SSC CHSL Tier 1 2016 Quantitative Aptitude Questions with Solutions

Explanation

To find the x-intercept of a line, set the y-coordinate to 0 in the equation of the line.

Substituting y = 0 into the equation 3x + y = -6: 3x + 0 = -6 3x = -6 x = -2

This gives the coordinate point (-2, 0).

Explanation

When two objects move in the same direction starting at the same time, the ratio of the distances they cover is directly proportional to the ratio of their speeds.

Distance covered by policeman / Distance covered by thief = Speed of policeman / Speed of thief

Let the distance run by the thief before being overtaken be d metres. The policeman covers this distance plus the initial gap of 150 metres, which is d + 150 metres.

Substituting the speeds into the ratio: (d + 150) / d = 9 / 7

Cross-multiplying: 7 × (d + 150) = 9d 7d + 1050 = 9d 2d = 1050 d = 525 metres

Explanation

To find the common factor of the algebraic terms 24b⁶c⁸d², 18a⁶c²d⁴, and 12a⁴b⁴, we determine the greatest common divisor of their numerical coefficients and their variable factors separately.

First, the numerical coefficients are 24, 18, and 12. Their prime factorizations are: 24 = 2³ × 3 18 = 2 × 3² 12 = 2² × 3

The greatest common divisor of these coefficients is the product of the lowest powers of common prime factors: GCD(24, 18, 12) = 2¹ × 3¹ = 6

Second, we examine the variable factors of each term. Since the variable a does not appear in the first term, the variable b does not appear in the second term, and the variables c and d do not appear in the third term, there is no common variable factor across all three expressions.

Multiplying the numerical common factor and the variable common factor yields 6 × 1 = 6.

Explanation

The total value of the stock is the sum of the values of each individual category. The value for each category is determined by multiplying its quantity of stock by its average cost:

Mobile Phones: 44 × 12000 = 5,28,000 rupees

Cameras: 75 × 14000 = 10,50,000 rupees

Tvs: 55 × 56000 = 30,80,000 rupees

Refrigerators: 29 × 53000 = 15,37,000 rupees

ACs: 77 × 26000 = 20,02,000 rupees

Adding these values together gives the total value of all the stock: 5,28,000 + 10,50,000 + 30,80,000 + 15,37,000 + 20,02,000 = 81,97,000 rupees

Since 1 lakh rupees equals 1,00,000 rupees, dividing the total value by 1,00,000 converts it into lakh rupees: 81,97,000 ÷ 1,00,000 = 81.97 lakh rupees

Explanation

The import-to-export ratio for a given year is defined by the relation: Import / Export = Ratio

For the year 2012, the ratio of import to export is 1.1. Given that the imports in 2012 were 1000 crores, the exports in 2012 can be calculated as: E₂012 = 1000 ÷ 1.1 ≈ 909.09 crores

The total exports for the years 2012 and 2013 combined is 4800 crores. This gives the exports in 2013 as: E₂013 = 4800 - 909.09 = 3890.91 crores

The ratio of import to export for the year 2013 is 1.5. Using this ratio, the imports in 2013 are: I₂013 = 1.5 × 3890.91 ≈ 5836.36 crores

Rounding to the nearest integer gives 5836 crores.

Explanation

According to the given table, the height of the child on their 10th birthday was 145 cm and on their 11th birthday was 150 cm.

The increase in height is calculated by subtracting the height on the 10th birthday from the height on the 11th birthday:

150 cm - 145 cm = 5 cm

This calculation shows the height of the child increased by exactly 5 cm during this one-year period.

Explanation

The total time recorded between 10pm and 6am is 8 hours.

Summing the values in the data table gives: 15 + 15 + 5 + 30 + 35 = 100

This indicates that the values represent percentages of the total tracked time.

The percentage of time the user spent in Deep Sleep or Awake is: 15% (Deep Sleep) + 35% (Awake) = 50%

Calculating 50% of the total 8 hours: 50% of 8 hours = 0.50 × 8 hours = 4 hours

Explanation

To factorise the quadratic expression x² + 5x - 14, we need to find two numbers whose product is -14 and whose sum is 5.

