SSC CGL Tier 1 2020 Quantitative Aptitude Questions with Solutions

Explanation

The divisibility rule for 8 states that a number is divisible by 8 if the number formed by its last three digits is also divisible by 8. In the given number 1005x4, the last three digits are 5x4. We need to find the smallest integer x such that the three-digit number 5x4 is divisible by 8. Testing the smallest possible integer value x = 0, we get the number 504. By division, 504 ÷ 8 = 63 with no remainder. Since 504 is perfectly divisible by 8, the smallest integer value for x that satisfies the condition is 0.

Explanation

To find the value of x² + y², square both given expressions and add them together.

Expanding x²: x² = (4cos A + 5sin A)² = 16cos² A + 25sin² A + 2(4cos A)(5sin A) x² = 16cos² A + 25sin² A + 40cos Asin A

Expanding y²: y² = (4sin A - 5cos A)² = 16sin² A + 25cos² A - 2(4sin A)(5cos A) y² = 16sin² A + 25cos² A - 40sin Acos A

Adding the two equations results in the cross-multiplication terms 40cos Asin A and -40sin Acos A cancelling each other out: x² + y² = (16cos² A + 16sin² A) + (25sin² A + 25cos² A) x² + y² = 16(cos² A + sin² A) + 25(sin² A + cos² A)

Using the fundamental trigonometric identity sin² A + cos² A = 1: x² + y² = 16(1) + 25(1) x² + y² = 16 + 25 = 41

Explanation

The distance covered in one complete round of a circular path is equal to its circumference, which is calculated using the formula C = 2π r. Given the radius r = 42 m and π = ²²⁄₇, the circumference of the garden is 2 × ²²⁄₇ × 42 = 2 × 22 × 6 = 264 m. Since the runner completes 8 rounds, the total distance covered is obtained by multiplying the circumference by the number of rounds. Therefore, the total distance is 8 × 264 = 2112 m.

Explanation

To find the total work, we take the Least Common Multiple (LCM) of the individual completion times for A, B, and C, which are 24, 15, and 12 days. The LCM(24, 15, 12) is 120 units.

The efficiency of each person is calculated as Total Work divided by Time: Efficiency of A = 120 / 24 = 5 units/day Efficiency of B = 120 / 15 = 8 units/day Efficiency of C = 120 / 12 = 10 units/day

B and C started the work together and worked for 3 days. Their combined efficiency is 8 + 10 = 18 units/day. The work completed by B and C in 3 days is 18 × 3 = 54 units.

The remaining work to be completed by A is 120 - 54 = 66 units. The number of days required by A to finish this remaining work is: Time = Remaining Work / Efficiency of A Time = 66 / 5 = 13 ⅕ days

Explanation

The value of the expression x³ - y³ can be calculated directly using the algebraic identity x³ - y³ = (x - y)³ + 3xy(x - y). By substituting the given values x - y = 4 and xy = 45 into this formula, we have:

x³ - y³ = (4)³ + 3(45)(4) x³ - y³ = 64 + 12(45) x³ - y³ = 64 + 540 x³ - y³ = 604

Alternatively, through inspection of the factors of xy = 45, we find that 9 × 5 = 45 and 9 - 5 = 4. Thus, setting x = 9 and y = 5, we can compute the value as 9³ - 5³ = 729 - 125 = 604.

Explanation

Let the Cost Price (CP) of the article be 100%. Initially, the article is sold at a 10% loss, so the original selling price is 100% - 10% = 90% of the CP. If the article were sold for ₹332 more, the profit would be 20%, making the new selling price 100% + 20% = 120% of the CP.

The difference between the two selling prices in percentage terms is 120% - 90% = 30% of the CP. According to the problem, this 30% difference corresponds to an absolute value of ₹332.

To find the original selling price, which is 90% of the CP, we can use the relation: 30% of CP = 332 Multiplying both sides by 3 to reach 90%: 90% of CP = 332 × 3 = 996

Explanation

The volume of a cylinder is calculated using the formula V = π r² h, where r is the base radius and h is the height. To find the ratio of the volumes of two cylinders, we can set up the ratio of their respective volume formulas:

(V₁)/(V₂) = (π r₁² h₁)/(π r₂² h₂) = ((r₁)/(r₂))² × ((h₁)/(h₂))

Given that the ratio of the radii r₁ : r₂ is 3 : 4 and the ratio of the heights h₁ : h₂ is 4 : 9, we substitute these values into the equation:

(V₁)/(V₂) = (¾)² × ⁴⁄₉

(V₁)/(V₂) = ⁹⁄₁₆ × ⁴⁄₉

Canceling the 9 in the numerator and denominator and simplifying ⁴⁄₁₆ results in:

(V₁)/(V₂) = ⁴⁄₁₆ = ¼

Therefore, the ratio of the volumes is 1:4.

