SSC CGL Tier 1 2015 Quantitative Aptitude Questions with Solutions

Explanation

Triangles sharing a common vertex and having their bases on the same straight line have areas proportional to the lengths of their bases. This is because they share the same altitude h dropped from the common vertex to the line containing the bases. In Δ ABC, Δ ABD and Δ ADC share the vertex A and their bases BD and DC lie on the line BC.

The ratio of their areas is:

(Area(Δ ABD))/(Area(Δ ADC)) = (1)/(2) × BD × h{½ × DC × h} = (BD)/(DC)

Given BD : DC = 4 : 5 and Area(Δ ABD) = 60 cm², we substitute these into the relation:

(60)/(Area(Δ ADC)) = ⅘

Area(Δ ADC) = (60 × 5)/(4) = 15 × 5 = 75 cm²

Explanation

The radius of a circle is always perpendicular to the tangent at the point of contact. This geometric property creates a right-angled triangle where the radius r = 6 cm and the tangent length L are the two legs, and the distance from the center to the external point d = 10 cm is the hypotenuse.

Applying the Pythagoras theorem: r² + L² = d² 6² + L² = 10² 36 + L² = 100 L² = 100 - 36 L² = 64 L = √(64) = 8 cm

Alternatively, this follows the standard Pythagorean triplet (6, 8, 10), which is a multiple of the basic (3, 4, 5) triplet. Since two sides are 6 and 10, the third side must be 8.

Explanation

The movement of the ship forms a right-angled triangle where the southward journey of 12 km and the eastward journey of 5 km serve as the two legs (perpendicular and base). The straightway distance from the starting point represents the hypotenuse of this triangle. According to the Pythagoras Theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Let d be the straightway distance: d² = 12² + 5² d² = 144 + 25 d² = 169 d = √(169) = 13 km

This calculation relies on the Pythagorean triple (5, 12, 13), which describes a right triangle where the sides are in the ratio 5:12:13. Therefore, the direct distance from the starting point is 13 km.

Explanation

The ratio of the sides 3:4:5 indicates that the triangle is a right-angled triangle because it satisfies the Pythagorean theorem 3² + 4² = 5². Let the sides of the triangle be 3x, 4x, and 5x, where 3x and 4x are the base and the height.

The area of a right-angled triangle is calculated using the formula Area = ½ × base × height. Substituting the given values: ½ × 3x × 4x = 7776 6x² = 7776 x² = ⁷⁷⁷⁶⁄₆ = 1296 x = √(1296) = 36

The perimeter of the triangle is the sum of all its sides: Perimeter = 3x + 4x + 5x = 12x By substituting the value of x: Perimeter = 12 × 36 = 432 cm.

Explanation

The length of a chord subtending an angle θ at the centre of a circle of radius r is given by the formula L = 2r sin(θ/2). For the first chord of length a unit making an angle of 60° at the centre, we have a = 2r sin(60°/2) = 2r sin 30°. Since sin 30° = 1/2, it follows that a = 2r × (1/2) = r.

For the second chord of length b unit making an angle of 90° at the centre, the relation is b = 2r sin(90°/2) = 2r sin 45°. Given that sin 45° = 1/√2, we get b = 2r × (1/√2) = √2r. Substituting the value r = a from the first relation into the second equation, we obtain the relation b = √2a.

Explanation

In a parallelogram PQRS, adjacent angles are supplementary, meaning their sum is 180°. Given that ∠ P = 4∠ Q, we can write the equation ∠ P + ∠ Q = 180° as 4∠ Q + ∠ Q = 180°. This simplifies to 5∠ Q = 180°, which yields ∠ Q = 36°. In any parallelogram, opposite angles are equal, so ∠ R = ∠ P. Substituting the value of ∠ Q into the initial relationship, we find ∠ R = ∠ P = 4 × 36° = 144°.

Explanation

Let the four numbers be x₁, x₂, x₃, and x₄. The problem states that their sum is 48, and after the specified transformations, they all equal the same constant value k. We can express the relationships as follows: x₁ + 5 = k, x₂ + 1 = k, x₃ - 3 = k, and x₄ - 7 = k. Solving for each variable in terms of k gives x₁ = k - 5, x₂ = k - 1, x₃ = k + 3, and x₄ = k + 7.