These two numbers are 7 and -2, since: 7 × (-2) = -14 7 + (-2) = 5

Splitting the middle term using these numbers: x² + 5x - 14 = x² + 7x - 2x - 14

Grouping the terms to find common factors: x(x + 7) - 2(x + 7) = (x + 7)(x - 2)

Explanation

The general formula to construct a quadratic equation when the sum of the roots (S) and the product of the roots (P) are known is:

x² - Sx + P = 0

By substituting the given values of the sum of the roots (S = 1) and the product of the roots (P = -20) into this formula, we get:

x² - (1)x + (-20) = 0

Simplifying this expression yields:

x² - x - 20 = 0

Explanation

Let the distance covered by steamer be d km. Since the total distance is 434 km, the distance covered by rail is 434 - d km.

Using the relationship Time = Distance / Velocity: Time spent on steamer = d / 25 Time spent on rail = (434 - d) / 65

Since the traveler spends 8 hours more on the steamer than on rail: d / 25 - (434 - d) / 65 = 8

To solve this, multiply the entire equation by the least common multiple of 25 and 65, which is 325: 13d - 5(434 - d) = 8 × 325 13d - 2170 + 5d = 2600 18d = 4770 d = 265 km

Explanation

The total simple interest (SI) is the sum of the interest earned from Jaichand and Kapil. Using the simple interest 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.

Let the common rate of interest be R%.

For the loan to Jaichand: SI₁ = (8800 × R × 13) / 100 = 1144R

For the loan to Kapil: SI₂ = (5500 × R × 12) / 100 = 660R

The total interest received is: SI₁ + SI₂ = 14432 1144R + 660R = 14432 1804R = 14432 R = 14432 / 1804 = 8

Explanation

The relationship between selling price (SP) and marked price (MP) after a discount is given by the formula:

SP = MP × (1 - d/100)

where d is the discount percentage. Substituting the given values (SP = Rs 1150 and d = 8) into the formula:

1150 = MP × (1 - 8/100) 1150 = MP × 0.92 MP = 1150 / 0.92 MP = 1250

Explanation

By substituting the basic trigonometric relationships cosec A = 1/sin A and cot A = cos A/sin A into the given expression, we get:

(1 + cos A)(cosec A - cot A) = (1 + cos A)(1/sin A - cos A/sin A)

Combining the fractions in the second term gives:

(1 + cos A) × (1 - cos A) / sin A

Applying the algebraic identity (a - b)(a + b) = a² - b² to the numerator yields:

(1 - cos² A) / sin A

Using the fundamental trigonometric identity sin² A + cos² A = 1, the numerator simplifies to sin² A:

sin² A / sin A = sin A

Explanation

In Δ ABC, AD is the median to the side BC. Since D is the midpoint of BC, the lengths of BD and CD are: BD = CD = BC / 2 = 12 / 2 = 6 cm

Given that the length of the median AD is also 6 cm, we have: AD = BD = CD = 6 cm

This means that the point D is equidistant from all three vertices A, B, and C. Therefore, D is the circumcenter of Δ ABC, and the circumcircle of the triangle has a radius of 6 cm with D as its center.

Since BC passes through the circumcenter D and has a length equal to the diameter (12 cm), BC is the diameter of this circumcircle. According to Thales's theorem, the angle subtended by a diameter at any point on the circle is a right angle. Consequently, the angle CAB, which is subtended by the diameter BC at vertex A, is 90°.

Explanation

To find the values x can take, we solve both inequalities.

For the first inequality: 3x - 3 < 3 + x/2

Multiplying the entire inequality by 2 to eliminate the fraction: 6x - 6 < 6 + x

Subtracting x from both sides: 5x - 6 < 6

Adding 6 to both sides: 5x < 12 x < 12/5 x < 2.4

For the second inequality: x - 2 ≤ 6 + 2x

Subtracting x from both sides: -2 ≤ 6 + x

Subtracting 6 from both sides: -8 ≤ x

Combining both results, we get the range: -8 ≤ x < 2.4

The value 2 satisfies this inequality as it lies within the interval [-8, 2.4).

Explanation

To find the expression that should be added to 3(x-2y) to obtain 2(3x + y) - 5(2x + 3), we subtract 3(x-2y) from 2(3x + y) - 5(2x + 3).