Explanation

The time taken by a train to cross a pole is the time taken to cover its own length, whereas the time taken to cross a bridge is the time taken to cover the sum of its own length and the length of the bridge. Therefore, the difference in time is solely due to the distance of the bridge itself.

Speed of the train v = (Length of Bridge)/(Time to cross bridge - Time to cross pole)

Substituting the given values: v = (170)/(36 - 12) = ¹⁷⁰⁄₂₄ m/s

To convert the speed from m/s to km/h, we multiply by the factor ¹⁸⁄₅: v = ¹⁷⁰⁄₂₄ × ¹⁸⁄₅ v = ¹⁷⁰⁄₅ × ¹⁸⁄₂₄ v = 34 × ¾ v = 34 × 0.75 = 25.5 km/h

Explanation

To find the value of the expression, we substitute the operators as per the standard conversion rules for this shift: + becomes -, - becomes +, × becomes ÷, and ÷ becomes ×.

First, we substitute the signs in the numerator and solve using the BODMAS rule: 42 + 12 ÷ 3 - 8 × 2 - 15 42 + 4 - 16 - 15 46 - 31 = 15

Next, we substitute the signs in the denominator and solve: 8 ÷ 2 + 4 - 9 × 3 4 + 4 - 27 8 - 27 = -19

By dividing the resulting numerator by the denominator, we get the final value: ¹⁵⁄₋₁₉ = -¹⁵⁄₁₉

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) is determined by the formula (CP)/(MP) = (100 - d%)/(100 + g%), where d is the discount percentage and g is the gain percentage.

Given that the discount is 28%, the gain is 12%, and the marked price is ₹224, we substitute these values into the formula:

(CP)/(224) = (100 - 28)/(100 + 12)

(CP)/(224) = ⁷²⁄₁₁₂

To find the cost price, rearrange the equation:

CP = ⁷²⁄₁₁₂ × 224

Since 224 is exactly 2 times 112, the calculation simplifies to:

CP = 72 × 2 = 144

Explanation

The simple interest I earned in the first scenario is the difference between the amount and the principal: I = 4644 - 4300 = 344. Using the simple interest formula I = (P × R × T)/(100), we can determine the annual interest rate R: 344 = (4300 × R × 2)/(100). This simplifies to 344 = 86R, which gives R = 4% per annum.

For the second scenario, the relationship between the principal P and the final amount A is given by A = P(1 + (RT)/(100)). Substituting the given values where A = 10104, R = 4, and T = 5, we get 10104 = P(1 + (4 × 5)/(100)). This further simplifies to 10104 = P(1 + 0.2), or 10104 = 1.2P. Solving for the principal, P = (10104)/(1.2) = 8420.

Explanation

Let the sixth number be represented by x. According to the given condition, the sum of the first five numbers is 7x. The total sum of all six numbers is the sum of the first five numbers plus the sixth number, which can be expressed as 7x + x = 8x.

The average of the six numbers is 136, so the total sum is also equal to 136 × 6. By equating the two expressions for the total sum, we get the equation 8x = 136 × 6. Solving for x gives x = (136 × 6)/(8). Dividing 136 by 8 yields 17, and multiplying 17 by 6 results in 102. Thus, the sixth number is 102.

Explanation

When A + B = 45°, a standard trigonometric identity states that (1 + tan A)(1 + tan B) = 2. This is derived from the formula for the tangent of a sum:

tan(A + B) = (tan A + tan B)/(1 - tan A tan B)

Since A + B = 45° and tan 45° = 1, we have:

(tan A + tan B)/(1 - tan A tan B) = 1 tan A + tan B = 1 - tan A tan B tan A + tan B + tan A tan B = 1

Adding 1 to both sides of the equation:

1 + tan A + tan B + tan A tan B = 1 + 1 (1 + tan A)(1 + tan B) = 2

Substituting this value into the expression 2(1 + tan A)(1 + tan B):

2 × 2 = 4

Alternatively, using the substitution method for a faster solution, let A = 45° and B = 0°. This satisfies the condition A + B = 45°. Substituting tan 45° = 1 and tan 0° = 0 into the given expression results in 2(1 + 1)(1 + 0) = 2 × 2 × 1 = 4.

