27°) · cot 45°

Since cot θ tan θ = 1 and the value of cot 45° = 1, the calculation becomes: 1 · 1 · 1 = 1

Explanation

In a sequence of n consecutive odd numbers where n is even, the average is the value exactly midway between the two middle terms. Here, for 8 numbers, the average of 48 is the midpoint between the 4th and 5th terms. Since consecutive odd numbers have a common difference of 2, the 4th term is 48 - 1 = 47 and the 5th term is 48 + 1 = 49.

To find the 7th term, we add the common difference twice to the 5th term. Thus, the 6th term is 49 + 2 = 51 and the 7th term is 51 + 2 = 53. The sum of the 5th and 7th numbers is 49 + 53 = 102.

Explanation

The expression (sin A + cos A)(1 - sin A cos A) can be simplified using the fundamental trigonometric identity sin² A + cos² A = 1. By substituting 1 with sin² A + cos² A, the original expression takes the form: (sin A + cos A)(sin² A + cos² A - sin A cos A)

Rearranging the terms in the second bracket gives: (sin A + cos A)(sin² A - sin A cos A + cos² A)

This matches the structure of the algebraic identity for the sum of cubes: x³ + y³ = (x + y)(x² - xy + y²). By letting x = sin A and y = cos A, the identity results in: sin³ A + cos³ A = (sin A + cos A)(sin² A - sin A cos A + cos² A)

This confirms that the given expression is equivalent to the sum of the cubes of the sine and cosine functions.

Explanation

The circumference of a circle is given by the formula C = 2π r, where r is the radius. Since 2π is a constant, the ratio of the circumferences is directly equal to the ratio of the radii. Let R be the radius of the outer circle and r be the radius of the inner circle. Given the ratio of circumferences is 5:4, we have R/r = 5/4, which implies R = 1.25r.

The width of the circular path is the difference between the outer radius and the inner radius, expressed as R - r = 50 metres. Substituting the value of R from the ratio into this equation gives 1.25r - r = 50. Simplifying this leads to 0.25r = 50. To find the inner radius, we calculate r = 50 / 0.25, which results in r = 200 metres. Alternatively, using the ratio parts, the difference of 1 unit (5 - 4) represents the width of 50 metres. Therefore, the inner radius, which represents 4 units, is 4 × 50 = 200 metres.

Explanation

To find the value of 4a² + 9b², use the algebraic identity (x + y)² = x² + y² + 2xy. By setting x = 2a and y = 3b, the square of the given expression 2a + 3b = 10 is:

(2a + 3b)² = (2a)² + (3b)² + 2(2a)(3b)

Substitute the values provided in the question into this equation:

10² = 4a² + 9b² + 12ab

100 = 4a² + 9b² + 12ab

Now, substitute ab = 3 into the equation:

100 = 4a² + 9b² + 12(3)

100 = 4a² + 9b² + 36

To isolate 4a² + 9b², subtract 36 from both sides:

4a² + 9b² = 100 - 36

4a² + 9b² = 64

Explanation

Using the algebraic identity (a - b)² = a² + b² - 2ab, let a = 4x and b = 7y. The expansion of the square of the given binomial is (4x - 7y)² = (4x)² + (7y)² - 2(4x)(7y). Simplifying this expression results in (4x - 7y)² = 16x² + 49y² - 56xy.

Substitute the known values (4x - 7y) = 11 and xy = 8 into the identity: 11² = 16x² + 49y² - 56(8) 121 = 16x² + 49y² - 448

To find the value of 16x² + 49y², add 448 to both sides of the equation: 16x² + 49y² = 121 + 448 16x² + 49y² = 569

Explanation

To find the smallest perfect square divisible by 16, 18, and 36, we first determine their Least Common Multiple (LCM). Prime factorization of the numbers gives 16 = 2⁴, 18 = 2 × 3², and 36 = 2² × 3². The LCM is calculated by taking the highest power of each prime factor present in the numbers, which results in LCM = 2⁴ × 3² = 16 × 9 = 144.

A perfect square must have even exponents for all its prime factors. In the prime factorization of 144, the exponents of 2 and 3 are 4 and 2 respectively, both of which are even. This confirms that 144 is already a perfect square (12² = 144). Since the LCM is the smallest number divisible by the given set, and 144 is a perfect square, it is the smallest perfect square that satisfies the condition.