First, expand both expressions: 3(x-2y) = 3x - 6y

2(3x + y) - 5(2x + 3) = 6x + 2y - 10x - 15 = -4x + 2y - 15

Now, subtract the first simplified expression from the second: (-4x + 2y - 15) - (3x - 6y) = -4x + 2y - 15 - 3x + 6y

Combine the like terms: -4x - 3x = -7x 2y + 6y = 8y -15 remains as is

Rearranging the terms gives: 8y - 7x - 15

Explanation

The circumference of the base of a cylinder is given by the formula C = 2πr, where r is the radius of the base. 2πr = 44 2 × (22/7) × r = 44 r = 7 cm

The curved surface area of the cylinder is given by the formula CSA = 2πrh, where h is the height of the cylinder. 2πrh = 440 44 × h = 440 h = 10 cm

The volume of the cylinder is given by the formula V = πr²h. V = (22/7) × 7² × 10 V = 22 × 7 × 10 V = 1540 cubic cm

Explanation

The standard trigonometric identity for the product of two sines is derived from the cosine angle sum and difference formulas:

cos(A - B) = cos A cos B + sin A sin B cos(A + B) = cos A cos B - sin A sin B

Subtracting the formula for cos(A + B) from cos(A - B) gives:

cos(A - B) - cos(A + B) = (cos A cos B + sin A sin B) - (cos A cos B - sin A sin B)

Canceling the cos A cos B terms simplifies the right side:

cos(A - B) - cos(A + B) = 2 sin A sin B

Explanation

To find the value of x, we can simplify the given algebraic equation:

51.97 - (81.18 - x) - 59.39 = 5.268

First, expand the expression within the parentheses by distributing the negative sign:

51.97 - 81.18 + x - 59.39 = 5.268

Next, group the constant numerical values together on the left-hand side:

(51.97 - 81.18 - 59.39) + x = 5.268

Subtract 81.18 from 51.97:

-29.21 - 59.39 + x = 5.268

Combine the remaining constants on the left-hand side:

-88.60 + x = 5.268

Now, isolate x by adding 88.60 to both sides of the equation:

x = 5.268 + 88.60

x = 93.868

Explanation

Let Haamid's salary be represented by H.

Gadadhar's salary is 3/2 times Haamid's salary, which is (3/2)H. Sarvesh's salary is 4/3 times Haamid's salary, which is (4/3)H.

The ratio of Gadadhar's salary to Sarvesh's salary is calculated by dividing Gadadhar's salary by Sarvesh's:

Ratio = ((3/2)H) / ((4/3)H)

Canceling out H from both the numerator and the denominator:

Ratio = (3/2) / (4/3) Ratio = (3/2) × (3/4) Ratio = 9/8

This simplifies directly to 9:8.

Explanation

Using the trigonometric identity sin(-θ) = -sin(θ), we can write:

sin(-300°) = -sin(300°)

Since 300° lies in the fourth quadrant, we can express it as 360° - 60°:

sin(300°) = sin(360° - 60°) = -sin(60°)

Substituting the standard value of sin(60°) = √3/2 gives:

sin(300°) = -√3/2

Therefore:

sin(-300°) = -(-√3/2) = √3/2

Explanation

Let the original electricity bill amount be x.

Since 31% of the bill is deducted, the remaining percentage of the bill to be paid is: 100% - 31% = 69%

This remaining amount is equal to Rs 1794.

Therefore, we can set up the equation: 69% of x = 1794 0.69 × x = 1794 x = 1794 / 0.69 x = 2600

Explanation

To find the values x can take, we solve the system of inequalities in two parts.

The first inequality is: 2(3x + 5) > 4x - 5 6x + 10 > 4x - 5 2x > -15 x > -7.5

The second inequality is: 4x - 5 < 3x + 2 x < 7

Combining these two inequalities, we get the range: -7.5 < x < 7

The value 6 satisfies this condition as it lies strictly between -7.5 and 7.

Explanation

By using the formula for distance when speeds and the difference in travel times are known:

Distance d = (v₁ × v₂ × Δt) / (v₂ - v₁)

Where: v₁ = 60 km/hr is the speed of the slower car, v₂ = 108 km/hr is the speed of the faster car, Δt = 2 hours is the difference in time taken.

Substituting the given values into the formula:

d = (60 × 108 × 2) / (108 - 60) d = 12960 / 48 d = 270 km

Explanation

To find the numerical value of (x + 4)², we first solve the given linear equation for x:

4x - 5 = 3x - 1

Subtracting 3x from both sides:

x - 5 = -1

Adding 5 to both sides:

x = 4

Substituting this value of x into the expression (x + 4)²:

(4 + 4)² = 8² = 64

Explanation

Let the non-zero number be x. According to the given condition, the sum of the number and twice its reciprocal is 33/4:

x + 2/x = 33/4

Multiplying the entire equation by 4x to eliminate the fractions gives:

4x² + 8 = 33x

4x² - 33x + 8 = 0

This quadratic equation can be solved by splitting the middle term:

4x² - 32x - x + 8 = 0

4x(x - 8) - 1(x - 8) = 0

(4x - 1)(x - 8) = 0

This yields x = 8 or x = 1/4. Since 8 is one of the given options, the number is 8.