The sum of these original numbers is x₁ + x₂ + x₃ + x₄ = 48. Substituting the expressions involving k into this equation yields (k - 5) + (k - 1) + (k + 3) + (k + 7) = 48. Combining the terms results in 4k + 4 = 48, which simplifies to 4k = 44, giving a value of k = 11.

By substituting k = 11 back into the individual equations, we find the values of the four numbers: x₁ = 11 - 5 = 6 x₂ = 11 - 1 = 10 x₃ = 11 + 3 = 14 x₄ = 11 + 7 = 18

Explanation

To find the least number that must be added to 2055 to make it exactly divisible by 27, we first divide 2055 by 27 to find the remainder.

2055 = 27 × 76 + 3

Performing the division, 2055 ÷ 27 gives a quotient of 76 and a remainder of 3. The remainder represents how much the number exceeds the previous multiple of 27. To reach the next multiple of 27, we need to add the difference between the divisor and the remainder.

Required number to be added = Divisor - Remainder Required number to be added = 27 - 3 = 24

By adding 24 to 2055, we get 2079, which is exactly divisible by 27 (2079 = 27 × 77).

Explanation

The efficiency or daily work rate of A and B together is 1/6. Given that A can complete the work alone in 18 days, A's daily work rate is 1/18.

The daily work rate of B is found by subtracting A's rate from the combined rate of A and B: Rate of B = 1/6 - 1/18 To subtract these fractions, use 18 as the common denominator: Rate of B = 3/18 - 1/18 = 2/18 = 1/9

Since B completes 1/9 of the work in one day, the total number of days required for B to finish the entire work is the reciprocal of this rate: Days for B = 1 / (1/9) = 9 days.

Alternatively, this can be solved using the product-difference formula for work: Time_B = (Time_A × Time_A₊_B) / (Time_A - Time_A₊_B) Time_B = (18 × 6) / (18 - 6) Time_B = 108 / 12 = 9 days.

Explanation

The first pipe fills the tank at a rate of 1/x units per hour, while the second pipe empties the tank at a rate of 1/y units per hour. When both pipes are operated together, the net rate at which the tank is being filled is the difference between the filling rate and the emptying rate, which is 1/x - 1/y. To simplify this expression, we use a common denominator of xy, resulting in a net filling rate of (y - x) / xy units per hour. The total time taken to fill the tank is the reciprocal of this net filling rate, which is xy / (y - x) hours. Since the condition y > x is given, the filling rate is greater than the emptying rate, resulting in a positive net filling rate that allows the tank to be completed.

Explanation

To find the distance between the tops of the poles, we can model the scenario as a right-angled triangle. Let the two poles be AB and CD with heights h₁ = 7 m and h₂ = 12 m respectively. The horizontal distance between their feet is d = 12 m.

By drawing a horizontal line from the top of the shorter pole (A) to the taller pole (CD), we create a right-angled triangle where the base is equal to the distance between the feet (12 m) and the height is the difference between the heights of the two poles.

The vertical height of this triangle is h₂ - h₁ = 12 - 7 = 5 m. According to the Pythagoras theorem, the distance between the tops (D) is the hypotenuse of this triangle:

D = √(12²⁾⁺⁵^^^² D = √(144 + 25) D = √(169) D = 13 m

The result follows the standard Pythagorean triplet (5, 12, 13), making the distance between the tops 13 metres.

Explanation

For a trigonometric expression of the form a sin Θ + b cos Θ, the maximum value is given by the formula √(a² + b²). In the expression sin Θ + cos Θ, the coefficients are a = 1 and b = 1. Substituting these values into the formula gives √(1² + 1²), which simplifies to √2. This result can also be derived by rewriting the expression as √2 ((1)/(√(2)) sin Θ + (1)/(√(2)) cos Θ), which is equivalent to √2 sin(Θ + (π)/(4)). Since the maximum value of any sine function is 1, the maximum value of √2 sin(Θ + (π)/(4)) is √2 × 1 = √2.

Explanation

The value of the expression can be found using the trigonometric identity tan θ · tan(90° - θ) = 1, which is derived from the fact that tan(90° - θ) = cot θ and tan θ · cot θ = 1.

In the given expression tan 4° tan 43° tan 47° tan 86°, we can pair the terms whose angles sum to 90°.

First, consider the pair tan 4° and tan 86°. Since 86° = 90° - 4°, we have tan 86° = cot 4°. Therefore, tan 4° · tan 86° = tan 4° · cot 4° = 1.