Explanation

The trigonometric identity for the triple angle of cosine is cos(3θ) = 4cos³ θ - 3cos θ. The given expression 3cos θ - 4cos³ θ is the negative of this identity, which simplifies to -cos(3θ).

Given that sin θ = ½, the value of θ is 30° (for the principal range). Substituting this value into the simplified expression, we get -cos(3 × 30°) = -cos(90°). Since cos(90°) = 0, the value of the expression is 0.

Alternatively, if sin θ = ½, then cos θ = √(1 - sin² θ) = √(1 - ¼) = √3{2}. Substituting this directly into the expression: 3(√3{2}) - 4(√3{2})³ = 3√3{2} - 4(3√3{8}) = 3√3{2} - 3√3{2} = 0.

Explanation

The question contains a typographical error where the first term should be (a³ + b³) / (a - b). Calculating the mean proportion with this term gives the square root of [(a³ + b³)/(a - b)] * [(a² - b²)/(a² - ab + b²)], which simplifies to the square root of (a + b)², resulting in a + b.

Explanation

The numbers divisible by 9 between 43 and 481 form an arithmetic progression. The first term a is the smallest multiple of 9 greater than 43, which is 9 × 5 = 45. The last term l is the largest multiple of 9 less than 481, which is 9 × 53 = 477. Since the numbers must be divisible by 9, the common difference d is 9.

To find the total number of terms n in this sequence, we use the formula for the nᵗʰ term of an arithmetic progression: l = a + (n - 1)d. Substituting the identified values into the formula:

477 = 45 + (n - 1) × 9

Subtracting 45 from both sides:

432 = (n - 1) × 9

Dividing both sides by 9:

48 = n - 1

Solving for n:

n = 48 + 1 n = 49

The calculation determines that there are 49 such integers within the specified range.

Explanation

Total work is calculated as the Least Common Multiple of the time taken by A and B, which is LCM(10, 16) = 80 units. The efficiency of A is 80 / 10 = 8 units per day, and the efficiency of B is 80 / 16 = 5 units per day.

In a single cycle of 2 days where A and B work alternately, the total work completed is 8 + 5 = 13 units. To find the number of full cycles, we divide the total work by the work done in one cycle: 80 / 13 = 6 cycles with a remainder. The work completed in 6 cycles (12 days) is 13 × 6 = 78 units.

The remaining work is 80 - 78 = 2 units. On the 13th day, it is A's turn to work. Since A's efficiency is 8 units per day, the time taken by A to complete the remaining 2 units is 2 / 8 = 1/4 day. Therefore, the total time required to complete the whole work is 12 + 1/4 = 12¼ days.

Explanation

The volume of a sphere is calculated using the formula V = ⁴⁄₃ π r³. Given the volume is 179 ⅗ m³, which is approximately 179.6 m³, we find the radius r. For calculation purposes in competitive exams, this value is a close approximation of ⁵³⁹⁄₃ m³ (which is 179 ⅔ m³), yielding an exact radius:

⁴⁄₃ × ²²⁄₇ × r³ = ⁵³⁹⁄₃ r³ = (539 × 21)/(88 × 3) = (539 × 7)/(88) = (49 × 7)/(8) r³ = ³⁴³⁄₈ r = ³√(³⁴³⁄₈) = ⁷⁄₂ = 3.5 m

The surface area of a sphere is given by the formula A = 4 π r². Substituting the radius r = 3.5 m:

A = 4 × ²²⁄₇ × (⁷⁄₂)² A = 4 × ²²⁄₇ × ⁴⁹⁄₄ A = 22 × 7 = 154 m²

Explanation

Let x = sec θ. Given that cos θ = (1)/(sec θ), the equation sec θ - 2cos θ = ⁷⁄₂ can be expressed in terms of x as:

x - ²⁄ₓ = ⁷⁄₂

To solve for x, multiply the entire equation by 2x to eliminate the denominators:

2x² - 4 = 7x

Rearranging the terms into a standard quadratic equation ax² + bx + c = 0 gives:

2x² - 7x - 4 = 0

We can solve this by factoring the quadratic expression. Splitting the middle term:

2x² - 8x + x - 4 = 0

2x(x - 4) + 1(x - 4) = 0

(2x + 1)(x - 4) = 0

This yields two possible values for x: x = 4 or x = -½.

