SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The expression involves trigonometric terms that are simplified using the fundamental identity relating cosecant and cotangent: csc² θ - cot² θ = 1. This identity can be factored using the difference of squares as (csc θ - cot θ)(csc θ + cot θ) = 1.

By rearranging this factored form, the reciprocal of the difference is equivalent to the sum: (1)/(csc θ - cot θ) = csc θ + cot θ

Substituting this into the first part of the given expression: (csc θ + cot θ) - (1)/(sin θ)

The second term, (1)/(sin θ), is the definition of the cosecant function, csc θ. Replacing it in the expression gives: csc θ + cot θ - csc θ

By subtracting the identical csc θ terms, the expression simplifies to: cot θ

Explanation

The given equation is a + b + c = 0. By rearranging this equation to isolate the terms a and b on one side, we obtain a + b = -c. The expression to be evaluated, a² + b² + 2ab, is the standard expansion of the binomial square (a + b)². Substituting the value of a + b from the rearranged equation into this identity, we get (a + b)² = (-c)². Since the square of any real number is non-negative, (-c)² = c². Therefore, a² + b² + 2ab = c².

Explanation

The simple interest SI is calculated using the formula SI = (P × R × T)/(100), where P is the principal, R is the rate of interest per annum, and T is the time in years. The problem states that T = 8 and SI = ⅗P. Substituting these values into the formula gives the equation ⅗P = (P × R × 8)/(100). By dividing both sides by P, we get ⅗ = (8R)/(100). Solving for R yields R = (3 × 100)/(5 × 8), which simplifies to R = ³⁰⁰⁄₄₀. This results in R = 7.5, which is equivalent to 7½%.

Explanation

Let the ages of Ram, Ravi, Mayadevi, and Soham be R, V, M, and S respectively. From the problem, we have the following relations: M = 3R V = 4S ⟹ S = (V)/(4) |R - V| = 7 |M - S| = 65

Substitute M and S into the difference equation: |3R - (V)/(4)| = 65 |12R - V| = 260

Assuming Ram is older than Ravi, R - V = 7 ⟹ V = R - 7. Substituting this into the absolute value equation: |12R - (R - 7)| = 260 |11R + 7| = 260 11R + 7 = 260 ⟹ 11R = 253 ⟹ R = 23

Mayadevi's age M = 3R = 3 × 23 = 69. To verify, if Ram is 23, then Ravi is 23 - 7 = 16. Soham's age S = 16/4 = 4. The difference between Mayadevi and Soham is 69 - 4 = 65 years, which satisfies the given condition.

Explanation

The angle-angle-angle (AAA) criterion establishes similarity between two triangles but is insufficient to prove congruence. While it ensures that the triangles have the exact same shape, it does not provide any information regarding their actual physical dimensions or scale. For two triangles to be congruent, they must be identical in both shape and size. Since triangles with identical corresponding angles can have proportional sides of different lengths—such as two equilateral triangles where one has sides of 2 units and the other has sides of 5 units—the AAA condition cannot guarantee that the triangles are equal in all respects.

Explanation

Let the number of coins of denominations ₹1, ₹5, and ₹10 be 3x, 5x, and 7x respectively, following the given ratio 3:5:7. The total value of the money in the bag is the sum of the products of each denomination and its corresponding number of coins. This gives the equation 1(3x) + 5(5x) + 10(7x) = 980. Simplifying this expression, we have 3x + 25x + 70x = 980, which combines to 98x = 980. Dividing both sides by 98, we find x = 10. The number of ₹10 coins is represented by 7x, so substituting the value of x gives 7 × 10 = 70.

Explanation

Let the ages of A and B three years ago be 5x and 7x respectively. The time gap between 3 years ago and 13 years from now is 3 + 13 = 16 years.

According to the ratio method, the increase in both parts of the ratio from 5:7 to 9:11 is 9 - 5 = 4 and 11 - 7 = 4. Since the difference is the same, we can equate these 4 units to the total time interval of 16 years. Therefore, 4 units = 16 years, which gives 1 unit = 4 years.

The age of B three years ago was 7 units = 7 × 4 = 28 years. Consequently, the present age of B is 28 + 3 = 31 years. The present age of C is given as 11 years less than that of B. Thus, the present age of C is 31 - 11 = 20 years.