Next, consider the pair tan 43° and tan 47°. Since 47° = 90° - 43°, we have tan 47° = cot 43°. Therefore, tan 43° · tan 47° = tan 43° · cot 43° = 1.

Multiplying these results together gives 1 · 1 = 1.

Explanation

Let the center of the quarter-circle be the origin O(0,0) and its two radii lie along the positive x and y axes. Let the four vertices of the square be A, B, C, and D.

Two adjacent vertices A and B lie on the radii at equal distance a from the center. Thus, A = (a, 0) and B = (0, a). The side length of the square is x, which corresponds to the distance between A and B: x = √((a - 0)² + (0 - a)²) = √(2a²) = a√2 a = (x)/(√(2))

The other two vertices C and D lie on the circular arc. Because the square is symmetric about the line y = x, we can determine the coordinates of C by moving from B(0, a) in a direction perpendicular to AB. The vector AB = (a, -a), so a perpendicular vector of length x pointing toward the arc is (x/√2, x/√2). The coordinates of C are: C = (0, a) + ((x)/(√(2)), (x)/(√(2))) = ((x)/(√(2)), (x)/(√(2)) + (x)/(√(2))) = ((x)/(√(2)), √2x)

The radius R of the circle is the distance from the origin O to any point on the arc, such as vertex C: R² = ((x)/(√(2)))² + (√2x)² R² = (x²)/(2) + 2x² = (5x²)/(2) R = √((5x²)/(2)) = √5x{√2}

Explanation

The equivalent single discount for two successive discounts of x% and y% is given by the formula (x + y - ˣʸ⁄₁₀₀)%. Applying this to the given values where x = 10 and y = 20:

Effective Discount = (10 + 20 - (10 × 20)/(100))% Effective Discount = (30 - ²⁰⁰⁄₁₀₀)% Effective Discount = (30 - 2)% Effective Discount = 28%

This can also be calculated by considering an initial value of 100. A 10% discount reduces the value to 90 (100 - 10). Then, a 20% discount on the remaining 90 reduces it by 18 (90 × 0.20), resulting in a final value of 72. the total reduction from the original 100 is 100 - 72 = 28, which is equivalent to a 28% discount.

Explanation

The marked price (MP) of the watch is Rs. 720. The first discount is 10%. The price after the first discount is calculated as: Price₁ = 720 - (10% of 720) = 720 - 72 = 648

The final selling price (SP) is Rs. 550.80. The second discount is applied to the intermediate price of Rs. 648. The amount of the second discount is the difference between this intermediate price and the final price: Discount Amount = 648 - 550.80 = 97.20

The rate of the second discount is determined by dividing the discount amount by the price it was applied to: Second Discount Rate = (97.20)/(648) × 100 Second Discount Rate = ⁹⁷²⁰⁄₆₄₈ = 15%

Explanation

When two taps operate simultaneously to perform the same task (in this case, emptying a tank), their combined time T can be calculated using the formula T = (x × y)/(x + y), where x and y are the individual times taken by each tap.

Given that the first tap empties the tank in x = 30 minutes and the second tap empties it in y = 45 minutes, substituting these values into the formula gives:

T = (30 × 45)/(30 + 45)

T = ¹³⁵⁰⁄₇₅

T = 18 minutes

Alternatively, using the efficiency method, let the total capacity of the tank be the Least Common Multiple (LCM) of 30 and 45, which is 90 units. The efficiency of the first tap is 90 / 30 = 3 units per minute, and the efficiency of the second tap is 90 / 45 = 2 units per minute. Their combined efficiency is 3 + 2 = 5 units per minute. The time required for both taps to empty the 90-unit tank together is 90 / 5 = 18 minutes.

Explanation

The perimeter of a square face of a cube is given by the formula P = 4a, where a represents the length of one side. Given that the perimeter is 20 cm, we find the side length by solving 4a = 20, which yields a = 5 cm. The volume V of a cube is calculated using the formula V = a³. Substituting the side length into this equation, we get V = 5³ = 125 cm³.

Explanation

The area of a circle is given by the formula A = π r² and the circumference is given by C = 2π r. By multiplying the circumference C by the radius r, we get rC = r(2π r) = 2π r². Since A = π r², it follows that 2A = 2π r². Comparing these two expressions, we find that rC = 2A.

