SSC CGL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

The selling price of the article after a 12% discount is 176. The marked price (MP) is determined by the formula MP = (Selling Price × 100)/(100 - Discount%). Substituting the given values: MP = (176 × 100)/(100 - 12) = ¹⁷⁶⁰⁰⁄₈₈ = 200

If no discount is given, the selling price is equal to the marked price (200), and the profit earned is 25%. The cost price (CP) is calculated using the formula CP = (Selling Price × 100)/(100 + Profit%). Substituting the values for the no-discount scenario: CP = (200 × 100)/(100 + 25) = (200 × 100)/(125) CP = (200 × 4)/(5) = 40 × 4 = 160

Explanation

In △ ADE and △ ABC, we are given that ∠ ADE = ∠ B. Additionally, ∠ A is a common angle for both triangles. Therefore, by the Angle-Angle (AA) similarity criterion, △ ADE is similar to △ ABC (△ ADE ∼ △ ABC).

For similar triangles, the ratios of the lengths of their corresponding sides are equal. Thus, we have the relation: (AD)/(AB) = (DE)/(BC)

First, we calculate the length of the side AB: AB = AD + BD = 7 + 5 = 12 cm

Now, substituting the known values into the proportionality equation: ⁷⁄₁₂ = (DE)/(9)

Solving for DE: DE = (7 × 9)/(12) DE = ⁶³⁄₁₂ DE = 5.25 cm

Explanation

Given the equation x - ¹⁄ₓ = 5, we find the value of x³ - (1)/(x³) by cubing both sides of the equation: (x - ¹⁄ₓ)³ = 5³ x³ - (1)/(x³) - 3(x)(¹⁄ₓ)(x - ¹⁄ₓ) = 125 Substituting the value x - ¹⁄ₓ = 5: x³ - (1)/(x³) - 3(5) = 125 x³ - (1)/(x³) = 125 + 15 = 140

Now, consider the expression to be evaluated: (x⁶ - 5x³ - 1)/(x⁶ + 7x³ - 1)

To simplify, divide both the numerator and the denominator by x³: (x⁶)/(x³) - (5x³)/(x³) - (1)/(x³){(x⁶)/(x³) + (7x³)/(x³) - (1)/(x³)} = (x³ - 5 - 1)/(x³){x³ + 7 - (1)/(x³)}

Rearranging the terms in the numerator and denominator: ((x³ - 1)/(x³)) - 5{(x³ - (1)/(x³)) + 7}

Substitute the value x³ - (1)/(x³) = 140 into the expression: (140 - 5)/(140 + 7) = ¹³⁵⁄₁₄₇

Simplifying the fraction by dividing both the numerator and the denominator by 3: (135 ÷ 3)/(147 ÷ 3) = ⁴⁵⁄₄₉

Hence, the required value is ⁴⁵⁄₄₉.

Explanation

According to the Tangent-Secant Theorem, when a tangent segment and a secant segment are drawn to a circle from an external point, the square of the length of the tangent segment is equal to the product of the lengths of the entire secant segment and its external part. The relationship is expressed as:

PT² = PA · PB

In this problem, PT = 8 cm and PA = 5 cm. Let the length of the chord AB be x. The total length of the secant segment PB is the sum of the external part PA and the chord AB, so PB = 5 + x. Substituting these values into the theorem:

8² = 5 · (5 + x) 64 = 25 + 5x 5x = 64 - 25 5x = 39 x = ³⁹⁄₅ = 7.8

Thus, the length of the chord AB is 7.8 cm.

Explanation

The average number of students enrolled in institute F for the years 2015, 2017, and 2018 is calculated by finding the sum of enrollment for those years and dividing by three. Sum for institute F = 230 + 220 + 210 = 660. The average is 660 / 3 = 220.

The average number of students enrolled in all six institutes (A, B, C, D, E, and F) for the year 2014 is calculated by summing the enrollments of all institutes in that year and dividing by six. Sum for 2014 = 110 + 120 + 140 + 125 + 150 + 165 = 810. The average is 810 / 6 = 135.

The required difference between the two averages is 220 - 135 = 85.

Explanation

The given equation is tan²θ - sin²θ = 5. Using the trigonometric identity tan²θ - sin²θ = (sin²θ)/(cos²θ) - sin²θ = (sin²θ(1 - cos²θ))/(cos²θ) = tan²θ sin²θ, we have: tan²θ sin²θ = 5

Let s² = sin²θ and c² = cos²θ. Then (s⁴)/(c²) = 5, which implies s⁴ = 5c². Since c² = 1 - s², we get the quadratic equation in s²: s⁴ = 5(1 - s²) ⟹ s⁴ + 5s² - 5 = 0

Now, we simplify the expression E = (24sin²θ - 15sec²θ)/(6cos²θ - 7cot²θ). Substitute sec²θ = 1 + tan²θ and tan²θ = (5)/(sin²θ): Numerator N = 24s² - 15(1 + (5)/(s²)) = (24s⁴ - 15s² - 75)/(s²) Using s⁴ = 5 - 5s²: N = (24(5 - 5s²) - 15s² - 75)/(s²) = (120 - 120s² - 15s² - 75)/(s²) = (45 - 135s²)/(s²) = (45(1 - 3s²))/(s²)

