SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

When a solid sphere is cut into two equal halves, two identical hemispheres are formed. The increase in the total surface area is equal to the sum of the areas of the two newly created circular cross-sections (the bases of the hemispheres). Given the diameter d = 7 cm, the radius r is 7/2 = 3.5 cm.

The area of one circular face is π r². Since two such faces are created by the cut, the total increase in surface area is calculated as: Δ A = 2 × π r² Δ A = 2 × ²²⁄₇ × (⁷⁄₂) × (⁷⁄₂) Δ A = (22 × 7)/(2) Δ A = 11 × 7 = 77 cm²

Explanation

The relationship between the marked price and the selling price after a discount is given by the formula SP = MP × (1 - D/100), where SP is the selling price, MP is the marked price, and D is the discount percentage. Substituting the given values, we have 1848 = MP × (1 - 23/100), which simplifies to 1848 = MP × (77/100). To solve for the marked price, the equation is rearranged as MP = (1848 × 100) / 77. Dividing 1848 by 77 yields 24, and multiplying this quotient by 100 results in a marked price of Rs 2400.

Explanation

Given the equation ⅗P = ⁷⁄₂Q = ⁷⁄₅R, we can find the ratio P : Q : R by taking the reciprocal of the coefficients. If aP = bQ = cR, then the ratio P : Q : R is ¹⁄ₐ : (1)/(b) : (1)/(c).

Applying this, we get: P : Q : R = ⁵⁄₃ : ²⁄₇ : ⁵⁄₇

To convert these fractions into whole numbers, multiply each term by the Least Common Multiple (LCM) of the denominators 3 and 7, which is 21: P = ⁵⁄₃ × 21 = 35 Q = ²⁄₇ × 21 = 6 R = ⁵⁄₇ × 21 = 15

Thus, the ratio P : Q : R is 35 : 6 : 15.

Explanation

The average of the first n natural numbers is given by the formula (n + 1)/(2). This result is derived from the sum of the first n natural numbers, which is Sₙ = (n(n + 1))/(2). Since the average is the sum divided by the number of terms n, the expression simplifies to (n + 1)/(2).

For the first 93 natural numbers, we substitute n = 93 into the formula:

Average = (93 + 1)/(2)

Average = ⁹⁴⁄₂

Average = 47

In an arithmetic progression where the difference between consecutive terms is constant, the average is also the middle term. For a sequence of 93 terms, the middle term is the 47th term, which is 47.

Explanation

When two items are sold at the same selling price, one at a gain of x% and the other at a loss of x%, the overall result is always a loss. The net loss percentage is calculated using the formula L = x²{100}. For a gain and loss of 15%, the net loss percentage is 15²{100} = ²²⁵⁄₁₀₀ = 2.25%.

The total selling price (SP) for both items is 400 + 400 = 800 Rs. The relationship between the total cost price (CP) and the total selling price is SP = CP × (1 - (L)/(100)). Substituting the values gives 800 = CP × (1 - 0.0225), which simplifies to 800 = CP × 0.9775. Solving for the total cost price, CP = (800)/(0.9775) ≈ 818.4143 Rs.

The value of the loss in rupees is the difference between the total cost price and the total selling price. Loss = 818.4143 - 800 = 18.4143 Rs. Rounding to two decimal places, the total loss is Rs 18.41.

Explanation

The difference between 30% and 25% of the same number is given as 27. In percentage terms, this difference is calculated as:

30% - 25% = 5%

This means that 5% of the number corresponds to the value 27. To determine the original number (100%), we can use the following relationship:

5% of the number = 27

To find 100%, we multiply both sides by 20, since 5 × 20 = 100:

100% of the number = 27 × 20 100% of the number = 540

Alternatively, using the algebraic form: ⁵⁄₁₀₀ × x = 27 x = (27 × 100)/(5) x = 27 × 20 x = 540

By solving the linear relationship where 5% represents 27, the value of the number is determined to be 540.

Explanation

The distance between trees is divided into equal intervals. To reach the 13th tree from the 1st tree, the car must cover 13 - 1 = 12 intervals. The time taken to cover these 12 intervals is 20 seconds.

The total number of trees is 37, so the total number of intervals from the 1st tree to the last tree is 37 - 1 = 36. The car has already reached the 13th tree, meaning it has 12 intervals behind it and 36 - 12 = 24 intervals remaining to reach the last tree.

Since the speed is constant and the intervals are equal, we use the unitary method: Time for 12 intervals = 20 seconds Time for 1 interval = ²⁰⁄₁₂ seconds Time for the remaining 24 intervals = ²⁰⁄₁₂ × 24 = 20 × 2 = 40 seconds

Explanation

For a sum of money to double itself under simple interest, the interest earned must be equal to the principal amount P. Therefore, SI = P. The time duration provided is 7 years and 8 months, which can be converted into years as follows: T = 7 + ⁸⁄₁₂ = 7 + ⅔ = ²³⁄₃ years.

