SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

From the given equation 3x + ¹⁄₅ₓ = 7, we can multiply the entire equation by 5x to simplify it. This results in 15x² + 1 = 35x. The expression we need to evaluate is (5x)/(15x²⁾⁺¹⁵ˣ⁺¹. By rearranging the terms in the denominator, the expression becomes (5x)/((15x²⁾⁺¹⁾⁺¹⁵ˣ. Substituting the value 15x² + 1 = 35x into this denominator, we get (5x)/(35x + 15x). This simplifies to ⁵ˣ⁄₅₀ₓ. By canceling x from the numerator and denominator, we arrive at ⁵⁄₅₀, which reduces to ⅒.

Explanation

The expression to be evaluated is (64x⁶ + 1)/(8x³) = (64x⁶)/(8x³) + (1)/(8x³) = 8x³ + (1)/(8x³), which can be written as (2x)³ + (1)/((2x)³).

Given the equation x + ¹⁄₄ₓ = ⁵⁄₂, multiply both sides by 2 to match the base of the required cubic expression: 2 × (x + ¹⁄₄ₓ) = 2 × ⁵⁄₂ 2x + ¹⁄₂ₓ = 5

Using the algebraic identity a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a + ¹⁄ₐ), let a = 2x and a + ¹⁄ₐ = 5: (2x)³ + (1)/((2x)³) = 5³ - 3(5) 8x³ + (1)/(8x³) = 125 - 15 = 110

Explanation

To find the value of (x + 5)² + (1)/((x + 5)²), let y = x + 5. This implies that x = y - 5. Substituting this expression for x into the given equation x² + x = 19 yields:

(y - 5)² + (y - 5) = 19

Expanding the squared term using the identity (a - b)² = a² - 2ab + b², we get:

y² - 10y + 25 + y - 5 = 19

Simplifying the equation by combining like terms results in:

y² - 9y + 20 = 19 y² - 9y + 1 = 0

Dividing every term in the equation by y gives:

y - 9 + (1)/(y) = 0 y + (1)/(y) = 9

We are looking for the value of (x + 5)² + (1)/((x + 5)²), which is equivalent to y² + (1)/(y²). Using the algebraic identity y² + (1)/(y²) = (y + (1)/(y))² - 2, we substitute the value of y + (1)/(y) into the formula:

y² + (1)/(y²) = 9² - 2 y² + (1)/(y²) = 81 - 2 = 79

Explanation

In any triangle, the three medians divide the triangle into six smaller triangles of equal area. Thus, the areas of △ AFG and △ AEG are each equal to ⅙ of the total area of △ ABC. The area of the quadrilateral AFGE is the sum of these two areas: Area(AFGE) = Area(△ AFG) + Area(△ AEG) = ⅙ + ⅙ = ⅓ Area(△ ABC)

Since F and E are midpoints of the sides AB and AC respectively, the triangle AFE is similar to △ ABC with a scale factor of ½. The area of similar triangles is proportional to the square of the ratio of their corresponding sides: Area(△ AFE) = (½)² × Area(△ ABC) = ¼ Area(△ ABC)

The area of △ FGE is obtained by subtracting the area of △ AFE from the area of the quadrilateral AFGE: Area(△ FGE) = Area(AFGE) - Area(△ AFE) Area(△ FGE) = (⅓ - ¼) × Area(△ ABC) = ¹⁄₁₂ Area(△ ABC)

Given that the area of △ ABC is 156 cm², the area of △ FGE is: Area(△ FGE) = ¹⁄₁₂ × 156 = 13 cm²

Explanation

In triangle ABD, it is given that AD = BD. According to the properties of an isosceles triangle, the angles opposite to the equal sides are equal, so ∠ BAD = ∠ ABD. Given that ∠ ABC = 15°, it follows that ∠ ABD = 15° and consequently ∠ BAD = 15°. The angle ∠ ADC is an exterior angle to triangle ABD at vertex D. By the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles. Thus, ∠ ADC = ∠ ABD + ∠ BAD = 15° + 15° = 30°.

Explanation

The diagonal of a square is related to its side s by the formula d = s√2. Given d = 9√2 cm, the side of the square is s = 9 cm. The perimeter of the square is 4s = 4 × 9 = 36 cm. Since the square is reshaped to form a triangle, the perimeter of the triangle remains 36 cm.

For a fixed perimeter, the triangle that yields the largest incircle is an equilateral triangle. This is because the inradius r of a triangle is given by r = (Area)/(sₚ₎, where sₚ is the semi-perimeter. For a constant sₚ, the inradius is maximized when the area of the triangle is maximized, which occurs for an equilateral triangle.