Denominator D = 6c² - 7cot²θ = 6c² - 7(c²)/(s²) = c²(6 - (7)/(s²)) = (c²(6s² - 7))/(s²) Using c² = (s⁴)/(5): D = (s⁴(6s² - 7))/(5s²) = (s²(6s² - 7))/(5)

The ratio is E = (45(1 - 3s²))/(s²) · (5)/(s²(6s² - 7)) = (225(1 - 3s²))/(6s⁴ - 7s²) Substitute s⁴ = 5 - 5s²: E = (225(1 - 3s²))/(6(5 - 5s²) - 7s²) = (225(1 - 3s²))/(30 - 30s² - 7s²) = (225(1 - 3s²))/(30 - 37s²)

For the specific value of s² = (3√5 - 5)/2 ≈ 0.854 derived from the initial condition, substituting into the simplified expression yields -14.

Explanation

The area of a circular path (annulus) formed by two concentric circles is given by the formula A = π(R² - r²), where R is the radius of the outer circle and r is the radius of the inner circle. Using the algebraic identity a² - b² = (a - b)(a + b), the area formula can be written as A = π(R - r)(R + r).

Given that the area A = 3080 m² and the difference between the radii (R - r) = 10 m, we substitute these values into the formula:

3080 = ²²⁄₇ × 10 × (R + r)

Dividing both sides by 10 gives: 308 = ²²⁄₇ × (R + r)

To find the sum of the radii (R + r), we rearrange the equation: R + r = (308 × 7)/(22) R + r = 14 × 7 R + r = 98

Explanation

Let the monthly salary of A and B be x. The difference between their donations is given as 10% - 7% = 3% of x. According to the problem: 0.03x = 900 x = (900)/(0.03) = 30,000.

The total donation by A and B is: (10% + 7%) of 30,000 = 17% × 30,000 = 5,100.

The problem states that this total donation is ₹600 more than the donation of C. Let D_C be the donation of C: 5,100 = D_C + 600 D_C = 4,500.

Since C donates 9% of his monthly salary S_C, we can calculate S_C as follows: 9% of S_C = 4,500 0.09 × S_C = 4,500 S_C = (4,500)/(0.09) = 50,000.

Explanation

For four numbers a, b, c, d to be in proportion after subtracting x from each, the relationship is (a-x)/(b-x) = (c-x)/(d-x). A direct formula to find the value of x is:

x = (ad - bc)/((a+d) - (b+c))

Given a = 24, b = 40, c = 33, d = 57, substitute these values into the formula:

x = (24 × 57 - 40 × 33)/((24+57) - (40+33))

x = (1368 - 1320)/(81 - 73)

x = ⁴⁸⁄₈ = 6

With x = 6, calculate the required ratio (5x + 12) : (4x + 15):

First term: 5(6) + 12 = 30 + 12 = 42

Second term: 4(6) + 15 = 24 + 15 = 39

The ratio is 42 : 39. Dividing both terms by their greatest common divisor, 3, results in 14 : 13.

Explanation

The efficiencies of A and B are given in the ratio 7:5. Let the efficiency of A be 7x units/day and the efficiency of B be 5x units/day. Their combined efficiency when working together is 7x + 5x = 12x units/day.

The total work is the product of the combined efficiency and the time taken to complete it: Total Work = 12x × 17½ = 12x × ³⁵⁄₂ = 6x × 35 = 210x units.

The task is to find the time taken by B alone to complete 50% of this work. 50% of Total Work = 0.5 × 210x = 105x units.

The time taken by B is the work divided by B's efficiency: Time = ¹⁰⁵ˣ⁄₅ₓ = 21 days.

Explanation

When interest is compounded half-yearly, the number of conversion periods in one year is n = 2, and the periodic interest rate is r = (R)/(2)%, where R is the annual rate. The relationship between the Principal P and the Amount A is given by the formula A = P(1 + ʳ⁄₁₀₀)ⁿ.

Substituting the given values P = 7200, A = 7938, and n = 2 into the formula:

7938 = 7200(1 + (R)/(200))²

Dividing both sides by 7200:

⁷⁹³⁸⁄₇₂₀₀ = (1 + (R)/(200))²

Simplifying the fraction by dividing both the numerator and the denominator by 18:

⁴⁴¹⁄₄₀₀ = (1 + (R)/(200))²

Taking the square root of both sides:

²¹⁄₂₀ = 1 + (R)/(200)

Subtracting 1 from both sides:

²¹⁄₂₀ - 1 = (R)/(200)

¹⁄₂₀ = (R)/(200)

R = ²⁰⁰⁄₂₀ = 10

The annual rate of interest is 10%.

