SSC CGL Tier 1 2023 Quantitative Aptitude Questions with Solutions

Explanation

Based on the given condition, A = 95% of B, which can be written mathematically as A = ⁹⁵⁄₁₀₀B = ¹⁹⁄₂₀B.

To find what percent of A is B, we need to calculate ((B)/(A) × 100)%.

From the relationship A = ¹⁹⁄₂₀B, we can express B in terms of A as B = ²⁰⁄₁₉A.

Substituting this into the percentage formula: Percentage = (20)/(19)A{A} × 100 = ²⁰⁄₁₉ × 100 = ²⁰⁰⁰⁄₁₉%.

Performing the division ²⁰⁰⁰⁄₁₉: 2000 = 19 × 105 + 5.

Thus, the fraction ²⁰⁰⁰⁄₁₉ expressed as a mixed number is 105⁵⁄₁₉%.

Explanation

Let the cost price of the mustard oil be CP = 100. The marked price is 25% more than the cost price, so MP = 100 + 25 = 125. To have no profit and no loss, the selling price must be equal to the cost price, meaning SP = 100.

The discount needed from the marked price to reach the selling price is MP - SP = 125 - 100 = 25. To find the percentage reduction from the marked price, we use the formula:

Percentage less than marked price = (Reduction)/(MP) × 100

Substituting the values, we get:

²⁵⁄₁₂₅ × 100 = ⅕ × 100 = 20%

Explanation

Let the total work be the Least Common Multiple (LCM) of the individual time periods. The LCM of 25 and 30 is 150 units. Based on this, the efficiency of A is 150 / 25 = 6 units per day, and the efficiency of B is 150 / 30 = 5 units per day.

The work is performed on an alternate basis starting with A for 5 days. The work schedule for these 5 days is: Day 1: A works (6 units) Day 2: B works (5 units) Day 3: A works (6 units) Day 4: B works (5 units) Day 5: A works (6 units)

The total work completed in these 5 days is 6 + 5 + 6 + 5 + 6 = 28 units. The remaining work is 150 - 28 = 122 units. After 5 days, A leaves and B finishes the rest of the work. Since B's efficiency is 5 units per day, the time required for B to complete the remaining task is: Days = ¹²²⁄₅ = 24 ⅖

Explanation

Average speed is defined as the total distance traveled divided by the total time taken for the journey. The journey consists of three segments. In the first segment, the person travels 120 km at 80 km/h, taking t₁ = 120 / 80 = 1.5 hours. In the second segment, the person travels 100 km at 40 km/h, taking t₂ = 100 / 40 = 2.5 hours. For the return journey, the person travels the entire distance back to the starting point, which is 120 + 100 = 220 km, at a speed of 75 km/h, taking t₃ = 220 / 75 = 44/15 ≈ 2.933 hours.

The total distance for the entire journey is 120 + 100 + 220 = 440 km. The total time taken is t₁ + t₂ + t₃ = 1.5 + 2.5 + 44/15 = 4 + 44/15 = 104/15 hours. Applying the formula for average speed:

Average Speed = (Total Distance)/(Total Time) = (440)/(104/15) = (440 × 15)/(104) = ⁶⁶⁰⁰⁄₁₀₄ ≈ 63.46 km/h

Explanation

Total work is constant and can be defined by the product of the number of workers, their individual efficiency, and the time taken. Let M and W represent the efficiency of one man and one woman, respectively. Based on the problem, 8 × M × 45 = 8 × W × 18. Simplifying this equation by dividing both sides by 8 gives 45M = 18W, which establishes the efficiency ratio M : W = 18 : 45, or M : W = 2 : 5.

Assigning M = 2 units and W = 5 units, the total work is 8 × 2 × 45 = 720 units. The collective efficiency of the group consisting of 5 men and 8 women is (5 × 2) + (8 × 5) = 10 + 40 = 50 units per day. The number of days required to complete the total work is the total work divided by the daily efficiency of the group, which is ⁷²⁰⁄₅₀ = ⁷²⁄₅. Expressed as a mixed fraction, this is 14 ⅖ days.

Explanation

To find which number divides the expression 6²⁵ + 6²⁶ + 6²⁷ + 6²⁸, we factor out the common term 6²⁵:

6²⁵(6⁰ + 6¹ + 6² + 6³)

Now, we calculate the sum within the parentheses:

6⁰ = 1 6¹ = 6 6² = 36 6³ = 216

Sum = 1 + 6 + 36 + 216 = 259

The expression simplifies to 6²⁵ × 259. Since 259 is a direct factor of the product, the entire expression is divisible by 259.

