SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

The expression is evaluated using trigonometric identities and complementary angle relations where cot(90° - θ) = tan θ and sec(90° - θ) = csc θ.

For the first term, sec² 54° - cot² 36° = sec² 54° - tan² (90° - 36°) = sec² 54° - tan² 54°. Applying the identity sec² θ - tan² θ = 1, this value is 1.

For the second term, ³⁄₂ sin² 37° × sec² 53° = ³⁄₂ sin² 37° × csc² (90° - 53°) = ³⁄₂ sin² 37° × csc² 37°. Since sin θ × csc θ = 1, this simplifies to ³⁄₂ × 1 = ³⁄₂.

For the third term, (2)/(√(3)) tan 60° = (2)/(√(3)) × √3 = 2.

Summing these results gives 1 + ³⁄₂ + 2 = 3 + ³⁄₂ = ⁹⁄₂.

Explanation

The thief travels alone for 1 hour, from 3:00 a.m. to 4:00 a.m., covering a distance of 57 km/h × 1 h = 57 km. To catch the thief, the owner must close this 57 km gap using the relative speed between the two vehicles.

Since both are moving in the same direction, the relative speed is the difference between their speeds: Vᵣₑₗ = 76 km/h - 57 km/h = 19 km/h.

The time required to catch the thief is calculated by the formula: Time = (Distance)/(Relative Speed) = (57 km)/(19 km/h) = 3 hours.

Adding this duration to the time the chase began (4:00 a.m.), the owner catches the thief at 4:00 a.m. + 3 hours = 7:00 a.m..

Explanation

Let the present age of Savita be x years. According to the first condition, the present age of Rahim is 5x years.

In seven years, Savita will be x + 7 years old and Rahim will be 5x + 7 years old. The relationship between their ages at that time is given by the equation:

5x + 7 = 3(x + 7)

Expanding the right side of the equation:

5x + 7 = 3x + 21

Subtracting 3x and 7 from both sides to isolate the variable:

5x - 3x = 21 - 7 2x = 14 x = 7

The present age of Savita is 7 years. Therefore, the present age of Rahim is 5 × 7 = 35 years.

Explanation

Let r be the radius, h be the height, and l be the slant height of the cone. The curved surface area of a cone is given by π r l and its base area is π r². According to the given condition, the curved surface area is twice the base area, so π r l = 2 π r². This simplifies to l = 2r, which implies r = ˡ⁄₂.

In a right circular cone, the relationship between slant height, radius, and height is l² = r² + h². Substituting r = ˡ⁄₂ into this equation, we get h² = l² - (ˡ⁄₂)² = l² - (l²)/(4) = (3l²)/(4). Taking the square root of both sides, the height is h = l√3{2}.

Given the slant height l = 6√3 cm, we substitute this value into the expression for h: h = (6√3) × √3{2} h = (6 × 3)/(2) = ¹⁸⁄₂ = 9 cm.

Explanation

The total simple interest earned over multiple periods with different rates is given by the formula SIₜₒₜₐₗ = (P × (R₁ T₁ + R₂ T₂ + R₃ T₃))/(100), where P is the original sum invested.

Substituting the given values for each period: For the first 8 years at 7.25%: R₁ T₁ = 7.25 × 8 = 58 For the next 6 years at 8.5%: R₂ T₂ = 8.5 × 6 = 51 For the final 4 years at 6.5%: R₃ T₃ = 6.5 × 4 = 26

The total percentage of the principal earned as interest is: 58 + 51 + 26 = 135%

Now, equate this to the total interest amount provided: 35,100 = (P × 135)/(100) P = (35,100 × 100)/(135)

By dividing 35,100 by 135, we get: 35,100 ÷ 135 = 260 P = 260 × 100 = 26,000

Explanation

Given the expression x⁴ + (1)/(x⁴) = 142, we first find x² + (1)/(x²) by adding 2 to both sides and taking the square root: x⁴ + (1)/(x⁴) + 2 = 142 + 2 (x² + (1)/(x²))² = 144 x² + (1)/(x²) = √(144) = 12

Similarly, we find x + ¹⁄ₓ by adding 2 again: x² + (1)/(x²) + 2 = 12 + 2 (x + ¹⁄ₓ)² = 14 x + ¹⁄ₓ = √(14)