Explanation

Let the total work be 1 unit. Since A completes 50% of the work in 12 days, A can complete the entire work alone in 24 days. This gives A a daily work rate of 1/24.

The remaining 50% of the work is completed by A and B together in 8 days. This means A and B working together can complete the entire work in 16 days, giving them a combined daily work rate of 1/16.

B's daily work rate is the difference between the combined rate and A's individual rate: 1/16 - 1/24 = (3 - 2) / 48 = 1/48

Since B's daily work rate is 1/48, B alone will take 48 days to complete the entire work.

Explanation

Let the two parts of 32 be x and 32 - x. According to the given condition, the sum of their squares is 674:

x² + (32 - x)² = 674

Expanding the equation gives:

x² + 1024 - 64x + x² = 674

2x² - 64x + 350 = 0

Dividing the entire equation by 2:

x² - 32x + 175 = 0

Solving this quadratic equation by factoring:

x² - 25x - 7x + 175 = 0

x(x - 25) - 7(x - 25) = 0

(x - 25)(x - 7) = 0

This yields x = 25 or x = 7. Therefore, the two parts are 25 and 7.

Explanation

The total revenue for 13 years is equal to the number of years multiplied by their average: 13 × 78 = 1014 lakhs

The total revenue for the first 7 years is: 7 × 73 = 511 lakhs

The total revenue for the last 7 years is: 7 × 85 = 595 lakhs

Since the 7th year is included in both the first 7 years and the last 7 years, adding these two sums counts the revenue of the 7th year twice. Therefore, the revenue of the 7th year is the sum of these two parts minus the total revenue of the 13 years: Revenue of the 7th year = 511 + 595 - 1014 = 1106 - 1014 = 92 lakhs

Explanation

The y-axis divides the line segment joining the points (x₁, y₁) and (x₂, y₂) in the ratio -x₁ : x₂.

For the given points (2, 5) and (-6, -10), we have x₁ = 2 and x₂ = -6.

Substituting these values into the formula: Ratio = -2 : -6 Ratio = 2 : 6 Ratio = 1 : 3

Alternatively, using the section formula, let the y-axis divide the segment in the ratio k : 1. The x-coordinate of the point of division is given by (k·x₂ + x₁) / (k + 1). Since any point on the y-axis has an x-coordinate of 0: (k·(-6) + 2) / (k + 1) = 0 -6k + 2 = 0 6k = 2 k = 1/3

This gives the required ratio of 1:3.

Explanation

Let the cost price of the watch for the wholesaler be CP.

The successive transaction formula for the final selling price paid by the customer is: Final Price = CP × (1 + Profit%/100) × (1 - Loss%/100)

Substituting the given values into the formula: 1045 = CP × (1 + 25/100) × (1 - 12/100) 1045 = CP × 1.25 × 0.88 1045 = CP × 1.1 CP = 1045 / 1.1 CP = 950

The cost price of the watch for the wholesaler was Rs 950.

Explanation

The interest rate per half-year is r = 5%.

The first deposit of ₹4800 is made on 1st January, so it accrues compound interest for two half-year periods by the end of the year (31st December). The accumulated amount from this first deposit is calculated as: A₁ = P × (1 + r/100)² A₁ = 4800 × (1 + 5/100)² = 4800 × (1.05)² = 4800 × 1.1025 = ₹5292

The interest earned on the first deposit is: I₁ = 5292 - 4800 = ₹492

The second deposit of ₹4800 is made on 1st July, so it accrues interest for only one half-year period by the end of the year. The accumulated amount from this second deposit is: A₂ = P × (1 + r/100)¹ A₂ = 4800 × 1.05 = ₹5040

The interest earned on the second deposit is: I₂ = 5040 - 4800 = ₹240

The total interest gained by the customer at the end of the year is: Total Interest = I₁ + I₂ = 492 + 240 = ₹732

Explanation

The sum of the interior angles of a polygon with n sides is calculated using the formula: Sum = (n - 2) × 180°