Using the simple interest formula SI = (P × R × T)/(100), we substitute the known values: P = (P × R × 23)/(3){100} 1 = (23R)/(300) R = ³⁰⁰⁄₂₃

By dividing 300 by 23, we get a quotient of 13 and a remainder of 1. Expressed as a mixed fraction, the rate of interest is 13 ¹⁄₂₃ % per annum.

Explanation

The given expression is (1)/(x⁽ᵖ⁻^q⁾⁾⁺¹ + (1)/(x⁽^q⁻ᵖ⁾⁾⁺¹. Using the negative exponent rule a⁻ⁿ = (1)/(aⁿ), the term x⁽^q⁻ᵖ⁾ can be written as x⁻⁽ᵖ⁻^q⁾, which is equivalent to (1)/(x⁽ᵖ⁻^q⁾⁾. Substituting this into the second fraction:

(1)/(x⁽ᵖ⁻^q⁾⁾⁺¹ + (1)/(1){x⁽ᵖ⁻^q⁾ + 1}

Simplifying the denominator of the second fraction:

(1)/(1 + x⁽ᵖ⁻^q⁾⁾{x⁽ᵖ⁻^q⁾} = x⁽ᵖ⁻^q⁾{1 + x⁽ᵖ⁻^q⁾}

Now, substituting this back into the original expression:

(1)/(x⁽ᵖ⁻^q⁾⁾⁺¹ + x⁽ᵖ⁻^q⁾{x⁽ᵖ⁻^q⁾ + 1}

Since the denominators are identical, the numerators are added directly:

1 + x⁽ᵖ⁻^q⁾{1 + x⁽ᵖ⁻^q⁾} = 1

Alternatively, setting p = q simplifies the expression to (1)/(x⁰+1) + (1)/(x⁰+1) = ¹⁄₁₊₁ + ¹⁄₁₊₁ = ½ + ½ = 1.

Explanation

To find √x, we express x = 8 + 2√(15) in the form (a + b)² = a² + b² + 2ab. Comparing the terms, we have 2ab = 2√(15), which gives ab = √(15). Possible factors of √(15) are √5 and √3. Checking the sum of their squares, (√5)² + (√3)² = 5 + 3 = 8, which matches the constant term. Thus, x = (√5 + √3)², and √x = √5 + √3.

Next, we calculate the reciprocal: (1)/(√(x)) = (1)/(√(5) + 3)

Rationalizing the denominator by multiplying the numerator and denominator by the conjugate (√5 - √3): (1)/(√(5) + 3) × √5 - √3{√5 - √3} = √5 - √3{5 - 3} = √5 - √3{2}

Now, we sum the two components: √x + (1)/(√(x)) = (√5 + √3) + √5 - √3{2} = 2(√5 + √3) + √5 - √3{2} = 2√5 + 2√3 + √5 - √3{2} = 3√5 + √3{2}

Explanation

The expression can be simplified by evaluating each fraction independently. First, consider the term (1+a)/(a¹^/²⁾⁺ᵃ^^^⁻^¹^^^/^². By rewriting the denominator as a¹^/² + (1)/(a¹^/²⁾, it becomes (a+1)/(a¹^/²⁾. Substituting this back into the first term: (1+a)/(a+1){a¹^/²} = (1+a) · a¹^/²{a+1} = a¹^/²

Next, simplify the second term a¹^/² + a⁻¹^/²{1+a}. Using the same denominator simplification for the numerator: (a+1)/(a¹^/²⁾{1+a} = (a+1)/(a¹^/²⁾⁽¹⁺ᵃ⁾ = (1)/(a¹^/²⁾ = a⁻¹^/²

Now, substitute these simplified values back into the original expression: a¹^/² - a⁻¹^/² + a⁻¹^/²

The terms -a⁻¹^/² and +a⁻¹^/² cancel each other out, leaving: a¹^/² = √a

Explanation

Given the ratio (p)/(q) = ˣ⁺³⁄ₓ₋₃, we square both sides to relate the terms to the required expression: (p²)/(q²) = ((x+3)²)/((x-3)²) = (x² + 6x + 9)/(x² - 6x + 9)

To find (p² + q²)/(p² - q²), we apply the rule of Componendo and Dividendo, which states that if (a)/(b) = (c)/(d), then (a+b)/(a-b) = (c+d)/(c-d). Substituting our values into this formula: (p² + q²)/(p² - q²) = ((x² + 6x + 9) + (x² - 6x + 9))/((x² + 6x + 9) - (x² - 6x + 9))

In the numerator, the terms 6x and -6x cancel out, leaving 2x² + 18. In the denominator, the terms x² and 9 cancel out, leaving 12x. The expression becomes: (p² + q²)/(p² - q²) = (2x² + 18)/(12x)

Factoring out a 2 from the numerator and simplifying the fraction gives: (2(x² + 9))/(12x) = (x² + 9)/(6x)

Explanation

Let the side of the square PQRS be a cm. Since M is the mid-point of side PQ, the length of MQ = a/2. Given that NR = 2/3 of QR, the remaining segment QN = QR - NR = a - (2/3)a = a/3.