Explanation

When one person in a group is replaced by another, the change in the total weight of the group is equal to the number of members multiplied by the change in the average weight. In this case, the total weight of the 15 oarsmen increases because the average weight increases. The total increase in weight is calculated as 15 × 1.6 = 24 kg. This increase occurs because the weight of the new man is greater than the weight of the man he replaced. Using the relation Wₙₑ_w = Wₒₗ_d + (n × Δ Average), where Wₒₗ_d is 42 kg, n is 15, and Δ Average is 1.6 kg, the weight of the new man is 42 + 24 = 66 kg.

Explanation

The arithmetic mean of a set of numbers is defined as the sum of the numbers divided by the total count of those numbers. The first n natural numbers form an arithmetic progression: 1, 2, 3, ..., n. The sum of the first n natural numbers is calculated using the formula:

S = (n(n + 1))/(2)

The number of terms in this sequence is n. Therefore, the arithmetic mean is determined by dividing the sum by n:

Arithmetic Mean = (n(n + 1))/(2){n}

By simplifying the expression and canceling the common factor n from the numerator and the denominator, the formula becomes:

Arithmetic Mean = (n + 1)/(2)

This result indicates that for any sequence of the first n natural numbers, the mean is always the average of the first and the last terms.

Explanation

The total cost price (CP) of the rice is the sum of the costs of the two individual brands. For the first brand, the cost is 30 kg × Rs. 70/kg = Rs. 2100. For the second brand, the cost is 20 kg × Rs. 70.75/kg = Rs. 1415. Therefore, the total CP equals 2100 + 1415 = Rs. 3515.

The total quantity of the mixture is 30 kg + 20 kg = 50 kg. The total selling price (SP) is determined by multiplying this total quantity by the selling rate: 50 kg × Rs. 80.50/kg = Rs. 4025. The shopkeeper's total gain is the difference between the total SP and the total CP, which is 4025 - 3515 = Rs. 510.

Explanation

The relationship between the marked price (MP) and the selling price (SP) after successive discounts is given by the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)).

Substituting the given values SP = 3060, d₁ = 20%, and d₂ = 15% into the formula:

3060 = MP × (1 - ²⁰⁄₁₀₀) × (1 - ¹⁵⁄₁₀₀)

3060 = MP × ⁸⁰⁄₁₀₀ × ⁸⁵⁄₁₀₀

3060 = MP × ⅘ × ¹⁷⁄₂₀

3060 = MP × ⁶⁸⁄₁₀₀

MP = (3060 × 100)/(68)

Dividing 3060 by 68, we get 45. Therefore:

MP = 45 × 100 = 4500

Explanation

Let the present ages of A and B be 5x and 7x respectively, following their present ratio of 5:7. Eighteen years ago, their ages were 5x - 18 and 7x - 18. The problem states that the ratio at that time was 8:13, which gives the following equation:

(5x - 18)/(7x - 18) = ⁸⁄₁₃

Cross-multiplying the terms to solve for x:

13(5x - 18) = 8(7x - 18)

65x - 234 = 56x - 144

Rearrange the equation to group the x terms on one side and the constants on the other:

65x - 56x = 234 - 144

9x = 90

x = 10

Since A's present age is represented by 5x, substitute the value of x:

A's present age = 5 × 10 = 50 years.

Explanation

In the initial mixture of 729 ml, the total parts for the ratio 7:2 are 7 + 2 = 9 units. The value of one unit is calculated as 729 / 9 = 81 ml. The ratio of milk to water changes from 7:2 to 7:3 after adding water. Since the quantity of milk remains constant and corresponds to 7 units in both the initial and final ratios, the change in the mixture is entirely due to the increase in water. The water component increases from 2 units to 3 units, which is an addition of 3 - 2 = 1 unit. Given that 1 unit equals 81 ml, the volume of water added to the mixture is 81 ml.

Explanation

For a sum of money to double itself under simple interest, the interest earned must be equal to the principal amount. Let the principal be P. Then the Simple Interest (SI) is also P. The rate of interest is R = 6 ¼% = ²⁵⁄₄% per annum.