Since θ is a positive acute angle (0 < θ < 90°), sec θ must be positive and greater than or equal to 1. Therefore, we discard x = -½ and conclude that sec θ = 4.

Explanation

For any two numbers x and y, the product of the numbers is equal to the product of their Highest Common Factor (HCF) and Lowest Common Multiple (LCM). This is expressed by the fundamental relation: x · y = HCF × LCM

Given that the HCF is 3 and the LCM is 54, the product of the two numbers is: x · y = 3 × 54 = 162

The sum of the numbers is given as: x + y = 18

The sum of their reciprocals is defined as ¹⁄ₓ + (1)/(y). By taking the common denominator, this expression simplifies to: ¹⁄ₓ + (1)/(y) = (x + y)/(x · y)

Substituting the known values for the sum and the product into the formula: Sum of reciprocals = ¹⁸⁄₁₆₂

Dividing both the numerator and the denominator by 18, we get: (18 ÷ 18)/(162 ÷ 18) = ⅑

Explanation

Let the marks scored by Rohan in English, Science, and Mathematics be represented by E, S, and M respectively. From the given conditions, the marks in English are twice those in Science, which gives the ratio E:S = 2:1. The ratio of marks in English and Mathematics is given as E:M = 2:3.

Since the value for E is common in both ratios, the combined ratio of marks in English, Science, and Mathematics is E:S:M = 2:1:3. Let the marks be 2x, x, and 3x. The total marks are the sum of these parts:

2x + x + 3x = 126 6x = 126 x = ¹²⁶⁄₆ = 21

The marks in English are represented by 2x. Substituting the value of x: E = 2 × 21 = 42

Explanation

The area of a square A is related to its diagonal d by the formula A = (d²)/(2). Since the area is directly proportional to the square of the diagonal, if the area is multiplied by a factor k, the diagonal is multiplied by a factor of √k.

Given the diagonal of the first square d₁ = 8√2 cm and the second square has triple the area (k=3), the diagonal of the second square d₂ is calculated as:

d₂ = d₁ × √3

Substituting the value of d₁:

d₂ = 8√2 × √3 = 8√(2 × 3) = 8√6 cm

Alternatively, the area of the first square is A₁ = (8√2)²{2} = (64 × 2)/(2) = 64 cm². The area of the second square is A₂ = 3 × 64 = 192 cm². Using the area formula for the second square:

192 = (d₂²)/(2) d₂² = 384 d₂ = √(384) = √(64 × 6) = 8√6 cm

Explanation

The given equation x + ¹⁄ₓ = 2 can be rewritten by multiplying throughout by x to get x² + 1 = 2x. Rearranging the terms results in the quadratic equation x² - 2x + 1 = 0, which is the expansion of the perfect square (x - 1)² = 0. Solving this gives the unique value x = 1. Substituting x = 1 into the required expression x⁵⁷ + (1)/(x⁵⁷⁾, we get 1⁵⁷ + (1)/(1⁵⁷⁾. Since 1 raised to any power remains 1, the calculation becomes 1 + ¹⁄₁, resulting in 2.

Explanation

The value is determined by evaluating the tangent of the sum of the angle components provided in the expression.

Summing the given numerical values: 30° + 15° = 45°

According to standard trigonometric tables, the value of tan 45° is: tan 45° = 1

This matches the value provided in the options.

Explanation

The overall profit or loss percentage on a consignment is the weighted average of the profit/loss percentages of its individual parts. Let the total value of the consignment be V.

The first part is ¾ of the consignment, sold at an 8% profit. The remaining part is 1 - ¾ = ¼ of the consignment, sold at a 4% loss. The net profit percentage P is calculated as:

P = (¾ × 8%) + (¼ × -4%)

P = 6% - 1% = 5%

We are given that the overall profit is Rs. 600, which corresponds to this 5% of the total value. Therefore:

5% of V = 600

⁵⁄₁₀₀ × V = 600

V = (600 × 100)/(5) = 600 × 20 = 12,000

Explanation

A tangent to a circle is defined as a line that touches the circle at exactly one point. A fundamental property of a circle is that the tangent at any point is perpendicular to the radius through the point of contact. For a given tangent at a point P, any line parallel to it must also be perpendicular to the diameter passing through P. Since a diameter intersects the circle at exactly two points, there are only two points on the circle where tangents can be parallel to a specific direction. One of these points is the original point of tangency P. The other point is the diametrically opposite point Q. Consequently, for any given tangent, there is exactly 1 other tangent parallel to it.