The triangle MQN is a right-angled triangle at Q because PQRS is a square. The area of Δ MQN is given by:

Area = ½ × MQ × QN 48 = ½ × ᵃ⁄₂ × ᵃ⁄₃ 48 = (a²)/(12) a² = 48 × 12 = 576 a = √(576) = 24 cm

The length of the diagonal PR of a square with side a is given by the formula PR = a√2. Substituting the value of a:

PR = 24√2 cm

Explanation

The expression (1 - sin A cos A)(sin A + cos A) can be simplified using the algebraic identity for the sum of cubes: a³ + b³ = (a + b)(a² - ab + b²). By setting a = sin A and b = cos A, we apply the identity to sin³ A + cos³ A:

sin³ A + cos³ A = (sin A + cos A)(sin² A - sin A cos A + cos² A)

Applying the Pythagorean trigonometric identity, sin² A + cos² A = 1, the second factor in the parentheses simplifies as follows:

sin³ A + cos³ A = (sin A + cos A)(1 - sin A cos A)

The resulting product is exactly the expression provided in the question, demonstrating that it simplifies to sin³ A + cos³ A.

Explanation

The expression is simplified by rationalizing the term within the square root. By multiplying both the numerator and the denominator by (1 - sin A), we obtain:

√((1 - sin A)/(1 + sin A) · (1 - sin A)/(1 - sin A)) = √(((1 - sin A)²)/(1 - sin² A))

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

√(((1 - sin A)²)/(cos² A)) = (1 - sin A)/(cos A)

This expression can be separated into two terms:

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

Applying the definitions of the secant and tangent functions, where sec A = (1)/(cos A) and tan A = (sin A)/(cos A), the final simplified form is:

sec A - tan A

Explanation

The given expression is √((1)/(sin² A) + (1)/(cos² A)). By taking the common denominator inside the square root, it becomes: √((cos² A + sin² A)/(sin² A cos² A)) Applying the fundamental trigonometric identity sin² A + cos² A = 1, the expression simplifies to: √((1)/(sin² A cos² A)) = (1)/(sin A cos A) The expression in option b, tan A + cot A, can be rewritten using basic trigonometric ratios: tan A + cot A = (sin A)/(cos A) + (cos A)/(sin A) = (sin² A + cos² A)/(sin A cos A) = (1)/(sin A cos A) Comparing the two results, the simplified value of the original expression is equivalent to tan A + cot A.

Explanation

To solve the expression (12)/(√(5) + 3) + (18)/(√(5) - 3), we use a common denominator which is the product of the two denominators: (√5 + √3)(√5 - √3). Applying the algebraic identity (a + b)(a - b) = a² - b², the denominator becomes (√5)² - (√3)² = 5 - 3 = 2.

Next, we cross-multiply to find the numerator: Numerator = 12(√5 - √3) + 18(√5 + √3) = 12√5 - 12√3 + 18√5 + 18√3 = 30√5 + 6√3

Dividing the simplified numerator by the common denominator: 30√5 + 6√3{2} = 15√5 + 3√3

By factoring out 3 from this result, we get 3(5√5 + √3).

Explanation

Total work is the Least Common Multiple of the individual times taken by S, T, and U. Total work = LCM(40, 48, 60) = 240 units. Daily work efficiency of S = 240 / 40 = 6 units per day. Daily work efficiency of T = 240 / 48 = 5 units per day. Daily work efficiency of U = 240 / 60 = 4 units per day.

Let the total work be completed in x days. According to the problem, S works for the entire duration of x days, T works for x - 2 days, and U works for x - 5 days. The sum of the work performed by each person equals the total work: 6x + 5(x - 2) + 4(x - 5) = 240 6x + 5x - 10 + 4x - 20 = 240 15x - 30 = 240 15x = 270 x = 18 days.

The total work done by S is calculated as: Work done by S = Efficiency of S × Total days S worked = 6 × 18 = 108 units.

The share of money is distributed based on the total units of work completed by each individual. Share of S = (Work done by S / Total work) × Total money Share of S = (108 / 240) × 10800 Share of S = 108 × 45 = 4860 rupees.