Using the simple interest formula SI = (P × R × T)/(100), we can derive the time T as: T = (100 × SI)/(P × R)

Substituting the known values: T = (100 × P)/(P × 25){4} T = (100)/(25){4} T = (100 × 4)/(25) T = 4 × 4 = 16 years

Using the shortcut for doubling time, T = (100)/(R), we get: T = (100)/(6.25) = 16 years

Explanation

To find the length of the portion of the line 3x + 4y = 12 intercepted between the coordinate axes, we first determine the x-intercept and the y-intercept. The x-intercept is found by setting y = 0, which gives 3x = 12, hence x = 4. The y-intercept is found by setting x = 0, which gives 4y = 12, hence y = 3.

The line intersects the x-axis at the point A(4, 0) and the y-axis at the point B(0, 3). The segment intercepted between the axes is the hypotenuse of a right-angled triangle formed by the origin O(0, 0) and these two points. The length L of this segment is calculated using the distance formula L = √((x₂ - x₁)² + (y₂ - y₁)²), or simply by the Pythagorean theorem since the axes are perpendicular. Substituting the values, we get L = √(4² + 3²) = √(16 + 9) = √(25) = 5.

Explanation

The expression is evaluated by rationalizing each individual term using the conjugate property of square roots. For any fraction of the form (1)/(√(a) - b), multiplying both the numerator and the denominator by the conjugate √a + √b yields √a + √b{a - b}.

In this series, the difference between the squares of the numbers in each denominator is 1 (3² - 8 = 1, 8 - 7 = 1, 7 - 6 = 1, 6 - 5 = 1, and 5 - 2² = 1). Thus, the rationalized form of each term simplifies as follows:

(1)/(3 - √(8)) = 3 + √8 (1)/(√(8) - 7) = √8 + √7 (1)/(√(7) - 6) = √7 + √6 (1)/(√(6) - 5) = √6 + √5 (1)/(√(5) - 2) = √5 + 2

Substituting these rationalized values back into the alternating series: (3 + √8) - (√8 + √7) + (√7 + √6) - (√6 + √5) + (√5 + 2)

Upon expanding the expression and removing the brackets, the intermediate square root terms cancel each other out: 3 + √8 - √8 - √7 + √7 + √6 - √6 - √5 + √5 + 2

This simplifies to 3 + 2, which equals 5.

Explanation

The population growth follows the compound interest formula P = P₀(1 + ʳ⁄₁₀₀)ⁿ, where P is the current population, P₀ is the population n years ago, and r is the annual growth rate.

Given P = 9261, r = 5, and n = 3, we substitute these values into the formula:

9261 = P₀(1 + ⁵⁄₁₀₀)³

Simplifying the fraction inside the parentheses, we get 1 + ¹⁄₂₀ = ²¹⁄₂₀.

9261 = P₀(²¹⁄₂₀)³

9261 = P₀ × ⁹²⁶¹⁄₈₀₀₀

To solve for P₀, we multiply both sides by the reciprocal:

P₀ = 9261 × ⁸⁰⁰⁰⁄₉₂₆₁

P₀ = 8000

Explanation

Let t be the time in hours spent travelling on foot. Given the total time is 9 hours, the time spent on the bicycle is 9 - t. Using the relation Distance = Speed × Time, we can represent the total distance of 61 km as: 4(t) + 9(9 - t) = 61 4t + 81 - 9t = 61 -5t = 61 - 81 -5t = -20 t = 4 hours

The distance travelled on foot is then determined by multiplying the speed on foot by the time spent on foot: Distance = 4 km/hr × 4 hours = 16 km

Explanation

The distance covered is constant, so we use the inverse relationship between speed and time: s₁ × t₁ = s₂ × t₂. First, convert the initial time into hours: 2 hrs 45 min = 2 + ⁴⁵⁄₆₀ = 2.75 hours.

The distance d is calculated as: d = 4 km/h × 2.75 hours = 11 km

Now, find the time taken at the running speed of 16.5 km/h: t₂ = (11)/(16.5) hours

To convert the time into minutes, multiply by 60: t₂ = (11)/(16.5) × 60 = ¹¹⁰⁄₁₆₅ × 60 = ⅔ × 60 = 40 minutes

Explanation

The algebraic identity for the given expression is x³ + y³ + z³ - 3xyz = ½(x + y + z)[(x - y)² + (y - z)² + (z - x)²]. In this problem, the values are given as x = 332, y = 332, and z = 332. Calculating the differences within the square brackets, we find that (x - y) = 332 - 332 = 0, (y - z) = 332 - 332 = 0, and (z - x) = 332 - 332 = 0. Substituting these into the formula results in ½(332 + 332 + 332)[0² + 0² + 0²]. Since the term multiplied by the sum is zero, the entire expression evaluates to zero. This is a standard result in algebra where the expression x³ + y³ + z³ - 3xyz vanishes whenever x = y = z.

