SSC CGL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

The expression (cosθ)/(1-sinθ) + (cosθ)/(1+sinθ) = 4 can be simplified by taking the common denominator (1-sinθ)(1+sinθ), which is equal to 1-sin²θ or cos²θ. Combining the terms, the numerator becomes cosθ(1+sinθ) + cosθ(1-sinθ). Expanding this, we get cosθ + cosθ + cosθ - cosθ, which simplifies to 2cosθ.

The equation now stands as (2cosθ)/(cos²θ) = 4. Further simplification leads to (2)/(cosθ) = 4, or 2secθ = 4. This results in secθ = 2, which means cosθ = ½. Within the given range of 0 < θ < 90°, the value of θ that satisfies cosθ = ½ is 60°.

Explanation

In compound interest, the principal grows by a constant factor over equal intervals of time. The sum of Rs. 2000 becomes Rs. 4000 in 2 years, which means the amount doubles in a 2-year period (4000 / 2000 = 2). In compound interest, if a sum becomes n times in t years, it will become nᵏ times in kt years.

Here, the amount needs to become Rs. 8000 from the initial Rs. 2000. The growth factor is 8000 / 2000 = 4. Since 4 = 2², and we know the sum doubles (n = 2) every 2 years (t = 2), we apply the relation with k = 2. The total time required is kt = 2 × 2 = 4 years. Alternatively, the sum doubles to Rs. 4000 in the first 2 years, and this Rs. 4000 doubles again to Rs. 8000 in the subsequent 2 years, totaling 4 years.

Explanation

Let the speed of the train be vₜ km/h and the speed of the car be v_c km/h. According to the first condition, the total time for the 600 km journey is 8 hours, with 120 km by train and 480 km by car. This gives the equation 120/vₜ + 480/v_c = 8. Dividing by 8, we get 15/vₜ + 60/v_c = 1.

In the second condition, the time taken is 8 hours and 20 minutes, which is 25/3 hours, for 200 km by train and 400 km by car. This gives 200/vₜ + 400/v_c = 25/3. Dividing by 25/3 or simplifying, we get 24/vₜ + 48/v_c = 1.

Since both expressions equal 1, we can equate them: 15/vₜ + 60/v_c = 24/vₜ + 48/v_c. Rearranging the terms to group vₜ and v_c, we get 60/v_c - 48/v_c = 24/vₜ - 15/vₜ, which simplifies to 12/v_c = 9/vₜ. Cross-multiplying to find the ratio vₜ/v_c, we get vₜ/v_c = 9/12 = 3/4. Therefore, the ratio of the speed of the train to that of the car is 3:4.

Explanation

Let the height of the lighthouse be h = 100m. When two objects are on opposite sides of a tower, the distance D between them is the sum of their individual horizontal distances from the base of the tower. For an angle of elevation θ, the horizontal distance x is calculated using the formula x = h cot θ.

For the first ship with an angle of elevation of 30°, the distance x₁ from the lighthouse is 100 cot 30° = 100√3m. For the second ship with an angle of elevation of 45°, the distance x₂ from the lighthouse is 100 cot 45° = 100(1) = 100m.

The total distance between the two ships is D = x₁ + x₂ = 100√3 + 100. Factoring out the height, we get D = 100(√3 + 1). Substituting the given value √3 = 1.73: D = 100(1.73 + 1) = 100(2.73) = 273m.

Explanation

The given equation is a + ¹⁄ₐ = 1. Multiplying both sides by a results in a² + 1 = a, which can be rearranged as a² - a + 1 = 0.

The expression to be evaluated is (a² - a + 1)/(a² + a + 1). By substituting the identity a² - a + 1 = 0 directly into the numerator, the expression becomes (0)/(a² + a + 1).

To verify the denominator is non-zero, we use the relation a² + 1 = a to simplify it: a² + a + 1 = (a² + 1) + a = a + a = 2a. Since a ≠ 0, the denominator 2a is non-zero. Therefore, the value of the entire expression is ⁰⁄₂ₐ = 0.

Explanation

Using the algebraic identity (m + n)³ = m³ + n³ + 3mn(m + n), we substitute the given condition m + n = 1 into the formula.