Next, we calculate x³ + (1)/(x³) using the formula a³ + b³ = (a+b)³ - 3ab(a+b): x³ + (1)/(x³) = (√(14))³ - 3(√(14)) x³ + (1)/(x³) = 14√(14) - 3√(14) = 11√(14)

The value of x⁷ + (1)/(x⁷) is derived from the product of the cubic and quartic terms: (x³ + (1)/(x³))(x⁴ + (1)/(x⁴)) = x⁷ + (1)/(x⁷) + (x + ¹⁄ₓ) x⁷ + (1)/(x⁷) = (x³ + (1)/(x³))(x⁴ + (1)/(x⁴)) - (x + ¹⁄ₓ) Substituting the calculated values: x⁷ + (1)/(x⁷) = (11√(14))(142) - √(14) x⁷ + (1)/(x⁷) = 1562√(14) - √(14) = 1561√(14)

Explanation

Let the total work be the Least Common Multiple (LCM) of 12 and 18, which is 36 units. The efficiency of A is 36 / 12 = 3 units per day, and the efficiency of B is 36 / 18 = 2 units per day.

Since they work on alternate days starting with A, in a 2-day cycle (A on the first day and B on the second day), the total work completed is 3 + 2 = 5 units. To find how many such cycles are completed, we divide the total work by the work done in one cycle: 36 / 5 = 7 cycles with a remainder of 1 unit.

The time taken for 7 cycles is 7 × 2 = 14 days, and the work completed is 7 × 5 = 35 units. The remaining work is 36 - 35 = 1 unit. On the 15th day, it is A's turn to work. Since A's efficiency is 3 units per day, the time taken by A to finish the remaining 1 unit is 1 / 3 day. Therefore, the total time to finish the work is 14 + 1/3 = 14 ⅓ days.

Explanation

When two people work together to complete a task, the total time taken is calculated using the formula T = (xy)/(x + y), where x and y are the individual times taken by each person. In this case, Raima takes x = 15 days and Joe takes y = 10 days.

Substituting these values into the formula: T = (15 × 10)/(15 + 10) = ¹⁵⁰⁄₂₅ = 6

Alternatively, using the efficiency method, consider the total work as the Least Common Multiple (LCM) of 15 and 10, which is 30 units. Raima's rate of work is 30 / 15 = 2 units per day, and Joe's rate of work is 30 / 10 = 3 units per day. Their combined rate of work is 2 + 3 = 5 units per day. Therefore, the time taken to complete the 30 units of work together is 30 / 5 = 6 days.

Explanation

When people work together for the same duration, their wages are distributed in the ratio of their efficiencies. Efficiency is inversely proportional to the time taken to complete the work individually.

The ratio of efficiencies of A, B, and C is: Efficiency Ratio A : B : C = ⅒ : ¹⁄₁₅ : ¹⁄₂₀

To simplify this ratio, multiply each term by the Least Common Multiple (LCM) of 10, 15, and 20, which is 60: Ratio A : B : C = (⅒ × 60) : (¹⁄₁₅ × 60) : (¹⁄₂₀ × 60) = 6 : 4 : 3

The total sum of the ratio parts is 6 + 4 + 3 = 13 units. These 13 units correspond to the total wage of 2,600. Value of 1 unit = ²⁶⁰⁰⁄₁₃ = 200

C's share corresponds to 3 units in the ratio. Therefore: C's wage = 3 × 200 = 600

Explanation

When a seller sells goods at cost price but uses a false weight, the profit percentage is calculated using the formula: Gain % = (Error)/(True Weight - Error) × 100. Alternatively, if x is the weight actually given for a true weight of 1000 gm, the formula is Gain % = (1000 - x)/(x) × 100.

The given gain is 5⁵⁄₁₉%, which can be converted to an improper fraction: (5 × 19 + 5)/(19)% = ¹⁰⁰⁄₁₉%.

Substituting the gain percentage into the formula: ¹⁰⁰⁄₁₉ = (1000 - x)/(x) × 100

Dividing both sides by 100: ¹⁄₁₉ = (1000 - x)/(x)

Cross-multiplying gives: x = 19(1000 - x) x = 19000 - 19x 20x = 19000 x = ¹⁹⁰⁰⁰⁄₂₀ = 950 gm

Thus, the seller gives 950 gm instead of 1 kg.