Explanation

From the given equation m · xᵐ - n · xⁿ = 0, we can establish the relationship: m · xᵐ = n · xⁿ ⟹ xᵐ = ⁿ⁄ₘ xⁿ

The expression to be evaluated is (1)/(xᵐ⁾⁺ˣ^^^ⁿ + (1)/(xᵐ⁾⁻ˣ^^^ⁿ. Simplifying this algebraic sum by taking a common denominator: (xᵐ - xⁿ) + (xᵐ + xⁿ){(xᵐ + xⁿ)(xᵐ - xⁿ)} = 2xᵐ{x²ᵐ - x²ⁿ}

Now, substitute xᵐ = ⁿ⁄ₘ xⁿ into the simplified expression: (2 (n)/(m) xⁿ){(ⁿ⁄ₘ xⁿ)² - (xⁿ)²} = (2n)/(m) xⁿ{(xⁿ)² (n²{m²} - 1)}

Cancel xⁿ from the numerator and denominator and simplify the denominator's fraction: (2n)/(m){xⁿ (n² - m²{m²})} = ²ⁿ⁄ₘ · m²{xⁿ(n² - m²)}

Canceling m results in the final value: (2mn)/(xⁿ⁾⁽ⁿ^^^² ⁻ ᵐ^²⁾

Explanation

Given the quadratic equation for a non-zero x: 5x² + 7x + 5 = 0

Dividing the entire equation by 5x to express it in terms of x + ¹⁄ₓ: 5x²{5x} + ⁷ˣ⁄₅ₓ + ⁵⁄₅ₓ = 0 x + ⁷⁄₅ + ¹⁄ₓ = 0 x + ¹⁄ₓ = -⁷⁄₅

To find the value of x³ + (1)/(x³⁾, we use the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Substituting a = x and b = ¹⁄ₓ: x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x)(¹⁄ₓ)(x + ¹⁄ₓ) x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ)

Now, substitute the value x + ¹⁄ₓ = -⁷⁄₅: x³ + (1)/(x³⁾ = (-⁷⁄₅)³ - 3(-⁷⁄₅) x³ + (1)/(x³⁾ = -³⁴³⁄₁₂₅ + ²¹⁄₅

To add the terms, use a common denominator of 125: x³ + (1)/(x³⁾ = -³⁴³⁄₁₂₅ + (21 × 25)/(5 × 25) x³ + (1)/(x³⁾ = -³⁴³⁄₁₂₅ + ⁵²⁵⁄₁₂₅ x³ + (1)/(x³⁾ = (525 - 343)/(125) = ¹⁸²⁄₁₂₅

Explanation

The total simple interest received from both B and C is the sum of the interests calculated for each individual loan. Using the formula for simple interest SI = (P × R × T)/(100), where P is the principal, R is the rate of interest, and T is the time in years, the equation is set up as follows:

(23000 × R × 3)/(100) + (19000 × R × 4)/(100) = 3625

By simplifying the expressions: 230 × 3 × R + 190 × 4 × R = 3625 690R + 760R = 3625

Combining the terms: 1450R = 3625

Solving for R: R = ³⁶²⁵⁄₁₄₅₀ R = 2.5

The annual rate of interest is 2.5 percent.

Explanation

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

(Perimeter of △ ABC)/(Perimeter of △ DEF) = (AB)/(DE)

Substituting the known values into the formula:

(Perimeter of △ ABC)/(25) = (9.1)/(6.5)

To simplify the calculation, the ratio (9.1)/(6.5) can be written as ⁹¹⁄₆₅. Dividing both the numerator and the denominator by their greatest common divisor, 13, the ratio simplifies to ⁷⁄₅.