Explanation

To find the value of x, first rationalize the denominator on the right side of the equation:

(1)/(2 + √(3)) = (1)/(2 + √(3)) × 2 - √3{2 - √3}

Using the algebraic identity (a + b)(a - b) = a² - b² in the denominator:

2 - √3{2² - (√3)²} = 2 - √3{4 - 3} = 2 - √3

Now, substitute this result back into the original equation:

2 + x√3 = 2 - √3

By subtracting 2 from both sides of the equation, we get:

x√3 = -√3

Dividing both sides by √3 yields:

x = -1

Explanation

Let the required angle be x. The complement of an angle x is defined as (90° - x), and its supplement is defined as (180° - x). Based on the given condition that the supplement is three times as large as the complement, the relationship is expressed by the equation:

180° - x = 3(90° - x)

Solving for x: 180° - x = 270° - 3x 3x - x = 270° - 180° 2x = 90° x = 45°

For an angle of 45°, the complement is 45° and the supplement is 135°. Since 135° is three times 45°, the value of the angle is confirmed.

Explanation

The expression m³ - 3m² + 3m + 3n + 3n² + n³ can be simplified by recognizing the algebraic identities for the cube of a binomial. The expansion of (a + b)³ is a³ + 3a²b + 3ab² + b³ and (a - b)³ is a³ - 3a²b + 3ab² - b³.

By grouping the terms related to m and n, we can rewrite the expression as: (m³ - 3m² + 3m) + (n³ + 3n² + 3n)

To complete the cubes, we add and subtract 1 for each group: (m³ - 3m² + 3m - 1 + 1) + (n³ + 3n² + 3n + 1 - 1) = (m - 1)³ + 1 + (n + 1)³ - 1 = (m - 1)³ + (n + 1)³

Substituting the given values m = -4 and n = -2 into this simplified form: = (-4 - 1)³ + (-2 + 1)³ = (-5)³ + (-1)³ = -125 - 1 = -126

Alternatively, direct substitution yields: (-4)³ - 3(-4)² + 3(-4) + 3(-2) + 3(-2)² + (-2)³ = -64 - 3(16) - 12 - 6 + 3(4) - 8 = -64 - 48 - 12 - 6 + 12 - 8 = -126

Explanation

For a system of linear equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 to have no solution, the lines must be parallel and distinct. This occurs when the coefficients satisfy the condition (a₁)/(a₂) = (b₁)/(b₂) ≠ (c₁)/(c₂).

In the given equations 2x - ky + 7 = 0 and 6x - 12y + 15 = 0, the coefficients are a₁ = 2, b₁ = -k, c₁ = 7, a₂ = 6, b₂ = -12, and c₂ = 15. Substituting these values into the ratio of the coefficients of x and y, we get:

²⁄₆ = ⁻ᵏ⁄₋₁₂

Simplifying the fractions results in:

⅓ = ᵏ⁄₁₂

Solving for k by cross-multiplication:

3k = 12 k = 4

The third ratio (c₁)/(c₂) = ⁷⁄₁₅ is not equal to ⅓, satisfying the condition for no solution.

Explanation

To compare the expressions, square both sides: (√6 + √2)² = 8 + 2√(12) and (√5 + √3)² = 8 + 2√(15). Since √(12) < √(15), the relation √6 + √2 < √5 + √3 is the only correct one (ii), meaning relations (i) and (iii) are incorrect. Therefore, the incorrect relations are (i) and (iii), which corresponds to option D.

Explanation

Given the equation x cos Θ - sin Θ = 1, we can rearrange it as x cos Θ = 1 + sin Θ. Squaring both sides of the equation gives x² cos² Θ = (1 + sin Θ)². Using the fundamental identity cos² Θ = 1 - sin² Θ, we substitute to get x² (1 - sin² Θ) = (1 + sin Θ)².

Factoring the left side using the difference of squares identity results in x² (1 - sin Θ)(1 + sin Θ) = (1 + sin Θ)². Dividing both sides by the common factor (1 + sin Θ) yields x² (1 - sin Θ) = 1 + sin Θ.