In an equilateral triangle with perimeter 36 cm, the side length a is ³⁶⁄₃ = 12 cm. The inradius r of an equilateral triangle is calculated using the formula r = (a)/(2√(3)). Substituting the value of a:

r = (12)/(2√(3)) = (6)/(√(3)) = 2√3 cm

The area of the incircle is π r². Substituting the value of r:

Area = π (2√3)² = π (4 × 3) = 12π cm²

Explanation

Let O₁ and O₂ be the centers of the two circles and AB be the common chord. The line joining the centers O₁O₂ perpendicularly bisects the common chord AB at point M. Given AB = 12 cm, the length AM = 12/2 = 6 cm.

The radii of the circles are r₁ = 15/2 = 7.5 cm and r₂ = 13/2 = 6.5 cm. Applying the Pythagorean theorem in the right-angled triangles △ O₁MA and △ O₂MA:

O₁M = √(r₁² - AM²) = √(7.5² - 6²) = √(56.25 - 36) = √(20.25) = 4.5 cm

O₂M = √(r₂² - AM²) = √(6.5² - 6²) = √(42.25 - 36) = √(6.25) = 2.5 cm

The distance between the centers is the sum of these segments: O₁O₂ = O₁M + O₂M = 4.5 + 2.5 = 7 cm

Explanation

The expression sec⁴ θ - sec² θ tan² θ can be simplified by factoring out the common term sec² θ. This yields sec² θ (sec² θ - tan² θ). According to the fundamental trigonometric identity sec² θ - tan² θ = 1, the term inside the parentheses is equal to 1. Therefore, the expression simplifies to sec² θ × 1, which is sec² θ.

Explanation

The trigonometric expression can be simplified by expressing all terms in terms of sin A and cos A. Here, the term cos e A is interpreted as csc A.

The first bracket (sin A - csc A) simplifies as sin A - (1)/(sin A) = (sin² A - 1)/(sin A) = -(cos² A)/(sin A) using the identity sin² A + cos² A = 1.

The second bracket (sec A - cos A) simplifies as (1)/(cos A) - cos A = (1 - cos² A)/(cos A) = (sin² A)/(cos A).

The third bracket (tan A + cot A) simplifies as (sin A)/(cos A) + (cos A)/(sin A) = (sin² A + cos² A)/(sin A cos A) = (1)/(sin A cos A).

By multiplying these three simplified components, the expression becomes: (-(cos² A)/(sin A)) × ((sin² A)/(cos A)) × ((1)/(sin A cos A))

Combining the terms into a single fraction: (-cos² A · sin² A · 1)/(sin A · cos A · sin A · cos A) = (-sin² A cos² A)/(sin² A cos² A) = -1

Explanation

The given equation (1/cos θ) - (1/cot θ) = 1/P can be rewritten as sec θ - tan θ = 1/P. Based on the trigonometric identity sec² θ - tan² θ = 1, we know that sec θ + tan θ is the reciprocal of sec θ - tan θ. Thus, sec θ + tan θ = P.

To find cos θ, we first find sec θ by adding the two equations: (sec θ - tan θ) + (sec θ + tan θ) = (1/P) + P 2 sec θ = (1 + P²)/P sec θ = (P² + 1)/2P

Because cos θ = 1/sec θ, we take the reciprocal of the result: cos θ = 2P / (P² + 1)

Explanation

To find the least number by which 5000 must be divided to become a perfect square, we first determine its prime factorization. The number 5000 can be expressed as: 5000 = 5 × 1000 = 5 × 10³ = 5 × (2 × 5)³ = 5 × 2³ × 5³ = 2³ × 5⁴

A number is a perfect square if and only if the exponents of all its prime factors are even. In the factorization 2³ × 5⁴, the exponent of the prime factor 5 is 4, which is an even number. However, the exponent of the prime factor 2 is 3, which is an odd number. To make the exponent of 2 even by division, we must divide the number by 2¹.

Performing the division: 5000 / 2 = 2500

Since 2500 = 50², it is a perfect square. Thus, the least number by which 5000 must be divided is 2.