Perimeter of △ ABC = 25 × ⁷⁄₅

Perimeter of △ ABC = 5 × 7 = 35 cm

Explanation

Let the daily work efficiency of one man be M and one woman be W. According to the problem, the total work can be expressed in two ways based on the groups given: (1M + 4W) × ⁶⁵⁄₄ = (3M + 4W) × ¹³⁄₂

To find the relationship between the efficiency of men and women, simplify the equation by dividing both sides by 13 and multiplying by 4: 5(M + 4W) = 2(3M + 4W) 5M + 20W = 6M + 8W M = 12W

This implies that the efficiency of one man is equal to that of 12 women. Now, calculate the total units of work by substituting M = 12W into the first expression: Total Work = (12W + 4W) × ⁶⁵⁄₄ = 16W × ⁶⁵⁄₄ = 4 × 65W = 260W

To find the time taken by 13 women to complete this work, divide the total work by their combined efficiency: Number of days = (260W)/(13W) = 20 days

Explanation

To find what percent the number of calculators sold by S5 is of the number sold by S1, we use the percentage formula: Percentage = (Value of S5)/(Value of S1) × 100

From the given table, the number of calculators sold by S5 is 80 and the number of calculators sold by S1 is 40. Substituting these values into the equation: Percentage = ⁸⁰⁄₄₀ × 100

Dividing 80 by 40 gives 2. Therefore: Percentage = 2 × 100 = 200%

This calculation shows that the sales of shopkeeper S5 are exactly double the sales of shopkeeper S1, which corresponds to 200 percent.

Explanation

To find the value of cos² 15°, we use the trigonometric power-reduction identity derived from the double-angle formula: cos 2θ = 2cos² θ - 1 ⟹ cos² θ = (1 + cos(2θ))/(2)

By substituting θ = 15° into the equation, we get: cos² 15° = (1 + cos(2 × 15°))/(2) cos² 15° = (1 + cos 30°)/(2)

The exact value of cos 30° is √3{2}. Substituting this value into the expression: cos² 15° = 1 + {√3{2}}{2}

To simplify the numerator, we find a common denominator: 1 + √3{2} = 2 + √3{2}

Now, dividing by the denominator 2: cos² 15° = {2 + √3{2}}{2} = 2 + √3{4}

Explanation

Distance HJ is covered by the train starting from point H. The time duration from 6:30 pm to 11:30 pm is t₁ = 5 hours. Using the formula Distance = Speed × Time, the distance HJ = 90 km/hr × 5 hours = 450 km.

Distance KJ is covered by the train starting from point K. The time duration from 7:30 pm to 11:30 pm is t₂ = 4 hours. Thus, the distance KJ = 72 km/hr × 4 hours = 288 km.

To find the ratio of the distances HJ and KJ: Ratio = (HJ)/(KJ) = ⁴⁵⁰⁄₂₈₈

Dividing both the numerator and the denominator by their common factor 18: 450 ÷ 18 = 25 288 ÷ 18 = 16

The resulting ratio is 25:16.

Explanation

Average speed is defined as the total distance covered divided by the total time taken: Vₐᵥ_g = Dₜₒₜₐₗ{Tₜₒₜₐₗ}

The total distance is the sum of the distances covered by all 4 team members. Since each member covers 5 km: Dₜₒₜₐₗ = 4 × 5 km = 20 km

The total time taken is 30 minutes. To express the average speed in km/hr, convert the time from minutes to hours: Tₜₒₜₐₗ = ³⁰⁄₆₀ hr = 0.5 hr

Substitute these values into the average speed formula: Vₐᵥ_g = (20 km)/(0.5 hr) = 40 km/hr

Explanation

The net amount payable after a trade discount is calculated by subtracting the discount value from the gross list price of the goods. Trade discounts are granted at the time of purchase and are generally deducted from the invoice itself, meaning the transaction is recorded in the books of accounts at the net value.

Using the net price formula: Net Amount = List Price × (1 - Discount Rate)

Substituting the given values: Trade Discount = 55,000 × 15% = 55,000 × 0.15 = 8,250

Amount Paid = 55,000 - 8,250 = 46,750

Alternatively, calculating the net percentage directly: Amount Paid = 55,000 × 85% = 55,000 × 0.85 = 46,750

Explanation

The average marked price is determined by dividing the total sum of the marked prices of all articles by the total number of articles. The marked prices for the seven articles D, E, F, G, H, I, and J are 1100, 700, 900, 600, 400, 500, and 1000 respectively. Summing these values gives 1100 + 700 + 900 + 600 + 400 + 500 + 1000 = 5200. Given that there are n = 7 articles, the average marked price is calculated using the formula Average = (Σ Marked Prices)/(n). Substituting the values, we get ⁵²⁰⁰⁄₇, which results in approximately 742.857. Rounding to two decimal places, the average is 742.85.