Explanation

Given the equation 3t - ¹⁄₃ₜ = 3, square both sides using the algebraic identity (a - b)² = a² + b² - 2ab.

(3t - ¹⁄₃ₜ)² = 3² 9t² + (1)/(9t²) - 2(3t)(¹⁄₃ₜ) = 9 9t² + (1)/(9t²) - 2 = 9 9t² + (1)/(9t²) = 11

To find the expression that equals 12, add 1 to both sides of the equation: 9t² + (1)/(9t²) + 1 = 11 + 1 = 12

This matches the expression 9t² + (1)/(9t²) + 1.

Explanation

The area of a rectangle is calculated using the standard formula: Area = Length × Breadth

By substituting the given dimensions into this relation: Area = 52m × 9m Area = 468m²

This product represents the total two-dimensional space enclosed within the boundary of the rectangle. The unit of measurement is square meters because it is the product of two linear dimensions measured in meters.

Explanation

Let the average age of all seven members be x years. The total age of all seven members is therefore 7x. According to the problem, the average age of six of the members is 42 years, making their combined age 6 × 42 = 252 years. The age of the seventh member is given as x + 36.

The sum of the ages of the first six members and the seventh member must equal the total age of the team. This gives the linear equation:

252 + (x + 36) = 7x

Combining the constant terms on the left side:

288 + x = 7x

Subtracting x from both sides results in:

6x = 288

Dividing by 6, we find the average age of the entire team:

x = 48

The age of the seventh member is x + 36. Substituting the value of x:

48 + 36 = 84

Explanation

At θ = 45°, the trigonometric values are sin 45° = (1)/(√(2)) and cos 45° = (1)/(√(2)). Substituting these into the given expression:

Numerator: sin² 45° - 2sin⁴ 45° = ((1)/(√(2)))² - 2((1)/(√(2)))⁴ = ½ - 2(¼) = ½ - ½ = 0.

Denominator: 2cos⁴ 45° + cos² 45° = 2((1)/(√(2)))⁴ + ((1)/(√(2)))² = 2(¼) + ½ = ½ + ½ = 1.

The value of the expression is ⁰⁄₁ = 0. Alternatively, using the identity cos 2θ = 1 - 2sin²θ, the numerator can be simplified to sin²θ(1 - 2sin²θ) = sin²θ cos 2θ. At θ = 45°, the term cos 2θ = cos 90° = 0, which makes the entire numerator zero. Since the denominator is non-zero at this angle, the resulting value is 0.

Explanation

The average score of the class is calculated using the weighted average formula A = n₁x₁ + n₂x₂{n₁ + n₂}, where n represents the number of students and x represents their respective average scores. Given that 12 students scored an average of 62 and the remaining 18 students (30 - 12) scored an average of 60, the calculation is as follows:

Total score of 12 students = 12 × 62 = 744 Total score of 18 students = 18 × 60 = 1080 Total score of the class = 744 + 1080 = 1824

Average score of the class = ¹⁸²⁴⁄₃₀ = 60.8

Using the deviation method, if we assume a base average of 60, the first group of 12 students contributes an excess of +2 marks each (62 - 60), totaling an excess of 12 × 2 = 24. Dividing this total excess by the entire class size of 30 gives an increase of ²⁴⁄₃₀ = 0.8 over the base average. Therefore, the class average is 60 + 0.8 = 60.8.

Explanation

The radius of a circle is always perpendicular to the tangent at the point of contact. This creates a right-angled triangle where the distance from the centre to the external point represents the hypotenuse, the radius is one leg, and the tangent length is the other leg.

Using the Pythagorean theorem, the length of the tangent l is given by the formula l = √(d² - r²), where d is the distance from the centre and r is the radius. Substituting the given values:

l = √(10² - 4²) l = √(100 - 16) l = √(84)

The square root of 84 can be simplified by factoring out the largest perfect square: √(84) = √(4 × 21) = 2√(21) cm.

Explanation

The largest 5-digit number is 99,999. To find the largest 5-digit number divisible by 47, we divide 99,999 by 47 to determine the remainder.

Performing the division: 99,999 ÷ 47 = 2127 with a remainder. 47 × 2 = 94; 99 - 94 = 5. Bringing down the next digit gives 59. 47 × 1 = 47; 59 - 47 = 12. Bringing down the next digit gives 129. 47 × 2 = 94; 129 - 94 = 35. Bringing down the final digit gives 359. 47 × 7 = 329; 359 - 329 = 30.