Explanation

The area A of a trapezium is given by the formula A = ½ × (a + b) × h, where a and b are the lengths of the parallel sides and h is the perpendicular distance between them. Substituting the given dimensions:

a = 2.56 yards b = 3.44 yards h = 1.44 yards

A = ½ × (2.56 + 3.44) × 1.44 A = ½ × 6.00 × 1.44 A = 3 × 1.44 = 4.32 square yards

The total cost of ploughing the land is the product of the area and the cost per square yard. Given the rate is Rs 1800 per square yard:

Total Cost = 4.32 × 1800 Total Cost = 432 × 18 Total Cost = 7776

By breaking down the multiplication, 432 × 18 = 432 × (20 - 2) = 8640 - 864 = 7776.

Explanation

Let the three consecutive natural numbers be n-1, n, and n+1. According to the algebraic identity (a+b+c)² = a² + b² + c² + 2(ab + bc + ca), the difference between the square of the sum and the sum of the squares is 2(ab + bc + ca).

Given this difference is 292, we have: 2((n-1)n + n(n+1) + (n-1)(n+1)) = 292 (n-1)n + n(n+1) + (n-1)(n+1) = 146 n² - n + n² + n + n² - 1 = 146 3n² - 1 = 146 3n² = 147 n² = 49 n = 7

The three consecutive natural numbers are 6, 7, and 8. Therefore, the largest number is 8.

Explanation

A takes 6 × 8 = 48 hours to complete the work, while B takes 4 × 10 = 40 hours. Let the total work be the least common multiple of 48 and 40, which is 240 units.

The efficiency of A is 240 / 48 = 5 units per hour, and the efficiency of B is 240 / 40 = 6 units per hour. Working together, their combined efficiency is 5 + 6 = 11 units per hour. To complete the total work of 240 units, the total time required by both working together is 240 / 11 hours.

Since the work must be completed in 5 days, the number of hours they need to work together each day is calculated by dividing the total required hours by the number of days: Daily hours = ²⁴⁰⁄₁₁ × ⅕ = ⁴⁸⁄₁₁ = 4 ⁴⁄₁₁ hours.

Explanation

The relationship between Marked Price (MP) and Cost Price (CP) is given by the formula MP/CP = (100 - Loss%) / (100 - Discount%). Substituting the given values where the discount is 25% and the loss is 15%:

MP / CP = (100 - 15) / (100 - 25) MP / CP = 85 / 75 MP / CP = 17 / 15

When the article is sold at the marked price, the Selling Price (SP) equals the Marked Price. The profit percentage is calculated using the formula:

Profit % = [(SP - CP) / CP] × 100 Profit % = [(MP / CP) - 1] × 100 Profit % = [(17 / 15) - 1] × 100 Profit % = (2 / 15) × 100 Profit % = 40 / 3 = 13.33%

Explanation

The Marked Price (MP) is determined by the relationship between the Selling Price (SP) and the discount rate. Since a discount of 18% is provided, the Selling Price represents 82% of the total Marked Price (100% - 18% = 82%).

The formula to calculate the Marked Price is: MP = (SP × 100)/(100 - Discount%)

Substituting the given values into the equation: MP = (1599 × 100)/(82)

By performing the division, 1599 ÷ 82 equals 19.5. Multiplying this result by 100 gives the final value: MP = 19.5 × 100 = 1950

Explanation

The total weight of the three varieties of wheat is 24 + 36 + 84 = 144 kg. When these are thoroughly mixed, the ratio of variety A in the entire mixture is determined by its original weight relative to the total weight.

Proportion of variety A in the mixture = ²⁴⁄₁₄₄ = ⅙

Since the mixture is uniform, every unit of the mixture contains variety A in this same proportion. The third box has a capacity of 84 kg and is filled completely with this mixture.

Weight of variety A in the third box = 84 × ⅙ = 14 kg

Explanation

The number of boys in the group remains constant because one boy leaves and another joins. Let n be the number of boys. The net change in the total weight of the group is the difference between the weight of the boy who joined and the boy who left, which is 30 - 42 = -12 kg.

The average weight of the group decreases from 36 kg to 35.7 kg, which is a reduction of 36 - 35.7 = 0.3 kg per boy. In any group, the total change in the sum of values is equal to the number of items multiplied by the change in the average. Using this relation: n × 0.3 = 12 n = (12)/(0.3) n = 40

Alternatively, this can be expressed by the equation for the total weight: 36n - 42 + 30 = 35.7n. Subtracting 35.7n from both sides and moving the constants results in 0.3n = 12, which yields n = 40.