(1)³ = m³ + n³ + 3mn(1)

1 = m³ + n³ + 3mn

Alternatively, this can be solved using the method of substitution by picking values for m and n that satisfy m + n = 1. If we let m = 1 and n = 0, the expression m³ + n³ + 3mn evaluates to 1³ + 0³ + 3(1)(0) = 1.

Explanation

Two circles are disjoint when the distance between their centers d is greater than the sum of their radii r₁ and r₂, mathematically expressed as d > r₁ + r₂. In this configuration, the circles do not touch or intersect at any point. There are two types of common tangents possible for such a pair. First, there are two direct common tangents, which are lines that touch both circles such that the circles lie on the same side of the line. Second, there are two transverse common tangents, which are lines that pass between the two circles, placing the circles on opposite sides of the line. The sum of these two types results in a maximum of 4 common tangents.

Explanation

The expression cos⁴θ - sin⁴θ can be factored as a difference of squares: cos⁴θ - sin⁴θ = (cos²θ + sin²θ)(cos²θ - sin²θ)

Using the fundamental trigonometric identity cos²θ + sin²θ = 1, the equation simplifies to: 1 · (cos²θ - sin²θ) = ⅓ cos²θ - sin²θ = ⅓

To find tan²θ, we can express both terms in the equation in terms of cos²θ and sin²θ. We have a system of two equations:

1. cos²θ + sin²θ = 1
2. cos²θ - sin²θ = ⅓

Adding the two equations: 2cos²θ = 1 + ⅓ = ⁴⁄₃ cos²θ = ⅔

Subtracting the second equation from the first: 2sin²θ = 1 - ⅓ = ⅔ sin²θ = ⅓

The value of tan²θ is given by the ratio: tan²θ = (sin²θ)/(cos²θ) = (1/3)/(2/3) = ½

Explanation

Any integer n can be expressed in the form 6k + r, where r ∈ {0, 1, 2, 3, 4, 5}. The remainder of the perfect square n² when divided by 6 is determined by the remainder of r² divided by 6. According to the question, the square is not divisible by 6, which means n² ≡ 0 6. This allows us to test the possible values of r from 1 to 5.

If n = 6k + 1, then n² = (6k+1)² ≡ 1² ≡ 1 6. If n = 6k + 2, then n² = (6k+2)² ≡ 2² ≡ 4 6. If n = 6k + 3, then n² = (6k+3)² ≡ 3² = 9 ≡ 3 6. If n = 6k + 4, then n² = (6k+4)² ≡ 4² = 16 ≡ 4 6. If n = 6k + 5, then n² = (6k+5)² ≡ 5² = 25 ≡ 1 6.

By evaluating these cases, the set of possible remainders for a perfect square not divisible by 6 is {1, 3, 4}. Simple verification with numerical examples also yields these results: 1² = 1 (remainder 1), 2² = 4 (remainder 4), 3² = 9 (remainder 3), 4² = 16 (remainder 4), and 5² = 25 (remainder 1).

Explanation

Let the average score after the 12th innings be A. Since the average increased by 5 after the 12th innings, the average after the 11th innings was A - 5. The total score is the product of the number of innings and the average score. Thus, the total score after 12 innings is 12A and the total score after 11 innings was 11(A - 5). The score in the 12th innings is the difference between these two totals.

12A - 11(A - 5) = 120 12A - 11A + 55 = 120 A + 55 = 120 A = 120 - 55 A = 65

Alternatively, the average after the nᵗʰ innings can be calculated using the formula: New Average = Score in nᵗʰ innings - (n - 1) × Increase in average. Substituting the given values where n = 12, Score = 120, and Increase = 5: A = 120 - (12 - 1) × 5 A = 120 - 11 × 5 A = 120 - 55 A = 65

Explanation

To simplify nested radicals of the form √(x ± 2y), we identify two numbers a and b such that a + b = x and ab = y. Then √(x ± 2y) = √a ± √b.

Beginning with the innermost radical, we simplify √(7 + 43) by expressing it as √(7 + 212). Since 4 + 3 = 7 and 4 × 3 = 12, we have √(7 + 212) = √4 + √3, which equals 2 + √3.

Substituting this result into the next part of the expression, we get √(3 + 8(2 + 3)). Expanding the terms gives √(3 + 16 + 83) = √(19 + 83). To simplify this, we rewrite it as √(19 + 248). Here, we seek two numbers that sum to 19 and multiply to 48, which are 16 and 3. Thus, √(19 + 248) = √(16) + √3, which equals 4 + √3.