Explanation

In a right-angled △ XYZ with ∠ X = 90° , the side YZ is the hypotenuse. The length of the side XY , which is adjacent to ∠ Y , is determined using the Pythagorean theorem: XY = √(YZ²⁾⁻^X^Z^^^² = √(15²⁾⁻¹²^^^² = √(225 - 144) = √(81) = 9 cm The side lengths 9, 12, and 15 form a Pythagorean triplet, being a 3 × multiple of the primitive 3, 4, 5 triplet. The trigonometric ratio for cos Y is defined as the length of the adjacent side divided by the length of the hypotenuse: cos Y = (Adjacent)/(Hypotenuse) = (XY)/(YZ) Substituting the known values: cos Y = ⁹⁄₁₅ = ⅗

Explanation

To find the LCM of the given expressions, we factorize each into its simplest algebraic components. The first expression is 8x³ + 80x² + 200x = 8x(x² + 10x + 25). Recognizing the quadratic as a perfect square, it becomes 8x(x + 5)².

The second expression is 4x⁴ + 16x³ - 20x² = 4x²(x² + 4x - 5). Factorizing the quadratic trinomial gives 4x²(x + 5)(x - 1).

The Least Common Multiple (LCM) is the product of the highest power of every factor appearing in the expressions. For the numerical coefficients 8 and 4, the LCM is 8. For the variable x, the highest power is x². For the binomial factor (x + 5), the highest power is (x + 5)². Finally, the factor (x - 1) appears with a maximum power of 1. Multiplying these highest powers together results in 8x²(x + 5)²(x - 1).

Explanation

The total work done is constant, allowing us to equate the two given scenarios using the formula Work = (n₁ M + n₂ W) × D. Equating the work for both cases gives (25M + 45W) × 15 = (15M + 60W) × 20. Dividing both sides by 5 results in 3(25M + 45W) = 4(15M + 60W). Expanding this yields 75M + 135W = 60M + 240W, which simplifies to 15M = 105W, or M = 7W. This indicates the efficiency of one man is equal to that of seven women.

The total work in terms of women-equivalent units is calculated by substituting M = 7 and W = 1 into the first equation: Total Work = (25 × 7 + 45 × 1) × 15 = (175 + 45) × 15 = 220 × 15 = 3300 units. The combined efficiency of the target group, 69 men and 67 women, is 69 × 7 + 67 × 1 = 483 + 67 = 550 units per day. The time required to complete the work is (Total Work)/(Group Efficiency) = ³³⁰⁰⁄₅₅₀ = 6 days.

Explanation

Let T be the time taken by vehicle P to run the 1500 m race. In the second scenario, when P gives Q a head start of 400 m, the race ends in a dead heat. This means that while P covers 1500 m, Q covers 1500 - 400 = 1100 m in the same time T. Thus, the speed of vehicle Q can be expressed as V_Q = (1100)/(T).

In the first scenario, P gives Q a head start of 200 m and wins by 8 seconds. This implies that P completes the 1500 m race in time T, while Q takes T + 8 seconds to cover the remaining 1500 - 200 = 1300 m. Therefore, the speed of vehicle Q can also be expressed as V_Q = (1300)/(T + 8).

By equating the two expressions for the speed of vehicle Q:

(1100)/(T) = (1300)/(T + 8)

(11)/(T) = (13)/(T + 8)

11(T + 8) = 13T

11T + 88 = 13T

2T = 88

T = 44 seconds

Vehicle P takes 44 seconds to complete the 1500 m race.

Explanation

The given equation is tan A - tan B - tan C = tan A tan B tan C. By rearranging the terms to isolate the components of the tangent addition formula, we get:

tan A - tan A tan B tan C = tan B + tan C

Factoring out tan A on the left side:

tan A (1 - tan B tan C) = tan B + tan C

Dividing both sides by (1 - tan B tan C) yields:

tan A = (tan B + tan C)/(1 - tan B tan C)

According to the standard trigonometric identity for the sum of two angles, tan(B + C) = (tan B + tan C)/(1 - tan B tan C). Substituting this identity into our equation results in:

tan A = tan(B + C)

By comparing the arguments of the tangent function, it follows that A = B + C.