The remainder is 30. To obtain the largest 5-digit number exactly divisible by 47, we subtract this remainder from 99,999: 99,999 - 30 = 99,969.

Explanation

Income is the sum of expenditure and savings, represented by the formula I = E + S. According to the problem, the amount spent (E) is 40% higher than the savings (S), which means E = S + 0.40S = 1.4S. Substituting this into the income formula gives I = 1.4S + S, or I = 2.4S.

Given the total income is ₹72,000, the equation becomes: 72,000 = 2.4S S = 72,000 / 2.4 S = 720,000 / 24 S = 30,000

Thus, the employee's savings amount to ₹30,000.

Explanation

The payment schedule forms an arithmetic progression where the first term a = 100, the common difference d = -5, and the total sum Sₙ = 975. The formula for the sum of an arithmetic progression is Sₙ = ⁿ⁄₂ [2a + (n - 1)d]. Substituting the values into the formula:

975 = ⁿ⁄₂ [2(100) + (n - 1)(-5)] 1950 = n [200 - 5n + 5] 1950 = n [205 - 5n] 1950 = 205n - 5n² 5n² - 205n + 1950 = 0 n² - 41n + 390 = 0

Solving the quadratic equation by factoring: n² - 15n - 26n + 390 = 0 n(n - 15) - 26(n - 15) = 0 (n - 15)(n - 26) = 0

This provides two mathematical solutions, n = 15 and n = 26. In a practical repayment scenario, the process concludes the first time the sum reaches the required amount. Additionally, for n = 26, the installments would eventually become negative values, which is not feasible for a customer paying a debt. Therefore, the time taken to pay the entire amount is 15 months.

Explanation

Let the selling price be S. The gain is given as ⅙ of the selling price, which is (S)/(6). To find the gain percentage, we first determine the cost price (C) using the relationship C = S - Gain: C = S - (S)/(6) = (5S)/(6)

The gain percentage formula is (Gain)/(C) × 100. Substituting the values into this formula: Gain Percentage = (S)/(6){(5S)/(6)} × 100 Gain Percentage = ⅕ × 100 = 20%

The specific numerical value of the selling price (₹31524) is not necessary for the calculation because the percentage is derived from the constant ratio between the gain and the cost price.

Explanation

The perpendicular dropped from the centre of a circle to a chord bisects the chord. Given the chord length is 14 cm, the length of the half-chord is 14 / 2 = 7 cm.

A right-angled triangle is formed by the radius r, the distance from the centre to the chord d = 24 cm, and the half-chord segment. Using the Pythagorean theorem:

r² = 24² + 7² r² = 576 + 49 r² = 625 r = √(625) = 25 cm

This follows the standard Pythagorean triple (7, 24, 25), where the radius represents the hypotenuse of the triangle.

Explanation

To find the value of 3x - 2y, we utilize the algebraic identity (a - b)² = (a + b)² - 4ab. By substituting a = 3x and b = 2y, the identity becomes:

(3x - 2y)² = (3x + 2y)² - 4(3x)(2y) (3x - 2y)² = (3x + 2y)² - 24xy

From the given information, we know that 3x + 2y = 10 and 2xy = 7. We can rewrite the term 24xy as 12(2xy) to substitute the known value. Therefore:

24xy = 12 × 7 = 84

Substituting these values into the identity:

(3x - 2y)² = (10)² - 84 (3x - 2y)² = 100 - 84 (3x - 2y)² = 16

Taking the square root of both sides gives 3x - 2y = ± 4. The problem specifies the condition 3x - 2y > 0, which requires us to take the positive result. Thus, 3x - 2y = 4.

Explanation

The altitude h of an equilateral triangle with side length a is determined by the formula h = √3{2}a. Given that each side a = 37√3 cm, substitute this value into the formula:

h = √3{2} × 37√3

Using the property that √3 × √3 = 3, the equation simplifies to:

h = (37 × 3)/(2)

h = ¹¹¹⁄₂

h = 55.5 cm

Explanation

In a triangle, the sum of all interior angles is always 180°. Given the relation 4∠ P = 5∠ Q = 20∠ R, we can express the angles in terms of their ratios by finding the least common multiple (LCM) of the coefficients 4, 5, and 20, which is 20. Dividing each term of the equation by 20, we obtain (∠ P)/(5) = (∠ Q)/(4) = (∠ R)/(1).