Expanding and rearranging the terms allows us to group sin Θ: x² - x² sin Θ = 1 + sin Θ x² - 1 = x² sin Θ + sin Θ x² - 1 = (1 + x²) sin Θ

We are asked to evaluate the expression x² - (1 + x²) sin Θ. By substituting the value of (1 + x²) sin Θ derived above, the expression becomes: x² - (x² - 1) = x² - x² + 1 = 1

Explanation

In the right-angled triangle formed by the ladder, the wall, and the ground, the ladder acts as the hypotenuse (h = 10 m). The distance from the foot of the ladder to the wall is the adjacent side (base) relative to the angle of inclination with the ground (θ = 30°). Using the trigonometric relation for cosine:

cos(θ) = (Base)/(Hypotenuse)

Substituting the known values:

cos(30°) = (Base)/(10)

Since cos(30°) = √3{2}, the equation becomes:

√3{2} = (Base)/(10)

Base = 10 × √3{2} = 5√3

Using the provided value √3 = 1.732:

Base = 5 × 1.732 = 8.66 m

Explanation

From the given equation sin Θ + sin²Θ = 1, we can rearrange it to find sin Θ = 1 - sin²Θ. Using the fundamental trigonometric identity sin²Θ + cos²Θ = 1, we know that 1 - sin²Θ = cos²Θ. Therefore, we establish the relation sin Θ = cos²Θ.

By squaring both sides of this relation, we obtain sin²Θ = cos⁴Θ. The expression to be evaluated is cos²Θ + cos⁴Θ. Substituting the derived equivalents into this expression, we get sin Θ + sin²Θ. Given that sin Θ + sin²Θ = 1, the value of cos²Θ + cos⁴Θ is also 1.

Explanation

The hour hand of a clock completes a full rotation of 360° in 12 hours. This means the angular speed of the hour hand is 360° / 12 = 30° per hour, which further simplifies to 30° / 60 = 0.5° per minute. From noon (12:00 PM) to 3:45 PM, the total time elapsed is 3 hours and 45 minutes. To find the total angle turned, we convert the entire duration into minutes: (3 × 60) + 45 = 225 minutes. Multiplying the total minutes by the hourly hand's rate per minute, we get 225 × 0.5° = 112.5°. Expressed as a mixed fraction, this value is 112 ½°.

Explanation

First, convert the sum of the angles from radians to degrees using the conversion factor 1 radian = 180 / π degrees. Substituting the given value π = 22/7, the sum S is calculated as: S = (22 / 9) × (180 / (22 / 7)) = (22 / 9) × (180 × 7 / 22) = (180 × 7) / 9 = 20 × 7 = 140°.

Let the two angles be x and y, where x > y. According to the problem: x + y = 140° x - y = 36°

To find the value of the smaller angle y, subtract the second equation from the first: (x + y) - (x - y) = 140 - 36 2y = 104 y = 104 / 2 = 52°

Explanation

The given expression p(p² + 3p + 3) can be expanded as p³ + 3p² + 3p. To simplify this, we use the algebraic identity for the cube of a binomial: (p + 1)³ = p³ + 3p² + 3p + 1. By subtracting 1 from both sides, the expression becomes p³ + 3p² + 3p = (p + 1)³ - 1. Substituting the value p = 99 into this formula, we get (99 + 1)³ - 1, which simplifies to 100³ - 1. Calculating the cube of 100 gives 1,000,000, and subtracting 1 results in 999,999.

Explanation

To find the least number that leaves a common remainder r when divided by a set of numbers, we use the formula N = LCM(a, b, c, …) + r. In this problem, the divisors are 6, 9, 12, 15, and 18, and the required remainder is 2.

First, determine the Least Common Multiple (LCM) of the divisors through prime factorization: 6 = 2 × 3 9 = 3² 12 = 2² × 3 15 = 3 × 5 18 = 2 × 3²

The LCM is the product of the highest powers of all prime factors involved: LCM(6, 9, 12, 15, 18) = 2² × 3² × 5 = 4 × 9 × 5 = 180

Since the division must leave a remainder of 2 in each case, we add this remainder to the LCM: Number = 180 + 2 = 182

Explanation

A straight line represented by the general equation ax + by + c = 0 passes through the origin (0, 0) if and only if the constant term c is equal to zero. This is because for the point (0, 0) to satisfy the equation, substituting x = 0 and y = 0 must yield a(0) + b(0) + c = 0, which simplifies to c = 0.