The complete expression is evaluated by substituting this simplified value into the outermost radical: √(-3) + (4 + √3). The terms -√3 and +√3 cancel out, leaving √4, which evaluates to 2.

Explanation

Let the length and width of the rectangle be l and w. The area is l × w = 60 and the perimeter is 2(l + w) = 34, which simplifies to l + w = 17. The length of the diagonal d is given by the formula d = √(l² + w²).

Using the algebraic identity (l + w)² = l² + w² + 2lw, we can substitute the known values: 17² = l² + w² + 2(60) 289 = l² + w² + 120 l² + w² = 289 - 120 = 169

The length of the diagonal is the square root of this sum: d = √(169) = 13 cm

Alternatively, we can identify that the factors of 60 which sum to 17 are 12 and 5. In a rectangle with sides 12 cm and 5 cm, the diagonal forms a right-angled triangle with the sides. According to the Pythagorean triple (5, 12, 13), the hypotenuse (diagonal) must be 13 cm.

Explanation

Let the original price of the article be 100. After a 10% decrease, the new price becomes 100 - 10 = 90. To restore the article to its former value of 100, the new price must be increased by 10.

The percentage increase required on the new price is calculated using the formula: Percentage Increase = (Increase)/(New Price) × 100

Substituting the values: Percentage Increase = ¹⁰⁄₉₀ × 100 = ¹⁰⁰⁄₉ = 11 ⅑ %

Alternatively, if a value is decreased by r%, the percentage increase required to restore the original value is given by the formula: Required Increase % = ( (r)/(100 - r) ) × 100

By substituting r = 10: Required Increase % = ( (10)/(100 - 10) ) × 100 = ¹⁰⁄₉₀ × 100 = ¹⁰⁰⁄₉ = 11 ⅑ %

Explanation

The given equation is pq(p + q) = 1. This can be rearranged to express the sum of the variables as p + q = (1)/(pq). To find the value of the required expression, we cube both sides of this equality: (p + q)³ = ((1)/(pq))³

Applying the algebraic identity (a + b)³ = a³ + b³ + 3ab(a + b) to the left side and simplifying the right side, we get: p³ + q³ + 3pq(p + q) = (1)/(p³q³)

Substitute the value from the original given condition pq(p + q) = 1 into the middle term of the equation: p³ + q³ + 3(1) = (1)/(p³q³) p³ + q³ + 3 = (1)/(p³q³)

Now, rearrange the terms to isolate the constant by moving p³ and q³ to the right side: 3 = (1)/(p³q³) - p³ - q³

Thus, the value of the expression is 3.

Explanation

Given the expression x⁴ + (1)/(x⁴) = 119, we can find the value of x - ¹⁄ₓ by using standard algebraic identities. First, we convert the fourth-degree expression into a quadratic form by adding 2 to both sides of the equation: x⁴ + (1)/(x⁴) + 2 = 119 + 2. This allows us to write the left side as a perfect square: (x² + (1)/(x²))² = 121. Taking the square root of both sides gives x² + (1)/(x²) = 11.

Next, to find the value of x - ¹⁄ₓ, we subtract 2 from both sides of this new equation: x² + (1)/(x²) - 2 = 11 - 2. This simplifies the left side to the square of a binomial: (x - ¹⁄ₓ)² = 9. Taking the square root of both sides, we get x - ¹⁄ₓ = 3.

Explanation

According to the Inscribed Angle Theorem, the angle subtended by an arc at the circumference is half the angle subtended by the same arc at the center. In a semicircle, the arc is a diameter, which subtends a straight angle of 180° at the center of the circle. By applying the formula inscribed = ½ × central, the angle subtended by the diameter at any point on the circumference is (180°)/(2) = 90°. This geometric principle is known as Thales's Theorem, which establishes that any triangle inscribed in a semicircle with the diameter as one side is a right-angled triangle.

Explanation

To find the value of x² + y², we square both the given equations and add them together.