Explanation

Let the cost price (CP) of the calculator be 100. A gain of 15% indicates that the original selling price (SP₁) is: SP₁ = 100 + 15 = 115

The question states that the calculator is sold at triple the original price. Therefore, the new selling price (SP₂) is: SP₂ = 3 × 115 = 345

The profit is the difference between the new selling price and the original cost price: Profit = SP₂ - CP = 345 - 100 = 245

To find the profit percentage, we use the formula: Profit Percentage = (Profit)/(CP) × 100 = ²⁴⁵⁄₁₀₀ × 100 = 245%

Explanation

Let the total number of votes polled be x. The winning candidate received 70% of the votes, which is 0.70x. Consequently, the losing candidate received the remaining 30%, which is 0.30x. The margin by which the winner won is the difference between their vote shares: 0.70x - 0.30x = 0.40x. According to the problem, this margin is 15400 votes. Therefore, we have the equation 0.40x = 15400. Solving for the total votes, x = (15400)/(0.40) = 38500. The number of votes polled for the losing candidate is 30% of this total, calculated as 0.30 × 38500 = 11550.

Explanation

Let the speed of the boat in still water be x km/hr and the speed of the stream be y km/hr. The downstream speed is D = x + y and the upstream speed is U = x - y. Based on the time taken for the journeys, we can set up the following equations:

4/U + 4/D = 1 3/U + 5/D = 55/60 = 11/12

To solve this system, multiply the first equation by 3 and the second equation by 4 to eliminate 1/U:

12/U + 12/D = 3 12/U + 20/D = 44/12 = 11/3

Subtracting the first modified equation from the second:

(20/D - 12/D) = 11/3 - 3 8/D = 2/3 D = 24 / 2 = 12 km/hr

Substitute D = 12 back into the first equation:

4/U + 4/12 = 1 4/U + 1/3 = 1 4/U = 2/3 U = 12 / 2 = 6 km/hr

The speed of the boat in still water is the average of the downstream and upstream speeds:

x = (D + U) / 2 = (12 + 6) / 2 = 18 / 2 = 9 km/hr

Explanation

The Simple Interest (SI) formula is 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 the interest earned over 10 years is 130% of the principal, which can be written as SI = ¹³⁰⁄₁₀₀P = 1.3P.

By substituting these values into the formula: 1.3P = (P × R × 10)/(100)

Since the principal P is present on both sides, it cancels out, showing that the rate depends only on the percentage growth and time: 1.3 = (10R)/(100) 1.3 = (R)/(10) R = 1.3 × 10 = 13

The rate of interest is 13% per annum.

Explanation

A quadratic expression in terms of x³ follows the identity (a - b)² = a² - 2ab + b². To make x⁶ - 18x³ + k a perfect square, we equate it to the expansion of (x³ - b)².

Rewriting the given expression as: (x³)² - 2(x³)(9) + k

Comparing this with the standard identity a² - 2ab + b², we identify a = x³ and b = 9. For the expression to be a perfect square, the constant term k must be equal to b².

k = 9² = 81

Substituting this value, the expression becomes x⁶ - 18x³ + 81, which is the perfect square (x³ - 9)².

Explanation

Apply the Componendo and Dividendo rule to the equation √(5+x) + √(5-x){√(5+x) - √(5-x)} = ³⁄₁. This rule states that if (a+b)/(a-b) = (c)/(d), then (a)/(b) = (c+d)/(c-d).

Using this relation, the equation simplifies to:

√(5+x){√(5-x)} = ³⁺¹⁄₃₋₁

√(5+x){√(5-x)} = ⁴⁄₂ = 2

To remove the square roots, square both sides of the equation:

⁵⁺ˣ⁄₅₋ₓ = 2²

⁵⁺ˣ⁄₅₋ₓ = 4

Now, perform cross-multiplication:

5 + x = 4(5 - x)

5 + x = 20 - 4x

Rearrange the terms to solve for x by adding 4x to both sides and subtracting 5 from both sides:

x + 4x = 20 - 5

5x = 15

x = ¹⁵⁄₅

x = 3

Explanation

The standard algebraic identity for the sum of cubes is x³ + y³ + z³ - 3xyz = (x + y + z)(x² + y² + z² - xy - yz - zx). To directly use the given values, the second factor is rewritten using the expansion (x + y + z)² = x² + y² + z² + 2(xy + yz + zx) to obtain the modified identity x³ + y³ + z³ - 3xyz = (x + y + z)[(x + y + z)² - 3(xy + yz + zx)].