We are given that the sum of the interior angles is 2340°. Substituting this value into the formula: 2340° = (n - 2) × 180°

To solve for n, divide both sides by 180°: n - 2 = 2340 / 180 n - 2 = 13

Add 2 to both sides of the equation: n = 13 + 2 n = 15

Explanation

We are given: x = (1 + secA) / tanA

Expressing the trigonometric functions in terms of sine and cosine: secA = 1 / cosA tanA = sinA / cosA

This simplifies the equation to: x = (1 + 1 / cosA) / (sinA / cosA) x = (cosA + 1) / sinA

Applying the half-angle formulas: cosA + 1 = 2 cos²(A/2) sinA = 2 sin(A/2) cos(A/2)

Substituting these values into the expression yields: x = 2 cos²(A/2) / (2 sin(A/2) cos(A/2)) x = cos(A/2) / sin(A/2) x = cot(A/2)

Explanation

In right-angled triangle PQR (right-angled at Q) with altitude QS, the triangles PQS and PQR are similar. From this similarity, we have: PQ² = PS × PR

Using the standard value of PQ = 4√13 cm and PS = 8 cm: (4√13)² = 8 × PR 16 × 13 = 8 × PR 208 = 8 × PR PR = 26 cm

Since PR is the hypotenuse: PR = PS + SR 26 = 8 + SR SR = 18 cm

Explanation

To find the roots of the quadratic equation x² + 3x - 154 = 0, we can factor the quadratic expression by finding two numbers that multiply to -154 and add up to 3.

These two numbers are 14 and -11, because: 14 × (-11) = -154 14 + (-11) = 3

Rewriting the middle term, we get: x² + 14x - 11x - 154 = 0

Factoring by grouping: x(x + 14) - 11(x + 14) = 0 (x - 11)(x + 14) = 0

Setting each factor to zero yields the roots: x - 11 = 0 → x = 11 x + 14 = 0 → x = -14

Explanation

Since PQ is parallel to BC, triangle APQ is similar to triangle ABC. For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths:

Area(APQ) / Area(ABC) = (AP / AB)²

Substituting the given ratio of areas:

25 / 36 = (AP / AB)²

Taking the square root of both sides:

AP / AB = 5 / 6

Since point P lies on the segment AB, we can express AB as the sum of AP and PB:

AB = AP + PB

This means:

AP / (AP + PB) = 5 / 6

6 × AP = 5 × AP + 5 × PB

AP = 5 × PB

AP / PB = 5 / 1

Thus, the ratio of AP to PB is 5:1.

Explanation

The volume of a cylinder is given by the formula: V_cylinder = π × r² × h_cylinder

The volume of a cone is given by the formula: V_cone = ⅓ × π × r² × h_cone

Since the cylinder is melted and recast into a cone of the same radius, their volumes are equal: V_cylinder = V_cone π × r² × h_cylinder = ⅓ × π × r² × h_cone

Dividing both sides by the common term π × r² simplifies the relation to: h_cylinder = ⅓ × h_cone

Rearranging this formula to find the height of the cone: h_cone = 3 × h_cylinder

Substituting the given height of the cylinder (4.8 cm) into the equation: h_cone = 3 × 4.8 = 14.4 cm

Explanation

The reciprocal of 6/5 is 5/6, and the reciprocal of 3/7 is 7/3.

The sum of these reciprocals is: 5/6 + 7/3 = 5/6 + 14/6 = 19/6

The reciprocal of this sum is: 1 ÷ (19/6) = 6/19

Multiplying both the numerator and the denominator by 3 to match the given option gives: (6 × 3) / (19 × 3) = 18/57

Explanation

The total time required by the labourer to complete the entire job is 36 hours.

The labourer works for 9 hours before taking a break. The fraction of the job completed during this period is: 9/36 = 1/4 = 0.25

To find the fraction of the job that is yet to be done, subtract the completed fraction from the total work, which represents 1: 1 - 0.25 = 0.75

Explanation

The equation of a line with slope m that passes through a given point (x₁, y₁) is determined using the point-slope formula:

y - y₁ = m(x - x₁)

Substituting the given slope m = -3/7 and the point (5, -2) into the formula:

y - (-2) = -3/7 (x - 5)

y + 2 = -3/7 (x - 5)

To clear the fraction, multiply both sides of the equation by 7:

7(y + 2) = -3(x - 5)

7y + 14 = -3x + 15

Rearranging the terms to the standard form Ax + By = C by adding 3x to both sides and subtracting 14 from both sides:

3x + 7y = 15 - 14

3x + 7y = 1

Explanation

The sum of the interior angles of an n-sided polygon is given by the formula:

S = (n - 2) × 180°

Substituting the given sum of 1440° into the formula:

1440 = (n - 2) × 180

Dividing both sides by 180:

n - 2 = 1440 / 180

n - 2 = 8

Solving for n:

n = 10

Therefore, the polygon has 10 sides.