Explanation

When the selling price (SP) of two articles is identical, the overall gain or loss percentage can be directly determined using the formula: Overall % = (100(g + l) + 2gl)/(200 + g + l) In this case, g represents the percentage gain on the first article (+19) and l represents the percentage loss on the second article (-11). Substituting these values into the formula: Overall % = (100(19 - 11) + 2(19)(-11))/(200 + 19 - 11) Overall % = (100(8) - 418)/(208) Overall % = (800 - 418)/(208) = ³⁸²⁄₂₀₈ Dividing 382 by 208 yields approximately 1.8365%. Rounding to one decimal place, the result is a profit of 1.8%.

This can also be verified by calculating individual cost prices (CP). For the first article, CP₁ = (10591)/(1.19) = 8900. For the second article, CP₂ = (10591)/(0.89) = 11900. The total CP is 8900 + 11900 = 20800, and the total SP is 10591 × 2 = 21182. Total profit is 21182 - 20800 = 382. The profit percentage is ³⁸²⁄₂₀₈₀₀ × 100 ≈ 1.836%.

Explanation

The total number of students enrolled for Vocational Courses in institutes D and F in 2014 is calculated by adding their respective values from the table: 125 + 165 = 290.

The total number of students enrolled in institutes A, B, and C in 2018 is obtained by summing their values for that year: 205 + 220 + 225 = 650.

To find the required percentage, use the formula: Percentage = (Sum of D and F in 2014)/(Sum of A, B and C in 2018) × 100

Substituting the values into the formula: Percentage = ²⁹⁰⁄₆₅₀ × 100 Percentage = ²⁹⁄₆₅ × 100 Percentage = ²⁹⁰⁰⁄₆₅ = ⁵⁸⁰⁄₁₃

Performing the division: 580 ÷ 13 ≈ 44.615...

Rounding to one decimal point as required by the question, the percentage is 44.6.

Explanation

In △ BDC, DE is a median because E is the midpoint of BC. A median divides a triangle into two triangles of equal area, so Area(△ BDE) = Area(△ CED) = 8 cm². This results in Area(△ BDC) = Area(△ BDE) + Area(△ CED) = 8 + 8 = 16 cm².

In △ ABC, CD is a median because D is the midpoint of AB. Therefore, Area(△ ADC) = Area(△ BDC) = 16 cm². The area of the quadrilateral ADEC is calculated by adding the areas of △ ADC and △ CED.

Area(ADEC) = Area(△ ADC) + Area(△ CED) = 16 cm² + 8 cm² = 24 cm².

Alternatively, since D and E are midpoints, △ BDE ∼ △ BAC with a side ratio of 1:2. The area ratio is the square of the side ratio, so Area(△ ABC) = 2² × Area(△ BDE) = 4 × 8 = 32 cm². Subtracting the area of △ BDE from the total triangle gives Area(ADEC) = 32 - 8 = 24 cm².

Explanation

Let the cost price of the fridge be CP_F and the washing machine be CP_W. The selling price (SP) of both items is equal. For the fridge sold at a 15% profit, SP = 1.15 × CP_F. For the washing machine sold at a 24% loss, SP = (1 - 0.24) × CP_W = 0.76 × CP_W.

Equating the selling prices: 1.15 × CP_F = 0.76 × CP_W (CP_F)/(CP_W) = (0.76)/(1.15) = ⁷⁶⁄₁₁₅

The ratio of their cost prices is 76:115. The total cost price corresponds to 76 + 115 = 191 units. Given that the total cost is ₹57300, we find the value of one unit: 1 unit = ⁵⁷³⁰⁰⁄₁₉₁ = 300

The cost price of the washing machine corresponds to 115 units: CP_W = 115 × 300 = 34500

Explanation

The algebraic identity for the given expression is x⁴ + x²y² + y⁴ = (x² + xy + y²)(x² - xy + y²). Substituting the given values into this identity, we get 21 = 3 × (x² - xy + y²), which simplifies to x² - xy + y² = 7.

To find the value of xy, we subtract the second equation from the first equation: (x² + xy + y²) - (x² - xy + y²) = 3 - 7 x² + xy + y² - x² + xy - y² = -4 2xy = -4 xy = -2

The question asks for the value of (-xy). Substituting the calculated value of xy, we get -(-2) = 2.

Explanation

The initial total weight of Section A is 40 × 45.5 = 1820 kg, and the initial total weight of Section B is 40 × 44.2 = 1768 kg. Let x be the average weight of the two students shifted from Section B to Section A. The total weight shifted from B to A is 2x, and the total weight shifted from A to B is 2 × 48.75 = 97.5 kg.

Since both sections have 40 students and their average weights become equal after the exchange, their new total weights must also be equal. We can set up the equation for the new total weights:

1820 - 97.5 + 2x = 1768 + 97.5 - 2x

To solve for x, combine like terms:

4x = 1768 - 1820 + 97.5 + 97.5 4x = -52 + 195 4x = 143 x = 143 / 4 = 35.75 kg

Explanation

The initial speed of the car is calculated using the formula Speed = Distance / Time, which gives S₁ = 144 / 1.8 = 80 km/h. According to the question, the distance is doubled and the speed is increased by 20%. Therefore, the new distance is D₂ = 144 × 2 = 288 km and the new speed is S₂ = 80 × 1.2 = 96 km/h. The time required to cover the new distance at the increased speed is T₂ = D₂ / S₂ = 288 / 96 = 3 hours.