Explanation

The area of a rectangle is the product of its length and breadth. When there are successive percentage changes in the dimensions of a figure, the net percentage change in the result can be calculated using the successive percentage formula: a + b + ᵃᵇ⁄₁₀₀.

In this case, the length increases by 40%, so a = +40. The breadth decreases by 20%, so b = -20.

Substituting these values into the formula gives the net percentage change in area as 40 - 20 + (40 × (-20))/(100). This simplifies to 20 - ⁸⁰⁰⁄₁₀₀, which equals 20 - 8 = 12%. Since the result is positive, the area increases by 12%, making the value of x equal to 12.

Explanation

In △ ABC, E and D are the midpoints of sides AB and BC respectively. According to the Midpoint Theorem, the segment ED is parallel to AC and ED = ½AC. This geometric relationship ensures that △ BDE is similar to △ BAC with a scale factor of 1:2. The property of similar triangles states that the ratio of their areas is equal to the square of the ratio of their corresponding sides. Therefore, Area(△ BDE) = Area(△ ABC) × ((BE)/(BA))². Substituting the given values, we get 44 × (½)² = 44 × ¼ = 11 cm². Geometrically, the four triangles formed by joining the midpoints of all three sides of a triangle are congruent and each occupies exactly one-fourth of the original triangle's area.

Explanation

Given the equation 6 tan A = 5, we find that tan A = ⅚. To evaluate the expression (8 sin A - 4 cos A)/(cos A + 2 sin A), we divide both the numerator and the denominator by cos A to express the terms using tan A:

(8 sin A)/(cos A) - (4 cos A)/(cos A){(cos A)/(cos A) + (2 sin A)/(cos A)} = (8 tan A - 4)/(1 + 2 tan A)

Now, substitute the value tan A = ⅚ into the simplified expression:

(8(5)/(6)) - 4{1 + 2(⅚)} = (40)/(6) - 4{1 + ¹⁰⁄₆}

Calculating the numerator and denominator separately:

Numerator: (40 - 24)/(6) = ¹⁶⁄₆

Denominator: (6 + 10)/(6) = ¹⁶⁄₆

Dividing the results: (16/6)/(16/6) = 1

Explanation

The initial number of boys and girls can be determined from the total of 640 students and the ratio 5:3. The sum of the ratio parts is 5 + 3 = 8. The number of boys is ⅝ × 640 = 400 and the number of girls is ⅜ × 640 = 240.

When 30 more girls are admitted, the new number of girls becomes 240 + 30 = 270. Let x be the number of additional boys required to make the ratio of boys to girls 14:9. The relationship is expressed as:

(400 + x)/(270) = ¹⁴⁄₉

Multiplying both sides by 270 to solve for x:

400 + x = ¹⁴⁄₉ × 270

400 + x = 14 × 30

400 + x = 420

x = 420 - 400 = 20

Explanation

In a circle with center O, the sum of angles around the center point is 360°. Given that the angles subtended by the chords AB and AC at the center are ∠ AOB = 110° and ∠ AOC = 130°, the angle subtended by the chord BC at the center is calculated as ∠ BOC = 360° - (110° + 130°) = 120°.

By the circle theorem, the angle subtended by an arc at the center is twice the angle subtended by it at any point on the remaining part of the circle. Since point A lies on the circumference, the angle ∠ BAC subtended by the arc BC is half of the angle subtended at the center. Therefore, ∠ BAC = ½ ∠ BOC = (120°)/(2) = 60°.

Alternatively, considering the isosceles triangles formed by the radii, in △ OAB, OA = OB, so ∠ OAB = (180° - 110°)/(2) = 35°. In △ OAC, OA = OC, so ∠ OAC = (180° - 130°)/(2) = 25°. Thus, ∠ BAC = ∠ OAB + ∠ OAC = 35° + 25° = 60°.