Explanation

To determine what percent the budget allocation for education in state K is relative to the budget allocation in state L, we use the standard percentage formula: Percentage = ( (Budget of State K)/(Budget of State L) ) × 100

According to the provided table, the budget allocation for state K is 175 and the budget allocation for state L is 350. Substituting these values into the formula gives: Percentage = ( ¹⁷⁵⁄₃₅₀ ) × 100

Since 175 is exactly half of 350, the fraction simplifies to 1/2 or 0.5. Multiplying this value by 100 results in 50%.

Explanation

The volume V of a hemisphere is given by the formula V = ⅔ π r³, where r is the radius. Given the diameter is 35 cm, the radius r = ³⁵⁄₂ = 17.5 cm. Substituting these values into the formula:

V = ⅔ × ²²⁄₇ × (17.5)³

V = ⅔ × ²²⁄₇ × ³⁵⁄₂ × ³⁵⁄₂ × ³⁵⁄₂

Simplifying the expression by canceling common factors:

V = ⅓ × 11 × 5 × ³⁵⁄₂ × 35

V = (11 × 5 × 1225)/(6)

V = ⁶⁷³⁷⁵⁄₆

V ≈ 11229.166... cm³

Rounding to two decimal places, the volume is 11229.17 cm³.

Explanation

The price of a mixture is calculated using the weighted average formula:

Pₐᵥ_g = w₁P₁ + w₂P₂ + w₃P₃{w₁ + w₂ + w₃}

Given the prices P₁ = 80, P₂ = 50 and the mixing ratio 1:2:3, we can assign the weights as w₁ = 1, w₂ = 2, and w₃ = 3. The average price Pₐᵥ_g is given as 75. Substituting these values into the formula:

75 = (80 × 1) + (50 × 2) + (P₃ × 3){1 + 2 + 3}

75 = 80 + 100 + 3P₃{6}

450 = 180 + 3P₃

270 = 3P₃

P₃ = 90

Thus, the price of the third variety of wheat is Rs. 90 per kg.

Explanation

The mean proportional M of two numbers a and b is calculated using the formula M = √(ab). Substituting the given values a = 6 and b = 54, the calculation is M = √(6 × 54) = √(324). The square root of 324 is 18. The problem asks for the difference between this mean proportional and the number 15. Subtracting 15 from 18 yields 18 - 15 = 3.

Explanation

Let the initial cost price of the item be CP₁ = 100. Since the initial profit percentage was 55%, the selling price is calculated as: SP = CP₁ × (1 + ⁵⁵⁄₁₀₀) = 100 × 1.55 = 155

According to the problem, the cost price increases by 24%, making the new cost price: CP₂ = 100 × (1 + ²⁴⁄₁₀₀) = 124

The selling price remains the same at 155. The new profit is the difference between the constant selling price and the new cost price: New Profit = SP - CP₂ = 155 - 124 = 31

The new profit percentage is calculated on the new cost price: New Profit Percentage = (New Profit)/(CP₂₎ × 100 = ³¹⁄₁₂₄ × 100

Simplifying the fraction ³¹⁄₁₂₄, we observe that 31 × 4 = 124, so the expression becomes: ¼ × 100 = 25%

Explanation

The outer radius R of the hemispherical bowl is the sum of the inner radius r and the thickness t of the steel. Given the inner radius is 6 cm and the thickness is 0.37 cm, the outer radius is: R = r + t = 6 + 0.37 = 6.37 cm

The formula for the curved surface area of a hemisphere is 2 π R². Using the value of π = ²²⁄₇, the outside curved surface area is calculated as: Outer CSA = 2 × ²²⁄₇ × (6.37)² Outer CSA = 2 × ²²⁄₇ × 6.37 × 6.37

Simplifying the calculation by dividing 6.37 by 7: Outer CSA = 2 × 22 × 0.91 × 6.37 Outer CSA = 44 × 0.91 × 6.37 Outer CSA = 40.04 × 6.37 Outer CSA = 255.0548 cm²

Explanation

The expression sec α + tan³ α csc α can be simplified using basic trigonometric identities. First, express tan α and csc α in terms of sine and cosine:

sec α + tan³ α csc α = (1)/(cos α) + (sin³ α)/(cos³ α) · (1)/(sin α)

Simplifying the second term:

(1)/(cos α) + (sin² α)/(cos³ α)

To add these fractions, find a common denominator of cos³ α:

(cos² α + sin² α)/(cos³ α)