Given: x = a cosθ + b sinθ y = b cosθ - a sinθ

Squaring x: x² = (a cosθ + b sinθ)² = a² cos²θ + b² sin²θ + 2ab cosθ sinθ

Squaring y: y² = (b cosθ - a sinθ)² = b² cos²θ + a² sin²θ - 2ab cosθ sinθ

Adding x² and y²: x² + y² = (a² cos²θ + b² sin²θ + 2ab cosθ sinθ) + (b² cos²θ + a² sin²θ - 2ab cosθ sinθ)

The terms 2ab cosθ sinθ and -2ab cosθ sinθ cancel each other out. Grouping the remaining terms: x² + y² = a² (cos²θ + sin²θ) + b² (sin²θ + cos²θ)

Applying the trigonometric identity sin²θ + cos²θ = 1: x² + y² = a²(1) + b²(1) x² + y² = a² + b²

Explanation

The given expression can be simplified by taking the common denominator: (cosθ(1+sinθ) + cosθ(1-sinθ))/((1-sinθ)(1+sinθ)) = 4 (cosθ + cosθ + cosθ - cosθ)/(1-sin²θ) = 4 (2cosθ)/(cos²θ) = 4 Using the trigonometric identity 1-sin²θ = cos²θ , the equation simplifies to: (2)/(cosθ) = 4 cosθ = ²⁄₄ = ½ For the given range 0° < θ < 90° , the value of θ for which cosθ = ½ is 60° .

Explanation

According to the exterior angle theorem, the exterior angle of a triangle is equal to the sum of the two opposite interior angles. In Δ ABC, when side BC is extended to D, the exterior angle ∠ ACD is formed. Therefore:

∠ ACD = ∠ A + ∠ B

Given that ∠ ACD = 112° and ∠ B = ¾ ∠ A, we can express ∠ A in terms of ∠ B as ∠ A = ⁴⁄₃ ∠ B. Substituting these values into the exterior angle equation:

112° = ⁴⁄₃ ∠ B + ∠ B

112° = ⁷⁄₃ ∠ B

To find ∠ B, rearrange the equation:

∠ B = (112° × 3)/(7)

∠ B = 16° × 3 = 48°

Explanation

Let the height of the lighthouse be h = 100m. Let the two ships be at distances x and y from the foot of the lighthouse on opposite sides. The total distance between the ships is D = x + y.

Using trigonometry for the first ship with an angle of elevation of 30°: tan 30° = (Height)/(Base) = 100/x (1)/(√3) = 100/x ⟹ x = 100√3

Using trigonometry for the second ship with an angle of elevation of 45°: tan 45° = (100)/(y) 1 = (100)/(y) ⟹ y = 100

The total distance between the ships is: D = x + y = 100√3 + 100 = 100(√3 + 1)

Substituting the given value √3 = 1.73: D = 100(1.73 + 1) = 100 × 2.73 = 273m

Explanation

In a right-angled triangle, the circumcenter always lies at the midpoint of the hypotenuse. This means the hypotenuse of the triangle acts as the diameter of its circumcircle. Given that the radius of the circumcircle R is 3 cm, the length of the hypotenuse c is calculated as c = 2R = 2 × 3 = 6 cm.

The area of any triangle is given by the formula ½ × base × height. By taking the hypotenuse as the base of △ABC, the altitude drawn from the opposite vertex to the hypotenuse serves as the height. Substituting the known values into the formula, the area is ½ × 6 cm × 2 cm = 6 sq cm.

Explanation

In a right-angled triangle formed by the tower, its shadow on the ground, and the line of sight to the top, the relationship between the height and the distance from the base is defined by the tangent of the angle of elevation θ. According to the formula tan θ = (Height)/(Distance), we substitute the given values:

tan θ = 50√3{50} tan θ = √3

The value of θ for which tan θ = √3 is 60°. Thus, the angle of elevation from the point on the ground to the top of the tower is 60°.

Explanation

For compound interest where the rate of interest varies each year, the final amount A is calculated using the formula A = P(1 + r₁/100)(1 + r₂/100), where P is the principal amount and r₁, r₂ are the interest rates for the first and second years respectively.

Given P = 10,000, r₁ = 10, and r₂ = 12, the calculation is as follows:

A = 10000 × (1 + 10/100) × (1 + 12/100) A = 10000 × (110/100) × (112/100) A = 10000 × 1.1 × 1.12 A = 11000 × 1.12 A = 12320

The interest earned in the first year is 10% of 10,000, which is 1,000, making the principal for the second year 11,000. The interest for the second year is 12% of 11,000, which is 1,320. Adding this to the intermediate principal gives the final amount of 12,320.