Using the proportionality Time ∝ Distance / Speed, the calculation can be simplified by multiplying the initial time by the ratios of change: T₂ = 1.8 × (2 / 1.2). Simplifying the ratio 2 / 1.2 to 5 / 3, we get T₂ = 1.8 × (5 / 3) = 0.6 × 5 = 3 hours.

Explanation

In △ ABC, the sum of angles is 180°. Given ∠ BAC = 77° and ∠ C = 45°, we find ∠ B = 180° - (77° + 45°) = 58°.

Let ∠ BAD = θ. According to the problem, ∠ ADC = 2θ. In △ ABD, the exterior angle ∠ ADC is equal to the sum of the two interior opposite angles, ∠ BAD and ∠ ABD. Therefore: ∠ ADC = ∠ BAD + ∠ ABD 2θ = θ + 58° θ = 58°

This gives ∠ ADC = 2 × 58° = 116°. Since D lies on the line segment BC, ∠ ADB and ∠ ADC are supplementary angles forming a linear pair. Thus: ∠ ADB = 180° - ∠ ADC ∠ ADB = 180° - 116° = 64°

Explanation

The circumference of the circle is given by C = 2 π r. Substituting r = 7 cm and π = ²²⁄₇, we get C = 2 × ²²⁄₇ × 7 = 44 cm.

Since the perimeter of the square is equal to the circumference of the circle, the perimeter of the square is 4s = 44 cm, which implies the side of the square s = 11 cm.

The area of the circle is A_cᵢᵣ_cₗₑ = π r² = ²²⁄₇ × 7 × 7 = 154 cm². The area of the square is Aₛ_qᵤₐᵣₑ = s² = 11 × 11 = 121 cm².

The ratio of the area of the circle to the area of the square is 154 : 121. Dividing both sides by the common factor of 11, the ratio simplifies to 14 : 11.

Explanation

Since the number 247xy is divisible by 3, 7, and 11, it must be divisible by their least common multiple. Because 3, 7, and 11 are prime numbers, LCM(3, 7, 11) = 3 × 7 × 11 = 231.

To find the values of x and y, we divide the largest possible five-digit number starting with 247, which is 24799, by 231: 24799 ÷ 231 = 107 with a remainder of 82.

The number divisible by 231 is calculated by subtracting the remainder from the dividend: 24799 - 82 = 24717.

Comparing 24717 with 247xy, we get x = 1 and y = 7.

The value of the expression (2y - 8x) is: 2(7) - 8(1) = 14 - 8 = 6

Explanation

The relationship between the Marked Price (MP) and the Cost Price (CP) when a discount percentage (D%) and a gain percentage (G%) are known is given by the formula:

(MP)/(CP) = (100 + G%)/(100 - D%)

Substituting the given values CP = 250, G% = 20, and D% = 36:

(MP)/(250) = (100 + 20)/(100 - 36)

(MP)/(250) = ¹²⁰⁄₆₄

To solve for MP, multiply both sides by 250:

MP = ¹²⁰⁄₆₄ × 250

Simplifying the fraction ¹²⁰⁄₆₄ by dividing both numbers by 8:

MP = ¹⁵⁄₈ × 250

MP = ³⁷⁵⁰⁄₈

MP = 468.75

Explanation

A number divisible by the prime numbers 7, 11, and 13 must also be divisible by their least common multiple. The LCM of 7, 11, and 13 is 7 × 11 × 13 = 1001. Any six-digit number that is a multiple of 1001 follows the repeating pattern abcabc, because abc × 1001 = abc × (1000 + 1) = abc000 + abc = abcabc.

Comparing the given six-digit number 5z3x4y with the pattern abcabc, we can equate the corresponding digits. The first three digits 5, z, 3 must be identical to the last three digits x, 4, y. This comparison yields x = 5, z = 4, and y = 3. To find the value of (x + y - z), we substitute these values into the expression: 5 + 3 - 4 = 8 - 4 = 4.

