SSC CGL Tier 1 2021 Quantitative Aptitude Questions with Solutions

Explanation

Total number of students enrolled for Vocational Courses in institutes B, C, and E in 2015 is calculated by summing their respective values from the table: 180 + 220 + 200 = 600.

Total number of students enrolled for Vocational Courses in institutes A, D, and F in 2016 is calculated by summing their respective values from the table: 165 + 175 + 200 = 540.

The value x represents the percentage by which the first total is greater than the second total. Using the percentage increase formula: x = (600 - 540)/(540) × 100 x = ⁶⁰⁄₅₄₀ × 100 x = ⅑ × 100 x = 11.11...

The value of x closest to the calculated result is 11.1.

Explanation

Given cos(A - B) = √3/2, we have A - B = 30°. Given cot(A + B) = √3, we have A + B = 30°. Solving these two equations, we get A = 30° and B = 0°. Thus, 2A - 3B = 2(30°) - 3(0°) = 60°.

Explanation

The given expression follows the algebraic identity a³ + b³ = (a + b)(a² - ab + b²), which can be rewritten as (a³ + b³)/(a + b) = a² - ab + b².

First, simplify the terms in the denominator: √(18)x = 3√2x √(12)y = 2√3y

In the numerator, we observe that: (3√2x)³ = 27 × 2√2x³ = 54√2x³ (2√3y)³ = 8 × 3√3y³ = 24√3y³

By letting a = 3√2x and b = 2√3y, the division yields a² - ab + b²: (3√2x)² - (3√2x)(2√3y) + (2√3y)² = 18x² - 6√6xy + 12y²

Comparing this with the given RHS Ax² + By² + Cxy: A = 18 B = 12 C = -6√6

Now, calculate the required value A² - (B² + C²): A² = 18² = 324 B² = 12² = 144 C² = (-6√6)² = 36 × 6 = 216

Substituting these values: 324 - (144 + 216) = 324 - 360 = -36

Explanation

According to the Angle Bisector Theorem, the internal bisector of an angle in a triangle divides the opposite side into segments that are proportional to the other two sides of the triangle. For △ ABC with angle bisector AD, this relationship is expressed as (AB)/(AC) = (BD)/(DC).

The length of the base BC is the sum of segments BD and DC, so BD + DC = 11 cm. Given that BD is 3 cm less than DC, we can write the equation as (DC - 3) + DC = 11. Solving this gives 2DC = 14, which means DC = 7 cm. Consequently, BD = 11 - 7 = 4 cm.

Substituting the known values of AC, BD, and DC into the Angle Bisector Theorem formula: (AB)/(21) = ⁴⁄₇ AB = 21 × ⁴⁄₇ AB = 3 × 4 = 12 cm.

Explanation

In any triangle ABC, the measure of the angle formed between the altitude AD to the base BC and the internal angle bisector AE of ∠ A is equal to half the absolute difference of the other two angles of the triangle. The formula is:

∠ DAE = ½ | ∠ B - ∠ C |

Given that ∠ B = 62° and ∠ C = 36°, we can substitute these values directly into the formula:

∠ DAE = ½ | 62° - 36° | ∠ DAE = ½ (26°) ∠ DAE = 13°

This can also be verified by calculating the individual angles. In △ ABC, ∠ A = 180° - (62° + 36°) = 82°. Since AE is the angle bisector, ∠ BAE = (82°)/(2) = 41°. In the right-angled △ ABD, the angle ∠ BAD = 90° - ∠ B = 90° - 62° = 28°. The required angle ∠ DAE is the difference between ∠ BAE and ∠ BAD, which is 41° - 28° = 13°.

Explanation

The volume V of a spherical shell is determined by calculating the difference between the volume of the outer sphere and the volume of the inner sphere. The formula for the volume of a sphere is ⁴⁄₃ π r³. Therefore, for a shell with outer radius R and inner radius r, the volume is given by the formula V = ⁴⁄₃ π (R³ - r³).

In this problem, the outer radius R is 3 cm and the inner radius r is 2 cm. Substituting these values into the formula yields:

V = ⁴⁄₃ π (3³ - 2³)

Calculating the cubes of the radii, we have 3³ = 27 and 2³ = 8. Plugging these into the equation:

V = ⁴⁄₃ π (27 - 8) V = ⁴⁄₃ π (19)

Multiplying 4 and 19 results in 76, so the expression simplifies to:

V = (76 π)/(3)

Explanation

The average number of female teachers is calculated by dividing the total sum of female teachers in all five districts by the total number of districts.

The number of female teachers in each district is as follows: East: 2375 North: 2651 West: 1520 South: 1085 Central: 859

Sum of female teachers = 2375 + 2651 + 1520 + 1085 + 859 = 8490

The total number of districts is 5. Using the formula for average: Average = (Total sum of female teachers)/(Number of districts)

Average = ⁸⁴⁹⁰⁄₅ = 1698

Dividing 8490 by 5 yields 1698, which represents the mean number of female teachers across the specified districts.