Substituting the given values x + y + z = 12, xy + yz + zx = 44, and xyz = 48 into this formula:

x³ + y³ + z³ - 3(48) = 12 × [12² - 3(44)]

x³ + y³ + z³ - 144 = 12 × (144 - 132)

x³ + y³ + z³ - 144 = 12 × 12

x³ + y³ + z³ = 144 + 144 = 288

Explanation

Given the expression x = 4√(ab){√a + √b}, we can rewrite it to isolate terms for the Componendo and Dividendo rule. First, divide by 2√a:

(x)/(2√(a)) = 2√b{√a + √b}

Applying Componendo and Dividendo: x + 2√a{x - 2√a} = 2√b + (√a + √b){2√b - (√a + √b)} = 3√b + √a{√b - √a}

Similarly, divide the original equation by 2√b: (x)/(2√(b)) = 2√a{√a + √b}

Applying Componendo and Dividendo: x + 2√b{x - 2√b} = 2√a + (√a + √b){2√a - (√a + √b)} = 3√a + √b{√a - √b}

To find the value of the required sum x + 2√a{x - 2√a} + x + 2√b{x - 2√b}, add the two derived expressions: 3√b + √a{√b - √a} + 3√a + √b{√a - √b}

Change the denominator of the first term by multiplying the numerator and denominator by -1: -(3√b + √a){√a - √b} + 3√a + √b{√a - √b} = -3√b - √a + 3√a + √b{√a - √b}

Simplifying the numerator: 2√a - 2√b{√a - √b} = 2(√a - √b){√a - √b} = 2

Explanation

Let the amount received by each son be x. Based on the given conditions, the amount received by each daughter is 1.5x and the amount received by the wife is x - 600. The total sum of Rs 19000 is distributed among 3 sons, 2 daughters, and 1 wife. The distribution can be represented by the following equation:

3(x) + 2(1.5x) + (x - 600) = 19000

Combining the terms involving x: 3x + 3x + x - 600 = 19000 7x - 600 = 19000 7x = 19600 x = 2800

Each son received Rs 2800. To find the total amount received by the 3 sons together, we multiply the individual share by the number of sons: 3 × 2800 = 8400

Explanation

In a right-angled triangle, the circumcentre lies at the midpoint of the hypotenuse. Consequently, the radius of the circumcircle R is equal to half the length of the hypotenuse. In triangle PQR, where ∠ Q = 90°, the side PR is the hypotenuse. Using the Pythagorean theorem, the length of PR is calculated as PR = √(PQ² + QR²) = √(12² + 5²) = √(144 + 25) = √(169) = 13 cm. The radius of the circumcircle is R = (PR)/(2) = ¹³⁄₂ = 6.5 cm.

Explanation

The angle subtended by a chord at the center of a circle is twice the angle it subtends at any point on the circumference. Given the angle at the circumference is 30°, the central angle θ is 2 × 30° = 60°. The length of a chord L in a circle of radius R is determined by the formula L = 2R sin(θ/2). Substituting the central angle into the formula, we have L = 2R sin(60°/2) = 2R sin(30°). Since sin(30°) = 1/2, the equation becomes L = 2R × (1/2) = R. Therefore, the length of the chord is equal to the radius of the circle, and the ratio of the radius to the length of the chord is R : R, which simplifies to 1 : 1. This can also be visualized through the triangle formed by the center of the circle and the endpoints of the chord; since the two radii form a 60° angle at the center, the triangle is equilateral, confirming that the chord length equals the radius.

Explanation

In a circle, the tangents drawn from an external point to the circle are equal in length, hence AP = BP = 6 cm. In the quadrilateral OAPB, the angles at the points of tangency are ∠ OAP = ∠ OBP = 90° because a tangent is perpendicular to the radius at the point of contact. Since the sum of the angles in a quadrilateral is 360°, the angle at the external point is ∠ APB = 360° - (∠ OAP + ∠ OBP + ∠ AOB) = 360° - (90° + 90° + 120°) = 60°. Because AP = BP and the vertex angle ∠ APB is 60°, triangle APB must be an equilateral triangle with all sides equal to 6 cm. The area of an equilateral triangle is calculated using the formula √3{4} × side². Substituting the side length of 6 cm, the area is √3{4} × 6² = √3{4} × 36 = 9√3 cm².

Explanation

The power of a point P with respect to a circle is given by the product of the lengths of the segments of any secant through P. For a point outside the circle at distance d from the center and a circle of radius r, the Power of Point theorem states that PQ × PR = d² - r².

Given the lengths PQ = 4.5 cm and QR = 3.5 cm, the total length of the secant segment PR is PQ + QR = 4.5 + 3.5 = 8 cm. The distance d from the center O to point P is 6.5 cm.

Substituting these values into the formula: 4.5 × 8 = (6.5)² - r² 36 = 42.25 - r² r² = 42.25 - 36 r² = 6.25 r = √(6.25) = 2.5 cm

Explanation

To find the value of [1/(1 - tan θ)] - [1/(1 + tan θ)], we simplify the expression by finding a common denominator for the two fractions. The common denominator is (1 - tan θ)(1 + tan θ), which equals 1 - tan² θ following the algebraic identity (a - b)(a + b) = a² - b².

By cross-multiplying, the numerator of the combined fraction becomes (1 + tan θ) - (1 - tan θ). Simplifying this numerator yields 1 + tan θ - 1 + tan θ, which equals 2 tan θ. Putting the numerator and denominator together, the expression results in the fraction (2 tan θ)/(1 - tan² θ).