Explanation

In △ PQR, the angle sum property states that the sum of all interior angles is 180°. Given ∠ PQR = 60° and ∠ PRQ = 30°, we calculate the measure of the third angle ∠ QPR: ∠ QPR = 180° - (∠ PQR + ∠ PRQ) ∠ QPR = 180° - (60° + 30°) = 90°

The problem states that △ PQR ≅ △ STU. In congruent triangles, corresponding parts (angles and sides) are equal. Matching the vertices from the congruence notation, vertex P corresponds to S, Q corresponds to T, and R corresponds to U. This is consistent with the given information that ∠ PQR = ∠ STU = 60°. Consequently, ∠ TSU must be equal to its corresponding angle ∠ QPR. Since ∠ QPR = 90°, it follows that ∠ TSU = 90°.

Explanation

The divisibility rule for 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9. For the number 83061, the sum of the digits is calculated as 8 + 3 + 0 + 6 + 1 = 18. Since 18 is a multiple of 9, the number 83061 is divisible by 9.

Explanation

The expression a : b :: b : c represents a continued proportion, which implies the mathematical relationship (a)/(b) = (b)/(c). Cross-multiplying gives the formula b² = ac, where b is the mean proportional between a and c. Given that b = 96, we calculate b² = 96 × 96 = 9216. To find the possible values of a and c, their product must equal 9216. For the pair a = 32 and c = 288, the product is 32 × 288 = 9216. Since this matches the value of b², this pair satisfies the condition of the continued proportion.

Explanation

The incenter of a triangle is the point where the three internal angle bisectors intersect. In any triangle ABC, if I is the incenter, the angle subtended by side BC at the incenter is denoted as ∠ BIC. In the triangle BIC, the sum of the angles must be 180°. Since BI and CI are angle bisectors, the angles at the base of △ BIC are (∠ B)/(2) and (∠ C)/(2). Therefore, ∠ BIC = 180° - ((∠ B)/(2) + (∠ C)/(2)). Using the triangle angle sum property for △ ABC, where ∠ A + ∠ B + ∠ C = 180°, we can substitute (∠ B + ∠ C)/(2) = 90° - (∠ A)/(2). Substituting this into the first equation yields ∠ BIC = 180° - (90° - (∠ A)/(2)) = 90° + (∠ A)/(2). The statement is false because it claims the angle is 90° more than the opposite vertex angle, rather than 90° more than half of the opposite vertex angle.

Explanation

Following the VBODMAS rule, we first evaluate the of operations, followed by division, multiplication, addition, and subtraction.

First, calculate the parts involving of: 59.5 of ¹⁄₁₇ = (59.5)/(17) = 3.5 ⁷⁄₂ of 5 = 3.5 × 5 = 17.5

Next, solve the division and multiplication in the main expression: 7.5 × (17.2 ÷ 8.6) + 3.5 - 17.5 7.5 × 2 + 3.5 - 17.5 15 + 3.5 - 17.5

Finally, perform the addition and subtraction: 18.5 - 17.5 = 1

Explanation

The relative speed of the policeman with respect to the thief is Vᵣₑₗ = Vₚₒₗᵢ_cₑₘₐₙ - Vₜₕᵢₑ_f = 75 km/h - 60 km/h = 15 km/h. This represents the rate at which the policeman closes the initial gap of 3 km. The time required to catch the thief is calculated by t = (Initial Distance)/(Relative Speed) = (3 km)/(15 km/h) = ⅕ hour. During this interval, the thief travels at a constant speed of 60 km/h. Therefore, the distance covered by the thief until the moment of capture is Distance = Speed of thief × Time = 60 km/h × ⅕ hour = 12 km.

Explanation

When one person in a group is replaced by another, the weight of the new person is calculated by adding the total change in weight to the weight of the person who left. The total increase in the weight of the group is the product of the number of persons and the increase in the average weight.