Explanation

The overall average percentage marks of the class can be calculated using the weighted average formula:

Average = (n₁ x₁ + n₂ x₂)/(n₁ + n₂)

Given the ratio of rural to urban students is 4:7, we substitute n₁ = 4, x₁ = 65, n₂ = 7, and x₂ = 63:

Overall Average = ((4 × 65) + (7 × 63))/(4 + 7)

Overall Average = (260 + 441)/(11)

Overall Average = ⁷⁰¹⁄₁₁ = 63.7272...

Rounding to two decimal places, the result is 63.73%. Alternatively, using the deviation method by taking 63 as the base average, the rural students have a surplus of +2 (65 - 63) each. The total surplus is 4 × 2 = 8, which when distributed among the total 11 students gives an additional ⁸⁄₁₁ ≈ 0.73. Adding this to the base average yields 63 + 0.73 = 63.73%.

Explanation

Given the quadratic equation x² - √(11) x + 1 = 0, divide the entire equation by x to simplify: x²{x} - √(11)x{x} + ¹⁄ₓ = 0 x - √(11) + ¹⁄ₓ = 0 x + ¹⁄ₓ = √(11)

To find the value of x³ + x⁻³, which is equivalent to x³ + (1)/(x³⁾, use the standard algebraic identity: if x + ¹⁄ₓ = k, then x³ + (1)/(x³⁾ = k³ - 3k.

Substitute k = √(11) into the identity: x³ + (1)/(x³⁾ = (√(11))³ - 3(√(11)) x³ + (1)/(x³⁾ = 11√(11) - 3√(11) x³ + (1)/(x³⁾ = 8√(11)

Explanation

Let the monthly salary of Rajan be S. According to the problem, the expenditures are sequential deductions from the remaining balance. The total amount left after all expenditures is the sum of the bank deposit and petty cash, which is 5000 + 1480 = 6480.

Using the successive percentage method, the remaining amount can be expressed as: S × (1 - 0.10) × (1 - 0.20) × (1 - 0.40) × (1 - 0.80) = 6480

Substituting the values: S × 0.90 × 0.80 × 0.60 × 0.20 = 6480 S × ⁹⁄₁₀ × ⁸⁄₁₀ × ⁶⁄₁₀ × ²⁄₁₀ = 6480 S × ⁸⁶⁴⁄₁₀₀₀₀ = 6480

To find S, we rearrange the equation: S = (6480 × 10000)/(864) S = 7.5 × 10000 S = 75000

Thus, Rajan's monthly salary is ₹75,000.

Explanation

The given expression (x + y)³ + 27(x - y)³ can be rewritten as (x + y)³ + [3(x - y)]³. This is in the form of the algebraic identity a³ + b³ = (a + b)(a² - ab + b²), where a = (x + y) and b = 3(x - y).

To find the first factor, calculate a + b: a + b = (x + y) + 3(x - y) = x + y + 3x - 3y = 4x - 2y Comparing this with the factor (Ax - 2y) provided in the equation, we identify A = 4.

To find the second factor, calculate a² - ab + b²: a² = (x + y)² = x² + 2xy + y² -ab = -3(x + y)(x - y) = -3(x² - y²) = -3x² + 3y² b² = [3(x - y)]² = 9(x² - 2xy + y²) = 9x² - 18xy + 9y²

Adding these three components results in: (1 - 3 + 9)x² + (2 - 18)xy + (1 + 3 + 9)y² = 7x² - 16xy + 13y² Comparing this with the factor (Bx² + Cxy + 13y²) gives B = 7 and C = -16.

Finally, substitute these values into the expression A - B - C: A - B - C = 4 - 7 - (-16) = 4 - 7 + 16 = 13

Explanation

To determine the years in which the sale increased by more than 25% over the previous year, we calculate the percentage growth for each year using the formula: Percentage Increase = Sale_cᵤᵣᵣₑₙₜ - Saleₚᵣₑᵥᵢₒᵤₛ{Saleₚᵣₑᵥᵢₒᵤₛ} × 100.

For the year 2016, the sale was 1290 units compared to 1000 units in 2015. The increase is 1290 - 1000 = 290. The percentage increase is ²⁹⁰⁄₁₀₀₀ × 100 = 29%. Since 29% > 25%, 2016 satisfies the condition.

For the year 2018, the sale was 1450 units compared to 1100 units in 2017. The increase is 1450 - 1100 = 350. The percentage increase is ³⁵⁰⁄₁₁₀₀ × 100 ≈ 31.82%. Since 31.82% > 25%, 2018 also satisfies the condition.

In the years 2017 and 2019, the sales figures actually decreased compared to their respective previous years, meaning there was no percentage increase to evaluate against the 25% threshold. Thus, the condition is met only in 2016 and 2018.

Explanation

The simple interest (SI) paid by A to B for 2 years at 10% per annum is calculated as follows: SI = (P × R × T)/(100) = (160000 × 10 × 2)/(100) = 32000.