Explanation

The total marks for the examination are 50. The question asks for the number of students scoring less than 20% marks in aggregate. First, we determine the mark equivalent of 20% of the total marks, which is 0.20 × 50 = 10.

According to the table, the row for Average (Aggregate) shows the cumulative number of students scoring within specific ranges. The column 10 and above indicates that 87 students scored 10 marks or more in aggregate. Since the total number of students is 100, the number of students who scored less than 10 marks is the difference between the total students and those who scored 10 or more. This is calculated as 100 - 87 = 13.

Explanation

Let R = 10 cm be the radius of the larger circle and r = 6 cm be the radius of the smaller circle. A chord of the larger circle that touches the smaller circle acts as a tangent to the smaller circle. The radius of the smaller circle drawn to the point of tangency is perpendicular to this chord and bisects it.

This geometry forms a right-angled triangle where the hypotenuse is the radius of the larger circle (R), one leg is the radius of the smaller circle (r), and the other leg is half the length of the chord (l). By the Pythagorean theorem:

l² + r² = R² l² + 6² = 10² l² + 36 = 100 l² = 64 l = √(64) = 8 cm

Since l represents half the length of the chord, the total length of the chord is 2 × l = 2 × 8 = 16 cm.

Explanation

The given equations utilize standard trigonometric ratios where sin 90° = 1 and cos 60° = ½. From sin(a + b) = 1, we obtain the first linear equation: a + b = 90°

From cos(a - b) = ½, we obtain the second linear equation: a - b = 60°

To find the value of a, we add the two equations together: (a + b) + (a - b) = 90° + 60° 2a = 150°

Dividing both sides by 2 results in: a = (150°)/(2) = 75°

Explanation

To calculate the time required for a train to clear a platform completely, the total distance covered is the sum of the train length and the platform length. The total distance D is 900 m + 900 m = 1800 m.

The speed of the train must be converted from km/h to m/s using the conversion factor ⁵⁄₁₈: Speed v = 108 × ⁵⁄₁₈ = 6 × 5 = 30 m/s.

Using the formula for time t = (D)/(v): t = ¹⁸⁰⁰⁄₃₀ = 60 s.

Explanation

The expression to be evaluated is 16b² + (1)/(49b²), which is (4b)² + ((1)/(7b))². To obtain this, transform the given equation 7b - (1)/(4b) = 7 by multiplying both sides by ⁴⁄₇:

⁴⁄₇ × (7b - (1)/(4b)) = ⁴⁄₇ × 7 4b - (1)/(7b) = 4

Squaring both sides of this equation using the identity (x - y)² = x² + y² - 2xy:

(4b - (1)/(7b))² = 4² (4b)² + ((1)/(7b))² - 2(4b)((1)/(7b)) = 16 16b² + (1)/(49b²) - ⁸⁄₇ = 16

To find the final value, move the constant term to the right side:

16b² + (1)/(49b²) = 16 + ⁸⁄₇ 16b² + (1)/(49b²) = (112 + 8)/(7) 16b² + (1)/(49b²) = ¹²⁰⁄₇

Explanation

The Side-Angle-Side (SAS) congruence theorem states that two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle. In the given triangles △ ABC and △ XYZ, we are provided with one pair of equal sides, AC = XZ, and one pair of equal angles, m ∠ C = m ∠ Z. For these triangles to be congruent by the SAS criterion, the angles must be the included angles between the corresponding equal sides. In △ ABC, ∠ C is the included angle between sides AC and BC. In △ XYZ, ∠ Z is the included angle between sides XZ and YZ. Given that AC = XZ, the necessary additional condition to satisfy the SAS congruence property is that the other sides forming the angles must be equal, which is BC = YZ.

Explanation

The discount percentage in a Buy X Get Y Free scheme is calculated using the formula: Discount Percentage = (Y)/(X + Y) × 100. This formula represents the ratio of free items to the total number of items received by the customer.

For Scheme A (Buy 3 get 4 free), the total number of items is 3 + 4 = 7. The discount percentage is ⁴⁄₇ × 100 ≈ 57.14%.

For Scheme B (Buy 5 get 6 free), the total number of items is 5 + 6 = 11. The discount percentage is ⁶⁄₁₁ × 100 ≈ 54.54%.