Explanation

The expression is evaluated using trigonometric identities for complementary angles, where α + β = 90°.

First, applying the identity tan(90° - θ) = cot θ, we convert tan 80°: tan 80° = tan(90° - 10°) = cot 10° The first term becomes cot 10° · tan 10°. Since tan θ · cot θ = 1, the value of this term is 1.

Second, applying the identity sin(90° - θ) = cos θ, the term sin 70° can be rewritten: sin 70° = sin(90° - 20°) = cos 20° sin² 70° = cos² 20°

According to the Pythagorean identity sin² θ + cos² θ = 1. In this context, the sum of the remaining trigonometric terms satisfies this relation: sin² 70° + sin² 20° = cos² 20° + sin² 20° = 1

Adding the results of the two parts of the expression: 1 + 1 = 2

Explanation

Let the smaller positive integer be x. Since the other number is twice the first, it can be represented as 2x. The product of these two integers is given as 2048.

The relationship can be expressed by the equation: x · (2x) = 2048 2x² = 2048

Dividing both sides by 2 gives: x² = 1024

Taking the square root of both sides to find the value of the smaller number: x = √(1024) x = 32

Since x represents the smaller of the two numbers, the result is 32.

Explanation

To find the time taken by A and C together, we first determine the total work using the Least Common Multiple (LCM) of the given time periods: 8, 12, and 6. Let the total work be 24 units.

The efficiency of (A + B) = 24 / 8 = 3 units/day. The efficiency of (B + C) = 24 / 12 = 2 units/day. The efficiency of (A + B + C) = 24 / 6 = 4 units/day.

We can calculate the individual efficiencies of A and C as follows: Efficiency of A = Efficiency (A + B + C) - Efficiency (B + C) = 4 - 2 = 2 units/day. Efficiency of C = Efficiency (A + B + C) - Efficiency (A + B) = 4 - 3 = 1 unit/day.

Combining their efforts, the total efficiency of (A + C) is 2 + 1 = 3 units/day. The time taken by A and C together to complete the work is calculated by dividing the total work by their combined efficiency: 24 / 3 = 8 days.

Explanation

The length of the diagonal d of a cuboid with length l, width w, and height h is determined by the formula d = √(l² + w² + h²). This formula is derived from applying the Pythagorean theorem in three dimensions.

Given the dimensions of the cuboid are l = 5 cm, w = 4 cm, and h = 3 cm, the calculation is as follows:

d = √(5² + 4² + 3²) d = √(25 + 16 + 9) d = √(50)

The value √(50) can be simplified by factoring out the largest perfect square, which is 25:

d = √(25 × 2) d = 5√2 cm

Explanation

The discount amount is the difference between the marked price and the selling price. Using the values provided: Discount = 540 - 496.80 = 43.20

The rate of discount is determined by expressing the discount as a percentage of the marked price. The formula is: Discount % = (Discount)/(Marked Price) × 100

By substituting the values into the formula: Discount % = (43.20)/(540) × 100 Discount % = ⁴³²⁰⁄₅₄₀ Discount % = ⁴³²⁄₅₄ = 8

Explanation

Let the three numbers be 5x, 7x, and 12x based on the given ratio 5:7:12. According to the problem, the sum of the first and third numbers exceeds the second number by 50. This relationship is represented by the equation (5x + 12x) - 7x = 50. Simplifying the expression inside the parentheses yields 17x - 7x = 50, which further simplifies to 10x = 50. Solving for the variable, we find that x = 5. The total sum of the three numbers is the sum of their ratio components, which is 5x + 7x + 12x = 24x. By substituting the value of x into this expression, the total sum is 24 × 5 = 120.

Explanation

The age of the tenth child is determined by calculating the difference between the total age of 10 children and the total age of 9 children using the formula: Total Age = Number of children × Average Age.

To simplify the calculation, convert the units into months: Average age of 10 children = 9 years 9 months = (9 × 12) + 9 = 117 months. Total age of 10 children = 10 × 117 = 1170 months.

Average age of 9 children = 8 years 11 months = (8 × 12) + 11 = 107 months. Total age of 9 children = 9 × 107 = 963 months.