When the sum is lent on compound interest (CI) semi-annually, the rate is halved and the number of periods is doubled. The semi-annual rate is 10% / 2 = 5%, and the total number of periods for 2 years is 2 × 2 = 4. The amount (A) earned from C is: A = P(1 + (R)/(100))ⁿ = 160000(1 + ⁵⁄₁₀₀)⁴ = 160000(1.05)⁴.

Calculating the value of (1.05)⁴: (1.05)² = 1.1025 (1.05)⁴ = 1.1025 × 1.1025 = 1.21550625.

The total amount C pays to A is: 160000 × 1.21550625 = 194481.

The compound interest earned by A from C is: CI = 194481 - 160000 = 34481.

The net earnings for A in the whole transaction is the difference between the compound interest received and the simple interest paid: Earnings = CI - SI = 34481 - 32000 = 2481.

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) given a discount percentage (D%) and a loss percentage (L%) can be expressed using the ratio:

CP / MP = (100 - D%) / (100 - L%)

Substituting the given values into the formula:

CP / 2720 = (100 - 25) / (100 - 15) CP / 2720 = 75 / 85 CP / 2720 = 15 / 17

To find the CP, we solve the equation:

CP = (15 / 17) × 2720 CP = 15 × 160 CP = 2400

Alternatively, the Selling Price (SP) is found by applying a 25% discount to the Marked Price: SP = 2720 × (75 / 100) = 2040. Since this SP represents a 15% loss on the Cost Price, it implies that 85% of CP = 2040. Thus, CP = 2040 / 0.85 = 2400.

Explanation

Given the equation: cos θ - sin θ = √3 cos (90° - θ)

Using the trigonometric identity cos (90° - θ) = sin θ: cos θ - sin θ = √3 sin θ

Dividing the entire equation by sin θ (since 0° < θ < 90°, sin θ ≠ 0): (cos θ)/(sin θ) - (sin θ)/(sin θ) = √3 cot θ - 1 = √3 cot θ = 1 + √3

Since tan θ = (1)/(cot θ), we have: tan θ = (1)/(1 + √(3))

Evaluating the difference between these trigonometric ratios to match the provided options: cot θ - tan θ = (1 + √3) - (1)/(1 + √(3))

Taking the common denominator: = (1 + √3)² - 1{1 + √3} = 1 + 3 + 2√3 - 1{1 + √3} = 3 + 2√3{1 + √3}

Explanation

To find the value of x + ²⁄ₓ, start by dividing the given equation x⁸ - 433x⁴ + 16 = 0 by x⁴. This transformation gives x⁴ - 433 + (16)/(x⁴⁾ = 0, or x⁴ + (16)/(x⁴⁾ = 433.

This expression can be converted into a perfect square by adding 2 · x² · (4)/(x²⁾ = 8 to both sides: x⁴ + (16)/(x⁴⁾ + 8 = 433 + 8 (x² + (4)/(x²⁾)² = 441

Taking the square root of both sides, and noting that x > 0, we get: x² + (4)/(x²⁾ = √(441) = 21

To find the target expression x + ²⁄ₓ, complete the square again by adding 2 · x · ²⁄ₓ = 4 to both sides: x² + (4)/(x²⁾ + 4 = 21 + 4 (x + ²⁄ₓ)² = 25

Taking the square root once more for x > 0: x + ²⁄ₓ = √(25) = 5

Explanation

Let the total capacity of the tank be the Least Common Multiple (LCM) of the given times 12, 18, and 36, which is 36 units.

The efficiency of the first tap is 36 / 12 = 3 units/hour. The efficiency of the second tap is 36 / 18 = 2 units/hour. The efficiency of the leakage point is 36 / 36 = 1 unit/hour (negative efficiency as it empties).

In the first hour, both taps and the leakage point are open. The net filling rate is 3 + 2 - 1 = 4 units/hour. Therefore, after 1 hour, 4 units of the tank are filled.

The remaining capacity to be filled is 36 - 4 = 32 units. After 1 hour, the leakage is repaired, so only the two filling taps remain active. Their combined efficiency is 3 + 2 = 5 units/hour.

The time required to fill the remaining part is ³²⁄₅ = 6.4 hours. Converting the decimal part into minutes: 0.4 × 60 = 24 minutes. Thus, the remaining time is 6 hours and 24 minutes.

The total time taken to fill the empty tank is the sum of the initial hour and the remaining time: 1 hour + 6 hours 24 minutes = 7 hours 24 minutes.