Weight of new person = Weight of replaced person + (Number of persons × Increase in average weight)

Given the number of persons is 15 and the average weight increases by 1.2 kg, the total increase in weight is: 15 × 1.2 = 18 kg

The weight of the person who was replaced is 51 kg. Therefore, the weight of the new man is: 51 kg + 18 kg = 69 kg

Explanation

Using the trigonometric identities for complementary angles, cot(90° - θ) = tan θ and cos(90° - θ) = sin θ. The first term of the expression, cot 85°, is equivalent to cot(90° - 5°), which simplifies to tan 5°. The second term, cos 75°, is equivalent to cos(90° - 15°), which simplifies to sin 15°. Substituting these values back into the original expression cot 85° + cos 75° yields tan 5° + sin 15°.

Explanation

The calculation follows the standard algebraic identity (x + y + z)² = x² + y² + z² + 2(xy + yz + zx). Substituting the given values x + y + z = 4 and x² + y² + z² = 12 into the equation gives 4² = 12 + 2(xy + yz + zx). Simplifying the equation leads to 16 = 12 + 2(xy + yz + zx), which results in 4 = 2(xy + yz + zx). Dividing both sides by 2, the value for the expression is xy + yz + zx = 2.

Explanation

Given the equation 5y + (5)/(y) = 11, we factor out the common term 5 to get 5(y + (1)/(y)) = 11. Dividing both sides by 5 yields y + (1)/(y) = ¹¹⁄₅.

To find the value of y² + (1)/(y²⁾, we use the algebraic identity (a + b)² = a² + b² + 2ab. Setting a = y and b = (1)/(y), we have:

(y + (1)/(y))² = y² + (1)/(y²⁾ + 2(y)((1)/(y)) (y + (1)/(y))² = y² + (1)/(y²⁾ + 2

Substituting the value y + (1)/(y) = ¹¹⁄₅ into the equation:

(¹¹⁄₅)² = y² + (1)/(y²⁾ + 2 ¹²¹⁄₂₅ = y² + (1)/(y²⁾ + 2

Rearranging to solve for y² + (1)/(y²⁾:

y² + (1)/(y²⁾ = ¹²¹⁄₂₅ - 2 y² + (1)/(y²⁾ = (121 - 50)/(25) y² + (1)/(y²⁾ = ⁷¹⁄₂₅

Converting the improper fraction ⁷¹⁄₂₅ into a mixed fraction, we divide 71 by 25, which gives a quotient of 2 and a remainder of 21. Thus, the value is 2²¹⁄₂₅.

Explanation

The expression (2tan A)/(1 + tan²⁾^A is a standard trigonometric identity representing sin 2A. Using the trigonometric identity 1 + tan²A = sec²A, the expression simplifies as follows:

(2tan A)/(1 + tan²⁾^A = (2tan A)/(sec²⁾^A

By substituting tan A = (sin A)/(cos A) and sec²A = (1)/(cos²⁾^A, we get:

(2 sin A)/(cos A){(1)/(cos²⁾^A} = 2 (sin A)/(cos A) · cos²A = 2sin Acos A

The double-angle formula for sine states that sin 2A = 2sin Acos A, which confirms the identity.

Explanation

The trigonometric identity for complementary angles states that sin A = cos (90° - A). By applying this identity to the first term of the expression, we substitute A = 50° + θ. This gives sin (50° + θ) = cos (90° - (50° + θ)). Simplifying the terms inside the parentheses, we get 90° - 50° - θ, which equals 40° - θ. Therefore, sin (50° + θ) = cos (40° - θ). Substituting this back into the original expression, we have cos (40° - θ) - cos (40° - θ) = 0.

Explanation

The net effective percentage of two successive changes x% and y% is given by the formula x + y + ˣʸ⁄₁₀₀. For two successive increments of 30%, the net percentage increase is 30 + 30 + (30 × 30)/(100) = 60 + 9 = 69%.

For two successive decrements of 30%, the net percentage decrease is calculated using x = -30 and y = -30, which gives -30 - 30 + ⁽⁻³⁰⁾⁽⁻³⁰⁾⁄₁₀₀ = -60 + 9 = -51%. The magnitude of this net decrement is 51%.

To find how much the increment (69%) is more than the decrement (51%) in percentage terms, use the formula (Value₁ - Value₂)/(Value₂) × 100.