This result matches the fundamental trigonometric double-angle formula tan 2θ = (2 tan θ)/(1 - tan² θ). Consequently, the entire expression simplifies to tan 2θ. In the given options, where the symbol θ is represented as a character similar to the digit zero, this corresponds to the value tan 2θ.

Explanation

Given the expression tan θ + cot θ = x. Since tan θ and cot θ are reciprocals, their product tan θ · cot θ = 1. Squaring both sides of the given equation, we apply the identity (a + b)² = a² + b² + 2ab:

(tan θ + cot θ)² = x² tan² θ + cot² θ + 2(tan θ · cot θ) = x² tan² θ + cot² θ + 2(1) = x² tan² θ + cot² θ = x² - 2

To find the value of tan⁴ θ + cot⁴ θ, we square the resulting equation once more:

(tan² θ + cot² θ)² = (x² - 2)² tan⁴ θ + cot⁴ θ + 2(tan² θ · cot² θ) = x⁴ - 4x² + 4 tan⁴ θ + cot⁴ θ + 2(1) = x⁴ - 4x² + 4 tan⁴ θ + cot⁴ θ = x⁴ - 4x² + 2

By factoring out x² from the first two terms of the expression, we get:

tan⁴ θ + cot⁴ θ = x²(x² - 4) + 2

Explanation

The equation tan² θ + cot² θ = 2 can be treated as an algebraic identity of the form x + ¹⁄ₓ = 2, where x = tan² θ. According to the Arithmetic Mean-Geometric Mean (AM-GM) inequality, for any positive real number x, the sum x + ¹⁄ₓ is always greater than or equal to 2, and equality holds if and only if x = 1. Therefore, tan² θ = 1, which implies tan θ = 1 and cot θ = 1 for acute angles. This occurs when θ = 45°.

Substituting θ = 45° into the expression 2 sec θ csc θ: 2 sec 45° csc 45° = 2 × √2 × √2 = 2 × 2 = 4

Alternatively, the expression can be simplified using trigonometric identities. Since sec θ = (1)/(cos θ) and csc θ = (1)/(sin θ), we have: 2 sec θ csc θ = (2)/(sin θ cos θ) Multiplying the numerator and denominator by 2 gives: (4)/(2 sin θ cos θ) = (4)/(sin 2θ) For θ = 45°, sin 2θ = sin 90° = 1. Thus, the value is ⁴⁄₁ = 4.

Explanation

To find the percentage increase in education loan disbursed by Bank 2 from 2010 to 2014, use the formula for percentage change:

Percentage Increase = (Final Value - Initial Value)/(Initial Value) × 100

According to the table, the amount disbursed by Bank 2 in 2010 was 65 crores and in 2014 it was 122 crores.

Initial Value = 65 Final Value = 122 Increase = 122 - 65 = 57

Percentage Increase = ⁵⁷⁄₆₅ × 100

Simplifying the fraction: ⁵⁷⁄₆₅ × 100 = ⁵⁷⁰⁰⁄₆₅

Dividing both numerator and denominator by 5: ¹¹⁴⁰⁄₁₃ ≈ 87.6923

Rounding to two decimal places, the percentage increase is 87.69.

Explanation

A continuous trend requires the disbursed loan amount to strictly increase or strictly decrease across all years from 2010 to 2014.

For Bank 1, the disbursement sequence is 265, 295, 317, 323, and 352 crores. Each subsequent year shows a higher value than the previous one (265 < 295 < 317 < 323 < 352), establishing a continuous increasing trend.

For Bank 4, the disbursement sequence is 109, 123, 125, 142, and 157 crores. Each year shows a growth in the loan amount (109 < 123 < 125 < 142 < 157), maintaining a continuous increasing trend.

For Bank 5, the disbursement sequence is 80, 103, 140, 143, and 158 crores. The loan amounts consistently rise over the five-year period (80 < 103 < 140 < 143 < 158), confirming a continuous increasing trend.

Explanation

To determine the relationship between the two ratios, we substitute the values from the table into the expressions. Ratio I is the loan amount disbursed by Bank 1 in 2011 divided by the amount disbursed by Bank 2 in 2014, which is ²⁹⁵⁄₁₂₂. Ratio II is the loan amount disbursed by Bank 3 in 2014 divided by the amount disbursed by Bank 4 in 2011, which is ²¹¹⁄₁₂₃.

By comparing the two fractions: Ratio I = ²⁹⁵⁄₁₂₂ ≈ 2.418 Ratio II = ²¹¹⁄₁₂₃ ≈ 1.715

Alternatively, using the benchmark of 2, we observe that 295 > 2 × 122 (244), so Ratio I is greater than 2. For Ratio II, 211 < 2 × 123 (246), so Ratio II is less than 2. This comparison confirms that I > II.