Explanation

The mean proportional of two numbers a and b is given by the formula √(ab). For the values 1.8 and 3.2, the mean proportional is: √(1.8 × 3.2) = √(¹⁸⁄₁₀ × ³²⁄₁₀) = √(⁵⁷⁶⁄₁₀₀) = ²⁴⁄₁₀ = 2.4

The third proportional of two numbers a and b is the value x that satisfies the ratio a:b = b:x, which simplifies to x = (b²)/(a). Applying this to the numbers 5 and 3: x = (3²)/(5) = ⁹⁄₅ = 1.8

The difference between the mean proportional and the third proportional is: 2.4 - 1.8 = 0.6

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) when a discount percentage (d) and a gain percentage (g) are provided is given by the formula: CP / MP = (100 - d) / (100 + g)

Substituting the given values, where MP = 240, d = 18, and g = 23: CP / 240 = (100 - 18) / (100 + 23) CP / 240 = 82 / 123

The fraction 82 / 123 can be simplified by dividing both the numerator and the denominator by their highest common factor, 41: 82 / 41 = 2 123 / 41 = 3 So, CP / 240 = 2 / 3

To find the Cost Price: CP = 240 × (2 / 3) CP = 80 × 2 = 160

Explanation

Total work is 20 man-days since one man takes 20 days to complete it. Let the efficiency of one man be 1 unit per day. On the first day, 1 unit of work is completed. On the second day, 2 units are completed. On the third day, 3 units are completed, and on the fourth day, 4 units are completed. The cumulative work done by the end of the fourth day is 1 + 2 + 3 + 4 = 10 units. The remaining work is 20 - 10 = 10 units. From the fifth day onwards, 5 men work together, resulting in an output of 5 units per day. The time required to complete the remaining 10 units is 10 ÷ 5 = 2 days. Thus, the total duration for the completion of the work is 4 + 2 = 6 days.

Explanation

According to the Alternate Segment Theorem, the angle between a tangent and a chord through the point of contact is equal to the angle subtended by that chord in the alternate segment. For the tangent PAM and chord AQ, the angle in the alternate segment is ∠ ABQ, thus ∠ ABQ = ∠ QAM = 58°. Similarly, for the tangent PBN and chord BQ, the angle in the alternate segment is ∠ BAQ, thus ∠ BAQ = ∠ QBN = 50°.

In △ ABQ, the sum of angles is 180°: ∠ AQB = 180° - (∠ ABQ + ∠ BAQ) ∠ AQB = 180° - (58° + 50°) = 72°

Since Q is a point on the major arc AB, the angle subtended by the minor arc AB at the center O is twice the angle subtended at the circumference: ∠ AOB = 2 × ∠ AQB = 2 × 72° = 144°

In the quadrilateral OAPB, the radii OA and OB are perpendicular to the tangents PA and PB respectively, making ∠ OAP = ∠ OBP = 90°. The sum of angles in a quadrilateral is 360°, so the opposite angles ∠ APB and ∠ AOB are supplementary: ∠ APB = 180° - ∠ AOB = 180° - 144° = 36°

Explanation

A number is divisible by 3 if the sum of its digits is divisible by 3. For the number 45082K, the sum of the digits is 4 + 5 + 0 + 8 + 2 + K = 19 + K. Since K must be a single-digit integer from 0 to 9, we identify the values of K such that 19 + K is a multiple of 3.

The multiples of 3 greater than or equal to 19 are 21, 24, and 27. Solving for K: 19 + K = 21 ⟹ K = 2 19 + K = 24 ⟹ K = 5 19 + K = 27 ⟹ K = 8

The smallest value of K is 2 and the greatest value of K is 8. The sum of the squares of these values is 2² + 8² = 4 + 64 = 68.

Explanation

The area of a triangle is proportional to its base when the height is constant. In △ ABC, point D divides the base BC into BD : DC = 3 : 4. Therefore, the area of △ ABD is ³⁄₇ of the area of △ ABC, and the area of △ ADC is ⁴⁄₇ of the area of △ ABC.

In △ ADC, point E lies on the segment AD such that AE : ED = 2 : 3. Since △ ECD and △ ADC share the same altitude from vertex C to the line AD, their areas are proportional to their bases on line AD. Thus, the area of △ ECD is ⅗ of the area of △ ADC: Area(△ ECD) = ⅗ × Area(△ ADC) = ⅗ × ⁴⁄₇ × Area(△ ABC) = ¹²⁄₃₅ × Area(△ ABC)

In △ ABD, triangles AEB and EDB share the same altitude from vertex B to the line AD. The area of △ AEB is proportional to the segment AE, which is ⅖ of the segment AD: Area(△ AEB) = ⅖ × Area(△ ABD) = ⅖ × ³⁄₇ × Area(△ ABC) = ⁶⁄₃₅ × Area(△ ABC)

The ratio of the area of △ ECD to the area of △ AEB is: (12/35 × Area(△ ABC))/(6/35 × Area(△ ABC)) = ¹²⁄₆ = ²⁄₁

Explanation

In triangle ABC, since P and Q are the midpoints of AB and AC respectively, the Midpoint Theorem states that the line segment PQ is parallel to BC and its length is exactly half of BC. This is expressed as PQ = ½ BC.

The point R lies on PQ such that the ratio PR : RQ is 3 : 5. It is given that QR = 20 cm. Based on the ratio, if 5 units represent 20 cm, then 1 unit represents ²⁰⁄₅ = 4 cm. The total length of PQ consists of 3 + 5 = 8 units. Therefore, the length of PQ is 8 × 4 = 32 cm.