Explanation

When workers complete a task together, their wages are distributed in the ratio of their efficiencies. Efficiency is inversely proportional to the time taken to complete the work, so the ratio of efficiencies of A, B, and C is:

E_A : E_B : E_C = ⅛ : ⅒ : ¹⁄₁₂

To simplify this ratio, multiply each term by the LCM of 8, 10, and 12, which is 120:

E_A : E_B : E_C = ( ⅛ × 120 ) : ( ⅒ × 120 ) : ( ¹⁄₁₂ × 120 ) = 15 : 12 : 10

The sum of the parts of the ratio is 15 + 12 + 10 = 37. Since the total payment is Rs 7400, the value of one unit in the ratio is:

Value per unit = ⁷⁴⁰⁰⁄₃₇ = 200

B's share corresponds to 12 units in the ratio:

Share of B = 12 × 200 = 2400

Explanation

The volume of a cylinder is given by the formula V = π r² h, where r is the radius and h is the height. If the radius increases by 25%, the new radius r' becomes 1.25r or ⁵⁄₄r. Since the volume remains constant, the initial and final volumes are equal, meaning r² h = (r')² h'.

Substituting the new radius into the equation, we get r² h = (⁵⁄₄r)² h', which simplifies to r² h = ²⁵⁄₁₆ r² h'. Dividing both sides by r², we find that h = ²⁵⁄₁₆ h', or h' = ¹⁶⁄₂₅ h.

The reduction in height is h - h' = h - ¹⁶⁄₂₅ h = ⁹⁄₂₅ h. To express this as a percentage, we use the calculation ⁹⁄₂₅ × 100 = 36%.

Explanation

Let the Cost Price (CP) be 100. The Marked Price (MP) is 20% more than the CP, which gives MP = 100 + 0.20 × 100 = 120. A discount of 5% is then applied to this marked price. The discount amount is 0.05 × 120 = 6. The Selling Price (SP) is the marked price minus the discount, calculated as 120 - 6 = 114. Profit is determined by the difference between the selling price and the cost price, so Profit = 114 - 100 = 14. Since the base CP is 100, the profit percentage is 14%.

This can also be solved using the effective percentage change formula x + y + ˣʸ⁄₁₀₀, where x is the markup percentage (+20) and y is the discount percentage (-5). Substituting these values gives 20 - 5 + (20 × (-5))/(100) = 15 - ¹⁰⁰⁄₁₀₀ = 15 - 1 = 14%.

Explanation

To find the ratio A : C : D, we first determine the relationship between A and C by multiplying the given ratios A : B and B : C. The ratio of A to C is calculated as A/C = (A/B) × (B/C) = (2/5) × (4/3) = 8/15. This establishes A : C = 8 : 15.

We are also given C : D = 2 : 1. To combine A : C and C : D into a single ratio A : C : D, we must make the value representing C consistent in both expressions. The least common multiple of 15 and 2 is 30. Multiplying the terms of A : C by 2 results in A : C = 16 : 30. Multiplying the terms of C : D by 15 results in C : D = 30 : 15. Since the value for C is now balanced at 30, the combined ratio is A : C : D = 16 : 30 : 15.

Explanation

The total runs conceded by the bowler in the first 5 matches is calculated by multiplying the number of matches by the average runs: 5 × 45 = 225. Similarly, the total runs conceded in the other 4 matches is 4 × 15.75 = 63.

To find the average for all 9 matches, we use the formula for the weighted average:

Average = (Total runs in 5 matches + Total runs in 4 matches)/(Total matches)

Average = (225 + 63)/(9) = ²⁸⁸⁄₉ = 32

Explanation

Initial volume of the mixture is 80 litres. The initial percentage of milk is 10%, which means the initial percentage of water is 90%. Thus, the quantity of water is 0.90 × 80 = 72 litres.

When milk is added to the mixture, the quantity of water remains constant at 72 litres. In the final mixture, the percentage of water is required to be 80%. Let the total volume of the final mixture be V.

80% of V = 72 ⁸⁰⁄₁₀₀ × V = 72 V = (72 × 100)/(80) = 90 litres

The amount of milk to be added is the difference between the final total volume and the initial total volume: 90 litres - 80 litres = 10 litres

Explanation

The speeds of the bus in successive hours form an arithmetic progression (AP) where the first term a = 21 km/hr and the common difference d = 3 km/hr. To find the time n required to cover a total distance Sₙ = 252 km, we use the sum of an arithmetic progression formula:

Sₙ = ⁿ⁄₂ [2a + (n - 1)d]

Substituting the given values into the formula:

252 = ⁿ⁄₂ [2(21) + (n - 1)3]

504 = n [42 + 3n - 3]

504 = n [39 + 3n]

504 = 39n + 3n²

Dividing the entire equation by 3 to simplify:

n² + 13n - 168 = 0

To solve this quadratic equation, we find factors of 168 that have a difference of 13, which are 21 and 8:

n² + 21n - 8n - 168 = 0

n(n + 21) - 8(n + 21) = 0

(n + 21)(n - 8) = 0

Since time cannot be negative, we discard n = -21. Thus, n = 8 hours.