Age of the tenth child = Total age of 10 children - Total age of 9 children Age of the tenth child = 1170 - 963 = 207 months.

To convert 207 months back to years and months, divide by 12: 207 ÷ 12 = 17 years and 3 months (since 17 × 12 = 204).

Explanation

Let the cost of the article at place A be x. Since the article costs 15% less at place B, its cost at B is 0.85x. Gopi's total expenditure to buy the article at place B, including travel expenses, is 0.85x + 150. The gain of 150 is the difference between the cost at A and the total expenditure at B:

x - (0.85x + 150) = 150 0.15x - 150 = 150 0.15x = 300 x = (300)/(0.15) = 2000

The cost of the article at place A is Rs. 2000. The profit percent is calculated relative to this original price at place A:

Profit% = (Gain)/(Cost at A) × 100 Profit% = ¹⁵⁰⁄₂₀₀₀ × 100 = 7.5%

Explanation

Let the income of Mahesh be 100. If Mohan's income is 150% higher than Mahesh's, then Mohan's income is 100 + 150 = 250.

To calculate the percentage by which Mahesh's income is less than Mohan's, we find the difference between their incomes relative to Mohan's income: Percentage less = (Mohan's income - Mahesh's income)/(Mohan's income) × 100

Substituting the values: Percentage less = (250 - 100)/(250) × 100 = ¹⁵⁰⁄₂₅₀ × 100 Percentage less = ⅗ × 100 = 60%

This can also be solved using the standard formula where if income is r% more, the reverse percentage less is given by (r)/(100 + r) × 100%. Here, r = 150, so the calculation is ¹⁵⁰⁄₂₅₀ × 100 = 60%.

Explanation

When a train crosses a platform, the total distance covered is the sum of the length of the train and the length of the platform. Given that the length of the train and the platform are equal, let the length be L meters. Thus, the total distance covered is D = L + L = 2L meters.

The speed of the train is 90 km/hr. To convert this into meters per second (m/s), multiply by ⁵⁄₁₈: Speed = 90 × ⁵⁄₁₈ = 25 m/s

The time taken to cross the platform is 1 minute, which is equal to 60 seconds. Using the basic formula Distance = Speed × Time: 2L = 25 × 60 2L = 1500 L = ¹⁵⁰⁰⁄₂ = 750

Explanation

The principal amount P can be calculated using the formula for Simple Interest SI = (P · R · T)/(100). Substituting the given values SI = 1600, R = 5%, and T = 2 years: 1600 = (P · 5 · 2)/(100) 1600 = (10P)/(100) = (P)/(10) P = 16000

The Compound Interest CI for 3 years at the same rate is calculated using the formula CI = P [ (1 + (R)/(100))ⁿ - 1 ]. Substituting P = 16000, R = 5, and n = 3: CI = 16000 [ (1 + ⁵⁄₁₀₀)³ - 1 ] CI = 16000 [ (²¹⁄₂₀)³ - 1 ] CI = 16000 [ ⁹²⁶¹⁄₈₀₀₀ - 1 ] CI = 16000 [ (9261 - 8000)/(8000) ] CI = 16000 · ¹²⁶¹⁄₈₀₀₀ CI = 2 · 1261 = 2522

Explanation

For any expression of the form x + ¹⁄ₓ = k, the value of x³ + (1)/(x³) is given by the algebraic identity k³ - 3k. This is derived from the expansion of (x + ¹⁄ₓ)³ = x³ + (1)/(x³) + 3(x)(¹⁄ₓ)(x + ¹⁄ₓ).

Given x + ¹⁄ₓ = √3, we substitute k = √3 into the formula:

x³ + (1)/(x³) = (√3)³ - 3(√3)

Calculating the terms: (√3)³ = √3 × √3 × √3 = 3√3 3(√3) = 3√3

Substituting these back into the expression: x³ + (1)/(x³) = 3√3 - 3√3 = 0

Explanation

Given the equation (a)/(b) + ᵇ⁄ₐ = 1, we can simplify it by taking the least common multiple of the denominators: (a² + b²)/(ab) = 1 Multiplying both sides by ab gives: a² + b² = ab Rearranging the terms to one side, we get: a² - ab + b² = 0

The algebraic identity for the sum of two cubes is: a³ + b³ = (a + b)(a² - ab + b²) Substituting the derived value a² - ab + b² = 0 into this identity: a³ + b³ = (a + b) × 0 a³ + b³ = 0

Explanation

To find the value of the expression (p)/(q) + (q)/(p), we first simplify it by finding a common denominator. By cross-multiplying the terms, the expression becomes:

(p · p + q · q)/(pq) = (p² + q²)/(pq)

The given equation states that p² + q² = 7pq. Substituting this relationship into the numerator of our expression, we have:

(7pq)/(pq)

Dividing both the numerator and the denominator by the common factor pq results in the constant value 7.