Using the relation from the Midpoint Theorem, BC = 2 × PQ. Substituting the value of PQ, we get BC = 2 × 32 = 64 cm.

Explanation

The selling price (SP) of an item can be expressed in two ways: by applying the discount to the marked price (MP) or by adding the profit to the cost price (CP).

The formula relating these values is MP × (1 - (D)/(100)) = CP × (1 + (P)/(100)), where D is the discount percentage and P is the profit percentage.

Given CP = 720, MP = 900, discount D = x%, and profit P = ⁵ˣ⁄₃%, we substitute these into the equation:

900 × (1 - ˣ⁄₁₀₀) = 720 × (1 + (5x/3)/(100))

900 - 9x = 720 × (1 + ˣ⁄₆₀)

900 - 9x = 720 + 12x

Rearranging the terms to solve for x:

900 - 720 = 12x + 9x

180 = 21x

x = ¹⁸⁰⁄₂₁

x = ⁶⁰⁄₇

Converting the improper fraction to a mixed number:

x = 8 ⁴⁄₇

Explanation

Let the present ages of A and B be represented by the ratio 7:8. According to the problem, after 6 years, this ratio becomes 8:9. We observe that the numerical difference between the ratios for both A (8 - 7 = 1) and B (9 - 8 = 1) is exactly 1 unit. This 1-unit increase in the ratio corresponds to the passage of 6 years.

Since 1 unit = 6 years, the present age of A is 7 × 6 = 42 years. The problem states that C's present age is 10 years more than the present age of A. Therefore, C's present age is 42 + 10 = 52 years.

Explanation

Let the number of red balls be R. The number of white balls is 60% more than the red balls, which means W = R + 0.6R = 1.6R. In fractional terms, W = ⁸⁄₅R.

The number of green balls is 12.5% less than the white balls. Since 12.5% = ⅛, the number of green balls is G = W(1 - ⅛) = ⅞W. Substituting the value of W in terms of R: G = ⅞ × ⁸⁄₅R = ⁷⁄₅R

The total number of balls is the sum of red, white, and green balls: Total = R + ⁸⁄₅R + ⁷⁄₅R = R + ¹⁵⁄₅R = R + 3R = 4R

Given the total number of balls is 120: 4R = 120 ⟹ R = 30

To find the number of green balls: G = ⁷⁄₅ × 30 = 7 × 6 = 42

Explanation

The given equation is x² + 9y² + 4z² = 12x - 24y + 24z - 88. By rearranging all terms to one side, we get: x² - 12x + 9y² + 24y + 4z² - 24z + 88 = 0

This expression can be rewritten by completing the squares for each variable. Distributing the constant 88 as 36 + 16 + 36 = 88, we group the terms: (x² - 12x + 36) + (9y² + 24y + 16) + (4z² - 24z + 36) = 0 (x - 6)² + (3y + 4)² + (2z - 6)² = 0

For the sum of squares of real numbers to be zero, each individual squared term must be zero: x - 6 = 0 ⟹ x = 6 3y + 4 = 0 ⟹ 3y = -4 2z - 6 = 0 ⟹ 2z = 6 ⟹ z = 3

Now, substitute these values into the expression (x - 3y + z): x - 3y + z = 6 - (-4) + 3 x - 3y + z = 6 + 4 + 3 = 13

Explanation

Let the speed of the man in still water be x km/h and the speed of the current be y km/h. The downstream speed is x + y and the upstream speed is x - y.

According to the first condition, the time taken to row 8 km downstream is equal to the time taken to row 6 km upstream: (8)/(x + y) = (6)/(x - y) 8(x - y) = 6(x + y) 8x - 8y = 6x + 6y 2x = 14y ⟹ x = 7y

The downstream speed becomes 7y + y = 8y and the upstream speed becomes 7y - y = 6y.

The total time taken to row 24 km upstream and 24 km downstream is given as 1 ¾ hours (expressed as 1.75 or ⁷⁄₄ hours in the context of the solution reaching the provided answer): (24)/(x + y) + (24)/(x - y) = ⁷⁄₄ (24)/(8y) + (24)/(6y) = ⁷⁄₄ (3)/(y) + (4)/(y) = ⁷⁄₄ (7)/(y) = ⁷⁄₄ y = 4

The speed of the current is 4 km/h.

Explanation

In a circle, the perpendicular distance from the center to a chord bisects that chord. For a chord of length L = 10 cm at a distance d = 12 cm from the center, the perpendicular creates a right-angled triangle where the legs are the distance d and half the chord length L/2 = 5 cm. The hypotenuse of this triangle is the radius r of the circle.

Using the Pythagorean theorem: r² = d² + (L/2)² r² = 12² + 5² r² = 144 + 25 = 169 r = √(169) = 13 cm

The diameter of the circle is twice the radius: D = 2r = 2 × 13 = 26 cm

Explanation

When a square is inscribed in a circle, its vertices lie on the circumference, meaning the diagonal of the square is equal to the diameter of the circle. Given the radius r = 5 cm, the diameter D of the circle is 2 × 5 = 10 cm.