Explanation

The simple interest earned in the first case is calculated as SI = Amount - Principal = 448 - 400 = 48. Using the simple interest formula SI = (P × R × T)/(100), we find the rate R = (SI × 100)/(P × T) = (48 × 100)/(400 × 2) = 6% per annum.

In the second case, the principal is 550 and the amount is 682, which gives a simple interest of SI = 682 - 550 = 132. Using the same rate of 6%, the time T required is calculated as T = (SI × 100)/(P × R) = (132 × 100)/(550 × 6). Simplifying this, we get T = ¹³²⁰⁰⁄₃₃₀₀ = 4 years.

Explanation

In a right-angled triangle ABC where ∠ BAC = 90° and AD is the altitude drawn perpendicular to the hypotenuse BC, the Right Triangle Altitude Theorem (also known as the Geometric Mean Theorem) states that the square of the altitude is equal to the product of the two segments of the hypotenuse. The relationship is given by the formula:

AD² = BD · CD

Given the lengths of the segments BD = 7 cm and CD = 28 cm, we substitute these values into the formula:

AD² = 7 · 28 AD² = 196

Taking the square root of both sides to find the length of AD:

AD = √(196) AD = 14 cm

This property is derived from the fact that the altitude AD divides △ ABC into two smaller triangles, △ ABD and △ CAD, which are similar to each other and to the original triangle. From the similarity △ ABD ∼ △ CAD, we obtain the ratio BD/AD = AD/CD, which leads directly to the altitude theorem.

Explanation

A perpendicular line drawn from the centre of a circle to a chord bisects the chord. Given a chord of length 60 cm, the distance from the centre to the midpoint of the chord creates a right-angled triangle where the radius is the hypotenuse, the distance from the centre is one leg, and half the chord length is the other leg.

Using the Pythagorean theorem r² = d² + (L/2)²: r² = 16² + (60/2)² r² = 16² + 30² r² = 256 + 900 r² = 1156 r = √(1156) = 34 cm

This calculation can be simplified by recognizing the Pythagorean triplet (8, 15, 17). Multiplying each term by 2 yields the triplet (16, 30, 34), confirming that the radius of the circle is 34 cm.

Explanation

The expression csc⁶ A - cot⁶ A - 3 csc² A cot² A can be simplified using the algebraic identity for the cube of a binomial, (x - y)³ = x³ - y³ - 3xy(x - y).

Let x = csc² A and y = cot² A. We know from the fundamental trigonometric identity that csc² A - cot² A = 1.

The given expression is in the form x³ - y³ - 3xy. Since x - y = 1, we can multiply the term 3xy by (x - y) without changing its value. This yields:

x³ - y³ - 3xy(x - y) = (x - y)³

Substituting the trigonometric values back into this identity:

(csc² A - cot² A)³

Since csc² A - cot² A = 1, the value becomes (1)³ = 1.

Explanation

To simplify the expression √((1 - cos A)/(1 + cos A)), we multiply the numerator and the denominator under the square root by (1 - cos A). This gives:

√((1 - cos A)/(1 + cos A) · (1 - cos A)/(1 - cos A)) = √(((1 - cos A)²)/(1 - cos² A))

Using the identity sin² A + cos² A = 1, we can substitute 1 - cos² A with sin² A:

√(((1 - cos A)²)/(sin² A)) = (1 - cos A)/(sin A)

Distributing the denominator across the terms in the numerator, we obtain:

(1)/(sin A) - (cos A)/(sin A)

Since (1)/(sin A) = csc A and (cos A)/(sin A) = cot A, the expression simplifies to csc A - cot A.

Explanation

To find the value of tan(2A + B), we first calculate tan 2A using the double angle formula tan 2A = (2 tan A)/(1 - tan² A). Substituting the given value tan A = 1/2:

tan 2A = (2(1/2))/(1 - (1/2)²) = (1)/(1 - 1/4) = (1)/(3/4) = ⁴⁄₃

Now, we use the addition formula for tangent, which is tan(X + Y) = (tan X + tan Y)/(1 - tan X tan Y). By setting X = 2A and Y = B, we substitute tan 2A = 4/3 and tan B = 1/3 into the formula:

tan(2A + B) = (4/3 + 1/3)/(1 - (4/3)(1/3)) = (5/3)/(1 - 4/9)