Let these ratios be equal to a common variable x, such that ∠ P = 5x, ∠ Q = 4x, and ∠ R = x. According to the angle sum property, ∠ P + ∠ Q + ∠ R = 180°. Substituting the ratio values into this equation gives 5x + 4x + x = 180°, which simplifies to 10x = 180°, or x = 18°. To find the specific value of ∠ Q, we use its corresponding ratio: ∠ Q = 4x = 4 × 18° = 72°.

Explanation

The given condition is expressed by the equation sin²(t) = ½ tan(t). Substituting tan(t) = (sin(t))/(cos(t)), the equation becomes sin²(t) = (sin(t))/(2cos(t)). For values of t where sin(t) ≠ 0, we can divide both sides by sin(t) to get sin(t) = (1)/(2cos(t)). Multiplying both sides by 2cos(t) results in 2 sin(t) cos(t) = 1. Applying the double-angle trigonometric identity sin(2t) = 2 sin(t) cos(t), the equation simplifies to sin(2t) = 1. The sine function equals 1 when the argument is 90°, so 2t = 90°, which gives t = 45°. Evaluation at t = 45° confirms that sin²(45°) = ((1)/(√(2)))² = ½ and ½ tan(45°) = ½(1) = ½.

Explanation

Given tan A = 1 / √(10), we represent this as the ratio of the perpendicular P to the base B in a right-angled triangle, where P = 1 and B = √(10).

The hypotenuse H is calculated using the Pythagorean theorem: H = √(P²⁾⁺^B^^^² = √(1²⁾⁺⁽^√⁽¹⁰⁾⁾^^^² = √(1 + 10) = √(11)

The trigonometric ratios for the required expression sin A + csc A are: sin A = P / H = 1 / √(11) csc A = H / P = √(11) / 1 = √(11)

Substituting these values into the expression: sin A + csc A = (1)/(√(11)) + √(11) = 1 + (√(11) · √(11)){√(11)} = (1 + 11)/(√(11)) = (12)/(√(11))

Explanation

The volume of a right circular cone is given by the formula V = ⅓ π r² h. Substituting the given volume V = 1232 cm³ and height h = 24 cm: 1232 = ⅓ × ²²⁄₇ × r² × 24 1232 = ¹⁷⁶⁄₇ r² r² = (1232 × 7)/(176) r² = 7 × 7 = 49 r = 7 cm

The slant height l of the cone is calculated using the relation l = √(r² + h²): l = √(7² + 24²) l = √(49 + 576) = √(625) l = 25 cm

The total surface area of a cone is calculated using the formula π r(r + l): TSA = ²²⁄₇ × 7 × (7 + 25) TSA = 22 × 32 TSA = 704 cm²

Explanation

To simplify the expression 0.07 × 0.28 ÷ 0.04 + 0.64 - 1.64 ÷ 0.04, we apply the BODMAS rule, which prioritizes division and multiplication before addition and subtraction.

First, calculate the division components within the expression: 0.28 ÷ 0.04 = 7 1.64 ÷ 0.04 = 41

Substitute these results back into the expression: 0.07 × 7 + 0.64 - 41

Next, perform the multiplication: 0.07 × 7 = 0.49

The expression now simplifies to: 0.49 + 0.64 - 41

Now, perform the addition and subtraction from left to right: 0.49 + 0.64 = 1.13 1.13 - 41 = -39.87

Explanation

The combined filling rate of pipes A and B is ¹⁄₂₀ + ¹⁄₃₀ = ⁵⁄₆₀ = ¹⁄₁₂ tank per hour. To fill the first ⅓ of the tank, the time taken is (1/3)/(1/12) = 4 hours. Once the tank is one-third full, a leak develops that removes ¼ of the water being supplied, meaning only ¾ of the combined rate is effective in filling the tank. The effective filling rate becomes ¾ × ¹⁄₁₂ = ¹⁄₁₆ tank per hour. The remaining volume to be filled is 1 - ⅓ = ⅔ of the tank. The time required to fill this remainder is (2/3)/(1/16) = ³²⁄₃ = 10 ⅔ hours. The total time taken to fill the tank is the sum of the two stages, which is 4 + 10 ⅔ = 14 ⅔ hours.