Using the Pythagorean identity sin² α + cos² α = 1, the expression becomes:

(1)/(cos³ α) = sec³ α

The squared value of this expression is (sec³ α)² = sec⁶ α, which can be written as (sec² α)³. According to the identity sec² α = 1 + tan² α, we can substitute the given value tan² α = 3 + Q²:

sec² α = 1 + (3 + Q²) = 4 + Q²

Substituting this back into the cubic form:

(sec² α)³ = (4 + Q²)³

Explanation

The area A of an equilateral triangle is related to its height h by the formula A = h²{√3}. Given the height h = 9√3 cm, substituting this value into the formula yields:

A = (9√3)²{√3} A = (81 × 3)/(√(3)) A = (243)/(√(3))

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

A = 243√3{3} A = 81√3 cm²

Alternatively, the side length a of the triangle can be found using the height formula h = √3{2}a. Substituting the given value, 9√3 = √3{2}a, results in a = 18 cm. The area is then calculated using the side-based formula A = √3{4}a² = √3{4} × 18² = √3{4} × 324 = 81√3 cm².

Explanation

The sum of the interior angles of any triangle is always 180°. Given that the angles are in the ratio of 1:2:3, we can represent them as x, 2x, and 3x. Using the angle sum property:

x + 2x + 3x = 180° 6x = 180° x = 30°

Substituting the value of x back into the expressions for the angles gives 30°, 60°, and 90°. Since one of the angles in the triangle is exactly 90°, the triangle is classified as a right-angle triangle.

Explanation

Let the sum lent be P. Using the simple interest formula SI = (P × R × T)/(100) with R = 7 and T = 11, the interest is SI = (P × 7 × 11)/(100) = (77P)/(100). The problem states that the interest received is Rs. 920 less than the sum lent, which is expressed as P - SI = 920. Substituting the expression for SI into the equation:

P - (77P)/(100) = 920

(100P - 77P)/(100) = 920

(23P)/(100) = 920

P = (920 × 100)/(23)

P = 40 × 100 = 4000

Explanation

The expression is simplified by applying the BODMAS rule, starting with the innermost parentheses and divisions.

First, simplify the second term: 4x²÷ (5x÷ 10) = 4x²÷ (ˣ⁄₂) = 4x² × ²⁄ₓ = 8x

Next, simplify the expression inside the curly braces: The division inside the parentheses is 3x²÷ x = 3x. The division following that is x³÷ 3x = x²{3}. The term 5⅓ is equal to ¹⁶⁄₃. Substituting these back into the brace term multiplied by 3 gives: 3{¹⁶⁄₃ - x²{3}} = 3 × 16 - x²{3} = 16 - x²

Now, substitute these simplified parts back into the original equation: 7x + 8x - (16 - x²) = 0 15x - 16 + x² = 0 x² + 15x - 16 = 0

This is a quadratic equation that can be factored: x² + 16x - x - 16 = 0 x(x + 16) - 1(x + 16) = 0 (x - 1)(x + 16) = 0

The two values of x are x₁ = 1 and x₂ = -16.

The difference between these two values is: |x₁ - x₂| = |1 - (-16)| = |1 + 16| = 17

Explanation

The given equation x² + y² + 2y + 4x + 5 = 0 can be rearranged by completing the squares as (x + 2)² + (y + 1)² = 0. This implies x + 2 = 0 and y + 1 = 0, giving x = -2 and y = -1, so x - y = -2 - (-1) = -1.

Explanation

The final number of masks produced in 2021 can be calculated using the successive percentage change formula. Let the initial number of masks in 2018 be X. The production changes are an increase of 30% in 2019, an increase of 15% in 2020, and a decrease of 40% in 2021.

The equation representing the final production is: X × (1 + ³⁰⁄₁₀₀) × (1 + ¹⁵⁄₁₀₀) × (1 - ⁴⁰⁄₁₀₀) = 179400

Simplifying the multipliers: X × 1.3 × 1.15 × 0.6 = 179400

First, multiply the factors: 1.3 × 0.6 = 0.78 0.78 × 1.15 = 0.897

Now, solve for X: 0.897 × X = 179400 X = (179400)/(0.897)

Removing the decimal: X = ¹⁷⁹⁴⁰⁰⁰⁰⁰⁄₈₉₇

Since 897 × 2 = 1794, the division yields: X = 200000