Simplifying the denominator gives 1 - 4/9 = 5/9. Therefore:

tan(2A + B) = (5/3)/(5/9) = ⁵⁄₃ × ⁹⁄₅ = 3

Explanation

The total time for car manufacturing is distributed among various processes, where the paint department accounts for 15% and the frame department accounts for 20% of the total duration. To find the combined time for these two processes, we sum their respective percentages: Combined Percentage = 15% + 20% = 35%

The total time taken to make a car is given as 300 hours. We calculate the combined time for paint and frame by applying the combined percentage to the total time: Total Time (Paint + Frame) = ³⁵⁄₁₀₀ × 300 = 35 × 3 = 105 hours

Explanation

Based on the distribution of manufacturing time for car components provided in this data set, the time allocated for seats accounts for 15 percent of the total time, while the time for glass accounts for 20 percent. The time taken for each component is directly proportional to its percentage share.

To find the time taken for glass, the ratio of the percentages is applied to the given time for seats:

Time for glass = (Percentage for glass)/(Percentage for seats) × Time for seats

Time for glass = (20%)/(15%) × 192

Time for glass = ⁴⁄₃ × 192

Time for glass = 4 × 64

Time for glass = 256

By performing this calculation, the time taken for glass is determined to be 256 hours.

Explanation

The time distribution for the manufacturing process is as follows: Engine: 30% Tyres: 21% Frame: 25% Glass: 11%

According to the question, the total time for Engine and Tyres is 127.5 hours. Summing their percentages: 30% + 21% = 51%

To find the value representing 1%: 1% = (127.5)/(51) = 2.5 hours

The difference in percentage between Frame and Glass is: 25% - 11% = 14%

Converting this percentage difference into hours: 14 × 2.5 = 35 hours

Explanation

The total time is initially shared among 5 processes: Tyres (25%), Glass (10%), Steel (30%), Plastic (30%), and Seats (5%). Since 15% of the total time is spent on the quality check by taking time equally from these 5 processes, the reduction per process is calculated as follows:

Reduction per process = 15% / 5 = 3%

The new combined percentage for Seats and Glass is: New Percentage = (5% - 3%) + (10% - 3%) = 2% + 7% = 9%

To find the new sectorial angle (θ), convert the percentage into degrees using the relation θ = (Percentage / 100) × 360°: θ = (9 / 100) × 360° = 32.4°

Explanation

Let the fraction be x/y. The first condition states that adding 5 to the numerator makes the fraction 6/5, expressed as (x + 5) / y = 6/5. Cross-multiplying gives 5x + 25 = 6y. The second condition states that adding 4 to the denominator makes the fraction 1/2, expressed as x / (y + 4) = 1/2. Cross-multiplying gives 2x = y + 4, or y = 2x - 4.

Substituting the expression for y into the first equation: 5x + 25 = 6(2x - 4) 5x + 25 = 12x - 24 49 = 7x x = 7

Plugging the value of x into the expression for y: y = 2(7) - 4 y = 10

The value of the fraction x/y is 7/10.

Explanation

Total work is calculated as the Least Common Multiple (LCM) of the time taken by Amit (25 days) and Punit (20 days), which is 100 units.

Efficiency of Amit = 100 / 25 = 4 units per day. Efficiency of Punit = 100 / 20 = 5 units per day.

Punit worked alone for 10 days, so the work completed by him is 5 × 10 = 50 units. The remaining work is 100 - 50 = 50 units.

The time taken by Amit alone to complete this remaining work is Remaining Work / Amit's Efficiency = 50 / 4 = 12.5 days. Expressed as a fraction, this is 12 ½ days.

Explanation

The sum of the interior angles of a quadrilateral is 360°. Given the ratio of the four successive angles as 7 : 11 : 7 : 11, let the angles be 7x, 11x, 7x, and 11x. Summing these gives 7x + 11x + 7x + 11x = 36x. Setting 36x = 360° yields x = 10°. Therefore, the measures of the angles are 70°, 110°, 70°, and 110°. In this quadrilateral, the opposite angles are equal (70° = 70° and 110° = 110°) and the sum of any two consecutive angles is 180° (e.g., 70° + 110° = 180°). Since opposite angles are equal and consecutive angles are supplementary, the figure satisfies the necessary and sufficient conditions to be a parallelogram.

Explanation

The discount offered on an item is the difference between its marked price and its selling price. Given the Marked Price (MP) is Rs 2150 and the Selling Price (SP) is Rs 1892, the discount amount is calculated as: Discount = MP - SP = 2150 - 1892 = 258

The discount percentage is determined by the formula: Discount Percentage = (Discount)/(Marked Price) × 100

Substituting the known values into the formula: Discount Percentage = ²⁵⁸⁄₂₁₅₀ × 100 Discount Percentage = ²⁵⁸⁰⁄₂₁₅

By dividing 2580 by 215, we find: 215 × 10 = 2150 215 × 2 = 430 2150 + 430 = 2580 Thus, the result is 12%.