Explanation

Let x²ᵃ = y²ᵇ = z²ᶜ = k. Expressing x, y, and z in terms of k: x = k^¹^⁄^₂^ₐ y = k⁽¹⁾^/⁽²ᵇ⁾ z = k⁽¹⁾^/⁽²ᶜ⁾

For the given exam question, the relationship between the variables is x² = yz. Substituting the values of x, y, and z into this equation: (k^¹^⁄^₂^ₐ)² = k⁽¹⁾^/⁽²ᵇ⁾ · k⁽¹⁾^/⁽²ᶜ⁾ k^²^⁄^₂^ₐ = k⁽¹⁾^/⁽²ᵇ⁾⁺⁽¹⁾^/⁽²ᶜ⁾ k^¹^⁄^ₐ = k⁽ᶜ⁺ᵇ⁾^/⁽²ᵇᶜ⁾

Equating the exponents: ¹⁄ₐ = (b + c)/(2bc) 2bc = a(b + c) 2bc = ab + ac

The expression to be evaluated is (ab + bc + ca)/(bc). Substitute ab + ac = 2bc into the numerator: ((ab + ac) + bc)/(bc) = (2bc + bc)/(bc) (3bc)/(bc) = 3

Explanation

The given data represents a cumulative frequency distribution of the heights of 40 students. To find the number of students with a height of 165 cm or more, we subtract the number of students whose height is less than 165 cm from the total number of students. According to the table, the total number of students (those with height less than 175 cm) is 40. The number of students with a height less than 165 cm is 25.

Number of students with height ≥ 165 cm = Total students - Number of students with height < 165 cm Number of students with height ≥ 165 cm = 40 - 25 = 15

Explanation

To find the value of a³ + (1)/(a³) when given a + ¹⁄ₐ = k, we use the algebraic shortcut formula a³ + (1)/(a³) = k³ - 3k. This formula is derived from the expansion of the identity (a + ¹⁄ₐ)³ = a³ + (1)/(a³) + 3(a)(¹⁄ₐ)(a + ¹⁄ₐ), which simplifies to a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a + ¹⁄ₐ).

Given that a + ¹⁄ₐ = 5, we substitute k = 5 into the shortcut formula:

a³ + (1)/(a³) = 5³ - 3(5)

Calculating the terms: 5³ = 125 3 × 5 = 15

Subtracting the values: 125 - 15 = 110

Explanation

To determine the number of workers earning less than ¥1,100, we must calculate the cumulative frequency for all class intervals up to the upper limit of 1,100. In a continuous frequency distribution, the class interval 1,000 to 1,100 includes all values starting from 1,000 up to, but not including, 1,100. Therefore, every worker counted in the intervals preceding 1,100 falls into the required category.

The sum of the workers in these intervals is calculated as follows: Number of workers earning less than 1,100 = (Workers in 700-800) + (Workers in 800-900) + (Workers in 900-1,000) + (Workers in 1,000-1,100) Substituting the given values: 4 + 15 + 10 + 10 = 39

Explanation

To find the time taken when Amit, Sunil, and Dinesh work together, we first determine the total work by taking the Least Common Multiple (LCM) of the given time periods. The LCM of 9, 12, and 18 is 36, so we assume the total work is 36 units.

The combined efficiency of Amit (A) and Sunil (S) is 36 / 9 = 4 units per day. The combined efficiency of Sunil (S) and Dinesh (D) is 36 / 12 = 3 units per day. The combined efficiency of Amit (A) and Dinesh (D) is 36 / 18 = 2 units per day.

By adding these three equations, we get the efficiency of all three individuals twice: 2(A + S + D) = 4 + 3 + 2 = 9 units per day. Therefore, the combined efficiency of Amit, Sunil, and Dinesh working together is 9 / 2 = 4.5 units per day.

The time taken by them to complete the work together is Total Work divided by their combined efficiency: 36 / 4.5 = 8 days.

Explanation

The product is most easily solved using the algebraic identity (a + b)(a - b) = a² - b². The given numbers 113 and 87 can be expressed as (100 + 13) and (100 - 13). Substituting these into the formula results in 100² - 13². This calculation yields 10000 - 169, which equals 9831.

Explanation

To find the ratio of the total number of boys to the total number of girls in the college, we calculate the sum for each category across all streams provided in the table.

Total number of boys = 18 (Science) + 47 (Arts) + 40 (Commerce) + 30 (Vocational) = 135

Total number of girls = 35 (Science) + 25 (Arts) + 45 (Commerce) + 10 (Vocational) = 115

The required ratio is calculated as: Ratio = Total Boys : Total Girls = 135 : 115

By dividing both sides of the ratio by their greatest common divisor, which is 5, we simplify the expression: Ratio = 27 : 23

Explanation

The manufacturing cost can be determined using the formula for successive profit percentages. If C is the manufacturing cost, the price after the manufacturer's 10% profit and the shopkeeper's 15% profit is given by the relation C × (1 + r₁/100) × (1 + r₂/100) = Final Price.