Explanation

The speed of the boat in still water u is the arithmetic mean of the downstream and upstream speeds. First, calculate the downstream speed using v_d = (Distance)/(Time_dₒ_wₙₛₜᵣₑₐₘ₎ = ⁵⁵⁄₅ = 11 km/h. Then, calculate the upstream speed using vᵤ = (Distance)/(Timeᵤₚₛₜᵣₑₐₘ₎ = ⁵⁵⁄₁₁ = 5 km/h. The speed of the boat in still water is given by the formula:

u = v_d + vᵤ{2}

Substituting the values into the formula:

u = (11 + 5)/(2) = ¹⁶⁄₂ = 8 km/h

Explanation

To find the single discount equivalent to a series of discounts, we calculate the final multiplier by subtracting each discount rate from 1 and multiplying them together. Let the initial price be 100.

The first discount of 60% leaves 100% - 60% = 40% of the value. The second discount of 70% leaves 100% - 70% = 30% of the remaining value. The third discount of 80% leaves 100% - 80% = 20% of the remaining value.

The final Selling Price (SP) as a fraction of the Marked Price (MP) is: SP = MP × (1 - 0.60) × (1 - 0.70) × (1 - 0.80) SP = 100 × 0.40 × 0.30 × 0.20 SP = 100 × 0.024 = 2.4

The single equivalent discount is the total reduction from the original price: Equivalent Discount = MP - SP Equivalent Discount = 100 - 2.4 = 97.6%

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3. For the number 4,29,714, the sum of the digits is 4 + 2 + 9 + 7 + 1 + 4 = 27. Since 27 is a multiple of 3, the number 4,29,714 is divisible by 3. A number is divisible by 6 if it is divisible by both 2 and 3. The number 4,29,714 is an even number because it ends in the digit 4, which means it is divisible by 2. Since it is divisible by both 2 and 3, it is also divisible by 6.

Explanation

In the first quadrant where 0°< t< 90°, sin(t) is a strictly increasing function and cos(t) is a strictly decreasing function. At the midpoint t = 45°, both functions are equal as sin(45°) = cos(45°) = (1)/(√(2)). For any angle t such that 0° < t < 45°, the value of sin(t) increases from 0 toward (1)/(√(2)), while the value of cos(t) decreases from 1 toward (1)/(√(2)). Consequently, in this interval, sin(t) remains strictly less than cos(t).

Explanation

Ramesh made two separate purchases which are settled on 19 November 2019. We calculate the discount for each transaction based on the terms provided.

For the first purchase of ₹9,800 made on 10 November 2019, the payment on 19 November occurs 9 days later. Since this is within the 10-day window, he receives a 2% discount. Amount 1 = 9800 × (1 - 0.02) = 9800 × 0.98 = 9604

For the second purchase of ₹5,480 made on 19 November 2019, the payment is made in cash at the time of purchase. This qualifies for the 5% cash discount. Amount 2 = 5480 × (1 - 0.05) = 5480 × 0.95 = 5206

The total amount paid by Ramesh on 19 November is the sum of these two settled bills: Total Payment = 9604 + 5206 = 14810

Explanation

Let the Selling Price (SP) be 6 units. Given that the gain is ⅙th of the selling price, the gain is 1 unit. The Cost Price (CP) is found by subtracting the gain from the selling price: CP = SP - Gain = 6 - 1 = 5 units

The gain percentage is calculated using the formula: Gain Percentage = ( (Gain)/(CP) ) × 100

Substituting the unit values: Gain Percentage = ⅕ × 100 = 20%

The specific value of the selling price (₹42564) is not needed for this calculation as the gain is expressed as a direct fraction of the selling price, allowing the percentage to be derived from the ratio of gain to cost price.

Explanation

Let the efficiency of the slower pipe be x units per minute. Since the faster pipe fills the tank 6 times faster, its efficiency is 6x units per minute. The combined efficiency of both pipes working together is x + 6x = 7x units per minute.