Explanation

For any arithmetic progression with n terms, first term a, and common difference d, the average is given by the formula: Average = a + ((n - 1)d)/(2)

For the first 3 consecutive even numbers, the common difference d is 2. Given the average is A: A = a + ((3 - 1) × 2)/(2) A = a + 2 This gives the first term of the sequence as a = A - 2.

When the next 5 even numbers are added to the set, the total number of terms n becomes 8. The common difference d remains 2. The new average for these 8 numbers is: New Average = a + ((8 - 1) × 2)/(2) New Average = a + 7

Substituting the value of the first term a = A - 2 into the equation: New Average = (A - 2) + 7 = A + 5

Explanation

The percentage increase in loan amount disbursed by each bank from 2010 to 2014 is calculated using the formula P = V₂₀₁₄ - V₂₀₁₀{V₂₀₁₀} × 100.

For Bank 1, the increase is (352 - 265)/(265) × 100 = ⁸⁷⁄₂₆₅ × 100 ≈ 32.83%. For Bank 2, the increase is (122 - 65)/(65) × 100 = ⁵⁷⁄₆₅ × 100 ≈ 87.69%. For Bank 3, the increase is (211 - 138)/(138) × 100 = ⁷³⁄₁₃₈ × 100 ≈ 52.89%. For Bank 4, the increase is (157 - 109)/(109) × 100 = ⁴⁸⁄₁₀₉ × 100 ≈ 44.03%. For Bank 5, the increase is (158 - 80)/(80) × 100 = ⁷⁸⁄₈₀ × 100 = 97.5%.

Comparing the calculated percentages, we have 97.5% > 87.69% > 52.89% > 44.03% > 32.83%. This establishes the descending order as Bank 5 > Bank 2 > Bank 3 > Bank 4 > Bank 1.

Explanation

The relationship between the initial selling price and the desired selling price can be expressed using the ratio of their respective percentages relative to the cost price (CP). If an article is sold at a loss of 50%, the selling price is 50% of the CP. To earn a profit of 50%, the new selling price must be 150% of the CP.

Using the direct formula: New Selling Price = Original Selling Price × (100 + Desired Profit%)/(100 - Initial Loss%)

Substituting the given values: SP = 18450 × (100 + 50)/(100 - 50) SP = 18450 × ¹⁵⁰⁄₅₀ SP = 18450 × 3 SP = 55350

Explanation

The population growth over time follows the compound growth 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. Substituting the given values P = 26010, r = 2, and n = 2, the equation becomes:

26010 = P₀(1 + 2/100)² 26010 = P₀(1 + 1/50)² 26010 = P₀(51/50)² 26010 = P₀(2601/2500)

To solve for P₀, rearrange the terms: P₀ = 26010 × (2500/2601) P₀ = 10 × 2500 P₀ = 25000

Explanation

When two trains cross each other, the total distance covered is the sum of their lengths. In this case, the distance D = 100 + 150 = 250 m. Since the trains are moving in opposite directions, the relative speed is the sum of their individual speeds (S₁ + S₂).

The relative speed Vᵣₑₗ in meters per second is calculated using the formula Vᵣₑₗ = (Distance)/(Time) = ²⁵⁰⁄₉ m/s. To convert this speed into km/hr, we multiply by the factor ¹⁸⁄₅: Vᵣₑₗ = ²⁵⁰⁄₉ × ¹⁸⁄₅ = 50 × 2 = 100 km/hr.

Given that the speed of the 150 m train is 40 km/hr, we substitute the values into the relative speed equation: S₁ + 40 = 100 S₁ = 100 - 40 = 60 km/hr.

Explanation

When interest is compounded half-yearly, the annual interest rate is halved and the number of compounding periods is doubled. Given an annual rate of 40%, the half-yearly rate r is 40% / 2 = 20%. For a period of 1 year, there are n = 2 half-yearly periods.

The compound interest CI can be calculated using the formula CI = P[(1 + ʳ⁄₁₀₀)ⁿ - 1]. Substituting the principal P = 20000, r = 20, and n = 2:

CI = 20000 × [(1 + ²⁰⁄₁₀₀)² - 1] CI = 20000 × [(⁶⁄₅)² - 1] CI = 20000 × (³⁶⁄₂₅ - 1) CI = 20000 × ¹¹⁄₂₅ CI = 800 × 11 = 8800

Alternatively, the effective interest rate for two periods of 20% can be found using the successive percentage change formula x + y + ˣʸ⁄₁₀₀. This gives 20 + 20 + (20 × 20)/(100) = 44%. Calculating 44% of the principal: CI = ⁴⁴⁄₁₀₀ × 20000 = 8800