In the equation 2x - 3y = 0, the constant term is zero. Substituting the coordinates of the origin: 2(0) - 3(0) = 0 - 0 = 0 Since the equation holds true, the line passes through the origin.

Explanation

The sum of the internal angles of a triangle is always 180°. According to the angle sum property, we set up the equation (x + 15) + (6x/5 + 6) + (2x/3 + 30) = 180. Combining the constant terms gives x + 6x/5 + 2x/3 + 51 = 180. Subtracting 51 from both sides results in x + 6x/5 + 2x/3 = 129. To solve for x, we find a common denominator of 15 for the fractions, which gives (15x + 18x + 10x)/15 = 129. This simplifies to 43x/15 = 129. Multiplying by 15 and dividing by 43 yields x = (129 × 15) / 43, which results in x = 3 × 15 = 45.

Substituting x = 45 back into the expressions for the three angles, we get the first angle as 45 + 15 = 60°. The second angle is 6(45)/5 + 6 = 54 + 6 = 60°. The third angle is 2(45)/3 + 30 = 30 + 30 = 60°. Because all three interior angles are equal to 60°, the triangle is equilateral.

Explanation

In the right-angled triangle formed by the kite, the string, and the ground, the height of the kite is the perpendicular (p = 75 m) and the string represents the hypotenuse (l). We are given cot θ = 8 / 15, and since cot θ = Base / Perpendicular, the ratio of the base (b) to the perpendicular (p) is 8:15.

According to the Pythagorean theorem, the hypotenuse (l) of a triangle with sides in the ratio 8 and 15 is √(8² + 15²) = √(64 + 225) = √(289) = 17. This identifies the Pythagorean triplet (8, 15, 17). Since the actual perpendicular height is 75 m, which is 5 × 15, the constant of proportionality for the triangle is 5. Therefore, the length of the string (hypotenuse) is 17 × 5 = 85 m.

Explanation

The passing marks required for the examination are the sum of the marks obtained by the student and the marks by which they failed. Since the student scored 190 marks and failed by 35 marks, the passing score is 190 + 35 = 225.

According to the question, the passing marks are 36% of the total marks. Let the total marks be T. The relationship is expressed as:

36% of T = 225

³⁶⁄₁₀₀ × T = 225

T = (225 × 100)/(36)

Simplifying the calculation by dividing both 225 and 36 by their greatest common divisor, 9:

T = (25 × 100)/(4)

T = 25 × 25 = 625

Explanation

By the Midpoint Theorem, the segments joining the midpoints of the sides of a triangle divide the triangle into four congruent triangles of equal area. Let D, E, F be the midpoints of BC, CA, AB respectively. These midpoints form four smaller triangles: Δ AFE, Δ FBD, Δ EDC, and Δ DEF. Let the area of each of these triangles be k.

The parallelogram DEFB is composed of two of these triangles, Δ FBD and Δ DEF. Therefore, the area of parallelogram DEFB = Area(Δ FBD) + Area(Δ DEF) = k + k = 2k.

The trapezium CAFD is composed of three of these triangles, Δ AFE, Δ DEF, and Δ EDC. Therefore, the area of trapezium CAFD = Area(Δ AFE) + Area(Δ DEF) + Area(Δ EDC) = k + k + k = 3k.

The ratio of the area of the parallelogram DEFB to the area of the trapezium CAFD is: (Area(DEFB))/(Area(CAFD)) = ²ᵏ⁄₃ₖ = ⅔

Explanation

Let M and W represent the efficiency of a man and a woman respectively. Since the work done in both cases is the same and completed in the same number of days (8), we equate the total effort:

(4M + 6W) × 8 = (2M + 9W) × 8

By cancelling 8 from both sides, we get: 4M + 6W = 2M + 9W 4M - 2M = 9W - 6W 2M = 3W M = ³⁄₂W

To find the total work in terms of woman-days, substitute M = ³⁄₂W into the first condition: Total Work = (4 × ³⁄₂W + 6W) × 8 Total Work = (6W + 6W) × 8 Total Work = 12W × 8 = 96W

The number of days required for 18 women to complete this work is: Days = (Total Work)/(Efficiency of 18 women) Days = (96W)/(18W) = ¹⁶⁄₃ Days = 5 ⅓ days