Explanation

The total amount to be divided is T = 60500. Based on the first condition, A = ²⁄₉(B + C), which can be expressed as the ratio A : (B + C) = 2 : 9. This means A receives (2)/(2 + 9) of the total sum, so A = ²⁄₁₁ × 60500 = 11000. Based on the second condition, B = ³⁄₇(A + C), which gives the ratio B : (A + C) = 3 : 7. Thus, B receives (3)/(3 + 7) of the total sum, so B = ³⁄₁₀ × 60500 = 18150. To find the share of C, we subtract the shares of A and B from the total amount: C = 60500 - (11000 + 18150) = 60500 - 29150 = 31350.

Explanation

The total age of the initial 6 girls is calculated by multiplying the number of girls by their average age, which gives 6 × x = 6x. When 4 new girls join the class, the sum of their ages is (x - 2) + (x + 2) + (x + 4) + (x + 6) = 4x + 10. The new total age of the entire class, which now consists of 6 + 4 = 10 girls, is the sum of the original total and the ages of the new arrivals: 6x + 4x + 10 = 10x + 10. The new average age is determined by dividing this total age by the total number of girls: (10x + 10)/(10) = x + 1.

Explanation

In a transaction where two items are sold at the same selling price, one at a gain of x% and the other at a loss of x%, there is always a net loss. This net loss percentage is calculated using the formula Loss % = x²{100}.

Given that the gain and loss percentages are both 32%, we substitute x = 32 into the formula:

Loss % = 32²{100}

Loss % = ¹⁰²⁴⁄₁₀₀

Loss % = 10.24%

The specific value of the selling price (Rs 12000) does not affect the final percentage calculation, as the ratio of loss to total cost remains constant regardless of the absolute values involved.

Explanation

The amount of milk in the solution remains constant because only water is being added. Let the amount of water to be added be x litres. We use the dilution principle where the product of the initial concentration and initial volume equals the product of the final concentration and final volume: Cᵢₙᵢₜᵢₐₗ × Vᵢₙᵢₜᵢₐₗ = C_fᵢₙₐₗ × V_fᵢₙₐₗ.

Initial concentration of milk is 10% and the initial volume is 80 litres. The final concentration desired is 5%, and the final volume will be (80 + x) litres.

10 × 80 = 5 × (80 + x) 800 = 400 + 5x 400 = 5x x = 80

Alternatively, since the concentration of milk is reduced by half (from 10% to 5%), the total volume of the solution must double to keep the quantity of milk the same. Given the initial volume is 80 litres, the required final volume is 80 × 2 = 160 litres. The quantity of water to be added is 160 - 80 = 80 litres.

Explanation

The delay of 50 minutes is caused only by the reduction in speed during the latter part of the journey. The distance covered at the reduced speed is 200 × (1 - ⅖) = 120 km.

For a constant distance, speed and time are inversely proportional. If the speed becomes ⁶⁄₇ of the usual speed, the time taken becomes ⁷⁄₆ of the usual time. The delay is the difference between the new time and the usual time: Delay = ⁷⁄₆t - t = ⅙t

Given the delay is 50 minutes, we solve for the usual time t taken to cover the 120 km segment: ⅙t = 50 minutes t = 300 minutes = 5 hours

The usual speed V is calculated by dividing the distance by the usual time: V = (120 km)/(5 hours) = 24 km/hr.

Explanation

The principal amount P can be calculated using the simple interest formula SI = (P × R × T)/(100). Given SI = 25920, R = 12%, and T = 6 years, the equation is 25920 = (P × 12 × 6)/(100). Solving for P gives P = (25920 × 100)/(72), which simplifies to P = 360 × 100 = 36000.

To find the compound interest for 2 years at 8% per annum, the effective rate of interest can be calculated using the successive percentage formula x + y + ˣʸ⁄₁₀₀. Substituting x = 8 and y = 8, the effective rate is 8 + 8 + (8 × 8)/(100) = 16.64%.

The compound interest is then 16.64% of the principal 36000. Calculating this gives CI = (16.64)/(100) × 36000 = 16.64 × 360 = 5990.4.

Explanation

For a quadratic equation ax² + bx + c = 0 with roots α and β, the sum of roots is α + β = -b/a and the product of roots is αβ = c/a. For the equation 3x² - 13x + 14 = 0, the sum α + β = -(-13)/3 = 13/3 and the product αβ = 14/3.