Substituting the given values: C × (1 + 10/100) × (1 + 15/100) = 7590 C × 1.10 × 1.15 = 7590 C × 1.265 = 7590

Solving for C: C = (7590)/(1.265)

Using fractional representation for calculation: C × ¹¹⁄₁₀ × ²³⁄₂₀ = 7590 C × ²⁵³⁄₂₀₀ = 7590 C = (7590 × 200)/(253)

As 253 × 30 = 7590: C = 30 × 200 C = 6000

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) is given by the ratio formula (CP)/(MP) = (100 - d)/(100 + p), where d is the discount percentage and p is the profit percentage. Substituting the given values, we get (CP)/(MP) = (100 - 10)/(100 + 8) = ⁹⁰⁄₁₀₈. Simplifying this fraction by dividing both the numerator and the denominator by 18 yields the ratio (CP)/(MP) = ⅚.

When no discount is allowed, the Selling Price (SP) is equal to the Marked Price (MP). Taking CP = 5 and SP = MP = 6, the gain percentage is calculated as Gain% = (SP - CP)/(CP) × 100. Substituting the ratio values into the formula, we have Gain% = (6 - 5)/(5) × 100 = ⅕ × 100 = 20%. Note that the actual value of the marked price (7,200) is not required to find the percentage change.

Explanation

The given expression is (sin 30° sin 60°)/(cos 60° cos 30°) - tan 45°. Using the trigonometric identity sin θ = cos (90° - θ), we can observe that sin 30° = cos 60° and sin 60° = cos 30°. Substituting these values into the numerator, the fraction becomes (cos 60° cos 30°)/(cos 60° cos 30°), which simplifies to 1. Alternatively, using standard values where sin 30° = 1/2, sin 60° = √3/2, cos 60° = 1/2, and cos 30° = √3/2, the expression evaluates to (1/2) · (√3/2){(1/2) · (√3/2)} - tan 45°. This simplifies to 1 - tan 45°. Since tan 45° = 1, the final calculation is 1 - 1 = 0.

Explanation

The calculation follows the BODMAS rule, which specifies the order of operations as Brackets, Orders, Division, Multiplication, Addition, and Subtraction. Since division and multiplication have the same level of precedence, they are evaluated from left to right as they appear in the expression.

First, perform the division: ⁴⁄₃ ÷ ⅙ is equivalent to multiplying the first fraction by the reciprocal of the second, which is ⁴⁄₃ × ⁶⁄₁. This simplifies to 4 × 2 = 8.

Next, proceed with the multiplication: 8 × 2 = 16.

Finally, perform the subtraction: 16 - 1 = 15. Applying these operations in the standard mathematical sequence results in the value 15.

Explanation

The teacher's contribution can be calculated by finding the difference between the total amount collected after the teacher's contribution and the total amount collected by the students initially.

Total contribution of 24 students = 24 × 50 = 1200 Total contribution of 25 people (including the teacher) = 25 × 56 = 1400 Teacher's contribution = Total with teacher - Total of students = 1400 - 1200 = 200

Alternatively, using the shortcut for inclusion in averages: Teacher's contribution = New Average + (Number of initial students × Increase in average) Teacher's contribution = 56 + (24 × (56 - 50)) Teacher's contribution = 56 + (24 × 6) Teacher's contribution = 56 + 144 = 200

Explanation

According to the Triangle Inequality Theorem, the length of the third side c of a triangle must be strictly greater than the absolute difference and strictly less than the sum of the other two sides a and b. This is expressed by the inequality:

|a - b| < c < a + b

Substituting the given side lengths 5 cm and 8 cm into the formula: Lower limit: 8 - 5 = 3 cm Upper limit: 8 + 5 = 13 cm

The measure of the third side must satisfy 3 < c < 13. Since 4 cm is greater than 3 cm and less than 13 cm, it is a valid length for the third side of the triangle.

Explanation

Let the distance between place A and place B be D km. The time taken by cyclist X to cover the distance is T_X = (D)/(6) hours. Cyclist Y travels at a speed of 8 km/h, so the actual cycling time for Y is (D)/(8) hours. Since Y stops for 15 minutes (¹⁵⁄₆₀ hours) and still reaches 10 minutes (¹⁰⁄₆₀ hours) before X, the total duration for Y's journey is (D)/(8) + ¹⁵⁄₆₀ hours.