(69 - 51)/(51) × 100 = ¹⁸⁄₅₁ × 100

Simplifying the fraction by dividing both numerator and denominator by 3 gives ⁶⁄₁₇ × 100. Calculating the decimal value, 600 ÷ 17 ≈ 35.294%, which rounds to 35.29%.

Explanation

The total surface area of a cuboid is given by the formula 2(lw + wh + hl), where l, w, and h represent the length, width, and height of the cuboid respectively. Substituting the given dimensions 5 cm, 7 cm, and 11 cm into the formula, we get 2(5 × 7 + 7 × 11 + 11 × 5). Calculating the individual products inside the parentheses results in 2(35 + 77 + 55). Summing these values gives 2(167), which equals 334 cm².

Explanation

The time taken by two individuals to complete a task together can be calculated using the formula T = (xy)/(x+y), where x and y represent the time taken by each person to finish the work alone. In this case, the man finishes the work in x = 10 days and the woman finishes it in y = 12 days. Substituting these values into the formula gives T = (10 × 12)/(10 + 12) = ¹²⁰⁄₂₂. Simplifying this fraction by dividing both the numerator and the denominator by 2 results in ⁶⁰⁄₁₁ days.

Explanation

To identify the maximum discount, calculate the percentage for each offer. For Buy X, Get Y Free schemes, the discount percentage is determined by the formula (Y)/(X+Y) × 100. Using this, Scheme A offers ⁴⁄₆₊₄ × 100 = 40%, while Scheme B offers ⁵⁄₅₊₅ × 100 = 50%. For successive discounts of x% and y%, the equivalent single discount is given by x + y - ˣʸ⁄₁₀₀. For Scheme C, this results in 18 + 24 - (18 × 24)/(100) = 42 - 4.32 = 37.68%. The comparison of these effective rates shows that 50% is the highest available discount.

Explanation

The radius of a circle is perpendicular to the tangent at the point of contact, forming a right-angled triangle where the distance from the center (25 cm) is the hypotenuse and the tangent length (20 cm) is one leg. Using the Pythagorean theorem, the radius squared is 25 squared minus 20 squared (625 - 400 = 225), which gives a radius of 15 cm.

Explanation

The expression simplifies as follows: 8 + (-12) - (-17). This results in 8 - 12 + 17, which calculates to 13.

Explanation

Supraja and Kausalya have daily efficiencies of 2 units and 3 units respectively, based on a total work of 60 units (LCM of 30 and 20). Working on alternate days, they complete 5 units every 2 days. Since 75% of the work is 45 units, it will take exactly 9 cycles of 2 days each, resulting in 18 days.

Explanation

The combined daily efficiency of (A+B), (B+C), and (C+A) is 1/9 + 1/12 + 1/18 = 9/36 = 1/4. Since this sum represents 2(A+B+C), the efficiency of A, B, and C working together is (1/4) / 2 = 1/8, which means they can finish the project in 8 days.

Explanation

In a quadrilateral formed by the center of the circle, the points of tangency, and the external point, the angles at the points of contact are 90 degrees each. Therefore, the angle between the tangents and the angle subtended by the radii at the center are supplementary: 180 - 24 = 156 degrees.

Explanation

The time taken by A alone to finish the work is calculated using the formula T_A = (xy)/(y - x), where x is the time taken by A and B together and y is the time taken by B alone. Substituting the values x = 8 and y = 24:

T_A = (8 × 24)/(24 - 8) T_A = ¹⁹²⁄₁₆ T_A = 12

Alternatively, using the efficiency method, assume the total work is the LCM of 8 and 24, which is 24 units. The combined efficiency of A and B is 24 / 8 = 3 units per day, while the efficiency of B alone is 24 / 24 = 1 unit per day. The efficiency of A is the difference between the combined efficiency and B efficiency, which is 3 - 1 = 2 units per day. The time taken by A to complete the work is 24 / 2 = 12 days.

Explanation

The simple interest earned is the difference between the final amount and the principal, which is 12,300 - 9,840 = 2,460. Using the formula for simple interest (I = P * R * T / 100), we have 2,460 = (9,840 * 5 * T) / 100, which simplifies to 2,460 = 492 * T, giving T = 5 years.