Explanation

The value is calculated using the algebraic identity x(x² + 3x + 3) = x³ + 3x² + 3x. By adding and subtracting 1, the expression can be rewritten as (x³ + 3x² + 3x + 1) - 1, which is the binomial expansion of (x + 1)³ - 1. Substituting the given value of x = 99 into this identity:

(99 + 1)³ - 1 = 100³ - 1

Calculating the power of 100:

100³ = 1,000,000

Subtracting 1 from this result:

1,000,000 - 1 = 999,999

Explanation

In any triangle, the centroid G is the point of intersection of its three medians. A fundamental geometric property of the centroid is that it divides the triangle into three smaller triangles of equal area: △ AGB, △ BGC, and △ CGA. Each of these triangles has an area exactly equal to one-third of the area of the original triangle △ ABC.

Given that the area of △ ABC = 72 sq. units, the area of △ BGC is calculated using the formula:

Area(△ BGC) = ⅓ × Area(△ ABC)

Substituting the given value:

Area(△ BGC) = ⅓ × 72 = 24 sq. units.

This equality of area occurs because the three medians divide the triangle into six smaller triangles of equal area, and the triangle formed by the centroid and two vertices (like △ BGC) is composed of two of those six smaller units.

Explanation

The orthocentre of a triangle is the point of concurrency where the three altitudes intersect. In an acute-angled triangle, every internal angle is less than 90°. When an altitude is drawn from a vertex to the opposite side in such a triangle, the perpendicular line segment necessarily falls within the interior space of the triangle. Since all three altitudes are contained within the boundary of the triangle, their mutual intersection point must also be located inside the triangle. This positional property is a fundamental characteristic of triangle centers, where the location of the orthocentre relative to the triangle is determined strictly by the nature of its angles.

Explanation

For similar triangles Δ PQR and Δ LMN, the ratios of their corresponding sides are equal:

(PQ)/(LM) = (QR)/(MN) = (PR)/(LN)

From the given equation 3PQ = LM, the ratio of side PQ to LM is determined as:

(PQ)/(LM) = ⅓

Substituting this ratio and the given length MN = 9 cm into the proportionality relationship for the side QR:

⅓ = (QR)/(9)

Solving for QR:

QR = ⁹⁄₃ = 3 cm

Explanation

In a circle, a radius perpendicular to a chord bisects that chord. Since OC is part of a radius perpendicular to chord AB at point P, it divides the chord into two equal segments, AP = PB = ⁶⁄₂ = 3 cm. By extending the segment CP through the center O to the opposite side of the circle at a point D, we form a diameter CD. According to the intersecting chords theorem, the product of the segments of one chord is equal to the product of the segments of the other. Thus, AP · PB = CP · PD. Substituting the given values, 3 · 3 = 2 · PD, which yields PD = 4.5 cm. The diameter of the circle is the total length of the segment CD, calculated as CP + PD = 2 + 4.5 = 6.5 cm.

Explanation

Substituting the trigonometric values, we get x² = (1/2)² + 4(1)² - (2)², which simplifies to x² = 1/4 + 4 - 4 = 1/4. Since the problem specifies that x > 0, the value of x is 1/2.

Explanation

Solving the given equation 7sin²θ + 3cos²θ = 4 by substituting cos²θ = 1 - sin²θ yields 4sin²θ = 1, or sinθ = 1/2. This corresponds to θ = 30°, and evaluating the expression secθ + cscθ gives sec 30° + csc 30° = 2/√3 + 2. Since this value is not equivalent to 2/3 + 2 or the other specific options provided, the correct choice is none of these.

Explanation

Let the height of the mountain be h = 500 m. The angles of depression of the two boats from the top are 45° and 30°, which are equal to the angles of elevation from the boats to the top of the mountain. Let x and y be the distances of the two boats from the foot of the mountain respectively.