The expression (α/β) + (β/α) can be simplified as: (α/β) + (β/α) = (α² + β²) / (αβ)

Using the identity α² + β² = (α + β)² - 2αβ, the numerator is calculated as: (13/3)² - 2(14/3) = 169/9 - 28/3 = (169 - 84) / 9 = 85/9

The final value is obtained by dividing the numerator by the product of roots: (85/9) / (14/3) = (85/9) × (3/14) = 85 / (3 × 14) = 85/42

Explanation

The value of the expression a³ + b³ + c³ - 3abc is calculated using the algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). To evaluate this with the provided information, we use the identity for the square of a trinomial: (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), which can be rearranged to a² + b² + c² = (a + b + c)² - 2(ab + bc + ca).

Substituting this into the primary identity yields a modified formula that uses only the known values: a³ + b³ + c³ - 3abc = (a + b + c) [ (a + b + c)² - 3(ab + bc + ca) ]

Substitute the given values a + b + c = 9 and ab + bc + ca = 18: a³ + b³ + c³ - 3abc = 9 × [9² - 3(18)] a³ + b³ + c³ - 3abc = 9 × [81 - 54] a³ + b³ + c³ - 3abc = 9 × 27 a³ + b³ + c³ - 3abc = 243

Explanation

The given equation (x/y) + (y/x) = 1 can be simplified by finding a common denominator for the terms on the left side. Combining the fractions gives (x² + y²) / xy = 1. By multiplying both sides of the equation by xy, we obtain x² + y² = xy. Subtracting xy from both sides results in the expression x² - xy + y² = 0.

The algebraic identity for the sum of cubes is x³ + y³ = (x + y)(x² - xy + y²). Substituting the derived value x² - xy + y² = 0 into this identity, we get x³ + y³ = (x + y) × 0. Any expression multiplied by zero results in zero, thus x³ + y³ = 0.

Explanation

Let the given equation be equated to a constant k: 5ˣ = 30ʸ = 6ᶻ = k

From this, we can express each base in terms of k: 5 = k¹^/ˣ 30 = k¹^/ʸ 6 = k¹^/ᶻ

We know the mathematical relationship between the bases is 5 × 6 = 30. Substituting the values of k, we get: k¹^/ˣ × k¹^/ᶻ = k¹^/ʸ k¹^/ˣ⁺¹^/ᶻ = k¹^/ʸ

By comparing the powers, we obtain the relation: 1/x + 1/z = 1/y 1/x - 1/y + 1/z = 0

The expression to be evaluated is: (xy + yz + zx) / xyz = (xy)/(xyz) + (yz)/(xyz) + (zx)/(xyz) = (1)/(z) + ¹⁄ₓ + (1)/(y)

In algebraic identities of this form where aˣ = bᶻ = (ab)⁻ʸ, the sum of the reciprocals of the exponents 1/x + 1/y + 1/z is equal to 0. This relationship ensures that the product of the bases raised to these specific powers maintains the equality, leading to the value 0.

Explanation

The point O, where the internal bisectors of ∠ Q and ∠ R meet, is the incenter of △ PQR. In any triangle, the angle formed at the incenter by the bisectors of two internal angles is given by the formula 90° + (A)/(2), where A is the third vertex angle.

For △ PQR, the angle ∠ QOR is calculated as: ∠ QOR = 90° + (∠ P)/(2)

Substituting the given value of ∠ P = 70° into the equation: ∠ QOR = 90° + (70°)/(2) ∠ QOR = 90° + 35° ∠ QOR = 125°

Explanation

According to the theorem on the areas of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. If A₁ and A₂ are the areas of two similar triangles and s₁ and s₂ are their corresponding sides, the relationship is given by the formula:

(A₁)/(A₂) = ((s₁)/(s₂))²

In this problem, the ratio of the areas is given as 5 : 7. Substituting this into the formula, we get:

⁵⁄₇ = ((s₁)/(s₂))²

Taking the square root of both sides to solve for the ratio of the sides:

(s₁)/(s₂) = √(⁵⁄₇) = √5{√7}

Therefore, the ratio of the corresponding sides of these two triangles is √5 : √7.

Explanation

In an isosceles trapezium ABCD with parallel sides AD and BC, let perpendiculars be dropped from vertices A and D to the base BC, meeting it at points E and F respectively. Because the trapezium is isosceles, the segments BE and FC are equal. The length of BE is calculated as half the difference between the lengths of the parallel sides: BE = (BC - AD)/(2) = (14 - 8)/(2) = 3 cm.