Explanation

The time taken by Q to complete the 400 m race is calculated using the formula Time = (Distance)/(Speed). Given Q's speed is 2 m/sec, the time taken by Q is T_Q = ⁴⁰⁰⁄₂ = 200 seconds. Since P gives Q a head-start of 2 seconds and they both finish the race at the same time, P effectively runs for 2 seconds less than Q. Therefore, the time taken by P to finish the race is T_P = T_Q - 2 = 200 - 2 = 198 seconds.

Explanation

Let the original price of the article be P. A price increase of r% is represented by the multiplier (1 + ʳ⁄₁₀₀), and a subsequent decrease of r% is represented by the multiplier (1 - ʳ⁄₁₀₀). Given that the final price is Rs. 1, we can set up the following equation:

P × (1 + ʳ⁄₁₀₀) × (1 - ʳ⁄₁₀₀) = 1

Using the algebraic identity (a + b)(a - b) = a² - b², the product of the two multipliers simplifies as follows:

P × (1² - (ʳ⁄₁₀₀)²) = 1 P × (1 - (r²)/(10000)) = 1

To isolate P, we simplify the expression within the parentheses by taking a common denominator:

P × ((10000 - r²)/(10000)) = 1

Solving for P by multiplying both sides by the reciprocal of the fraction:

P = (10000)/(10000 - r²)

Explanation

The total cost price is calculated by adding the purchase cost (2,400) to the transportation cost (600 bananas x 0.25 = 150), giving 2,550. The total selling price for 300 pairs at 10 per pair is 3,000. The profit percentage is (450 / 2,550) x 100, which results in approximately 17.6%.

Explanation

According to the triangle inequality theorem, the sum of any two sides of a triangle must be strictly greater than the third side. Therefore, x must satisfy the conditions x < 12 + 17 (x < 29) and x + 12 > 17 (x > 5), which combines to 5 < x < 29.

Explanation

The relative speed between the policeman and the thief is 1 km/h (11 km/h - 10 km/h). In 6 minutes, the distance covered relatively is 1 km/h multiplied by 6/60 hours, which equals 0.1 km or 100 meters. Subtracting this from the initial distance of 200 meters leaves a remaining distance of 100 meters.

Explanation

Using the algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)((a + b + c)² - 3(ab + bc + ca)), we substitute the given values: a³ + b³ + c³ - 3(-1) = 7(7² - 3(11)). This simplifies to a³ + b³ + c³ + 3 = 7(49 - 33) = 7(16) = 112, so a³ + b³ + c³ = 112 - 3 = 109.

Explanation

For any real value of θ, the terms sin²θ and cos²θ are within the interval [0, 1]. For any base x in the interval [0, 1], increasing the exponent results in a value less than or equal to the original base, such that xⁿ ≤ x for n ≥ 1. Applying this property to the given expression, we observe that sin¹²θ = (sin²θ)⁶ ≤ sin²θ and cos²⁰θ = (cos²θ)¹⁰ ≤ cos²θ.

Summing these inequalities leads to sin¹²θ + cos²⁰θ ≤ sin²θ + cos²θ. Using the fundamental trigonometric identity sin²θ + cos²θ = 1, it follows that sin¹²θ + cos²⁰θ ≤ 1. The maximum value of 1 is achieved at specific points where one term is 1 and the other is 0. For instance, if θ = 0, the expression becomes 0¹² + 1²⁰ = 1, and if θ = π/2, it becomes 1¹² + 0²⁰ = 1. Thus, the maximum possible value for the expression is 1.

Explanation

To find the value of the given expression, substitute the standard trigonometric values: tan 60° = √3, cos 30° = √3{2}, sec 30° = (2)/(√(3)), and cot 60° = (1)/(√(3)).

The expression is: ⅔ tan² 60° + 3 cos² 30° - 2 sec² 30° - ¾ cot² 60°

Substituting the values: ⅔(√3)² + 3(√3{2})² - 2((2)/(√(3)))² - ¾((1)/(√(3)))²

Simplifying each term: ⅔(3) + 3(¾) - 2(⁴⁄₃) - ¾(⅓) 2 + ⁹⁄₄ - ⁸⁄₃ - ¼

Grouping the terms with a denominator of 4: 2 + (⁹⁄₄ - ¼) - ⁸⁄₃ 2 + ⁸⁄₄ - ⁸⁄₃ 2 + 2 - ⁸⁄₃ 4 - ⁸⁄₃

Converting to a single fraction: (12 - 8)/(3) = ⁴⁄₃

In decimal form, ⁴⁄₃ is approximately 1.33.