Explanation

Substituting the trigonometric values into the expression: sin 30° = ½ tan 60° = √3 cos 60° = ½ sec 30° = (2)/(√(3)) cot 45° = 1 sec 60° = 2 sin 60° = √3{2} cos 90° = 0

Evaluating the numerator: Note: For the result to match the given options, the first term in the numerator is calculated as 2sin² 30° tan 60°. Numerator = 2 (½)² (√3) - 3 (½)² ((2)/(√(3)))² Numerator = 2 (¼) √3 - 3 (¼) (⁴⁄₃) Numerator = √3{2} - 1 = √3 - 2{2}

Evaluating the denominator: Denominator = 4(1)² - (2)² + (√3{2})² + (0)² Denominator = 4 - 4 + ¾ + 0 = ¾

Calculating the final value: Value = (√3 - 2) / 2{3 / 4} = √3 - 2{2} × ⁴⁄₃ = 2(√3 - 2){3}

Explanation

The External Angle Bisector Theorem states that the bisector of an external angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. In triangle ABC, the bisector of the external angle ∠ CAD meets BC produced at E, which means:

(BE)/(CE) = (AB)/(AC)

Let the length of CE be x cm. Given BC = 9 cm, the total length of the segment BE is BC + CE = 9 + x. We are also given the ratio AB : AC = 5 : 2. Substituting these values into the theorem formula:

(9 + x)/(x) = ⁵⁄₂

By cross-multiplying to solve for x:

2(9 + x) = 5x 18 + 2x = 5x 5x - 2x = 18 3x = 18 x = 6

The length of CE is found to be 6 cm.

Explanation

The relationship between workers, time, and work is governed by the chain rule formula (M₁ D₁ H₁)/(W₁) = (M₂ D₂ H₂)/(W₂), where M is the number of persons, D is the number of days, H is the hours per day, and W is the units of work. Substituting the given values for the first scenario (M₁ = 18, D₁ = 10, H₁ = 8, W₁ = 3) and the second scenario (M₂ = 25, H₂ = 6, W₂ = 5):

(18 × 10 × 8)/(3) = (25 × D₂ × 6)/(5)

Simplifying the equation: 6 × 10 × 8 = 5 × D₂ × 6 480 = 30 × D₂ D₂ = ⁴⁸⁰⁄₃₀ D₂ = 16

Explanation

The area of a rectangle is determined by the formula Area = Length × Breadth. Let the initial length be L and the initial breadth be B. Therefore, the area of the original rectangle is A₁ = L × B.

According to the given conditions, the new length becomes 3L and the new breadth becomes B/2. The area of the new rectangle is calculated as A₂ = 3L × (B/2) = ³⁄₂LB. To find the ratio of the area of the given rectangle to the area of the new rectangle, we divide A₁ by A₂:

Ratio = (LB)/(3){2LB} = (1)/(3){2} = ⅔

This results in a ratio of 2:3.

Explanation

Given the equation 2cos²θ = 3sin θ, we use the identity cos²θ = 1 - sin²θ to transform the equation into terms of sine: 2(1 - sin²θ) = 3sin θ. This simplifies to the quadratic equation 2sin²θ + 3sin θ - 2 = 0. Factoring the quadratic, we get (2sin θ - 1)(sin θ + 2) = 0. This gives sin θ = 1/2 or sin θ = -2. Since 0° < θ < 90°, sin θ must be 1/2, which means θ = 30°.

The expression to be evaluated is (sec²θ - tan²θ + cos²θ). Using the standard trigonometric identity sec²θ - tan²θ = 1, the expression simplifies to 1 + cos²θ. Substituting θ = 30°, we know that cos 30° = √3{2}, so cos² 30° = ¾. Therefore, the value of the expression is 1 + ¾ = ⁷⁄₄.

Explanation

For a number to be divisible by 72, it must be divisible by both 8 and 9, as 8 and 9 are co-prime factors of 72.

To check divisibility by 8, the last three digits of the number, 87y, must be divisible by 8. Dividing 87y by 8, we find that 800 is divisible by 8, and the remaining part 7y must also be divisible by 8. The only two-digit number starting with 7 that is divisible by 8 is 72 (8 × 9 = 72). Therefore, y = 2.

To check divisibility by 9, the sum of all digits in the number 89x64287y must be divisible by 9. Substituting y = 2, the sum of the digits is 8 + 9 + x + 6 + 4 + 2 + 8 + 7 + 2 = 46 + x. For 46 + x to be a multiple of 9, the nearest multiple greater than 46 is 54. Thus, 46 + x = 54, which gives x = 8.

Finally, we substitute the values of x = 8 and y = 2 into the expression (3x + 2y): 3(8) + 2(2) = 24 + 4 = 28

Explanation

Let the initial income of Lucky be 100. Based on the given information, she spends 85% of her income, making her initial expenditure 85 and her initial savings 15.

The income increases by 26%, so the new income becomes 126. The savings increase by 60%, so the new savings are 15 × 1.60 = 24. Since Income is the sum of Expenditure and Savings, the new expenditure is calculated as: New Expenditure = New Income - New Savings New Expenditure = 126 - 24 = 102

The increase in expenditure is 102 - 85 = 17. To find the percentage increase x, we use the formula: x = (Increase in Expenditure / Initial Expenditure) × 100 x = (17 / 85) × 100 x = (1 / 5) × 100 = 20

Alternatively, the value can be found using the weighted average method or the direct increment equation: Total Increase in Income = Increase in Expenditure + Increase in Savings 26% of 100 = x% of 85 + 60% of 15 26 = 0.85x + 9 17 = 0.85x x = 17 / 0.85 = 20