The total capacity of the tank is determined by multiplying the combined efficiency by the total time taken to fill it: Capacity = 7x × 40 = 280x units

The time taken by the slower pipe alone to fill the tank is calculated by dividing the total capacity by its individual efficiency: Time = ²⁸⁰ˣ⁄ₓ = 280 minutes

Explanation

A number is divisible by 3 if the sum of its digits is a multiple of 3. For the number 6745k2, the sum of the digits is 6 + 7 + 4 + 5 + k + 2 = 24 + k. For 24 + k to be divisible by 3, k must be such that the resulting sum belongs to the set {24, 27, 30, 33, ...}. Since 24 is already divisible by 3, the possible values for the single-digit k are 0, 3, 6, and 9. Among these possible values, the greatest value is 9. Substituting k = 9 gives a digit sum of 33, which is divisible by 3.

Explanation

Let Amit's current age be x. The statement that his father is aged five times more than Amit implies the father's age is x + 5x = 6x.

After 6 years, Amit's age will be x + 6 and his father's age will be 6x + 6. The problem states that at this point, the father is three and a half times as old as Amit: 6x + 6 = 3.5(x + 6) 6x + 6 = ⁷⁄₂(x + 6) 12x + 12 = 7x + 42 5x = 30 x = 6

Amit's current age is 6 years, and his father's current age is 6 × 6 = 36 years.

The question asks for the ratio after a further 9 years, which means 6 + 9 = 15 years from the present. At that time: Amit's age = 6 + 15 = 21 years Father's age = 36 + 15 = 51 years

The number of times the father's age will be relative to Amit's age is calculated as: ⁵¹⁄₂₁ = ¹⁷⁄₇ = 2³⁄₇ times

Explanation

The selling price of the mixture is ₹75 per kg with a profit of 20%. First, the cost price of the mixture (CPₘ) must be determined using the formula CP = (SP × 100)/(100 + Gain %).

CPₘ = (75 × 100)/(100 + 20) = ⁷⁵⁰⁰⁄₁₂₀ = 62.5

Using the rule of alligation, the ratio of the two quantities is calculated by comparing the individual cost prices with the mean cost price of the mixture: Ratio of Sugar 1 (CP₁ = 78) to Sugar 2 (CP₂ = 46) = (CPₘ - CP₂) : (CP₁ - CPₘ)

Substituting the values: Ratio = (62.5 - 46) : (78 - 62.5) Ratio = 16.5 : 15.5

To simplify, multiply both sides by 10 to remove the decimal: Ratio = 165 : 155

Dividing both terms by their highest common factor, 5: 165 / 5 = 33 155 / 5 = 31

The required ratio is 33:31.

Explanation

The distance covered by two objects starting at the same time and moving until they meet is directly proportional to their speeds. Let the distance covered by the policeman be dₚ and the distance covered by the thief be dₜ. The ratio of their speeds is vₚ : vₜ = 23 : 21.

Since time t is constant until the moment of overtaking, the ratio of distances covered is the same as the ratio of speeds: dₚ / dₜ = 23 / 21. The difference in the distances covered is the initial gap between them, which is 100 m. Therefore, dₚ - dₜ = 100 m.

Representing the distances in terms of a constant x, we have 23x - 21x = 100, which simplifies to 2x = 100, or x = 50. The distance the thief runs before being overtaken is dₜ = 21x = 21 × 50 = 1050 m.

Explanation

The expression (3x + 17 - 2y)(3x - 17 - 2y) can be rearranged as ((3x - 2y) + 17)((3x - 2y) - 17). This structure follows the difference of squares identity (a + b)(a - b) = a² - b², where a = 3x - 2y and b = 17.

Applying the identity: ((3x - 2y) + 17)((3x - 2y) - 17) = (3x - 2y)² - 17²

The term (3x - 2y)² is expanded using the square of a binomial formula (p - q)² = p² - 2pq + q²: (3x - 2y)² = (3x)² - 2(3x)(2y) + (2y)² = 9x² - 12xy + 4y²

The constant term is calculated as: 17² = 289

Combining these parts yields the simplified expression: 9x² - 12xy + 4y² - 289

Explanation

The expression is evaluated using the VBODMAS rule, which dictates the order of operations: brackets, of, division, multiplication, addition, and subtraction. Based on the provided expression 11 - {(3 ÷ 4) of 8 - 4} - 8:

First, evaluate the innermost parentheses: (3 ÷ 4) = ¾

Next, apply the of operator, which signifies multiplication: ¾ of 8 = ¾ × 8 = 6

Then, evaluate the contents within the curly braces: {6 - 4} = 2

Finally, perform the subtractions from left to right: 11 - 2 - 8 = 9 - 8 = 1

Explanation

The investment ratios are given by the relationship 2A = 4B = 6C. To find the ratio of their individual shares A:B:C, we divide each term by the least common multiple of 2, 4, and 6, which is 12:

(2A)/(12) = (4B)/(12) = (6C)/(12) ⟹ (A)/(6) = (B)/(3) = (C)/(2)

Thus, the ratio A:B:C is 6:3:2. The total parts in the ratio are 6 + 3 + 2 = 11. Given that the total profit is 44,000, we find the value of one part by dividing the total profit by the total parts:

One part = (44,000)/(11) = 4,000

Since A's share consists of 6 parts, A's profit share is 6 × 4,000 = 24,000$.

Explanation

To find the total gain or loss percentage, we first determine the cost price (CP) for each keyboard using the formula CP = (SP)/(1 ± r), where SP is the selling price and r is the profit or loss rate expressed as a decimal.

For the first keyboard sold at a 25% profit: CP₁ = (1260)/(1 + 0.25) = (1260)/(1.25) = 1008

For the second keyboard sold at a 10% loss: CP₂ = (1440)/(1 - 0.10) = (1440)/(0.9) = 1600

Now, calculate the total cost price and total selling price: Total CP = 1008 + 1600 = 2608 Total SP = 1260 + 1440 = 2700

Since the total selling price is greater than the total cost price, there is a total gain: Total Gain = 2700 - 2608 = 92

The total gain percentage is calculated as: Gain% = (Total Gain)/(Total CP) × 100 = ⁹²⁄₂₆₀₈ × 100 Gain% = ⁹²⁰⁰⁄₂₆₀₈ = ¹¹⁵⁰⁄₃₂₆ = ⁵⁷⁵⁄₁₆₃ = 3 ⁸⁶⁄₁₆₃ %

The resulting fraction 3 ⁸⁶⁄₁₆₃ % matches the gain and the numerical values provided in the selected option.

Explanation

The selling price (SP) after successive discounts can be calculated using the formula SP = MP × (1 - (d₁)/(100)) × (1 - (d₂)/(100)), where MP is the marked price and d₁, d₂ are the discount percentages.

Given MP = 15000, d₁ = 20, and d₂ = 12, we substitute these values into the formula: SP = 15000 × (1 - ²⁰⁄₁₀₀) × (1 - ¹²⁄₁₀₀) SP = 15000 × ⁸⁰⁄₁₀₀ × ⁸⁸⁄₁₀₀ SP = 15000 × 0.8 × 0.88

First, calculating the price after the first discount: 15000 × 0.8 = 12000

Now, applying the second discount of 12% to the reduced price: 12000 × 0.88 = 10560

Explanation

The distance d of a chord from the center of a circle can be found using the Pythagorean theorem, as the radius R, the distance d, and half the chord length L/2 form a right-angled triangle. According to the property of circles, a perpendicular drawn from the center to a chord bisects the chord.

In this triangle, the radius R is the hypotenuse, d is the perpendicular distance, and L/2 is the base. The relationship is given by: d = √(R²⁾⁻⁽^L^/²⁾^^^²

Substituting the given values R = 12.5 cm and L = 11 cm: d = √(12.5²⁾⁻⁽¹¹^/²⁾^^^² d = √(156.25 - 5.5²⁾ d = √(156.25 - 30.25) d = √(126)

To find the square root of 126, we note that 11² = 121 and 11.3² = 127.69. Calculating 11.2² = 125.44. Since 126 is closer to 125.44 than to 127.69, the value correct to one decimal place is 11.2 cm.

Explanation

The volume V of a right circular cylinder is given by the formula V = π r² h, where r is the radius and h is the height. Given that the volume V is 600 cm³ and the radius r is 30 cm, we can substitute these values into the formula to find the height:

600 = π × (30)² × h 600 = 900 π h

To solve for h, we divide both sides by 900 π:

h = (600)/(900 π) h = (2)/(3 π)

Using the value of π ≈ 3.14159:

h ≈ (2)/(3 × 3.14159) h ≈ (2)/(9.42477) h ≈ 0.2122 cm

Rounding to two decimal places, the height of the box is 0.21 cm.