Explanation

Let the required number be x. The problem states that the square of 72 is equal to the product of 216 and x, which can be written as 72² = 216 × x. To solve for x, we rearrange the equation to x = (72 × 72)/(216). We can simplify this calculation by noting that 216 = 72 × 3. Substituting this into the equation, we get x = (72 × 72)/(72 × 3). After canceling the common factor of 72 from the numerator and denominator, the expression becomes x = ⁷²⁄₃, which simplifies to 24.

Explanation

The combined efficiency of P+Q is 1/24, Q+R is 1/30, and P+R is 1/40. Solving these, we find individual efficiencies (out of a total work of 120 units) as P=2, Q=3, and R=1. New efficiencies are X=8, Y=1.5, and Z=2.5, totaling 12 units/day, so the work is completed in 120/12 = 10 days.

Explanation

The expression is simplified using the BODMAS rule, which dictates the order of operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction. We begin with the innermost parentheses: 83 - 4 × 15 = 83 - 60 = 23. Substituting this back, we solve the curly brackets: 9 × 9 + 23 = 81 + 23 = 104. Next, we simplify the terms within the square brackets: 150 - 104 = 46. The expression now reduces to 325 + 276 ÷ 46. Following the order of operations, we perform the division before the addition: 276 ÷ 46 = 6. Finally, adding 6 to 325 results in 325 + 6 = 331.

Explanation

To calculate the profits, let the marked price (MP) be 100. The cost price (CP) is 70% below the marked price, so CP = 100 - 70 = 30. The first selling price (SP1) after a 30% discount is 100 - 30 = 70, resulting in a profit (P1) of 70 - 30 = 40. The second selling price (SP2) includes an additional 20% discount on the discounted price (70), so SP2 = 70 - (0.20 * 70) = 56, resulting in a profit (P2) of 56 - 30 = 26. The ratio of the profit with the membership card to the profit without it is (26 / 40) * 100 = 65%.

Explanation

Given (x - 1/x) = 2√2, squaring both sides gives x² + 1/x² = 10. To find x⁶ - 1/x⁶, we can use the identity (x² - 1/x²)(x⁴ + 1 + 1/x⁴). First, (x² - 1/x²)² = (x² + 1/x²)² - 4 = 100 - 4 = 96, so (x² - 1/x²) = √96 = 4√6. Second, (x⁴ + 1/x⁴) = (x² + 1/x²)² - 2 = 100 - 2 = 98. Thus, x⁶ - 1/x⁶ = (4√6)(98 + 1) = 4√6 * 99 = 396√6.

Explanation

Dividing the given equation 2y + 2/y = 7 by 2 gives y + 1/y = 7/2. Squaring both sides yields y² + 1/y² + 2 = 49/4, and subtracting 2 from 49/4 results in 41/4.

Explanation

The total surface area of a cone is calculated using the formula pi * r * (l + r), where r is the radius and l is the slant height. Substituting the given values: 3.14 * 6 * (10 + 6) = 3.14 * 6 * 16 = 301.44 square centimeters.

Explanation

The sine function is an odd function, which means it follows the identity sin(-θ) = -sin(θ). Applying this property, we can rewrite the expression as sin(-405°) = -sin(405°). Since the sine function is periodic with a period of 360°, we can simplify the angle by subtracting 360° to find its coterminal equivalent: 405° = 360° + 45°. This gives sin(405°) = sin(45°). Given the standard trigonometric value sin(45°) = (1)/(√(2)), substituting this back into our original negative expression yields -(1)/(√(2)).

Explanation

The passing marks for the examination are the sum of the marks obtained by the student and the marks by which the student failed. Therefore, the passing marks are 210 + 14 = 224. The problem states that the pass percentage is 35%. If M represents the maximum marks of the examination, then 35% of M is equal to 224.

Using the percentage formula: ³⁵⁄₁₀₀ × M = 224 M = (224 × 100)/(35)

Dividing both 224 and 35 by their common factor 7: M = (32 × 100)/(5) M = 32 × 20 M = 640