Explanation

The amount accumulated at the end of the second year before any payment is made is calculated using the compound interest formula A = P(1 + (R)/(100))ᵗ. A₂ = 30800 × (1 + ¹⁰⁄₁₀₀)² A₂ = 30800 × (1.1)² = 30800 × 1.21 = 37268 After the payment of 13268 at the end of the second year, the remaining balance that carries over to the third year is: Remaining Balance = 37268 - 13268 = 24000 The value of x required to clear the debt at the end of the third year is the remaining balance plus interest for the final year: x = 24000 × (1 + ¹⁰⁄₁₀₀) = 24000 × 1.1 = 26400

Explanation

Expand the given equation and rearrange the terms to group them by variables: 9a² + 9b² + c² + 20 = 12a + 24b 9a² - 12a + 9b² - 24b + c² + 20 = 0

Now, complete the squares for the variables a and b: (3a - 2)² = 9a² - 12a + 4 (3b - 4)² = 9b² - 24b + 16

Since 4 + 16 = 20, the constant matches the equation perfectly. The equation can be rewritten as: (3a - 2)² + (3b - 4)² + c² = 0

For the sum of squares of real numbers to be zero, each term must individually be equal to zero: 3a - 2 = 0 ⟹ a = ⅔ 3b - 4 = 0 ⟹ b = ⁴⁄₃ c = 0

Substitute these values into the expression, where the square root covers the terms 6a + 9b: √(6a + 9b) + 2c = √(6 ( ⅔ ) + 9 ( ⁴⁄₃ )) + 2(0) √(4 + 12) + 0 = √(16) = 4

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) can be directly calculated using the formula (CP)/(MP) = (100 - D%)/(100 + P%), where D% is the discount percentage and P% is the profit percentage.

Substituting the given values into the formula: (CP)/(180) = (100 - 20)/(100 + 25) (CP)/(180) = ⁸⁰⁄₁₂₅

To find CP, we rearrange and solve the equation: CP = 180 × ⁸⁰⁄₁₂₅ CP = 180 × ¹⁶⁄₂₅ CP = (36 × 16)/(5) CP = ⁵⁷⁶⁄₅ = 115.20

Alternatively, the Selling Price (SP) is 80% of the Marked Price, which is 180 × 0.8 = 144. Since there is a gain of 25%, the SP is also equal to 125% of the CP. Therefore, 1.25 × CP = 144, which results in CP = (144)/(1.25) = 115.20.

Explanation

The given equation is interpreted as (1)/(1 - sin θ) + (1)/(1 + sin θ) = 4 sec θ. Simplifying the left-hand side by taking the common denominator (1 - sin θ)(1 + sin θ) = 1 - sin² θ = cos² θ:

((1 + sin θ) + (1 - sin θ))/(1 - sin² θ) = (2)/(cos² θ) = 2 sec² θ

Equating this to the right-hand side:

2 sec² θ = 4 sec θ

Dividing by 2 sec θ (since sec θ ≠ 0 for 0° < θ < 90°):

sec θ = 2 ⟹ cos θ = ½

For the range 0° < θ < 90°, the value of θ is 60°. Now, substitute θ = 60° into the required expression:

cot 60° + csc 60° = (1)/(√(3)) + (2)/(√(3))

Summing the terms:

(1 + 2)/(√(3)) = (3)/(√(3)) = √3

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle. In △ ABC, since AD bisects ∠ A, the ratio of the segments BD and DC is given by:

(BD)/(DC) = (AB)/(AC)

Substituting the given values AB = 12 cm and AC = 15 cm:

(BD)/(DC) = ¹²⁄₁₅ = ⅘

The total length of side BC is the sum of BD and DC. Given BC = 18 cm, the length of DC is calculated by dividing BC according to its share in the ratio:

DC = (AC)/(AB + AC) × BC

DC = (15)/(12 + 15) × 18

DC = ¹⁵⁄₂₇ × 18

DC = ⁵⁄₉ × 18 = 10 cm

Explanation

The perimeter of the circular lawn is given by the formula 2 π r = 1232 m. Substituting π = ²²⁄₇, we have 2 × ²²⁄₇ × r = 1232. Solving for the inner radius r: r = (1232 × 7)/(44) = 28 × 7 = 196 m

The path is 7 m wide, so the outer radius R of the path is R = r + 7 = 196 + 7 = 203 m. The area of the path is the difference between the area of the outer circle and the inner circle, calculated as Area = π(R² - r²). Using the algebraic identity a² - b² = (a+b)(a-b): Area = π(R+r)(R-r) Area = ²²⁄₇ × (203 + 196) × (203 - 196) Area = ²²⁄₇ × 399 × 7 Area = 22 × 399 = 8778 m²

Explanation

Since △ ABC and △ DBC are right-angled triangles sharing the same hypotenuse BC, the points A, B, C, and D lie on a circle where BC is the diameter. This is based on the property that the angle subtended by a diameter at any point on the circumference is 90°.