The difference in time between X and Y is given by the equation: T_X - T_Y = ¹⁰⁄₆₀ (D)/(6) - ((D)/(8) + ¹⁵⁄₆₀) = ¹⁰⁄₆₀ (D)/(6) - (D)/(8) = ¹⁰⁄₆₀ + ¹⁵⁄₆₀ (4D - 3D)/(24) = ²⁵⁄₆₀ (D)/(24) = ⁵⁄₁₂ D = (5 × 24)/(12) D = 10 km

Alternatively, the ratio of speeds of X and Y is 6:8 or 3:4. Since time is inversely proportional to speed for a constant distance, the ratio of time taken while moving is 4:3. Let the moving times be 4t for X and 3t for Y. The total time for Y is 3t + 15 minutes. The problem states X takes 10 minutes more than Y's total time, so 4t = (3t + 15) + 10, which results in t = 25 minutes. X's travel time is 4 × 25 = 100 minutes or ⁵⁄₃ hours. The distance is 6 × ⁵⁄₃ = 10 km.

Explanation

The binomial expansion of (a - b)³ is given by the formula a³ - 3a²b + 3ab² - b³. To find the coefficient of x in the expansion of (x - 3y)³, we identify a = x and b = 3y. The term containing x to the first power is 3ab². Substituting the values into this term, we get 3(x)(3y)² = 3(x)(9y²) = 27y²x. Thus, the coefficient of x is 27y².

Explanation

The area of a rhombus is determined by the formula A = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of its diagonals. Substituting the given values d₁ = 16 cm and d₂ = 12 cm into the formula, we get A = ½ × 16 × 12. Simplifying this calculation yields 8 × 12 = 96 cm². This relationship exists because the diagonals of a rhombus bisect each other at right angles, dividing the figure into four congruent right triangles, each with a base of d₁{2} and a height of d₂{2}. Summing the areas of these four triangles results in the standard area formula for the rhombus.

Explanation

The total number of notes is 12, and the ratio of ¥ 10 notes to ¥ 50 notes is given as 1:2. Let the number of ¥ 10 notes be x and the number of ¥ 50 notes be 2x. The sum of the notes is x + 2x = 12, which simplifies to 3x = 12, yielding x = 4. Consequently, there are 4 notes of ¥ 10 and 2 × 4 = 8 notes of ¥ 50. The total amount of money is the sum of the values of these notes: (4 × 10) + (8 × 50) = 40 + 400 = 440.

Explanation

To calculate the income other than salary for each employee, we use the relation Income_Oₜₕₑᵣ = Total Income - Salary.

For employee A, the income other than salary is 66,000 - 52,000 = 14,000, which is greater than 10,000. For employee B, the income other than salary is 59,000 - 48,500 = 10,500, which is greater than 10,000. For employee C, the income other than salary is 52,000 - 42,000 = 10,000. Since the question asks for more than 10,000, this value does not qualify. For employee D, the income other than salary is 40,000 - 31,000 = 9,000. For employee E, the income other than salary is 31,700 - 25,000 = 6,700.

By comparing the calculated values, only employees A and B received an income from non-salary sources that strictly exceeds ¥ 10,000. This gives a total count of 2 employees.

Explanation

The given quadratic equation x² - 4x + 4 = 0 is a perfect square expansion of the form (a - b)² = a² - 2ab + b². Thus, the equation can be written as: (x - 2)² = 0 Solving for x gives x = 2.

To find the value of the expression 16(x⁴ - (1)/(x⁴⁾), substitute x = 2 into it: 16(2⁴ - (1)/(2⁴⁾) = 16(16 - ¹⁄₁₆)

Distributing the 16 into the terms inside the parentheses: 16 × 16 - 16 × ¹⁄₁₆ = 256 - 1 = 255

Explanation

The total amount spent by the family in the year 2019 is calculated by summing the product of the quantity required for each commodity and its respective rate in 2019.

For Commodity A, the cost is 15 × 80 = 1200. For Commodity B, the cost is 20 × 60 = 1200. For Commodity C, the cost is 12 × 40 = 480. For Commodity D, the cost is 40 × 85 = 3400. For Commodity E, the cost is 8 × 72 = 576.

Total amount spent = 1200 + 1200 + 480 + 3400 + 576 = 6856.

Explanation

In any triangle, the sum of all internal angles is 180°. Given that the ratio of the angles is 3:5:4, we can represent the individual angles as 3x, 5x, and 4x degrees respectively. According to the angle sum property of a triangle:

3x + 5x + 4x = 180° 12x = 180° x = (180°)/(12) = 15°

The largest angle in the triangle corresponds to the highest ratio value, which is 5x, and the smallest angle corresponds to the lowest ratio value, which is 3x. To find the difference between the biggest and smallest angles, we calculate:

Difference = 5x - 3x = 2x

Substituting the value of x into this expression:

Difference = 2 × 15° = 30°

Explanation

We use the trigonometric identity tan 2A = tan((A + B) + (A - B)). Applying the tangent addition formula, we have tan 2A = (tan(A + B) + tan(A - B))/(1 - tan(A + B)tan(A - B)).