Using the trigonometric relation tan θ = (Opposite)/(Adjacent):

In the triangle formed by the boat closer to the mountain: tan 45° = 500/x 1 = 500/x ⟹ x = 500 m

In the triangle formed by the boat farther from the mountain: tan 30° = 500/y 1/√3 = 500/y ⟹ y = 500√3 m

Since both boats are on the same side of the mountain, the distance between them is the difference between their distances from the base: Distance = y - x = 500√3 - 500 Distance = 500(√3 - 1)

Taking the value of √3 ≈ 1.732: Distance = 500(1.732 - 1) = 500 × 0.732 = 366 m

Explanation

The relationship between the number of men (M) and the number of days (D) required to complete a fixed amount of work is inversely proportional, represented by the chain rule formula M₁ D₁ = M₂ D₂. Since the length of the wall remains constant at 140 meters in both scenarios, the work component cancels out, and we focus solely on the man-days required.

Given M₁ = 36 men, D₁ = 21 days, and D₂ = 14 days, we substitute these values into the equation to find the required number of men (M₂):

36 × 21 = M₂ × 14

To solve for M₂, rearrange the equation:

M₂ = (36 × 21)/(14)

Simplifying the fraction by dividing both 21 and 14 by 7:

M₂ = (36 × 3)/(2)

M₂ = 18 × 3 = 54

Explanation

Let the side of the square be s and the length and breadth of the rectangle be l and b respectively. Given that their perimeters are equal, 4s = 2(l + b), which implies s = (l + b)/(2). The area of the square is A = s² = ((l + b)/(2))² and the area of the rectangle is B = lb.

The relationship between the areas can be determined using the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that for any positive numbers l and b, (l + b)/(2) ≥ √(lb). Squaring both sides of this inequality yields ((l + b)/(2))² ≥ lb, which means A ≥ B. In a geometric context, for a fixed perimeter, the square is the specific rectangle that encloses the maximum possible area. Since a square and a rectangle are mentioned as distinct shapes in this comparison, l ≠ b, making the inequality strict. This is mathematically verified by the difference A - B = ((l + b)²)/(4) - lb = (l² + b² + 2lb - 4lb)/(4) = ((l - b)²)/(4). Because the square of any non-zero number is positive, A - B > 0, confirming that A > B.

Explanation

The difference in the discount percentages offered is 20% - 10% = 10%. This 10% difference in the discount corresponds to the ₹ 125 reduction in the purchase price. Let M represent the marked price of the watch. The relationship between the discount difference and the price difference is expressed as: 10% of M = 125 ¹⁰⁄₁₀₀ × M = 125 M = 125 × 10 M = 1250

Explanation

To find the cost price (CP) of the tape recorder when it is sold at a 4% gain for ₹ 1040, we use the formula CP = (SP × 100)/(100 + Gain%). Substituting the given values:

CP = (1040 × 100)/(100 + 4) = (1040 × 100)/(104) = 10 × 100 = ₹ 1000

Now, if the tape recorder is sold for a new selling price (SP) of ₹ 950, the loss incurred is calculated as Loss = CP - SP.

Loss = 1000 - 950 = ₹ 50

The loss percentage is determined by the formula Loss% = (Loss)/(CP) × 100.

Loss% = ⁵⁰⁄₁₀₀₀ × 100 = 5%

Explanation

The area of a square is determined by the formula A = s², where s represents the length of its side. When the side of a square is increased by a specific percentage, the resulting increase in area can be calculated using the successive percentage change formula: x + y + ˣʸ⁄₁₀₀. In the case of a square, since both dimensions (length and width) are equal and both increase by the same percentage, we substitute x = 10 and y = 10 into the formula.

Performing the calculation: Percentage increase = 10 + 10 + (10 × 10)/(100) Percentage increase = 20 + ¹⁰⁰⁄₁₀₀ Percentage increase = 20 + 1 = 21

This result can also be verified by assuming an initial side length. If the original side of the square is 10 units, the original area is 10 × 10 = 100 square units. After a 10% increase, the new side length becomes 11 units. The new area is 11 × 11 = 121 square units. The increase in area is 121 - 100 = 21 square units, which corresponds to a 21% increase relative to the original area of 100.