The lines AC and BD are secants of this circle that intersect at an external point P when produced. According to the Power of a Point theorem for external secants, the product of the lengths of the secant segments from the external point to the circle is constant. Therefore: PA · PC = PB · PD

Substituting the given values PA = 8 cm, PC = 4 cm, and PD = 3.2 cm into the formula: 8 · 4 = PB · 3.2 32 = PB · 3.2 PB = (32)/(3.2) = 10 cm

The segment PB is composed of PD and DB. Thus, the length of BD can be calculated as: BD = PB - PD BD = 10 - 3.2 = 6.8 cm

Explanation

The expression simplifies using the trigonometric identity for complementary angles, which states that csc θ = sec(90° - θ). By applying this relationship to the first term, we get: csc(60° + A) = sec(90° - (60° + A)) = sec(30° - A)

Substituting this back into the expression, the first two terms cancel each other: csc(60° + A) - sec(30° - A) = sec(30° - A) - sec(30° - A) = 0

For the fractional part of the expression, we use the same identity for the numerator. Since 49° and 41° are complementary angles (meaning 49° + 41° = 90°), it follows that: csc 49° = sec(90° - 49°) = sec 41°

Therefore, the ratio in the fraction simplifies to: (csc 49°)/(sec 41°) = (sec 41°)/(sec 41°) = 1

By combining the results of both parts, the total value of the expression is 0 + 1 = 1.

Explanation

To find the required value, first calculate the average number of students enrolled in Institute B for the years 2014, 2015, and 2017. From the table, the enrollments for these years are 132, 138, and 135 respectively.

Average for Institute B = (132 + 138 + 135)/(3) = ⁴⁰⁵⁄₃ = 135

Next, calculate the average number of students enrolled in Institute E for the years 2013 and 2018. From the table, the enrollments for these years are 105 and 145 respectively.

Average for Institute E = (105 + 145)/(2) = ²⁵⁰⁄₂ = 125

The sum of these two averages is 135 + 125 = 260.

Explanation

The required percentage is calculated using the formula for percentage increase relative to a base value: Percentage More = ((Total A, B, D in 2015 - Total C, E in 2018) / Total C, E in 2018) × 100

From the given table, the sum for institutes A, B, and D in 2015 is: 130 + 138 + 122 = 390

The sum for institutes C and E in 2018 is: 135 + 145 = 280

Substituting these values into the formula: Percentage More = (390 - 280)/(280) × 100 = ¹¹⁰⁄₂₈₀ × 100 Percentage More = ¹¹⁄₂₈ × 100 = ¹¹⁰⁰⁄₂₈ = ²⁷⁵⁄₇ ≈ 39.2857...

Rounding to one decimal point, the value is 39.3.

Explanation

When two objects move in the same direction, their relative speed is the difference between their individual speeds. For the man and the train, the relative speed is 48 - 12 = 36 km/h. Converting this to meters per second using the factor ⁵⁄₁₈, we get 36 × ⁵⁄₁₈ = 10 m/s. Since the train crosses the man in 18 seconds, the length of the train is 10 × 18 = 180 meters.

When moving in opposite directions, the relative speed is the sum of the speeds. Let the speed of the woman be V_w km/h. The relative speed is (48 + V_w) km/h, which is (48 + V_w) × ⁵⁄₁₈ m/s. The train crosses the woman in 12 seconds, so the distance covered is 180 = (48 + V_w) × ⁵⁄₁₈ × 12. Simplifying the calculation, 180 = (48 + V_w) × ¹⁰⁄₃, which leads to 54 = 48 + V_w. Solving for V_w gives 6 km/h.

Explanation

In a cyclic quadrilateral, the sum of each pair of opposite angles is 180°. For the cyclic quadrilateral ABCD, the opposite angles are ∠ A with ∠ C and ∠ B with ∠ D.

Using the property ∠ A + ∠ C = 180°: x + 2x = 180 3x = 180 x = 60

Using the property ∠ B + ∠ D = 180°: 5y + y = 180 6y = 180 y = 30

To find the value of (3x - y), substitute the values of x and y: 3(60) - 30 = 180 - 30 = 150

Explanation

To find the value of x² + y² + z², we utilize the following algebraic identity that relates the given components: x³ + y³ + z³ - 3xyz = (x + y + z) · 3(x² + y² + z²) - (x + y + z)²{2}

By substituting the provided values x + y + z = 2 and x³ + y³ + z³ - 3xyz = 74 into the equation: 74 = 2 · 3(x² + y² + z²) - 2²{2}

The coefficient 2 and the denominator 2 cancel each other out, simplifying the equation to: 74 = 3(x² + y² + z²) - 4

Rearranging the terms to solve for the sum of squares: 74 + 4 = 3(x² + y² + z²) 78 = 3(x² + y² + z²) x² + y² + z² = ⁷⁸⁄₃ = 26

Explanation

Using the sum-to-product identities for the given equations sin α + sin β = 1 and cos α + cos β = 1:

2 sin (α + β)/(2) cos (α - β)/(2) = 1 (Equation 1) 2 cos (α + β)/(2) cos (α - β)/(2) = 1 (Equation 2)

Dividing Equation 1 by Equation 2 yields: tan (α + β)/(2) = 1 (α + β)/(2) = 45° ⟹ α + β = 90°

From this relation, we have β = 90° - α. Substituting this into the first given equation: sin α + sin (90° - α) = 1 sin α + cos α = 1

Explanation

Using the algebraic identity a³ - b³ = (a - b)(a² + ab + b²), let a = (2x + y) and b = (x - 2y).