Explanation

In congruent triangles △ ABC ≅ △ PQR, corresponding angles are equal such that ∠ A = ∠ P, ∠ B = ∠ Q, and ∠ C = ∠ R. The sum of the internal angles of any triangle is 180°. For △ ABC, this is expressed as ∠ A + ∠ B + ∠ C = 180°. Substituting the given value ∠ B = 35° into the equation, we get ∠ A + 35° + ∠ C = 180°. Rearranging for the sum of the remaining angles, ∠ A + ∠ C = 180° - 35° = 145°. Because ∠ A = ∠ P by the CPCT (Corresponding Parts of Congruent Triangles) theorem, we can replace ∠ A with ∠ P in the sum, resulting in ∠ P + ∠ C = 145°.

Explanation

In the circle with center A, since PQ is the diameter, the points P, A, and Q are collinear. In △ MAQ, the sides AM and AQ are radii of the same circle, making them equal (AM = AQ). Therefore, △ MAQ is an isosceles triangle where the base angles opposite to the equal sides are also equal, implying m∠ AMQ = m∠ MQA.

According to the exterior angle theorem, the exterior angle ∠ MAP of △ MAQ is equal to the sum of the two interior opposite angles. Thus, m∠ MAP = m∠ AMQ + m∠ MQA. Substituting the known values and the equality of the base angles, we get 110° = 2 × m∠ MQA. Solving for the required angle gives m∠ MQA = 110° / 2 = 55°.

Explanation

The trigonometric ratio provided is sec θ = √2. Since cos θ = (1)/(sec θ), we have cos θ = (1)/(√(2)). This value corresponds to an angle of θ = 45° (or (π)/(4) radians) in the first quadrant. To find the value of csc θ, we use the relationship csc θ = (1)/(sin θ). At θ = 45°, the sine value is sin 45° = (1)/(√(2)). Therefore, csc 45° = (1)/(1/√(2)) = √2.

Alternatively, using the fundamental trigonometric identities, we know that 1 + tan² θ = sec² θ. Substituting the given value, we get 1 + tan² θ = (√2)² = 2, which implies tan² θ = 1. Since cot θ is the reciprocal of tan θ, then cot² θ = 1. Applying the identity 1 + cot² θ = csc² θ, we obtain csc² θ = 1 + 1 = 2. Taking the square root, we find csc θ = √2.

Explanation

The simple interest earned is calculated by subtracting the principal from the total amount: SI = A - P = 24,800 - 10,800 = 14,000. Using the simple interest formula SI = (P × R × T)/(100), where P is the principal, R is the rate of interest, and T is the time in years, we substitute the given values:

14,000 = (10,800 × 15.5 × T)/(100)

14,000 = 108 × 15.5 × T

14,000 = 1,674 × T

T = (14,000)/(1,674)

T ≈ 8.3632...

Rounding to two decimal places, the time required is 8.36 years.

Explanation

Let the total number of votes be V. According to the problem, 80% of the voters cast their votes, and 4% of those cast votes were declared invalid. This implies that 96% of the cast votes were valid. The candidate received 80% of these valid votes, which equals 15,360.

We can represent this relationship with the following equation: V × ⁸⁰⁄₁₀₀ × ⁹⁶⁄₁₀₀ × ⁸⁰⁄₁₀₀ = 15,360

Simplifying the fractions: V × ⅘ × ²⁴⁄₂₅ × ⅘ = 15,360 V × ³⁸⁴⁄₆₂₅ = 15,360

To find V, we isolate it on one side: V = (15,360 × 625)/(384)

Dividing 15,360 by 384: 15,360 ÷ 384 = 40

Now, multiply by 625: V = 40 × 625 V = 25,000

Explanation

The given expression 2sin²θ +3sin θ -2 = 0 is a quadratic equation in terms of sin θ. Let x = sin θ. The equation becomes 2x² + 3x - 2 = 0. Using the splitting the middle term method to factorize the quadratic: 2x² + 4x - x - 2 = 0 2x(x + 2) - 1(x + 2) = 0 (2x - 1)(x + 2) = 0

This yields two possible values for x: x = 1/2 or x = -2. Since the range of the sine function is restricted to [-1, 1], the value sin θ = -2 is mathematically impossible. Therefore, we have sin θ = 1/2. Within the given constraint 0 < θ < 90°, the only value that satisfies this condition is θ = 30° because sin 30° = 1/2.