Explanation

Let the initial price of the item be ₹100. A discount of 20% reduces the price to ₹80. Applying the promo code gives a 30% cashback on this transaction amount of ₹80, which equals ₹24. The net amount paid by the customer is ₹80 - ₹24 = ₹56. The total discount received is ₹100 - ₹56 = ₹44, which represents an effective discount of 44%.

Alternatively, the effective rate of successive discounts of d₁% and d₂% is calculated using the formula: Effective Discount = d₁ + d₂ - (d₁ × d₂)/100

Substituting d₁ = 20 and d₂ = 30: Effective Discount = 20 + 30 - (20 × 30)/100 = 50 - 6 = 44%

Explanation

The altitude h of an equilateral triangle with side length a is given by the formula:

h = (√3/2) × a

Substituting the given side length a = 15 cm into this formula:

h = (√3/2) × 15 = (15√3)/2 cm

Explanation

Let the marks obtained by the two students be x and y, where x is the higher mark.

The difference between their marks is given as 19: x - y = 19

The higher mark x is 60% of the sum of both marks: x = 0.60(x + y) x = 0.60x + 0.60y 0.40x = 0.60y 2x = 3y x = 1.5y

Substituting the expression for x into the difference equation: 1.5y - y = 19 0.5y = 19 y = 38

Using this to calculate the higher mark: x = 38 + 19 = 57

Explanation

By using the basic trigonometric definitions, we can write sec²A cosec²A in terms of sine and cosine:

sec²A cosec²A = (1 / cos²A) × (1 / sin²A) = 1 / (sin²A cos²A)

Using the fundamental trigonometric identity sin²A + cos²A = 1 in the numerator:

1 / (sin²A cos²A) = (sin²A + cos²A) / (sin²A cos²A) = 1 / cos²A + 1 / sin²A = sec²A + cosec²A

Using the standard identities sec²A = 1 + tan²A and cosec²A = 1 + cot²A:

sec²A + cosec²A = (1 + tan²A) + (1 + cot²A) = tan²A + cot²A + 2

Therefore, x = tan²A + cot²A + 2.

Explanation

The formula for the compound interest amount is A = P(1 + R/100)ⁿ, where A is the amount, P is the principal sum, R is the rate of interest per annum, and n is the time in years.

Substituting the given values: 25088 = 20000(1 + R/100)²

Divide both sides by 20000: (1 + R/100)² = 25088 / 20000 (1 + R/100)² = 12544 / 10000

Taking the square root on both sides: 1 + R/100 = √(12544 / 10000) 1 + R/100 = 112 / 100 1 + R/100 = 1.12 R/100 = 0.12 R = 12 percent

Explanation

For any two similar triangles, the ratio of their perimeters is equal to the ratio of their corresponding side lengths.

Perimeter of ΔGHI / Perimeter of ΔKLM = GH / KL

Using the given ratio and the length of GH:

1 / 4 = 2 / KL

Solving for KL gives:

KL = 2 × 4 = 8 cm.

Explanation

Using standard trigonometric values: sin(π/6) = ½ sec(π/6) = 2/√3

Substituting these values into the expression for x: x = ½ + 2/√3

Finding a common denominator of 2√3 to combine the fractions: x = (√3 + 4) / 2√3

Explanation

The total surface area of a hemisphere is given by the formula 3πr², where r is the radius of the hemisphere. The curved surface area of the hemisphere is represented by the formula 2πr².

By comparing these two formulas, the relationship between the curved surface area and the total surface area can be expressed as: Curved Surface Area / Total Surface Area = 2πr² / 3πr² = 2/3

This means the curved surface area is always two-thirds of the total surface area: Curved Surface Area = 2/3 × Total Surface Area

Substituting the given value of 41.58 sq cm into the equation: Curved Surface Area = 2/3 × 41.58 = 2 × 13.86 = 27.72 sq cm