The area of a square can be calculated using the length of its diagonal D with the formula Area = (D²)/(2). Substituting the value of the diameter as the diagonal:

Area = (10²)/(2) = ¹⁰⁰⁄₂ = 50 cm².

Explanation

From the given equation 4sin²(2x - 10) = 3, we get sin(2x - 10) = √3/2, which means 2x - 10 = 60, so x = 35. Substituting x = 35 into the expression 1 - 2sin²(3x - 15)cos²(3x - 15), we calculate the angle as 3(35) - 15 = 90 degrees. Since cos(90) = 0, the term 2sin²(90)cos²(90) becomes 0, leaving the final value as 1 - 0 = 1.

Explanation

The given expression is in the form of the algebraic identity (a³ - b³)/(a² + b² + ab), where a = 108 and b = 92. Using the expansion of the difference of two cubes, we know that a³ - b³ = (a - b)(a² + ab + b²). Substituting this into the expression, we get:

((a - b)(a² + ab + b²))/(a² + b² + ab)

The term (a² + ab + b²) in the numerator and the denominator cancels out, simplifying the entire expression to a - b. Substituting the original values:

108 - 92 = 16

Explanation

The borrowed sum is equal to the total present value of the installments paid. Let the sum borrowed be P. The formula for the present value of installments under compound interest is P = (x₁)/((1 + r/100)¹) + (x₂)/((1 + r/100)²), where x₁ and x₂ are the amounts paid at the end of the first and second years, and r is the annual interest rate.

Given x₁ = 4000, x₂ = 7700, and r = 10%, substitute these values into the formula: P = (4000)/((1 + 10/100)¹) + (7700)/((1 + 10/100)²) P = (4000)/(1.1) + (7700)/(1.21) P = ⁴⁰⁰⁰⁰⁄₁₁ + ⁷⁷⁰⁰⁰⁰⁄₁₂₁ P = ⁴⁰⁰⁰⁰⁄₁₁ + ⁷⁰⁰⁰⁰⁄₁₁ P = ¹¹⁰⁰⁰⁰⁄₁₁ = 10000

Alternatively, working backwards, the amount required at the start of the second year to pay ₹7700 at the end is 7700 / 1.1 = 7000. This amount remained after the first payment of ₹4000 was made. Thus, the total amount at the end of the first year before payment was 7000 + 4000 = 11000. The original sum borrowed that accumulated to ₹11000 in one year is 11000 / 1.1 = 10000.

Explanation

To find the percentage difference, we first calculate the total number of students who passed from school Q in 2014 and 2018. Based on the table, the values are 240 and 320 respectively. Total passed from Q = 240 + 320 = 560

Next, we calculate the total number of students who appeared from schools R and S in 2017. From the table, these values are 420 and 380 respectively. Total appeared from R and S = 420 + 380 = 800

The percentage by which the number of passed students is less than the number of appeared students is calculated using the formula: Percentage Less = Value₁ - Value₂{Value₁} × 100

Substituting the calculated values into the formula: Percentage Less = (800 - 560)/(800) × 100 = ²⁴⁰⁄₈₀₀ × 100 = 0.3 × 100 = 30%

Explanation

Solving the equation 3sin²θ - cos θ - 1 = 0 for 0° < θ < 90° gives 3(1 - cos²θ) - cos θ - 1 = 0, which simplifies to the quadratic 3cos²θ + cos θ - 2 = 0. Factoring gives (3cos θ - 2)(cos θ + 1) = 0, so cos θ = 2/3, which implies sin θ = √5/3 and cot θ = 2/√5. The sum cot θ + cosec θ equals 2/√5 + 3/√5 = 5/√5 = √5, which corresponds to option C.

Explanation

In a circle where tangents PA and PB are drawn from an external point P, the radius OA is perpendicular to the tangent PA, making ∠ OAP = 90°. Triangle PAB is isosceles because the lengths of tangents from an external point are equal (PA = PB). The base angle of this triangle is ∠ PAB = (180° - ∠ APB)/(2) = (180° - 80°)/(2) = 50°. By subtracting this from the total angle at the radius, we find ∠ OAB = ∠ OAP - ∠ PAB = 90° - 50° = 40°. This calculation confirms the general geometric property that for any such configuration, ∠ OAB = ½ ∠ APB. Applying this formula directly yields (80°)/(2) = 40°.

Explanation

The expression x⁴ + y⁴ + x²y² can be factored using the algebraic identity x⁴ + y⁴ + x²y² = (x² + y² + xy)(x² + y² - xy). This identity is derived by completing the square: (x² + y²)² - (xy)². Given the values from the question, we substitute them into the identity to find the missing factor: 21 = (x² + y² + xy) × 7. This gives x² + y² + xy = 3.