Given 0 < A, B < 45°, we have 0 < A + B < 90°. Since cos(A + B) = ²⁴⁄₂₅, we use the Pythagorean identity to find sin(A + B) = √(1 - (²⁴⁄₂₅)²) = √((625 - 576)/(625)) = ⁷⁄₂₅. Therefore, tan(A + B) = (sin(A + B))/(cos(A + B)) = ⁷⁄₂₄.

Similarly, since sin(A - B) = ¹⁵⁄₁₇ and the range of A - B is between -45° and 45°, the positive sine value implies A - B is in the first quadrant. We find cos(A - B) = √(1 - (¹⁵⁄₁₇)²) = √((289 - 225)/(289)) = ⁸⁄₁₇. Therefore, tan(A - B) = (sin(A - B))/(cos(A - B)) = ¹⁵⁄₈.

Substituting these into the formula for tan 2A: tan 2A = (7)/(24) + ¹⁵⁄₈{1 - (⁷⁄₂₄)(¹⁵⁄₈)}

Simplifying the numerator: ⁷⁄₂₄ + ⁴⁵⁄₂₄ = ⁵²⁄₂₄ = ¹³⁄₆. Simplifying the denominator: 1 - ¹⁰⁵⁄₁₉₂ = (192 - 105)/(192) = ⁸⁷⁄₁₉₂.

Final calculation: tan 2A = ¹³⁄₆ × ¹⁹²⁄₈₇ = (13 × 32)/(87) = ⁴¹⁶⁄₈₇.

Explanation

The algebraic identity relating a sum and its cubes is a³ + (1)/(a³) = (a + ¹⁄ₐ)³ - 3(a + ¹⁄ₐ). Let x = a + ¹⁄ₐ. Substituting the given value 52, we obtain the equation x³ - 3x = 52. By substituting x = 4 into this expression, we calculate 4³ - 3(4), which simplifies to 64 - 12 = 52. Because this satisfies the equation, the value of a + ¹⁄ₐ is 4.

Explanation

Given the equations: ax = b+c by = c+a cz = a+b

Add a to both sides of the first equation: ax + a = a+b+c a(x+1) = a+b+c 1/(x+1) = a/(a+b+c)

Similarly, for the other two equations: by + b = a+b+c ⟹ 1/(y+1) = b/(a+b+c) cz + c = a+b+c ⟹ 1/(z+1) = c/(a+b+c)

Summing these reciprocal terms: 1/(x+1) + 1/(y+1) + 1/(z+1) = a/(a+b+c) + b/(a+b+c) + c/(a+b+c) 1/(x+1) + 1/(y+1) + 1/(z+1) = (a+b+c)/(a+b+c) = 1

Applying the multiplier ⅑ to this sum yields: ⅑ [ 1/(x+1) + 1/(y+1) + 1/(z+1) ] = ⅑ × 1 = ⅑

Explanation

To find the percentage increase in the rate of commodity D, use the formula for percentage change:

Percentage Increase = (Rate in 2019 - Rate in 2016)/(Rate in 2016) × 100

From the table, the rate per kg for commodity D in 2016 is 75 and the rate in 2019 is 85.

Substituting these values into the formula:

Increase = 85 - 75 = 10

Percentage Increase = ¹⁰⁄₇₅ × 100

Percentage Increase = ²⁄₁₅ × 100 = ²⁰⁰⁄₁₅ = 13.33%

Explanation

According to the Basic Proportionality Theorem (Thales's Theorem), if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. Since DE ∥ BC in triangle ABC, the segments of the sides AB and AC are proportional such that (AD)/(DB) = (AE)/(EC).

Substituting the given algebraic expressions into the proportionality formula: ᵃ⁄ₐ₊₄ = ²ᵃ⁺³⁄₇ₐ

Cross-multiplying to solve for a: a(7a) = (a+4)(2a+3) 7a² = 2a² + 3a + 8a + 12 7a² = 2a² + 11a + 12

Rearranging the terms to form a quadratic equation: 5a² - 11a - 12 = 0

Solving the quadratic equation using the factorization method, we look for two numbers that multiply to -60 (from 5 × -12) and add to -11. Those numbers are -15 and 4: 5a² - 15a + 4a - 12 = 0 5a(a - 3) + 4(a - 3) = 0 (5a + 4)(a - 3) = 0

This gives two possible solutions for a: a = -⅘ or a = 3. Since the problem specifies that a > 0, we discard the negative result. Thus, the value of a is 3.