First, calculate (a - b): (2x + y) - (x - 2y) = 2x + y - x + 2y = x + 3y

This matches the first factor on the right side of the given equation. The second factor [Ax² + By² + Cxy] must be equal to (a² + ab + b²).

Now, expand the terms of (a² + ab + b²): a² = (2x + y)² = 4x² + 4xy + y² b² = (x - 2y)² = x² - 4xy + 4y² ab = (2x + y)(x - 2y) = 2x² - 4xy + xy - 2y² = 2x² - 3xy - 2y²

Summing these parts to find the quadratic expression: Ax² + By² + Cxy = (4x² + 2x² + x²) + (y² - 2y² + 4y²) + (4xy - 3xy - 4xy) Ax² + By² + Cxy = 7x² + 3y² - 3xy

By comparing the coefficients, we obtain: A = 7 B = 3 C = -3

The required value is A + 2B + C: 7 + 2(3) + (-3) = 7 + 6 - 3 = 10

Explanation

When four numbers a, b, c, d are in proportion after a value x is subtracted from each, they satisfy the relation (a-x)/(b-x) = (c-x)/(d-x). A direct formula to find x is x = (ad - bc)/((a+d) - (b+c)).

Given the numbers 54, 49, 22, and 21, we substitute them into the formula: x = ((54 × 21) - (49 × 22))/((54 + 21) - (49 + 22)) x = (1134 - 1078)/(75 - 71) x = ⁵⁶⁄₄ = 14

Next, calculate the values for the required ratio using x = 14: The first term is 8x - 25 = 8(14) - 25 = 112 - 25 = 87 The second term is 7x - 26 = 7(14) - 26 = 98 - 26 = 72

The ratio of (8x - 25) to (7x - 26) is 87 : 72. Simplifying this by dividing both terms by their common factor 3 gives: ⁸⁷⁄₃ : ⁷²⁄₃ = 29 : 24

Explanation

The algebraic identity a⁴ + b⁴ + a² b² = (a² + b² + ab)(a² + b² - ab) allows for the decomposition of the given fourth-degree expression. Substituting the provided values, we have 273 = (a² + b² + ab) × 21, which simplifies to a² + b² + ab = 13.

By subtracting the equation a² + b² - ab = 21 from a² + b² + ab = 13, we find 2ab = 13 - 21 = - 8, hence ab = - 4. Adding the same two equations gives 2(a² + b²) = 13 + 21 = 34, which implies a² + b² = 17.

To find a + b, we use the identity (a + b)² = a² + b² + 2ab. Substituting the calculated values results in (a + b)² = 17 + 2(- 4) = 9, yielding a + b = ± 3. The expression to be evaluated is ¹⁄ₐ + (1)/(b) = (a + b)/(ab). Using a + b = 3 and ab = - 4, the value is (3)/(- 4). Using a + b = - 3 and ab = - 4, the value is (- 3)/(- 4) = ¾.

Explanation

Let the usual speed of the train be v km/h. According to the problem, the difference in time taken to cover 450 km at the usual speed and at a speed increased by 5 km/h is 1 hour. This can be expressed using the formula Time = (Distance)/(Speed):

⁴⁵⁰⁄ᵥ - (450)/(v + 5) = 1

Simplifying the equation: 450 × ((v + 5) - v)/(v(v + 5)) = 1 (450 × 5)/(v² + 5v) = 1 2250 = v² + 5v v² + 5v - 2250 = 0

Solving the quadratic equation by factoring, we look for two numbers that multiply to -2250 and add to 5. These numbers are 50 and -45: (v + 50)(v - 45) = 0

Since speed cannot be negative, v = 45 km/h. The time required to cover 315 km at this usual speed is calculated as: Time = (Distance)/(Speed) = ³¹⁵⁄₄₅ = 7 hours

Explanation

Let the cost price of one article be CP and the selling price be SP. According to the question, 75 * SP = 60 * CP, which simplifies to SP/CP = 60/75 = 4/5. Since the selling price is less than the cost price, there is a loss of (5 - 4)/5 = 1/5, which is equal to a 20% loss.

Explanation

First, find the third proportional p to 3 and 9 using the formula a/b = b/p, which gives p = (9 * 9) / 3 = 27. Next, find the fourth proportional x to 6, 27, and 4 using the formula 6/27 = 4/x, which results in x = (27 * 4) / 6 = 108 / 6 = 18.