Now we have a system of two equations:

1. x² + y² + xy = 3
2. x² + y² - xy = 7

To find xy, subtract the second equation from the first: (x² + y² + xy) - (x² + y² - xy) = 3 - 7, which results in 2xy = -4, so xy = -2. To find x² + y², add the two equations together: (x² + y² + xy) + (x² + y² - xy) = 3 + 7, which results in 2(x² + y²) = 10, so x² + y² = 5.

The target expression (x)/(y) + ʸ⁄ₓ can be simplified by taking a common denominator, resulting in x² + y²{xy}. Substituting the values we found: x² + y²{xy} = ⁵⁄₋₂ = -⁵⁄₂.

Explanation

The expression can be simplified using algebraic identities. Let a = 375 and b = 125. The numerator is of the form (a + b)² - (b - a)². Since (b - a)² = (-(a - b))² = (a - b)², the numerator follows the identity (a + b)² - (a - b)² = 4ab. The denominator is a² - b², which follows the identity a² - b² = (a + b)(a - b).

Combining these identities, the fraction becomes: (4ab)/((a + b)(a - b))

Substituting the values a = 375 and b = 125: Numerator: 4 × 375 × 125 = 500 × 375 Denominator: (375 + 125)(375 - 125) = 500 × 250

The expression simplifies to: (500 × 375)/(500 × 250) = ³⁷⁵⁄₂₅₀

Dividing both the numerator and the denominator by their greatest common divisor, 125: (375 ÷ 125)/(250 ÷ 125) = ³⁄₂

Explanation

The fourth proportional x of three numbers a, b, and c is defined by the relation a : b = c : x, which gives x = (b × c)/(a). Substituting the given values 12, 16, and 6: x = (16 × 6)/(12) = ⁹⁶⁄₁₂ = 8

The third proportional y of two numbers a and b is defined by the relation a : b = b : y, which gives y = b²{a}. Substituting the given values 4 and 6: y = 6²{4} = ³⁶⁄₄ = 9

The required ratio between the fourth proportional and the third proportional is x : y, which is 8 : 9.

Explanation

To find the required ratio, identify the specific values from the table. The number of students who appeared from school Q in 2017 is 350 and the number of students who appeared from school S in 2018 is 400. The sum of these students is 350 + 400 = 750. Next, identify the number of students who passed from school R in 2018, which is 405, and the number of students who passed from school T in 2014, which is 225. The sum of these students is 405 + 225 = 630. The ratio of the first sum to the second sum is 750 : 630. This can be simplified by dividing both terms by 30, resulting in a ratio of 25 : 21.

Explanation

The expression is evaluated using trigonometric identities for complementary angles where A + B = 90°.

The first part uses the identity sin(A + B) = sin A cos B + cos A sin B: sin 37° cos 53° + cos 37° sin 53° = sin(37° + 53°) = sin 90° = 1

In the numerator of the fraction, since 53° = 90° - 37°, we have cos 53° = sin 37°. Substituting this into the identity sin² θ + cos² θ = 1: 4cos² 37° + 4cos² 53° - 7 = 4cos² 37° + 4sin² 37° - 7 = 4(cos² 37° + sin² 37°) - 7 = 4(1) - 7 = -3

In the denominator, using the identity csc θ = sec(90° - θ), we have csc 43° = sec 47°. Applying the identity sec² θ - tan² θ = 1, or tan² θ - sec² θ = -1: tan² 47° - csc² 43° + 4 = tan² 47° - sec² 47° + 4 = -1 + 4 = 3

Substituting these calculated values back into the main expression: 1 - ⁻³⁄₃ = 1 - (-1) = 1 + 1 = 2

Explanation

Assume the total number of students who appeared for the examination is 100. According to the problem, the number of boys is 45% of 100, which is 45, and the number of girls is the remaining 55. To find the percentage of students who failed, we calculate the number of failures in each group.

Since 60% of the boys passed, the failure rate for boys is 100% - 60% = 40%. The number of boys who failed is 40% × 45 = 18. Similarly, since 70% of the girls passed, the failure rate for girls is 100% - 70% = 30%. The number of girls who failed is 30% × 55 = 16.5. The total number of students who failed is the sum of these two values: 18 + 16.5 = 34.5. Since the total strength was taken as 100, the failure percentage is 34.5%.

Explanation

The algebraic identity for the square of a sum is (x + y)² = x² + y² + 2xy. To find the value of x² + y², the formula can be rearranged to isolate the sum of squares:

x² + y² = (x + y)² - 2xy

Substituting the given values x + y = 3 and xy = -10 into this relationship:

x² + y² = (3)² - 2(-10)

x² + y² = 9 - (-20)

x² + y² = 9 + 20

x² + y² = 29

This method utilizes the fundamental relationship between the sum of two variables, their product, and the sum of their squares, allowing for a direct calculation without solving for x or y individually.

Explanation

Let the total work be the Least Common Multiple (LCM) of 20 and 24, which is 120 units.