Explanation

The extra amount spent on commodities is calculated by finding the difference in rates between the two years and multiplying it by the quantity required per month.

For commodity B: Quantity = 15 kg Rate in 2016 = 50 Rate in 2019 = 60 Increase in rate = 60 - 50 = 10 per kg Extra amount spent on B = 20 × 10 = 200

For commodity C: Quantity = 12 kg Rate in 2016 = 35 Rate in 2019 = 40 Increase in rate = 40 - 35 = 5 per kg Extra amount spent on C = 12 × 5 = 60

Total extra amount spent on B and C = 200 + 60 = 260

Explanation

Total work can be assumed as the LCM of 20, 16, and 30, which is 240 units. The daily efficiency of each individual is calculated by dividing the total work by the number of days they take to complete it. Thus, the efficiency of A is 240 / 20 = 12 units/day, the efficiency of B is 240 / 16 = 15 units/day, and the efficiency of C is 240 / 30 = 8 units/day.

A and B work together for 4 days. Their combined efficiency is 12 + 15 = 27 units/day. In 4 days, they complete 27 × 4 = 108 units of work. The remaining work to be completed by C is 240 - 108 = 132 units.

The time taken by C to finish this remaining task is calculated by dividing the remaining units by C's efficiency. Therefore, time taken by C = 132 / 8 = 33 / 2 = 16 ½ days.

Explanation

First, convert the speed of the train from km/h to m/s by multiplying by ⁵⁄₁₈. The speed v = 72 × ⁵⁄₁₈ = 20 m/s.

When a train crosses a platform, the total distance covered is the sum of the length of the train L and the length of the platform. Using the formula Distance = Speed × Time, we have L + 180 = 20 × 60. Solving this gives L + 180 = 1200, so the length of the train L = 1020 m.

To cross an electric pole, the train must cover a distance equal to its own length. The time taken to cross the pole is calculated as Time = (Distance)/(Speed). Substituting the known values, Time = ¹⁰²⁰⁄₂₀ = 51 seconds.

Explanation

The total simple interest is the sum of interest earned during each period. Using the formula SI = (P × R × T)/(100), the total interest percentage over 10 years is calculated by summing the products of the rates and their respective durations.

For the first 4 years at 5%: 5 × 4 = 20% For the next 3 years at 8%: 8 × 3 = 24% For the remaining 3 years (from year 7 to year 10) at 10%: 10 × 3 = 30%

The cumulative interest rate is 20% + 24% + 30% = 74% of the principal P. Given that the total interest is 1,850, we set up the equation:

⁷⁴⁄₁₀₀ × P = 1,850

P = (1,850 × 100)/(74)

Dividing 1,850 by 74 yields 25. Therefore:

P = 25 × 100 = 2,500

Explanation

The expression is solved by applying the BODMAS rule, which dictates the order of operations: Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

For the numerator: 54 - (5 ÷ 2) × 8 + 13 First, perform the division inside the implied grouping or as the first step: 5 ÷ 2 = 2.5. Next, perform the multiplication: 2.5 × 8 = 20. Then, perform the addition and subtraction from left to right: 54 - 20 + 13 = 34 + 13 = 47.

For the denominator: 48 - 4 ÷ 3 × 8 - 2 First, perform the division: 4 ÷ 3 = 4/3. Next, perform the multiplication: 4/3 × 8 = 32/3. Then, perform the subtraction from left to right: 48 - 32/3 - 2 = 46 - 32/3. Convert 46 to a fraction with a denominator of 3: 46 = 138/3. Subtract: 138/3 - 32/3 = 106/3.

Combining the simplified numerator and denominator: 47 / (106/3) = 47 × 3/106 = 141/106

Explanation

The total sum of results for the 60 students is calculated using the formula Sum = Average × Count, which gives 60 × 38 = 2280. The sum of the first 22 students is 22 × 36 = 792, and the sum of the last 32 students is 32 × 32 = 1024. The number of remaining students is 60 - (22 + 32) = 6. The sum of the results for these remaining students is found by subtracting the known sums from the total sum: 2280 - (792 + 1024) = 2280 - 1816 = 464. Dividing this sum by the number of remaining students gives an average of 464 / 6 = 77.33. Assuming integer values for the options, this evaluates to 77.