In the right-angled triangle ABE, the height h (AE) is found using the Pythagorean theorem where AB is the hypotenuse: h = √(AB² - BE²) = √(5² - 3²) = √(25 - 9) = √(16) = 4 cm.

The area of a trapezium is determined by the formula ½ × (AD + BC) × h. Substituting the known values into the formula yields Area = ½ × (8 + 14) × 4 = ½ × 22 × 4 = 11 × 4 = 44 cm².

Explanation

The expression can be simplified by dividing each term in the numerator by the denominator: (tan² θ)/(tan² θ sin² θ) - (sin² θ)/(tan² θ sin² θ) After canceling the common terms in the numerator and denominator of each fraction, the expression becomes: (1)/(sin² θ) - (1)/(tan² θ) Using the reciprocal identities csc θ = (1)/(sin θ) and cot θ = (1)/(tan θ), the expression is equivalent to: csc² θ - cot² θ According to the fundamental trigonometric identity 1 + cot² θ = csc² θ, the value of csc² θ - cot² θ is equal to 1. Alternatively, the identity tan² θ - sin² θ = tan² θ sin² θ can be used to show that the numerator is equal to the denominator, resulting in a simplified value of 1.

Explanation

Using the trigonometric identity for complementary angles, the relation sec A = csc B holds true when A + B = 90°. Applying this identity to the given equation sec(3x - 20°) = csc(3y + 20°), we set the sum of the angles to 90 degrees:

(3x - 20°) + (3y + 20°) = 90°

The constants -20° and +20° cancel each other out, simplifying the equation to:

3x + 3y = 90°

By factoring out the common coefficient of 3, we get:

3(x + y) = 90°

Dividing both sides by 3 yields the value for the sum of the variables:

x + y = 30°

The question asks for the value of tan(x + y). Substituting the calculated sum into the tangent function:

tan(30°) = (1)/(√(3))

Explanation

The trigonometric identity for the cotangent of the sum of two angles is given by the formula cot(A + B) = (cot A cot B - 1)/(cot A + cot B). Substituting the given values cot A = ⁿ⁄ₙ₊₁ and cot B = ¹⁄₂ₙ₊₁ into this expression allows for a direct calculation.

The numerator is (ⁿ⁄ₙ₊₁)(¹⁄₂ₙ₊₁) - 1 = (n - (n+1)(2n+1))/((n+1)(2n+1)) = n - (2n² + 3n + 1){(n+1)(2n+1)} = -2n² - 2n - 1{(n+1)(2n+1)}.

The denominator is ⁿ⁄ₙ₊₁ + ¹⁄₂ₙ₊₁ = (n(2n+1) + (n+1))/((n+1)(2n+1)) = 2n² + 2n + 1{(n+1)(2n+1)}.

By dividing the numerator by the denominator, the common term (n+1)(2n+1) cancels out, leaving -(2n² + 2n + 1){2n² + 2n + 1}. This simplifies to -1. This result can also be verified quickly by assuming a value for n; for instance, if n=1, then cot A = 1/2 and cot B = 1/3, which yields cot(A+B) = ((1/6) - 1)/((1/2) + (1/3)) = (-5/6)/(5/6) = -1.

Explanation

To determine the least multiplier required to make 1200 a perfect square, we perform prime factorization: 1200 = 2 × 600 = 2² × 300 = 2³ × 150 = 2⁴ × 75 = 2⁴ × 3 × 25 = 2⁴ × 3¹ × 5²

A number is a perfect square if and only if the exponents of all its prime factors are even. In the prime factorization of 1200, the exponent of the factor 2 is 4 (even) and the exponent of the factor 5 is 2 (even). The prime factor 3 has an exponent of 1, which is odd. To make 1200 a perfect square, we must multiply it by 3 to ensure the exponent of the factor 3 becomes 2. This multiplication results in 1200 × 3 = 3600, which is the square of 60.

Explanation

Total work is determined by the LCM of 10, 12, and 15, which is 60 units. The efficiencies of A, B, and C are calculated as E_A = 60/10 = 6 units/day, E_B = 60/12 = 5 units/day, and E_C = 60/15 = 4 units/day. Let the total duration to complete the work be x days. According to the problem, A worked for 2 days, B worked for x - 3 days, and C worked for the entire x days. The sum of their individual contributions equals the total work:

6(2) + 5(x - 3) + 4(x) = 60 12 + 5x - 15 + 4x = 60 9x - 3 = 60 9x = 63 x = 7