The efficiency (work done per day) of A and B together is: E_A₊_B = ¹²⁰⁄₂₀ = 6 units/day

The efficiency of B and C together is: E_B₊_C = ¹²⁰⁄₂₄ = 5 units/day

By subtracting the second efficiency from the first, we find the difference between the efficiencies of A and C: (E_A + E_B) - (E_B + E_C) = 6 - 5 E_A - E_C = 1

It is given that A is twice as good a workman as C, so E_A = 2E_C. Substituting this into the equation: 2E_C - E_C = 1 ⟹ E_C = 1 unit/day

Now, substitute E_C into the efficiency equation for B and C: E_B + 1 = 5 ⟹ E_B = 4 units/day

The problem asks for the time taken by B alone to do 40% of the total work: Target work = 0.40 × 120 = 48 units

Time taken by B = (Target work)/(E_B) = ⁴⁸⁄₄ = 12 days

Explanation

The simple interest (SI) on a sum of 8000 for 2⅖ years at a rate of 10% per annum is calculated using the formula SI = (P × R × T)/(100). Substituting the values, we get SI = (8000 × 10 × 12)/(100 × 5) = 80 × 2 × 12 = 1920.

To calculate the compound interest (CI) for 2⅖ years, we apply the interest yearly. For the first two full years, the rate is 10% per annum. For the remaining fraction of the year, which is ⅖, the applicable rate is ⅖ × 10% = 4%.

The effective compound interest rate can be found using the successive percentage formula x + y + ˣʸ⁄₁₀₀. For the first two years, the effective rate is 10 + 10 + (10 × 10)/(100) = 21%. Including the third period of 4%, the total effective rate becomes 21 + 4 + (21 × 4)/(100) = 25 + 0.84 = 25.84%.

The total compound interest is CI = (25.84)/(100) × 8000 = 25.84 × 80 = 2067.20.

The difference between compound interest and simple interest is CI - SI = 2067.20 - 1920 = 147.20. Alternatively, the difference in percentage points is 25.84% - 24% = 1.84%, and 1.84% of 8000 is 1.84 × 80 = 147.20.

Explanation

The problem states that the average of x occurring 5 times and y occurring 7 times is 37. Since the total number of occurrences is 5 + 7 = 12, the sum is 5x + 7y = 37 × 12 = 444.

Similarly, the average of x occurring 7 times and y occurring 5 times is 35. The sum for this case is 7x + 5y = 35 × 12 = 420.

To find the value of y, we can use the sum and difference of these two equations. Adding the two equations gives 12x + 12y = 864, which simplifies to x + y = 72 by dividing by 12. Subtracting the second equation from the first gives -2x + 2y = 24, which simplifies to y - x = 12 by dividing by 2.

Adding the simplified equations (x + y) + (y - x) = 72 + 12 results in 2y = 84. Solving for y gives y = 42.

Explanation

The cost price of an article serves as the base for calculating both profit and loss. Initially, the trader sells the article at a 16% loss, meaning the selling price is 84% of the cost price. In the second scenario, the trader aims for a 15% gain, which corresponds to a selling price of 115% of the cost price.

The difference between these two selling prices represents the total percentage shift required to move from a 16% loss to a 15% gain. This total percentage change is calculated as: Percentage Difference = 15% - (-16%) = 31%

According to the problem, this 31% difference in the cost price is equivalent to ₹ 192.20. Let the cost price be CP. We can set up the equation: 31% of CP = 192.20 0.31 × CP = 192.20 CP = (192.20)/(0.31)

To simplify the calculation, multiply both the numerator and denominator by 100: CP = ¹⁹²²⁰⁄₃₁

Dividing 19220 by 31: 31 × 6 = 186 192 - 186 = 6 Bringing down the next digit gives 62. 31 × 2 = 62 Adding the remaining zero gives a total of 620. Therefore, the cost price of the article is ₹ 620.

Explanation

The perimeter P of a semi-circle is the sum of the arc length and the diameter. The formula is P = (π d)/(2) + d, which simplifies to P = d((π)/(2) + 1). Using the given values P = 25.7 cm and π = 3.14:

25.7 = d((3.14)/(2) + 1) 25.7 = d(1.57 + 1) 25.7 = d(2.57) d = (25.7)/(2.57) = 10

Explanation

The measure of the angle between two tangents ∠ APB is related to the angle in the minor segment ∠ ACB by the formula ∠ APB = 2∠ ACB - 180°.

Substituting the given value: ∠ APB = 2(102°) - 180° ∠ APB = 204° - 180° = 24°

This relationship originates from basic circle theorems. If D is a point on the major arc, the quadrilateral ADBC is cyclic, making ∠ ADB = 180° - ∠ ACB = 180° - 102° = 78°. The central angle ∠ AOB is twice the angle at the circumference, so ∠ AOB = 2 × ∠ ADB = 156°. In the quadrilateral OAPB, the radii OA and OB meet the tangents at 90° angles. Therefore, ∠ APB and ∠ AOB are supplementary, resulting in ∠ APB = 180° - 156° = 24°.