Comparing the two values, Scheme A offers a discount of approximately 57.14%, which is higher than the 54.54% discount offered by Scheme B. A quick way to compare the fractions ⁴⁄₇ and ⁶⁄₁₁ without full division is cross-multiplication: 4 × 11 = 44 and 6 × 7 = 42. Since 44 > 42, the fraction ⁴⁄₇ is larger, confirming that Scheme A provides the maximum discount percentage.

Explanation

The simple interest is calculated using the formula SI = (P × R × T)/(100). Since the interest produced in both cases is equal, we can equate the product of the principal, rate, and time for both scenarios: P₁ × R₁ × T₁ = P₂ × R₂ × T₂.

Substituting the given values: 10,000 × 4 × T₁ = 8,000 × 5 × 4 40,000 × T₁ = 160,000 T₁ = (160,000)/(40,000) T₁ = 4

Thus, the required time is 4 years.

Explanation

The time taken for multiple individuals working together can be calculated using their combined rate of work. Let the total work be the Least Common Multiple (LCM) of the individual time periods 10, 15, and 30, which is 30 units. The efficiency of each person is the total work divided by their respective time taken. Thus, the efficiency of the man is 30 / 10 = 3 units per day, the efficiency of the boy is 30 / 15 = 2 units per day, and the efficiency of the woman is 30 / 30 = 1 unit per day.

When all three work together, their combined efficiency is the sum of their individual efficiencies: 3 + 2 + 1 = 6 units per day. The total time required to complete the work is the total work divided by the combined efficiency, calculated as 30 / 6 = 5 days. This can also be expressed using the combined rate formula 1/T = 1/10 + 1/15 + 1/30, which simplifies to 1/T = (3 + 2 + 1) / 30 = 6/30 = 1/5, resulting in T = 5 days.

Explanation

The expression can be simplified using the basic trigonometric identity tan θ = (sin θ)/(cos θ). By substituting this identity into the given expression, we find:

tan² θ cos² θ = (sin² θ)/(cos² θ) · cos² θ = sin² θ

Using the Pythagorean identity sin² θ + cos² θ = 1, we can express the result in terms of cos θ:

sin² θ = 1 - cos² θ

Given that cos θ = √3{2}, we first find the value of cos² θ:

cos² θ = (√3{2})² = ¾

Now, substituting this value into our simplified expression:

sin² θ = 1 - ¾ = ¼

Alternatively, identifying the angle for which cos θ = √3{2} gives θ = 30°. Substituting this angle into the original expression:

tan² 30° cos² 30° = ((1)/(√(3)))² · (√3{2})² = ⅓ · ¾ = ¼

Explanation

To find the value of the expression 3x³ + 5x²y + 12xy² + 7y³, substitute x = -4 and y = -1 into each term:

The first term is 3(-4)³ = 3(-64) = -192. The second term is 5(-4)²(-1) = 5(16)(-1) = -80. The third term is 12(-4)(-1)² = 12(-4)(1) = -48. The fourth term is 7(-1)³ = 7(-1) = -7.

Summing these individual values gives -192 - 80 - 48 - 7. Combining the negative values results in -(192 + 80 + 48 + 7) = -327.

Explanation

For four numbers a, b, c, and d to be in proportion, the relation (a)/(b) = (c)/(d) must be satisfied. Substituting the given values 39, 117, 17, and y into this formula gives:

³⁹⁄₁₁₇ = (17)/(y)

The fraction on the left can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 39, since 39 × 3 = 117:

⅓ = (17)/(y)

By cross-multiplying to solve for the unknown variable y:

y = 17 × 3 y = 51

Explanation

The volume V of a sphere is calculated using the formula V = ⁴⁄₃ π r³. By substituting the given values r = 4.2 cm and π = ²²⁄₇ into the formula:

V = ⁴⁄₃ × ²²⁄₇ × 4.2 × 4.2 × 4.2

To simplify the arithmetic, dividing 4.2 by 7 yields 0.6, and dividing 0.6 by 3 yields 0.2. The expression simplifies to:

V = 4 × 22 × 0.2 × 4.2 × 4.2 V = 17.6 × 17.64 V = 310.464 cm³

A useful shortcut for such problems is to verify divisibility by 11, as the calculation involves the factor 22. For the value 310.464, the sum of the digits in the odd positions is 3 + 0 + 6 = 9 and the sum of the digits in the even positions is 1 + 4 + 4 = 9. Since the difference is 0, the number is divisible by 11.

Explanation

The algebraic identity for the square of a trinomial is (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Substituting the given values (a + b + c) = 16 and (a² + b² + c²) = 90 into this formula results in 16² = 90 + 2(ab + bc + ca). Calculating the square of 16 gives 256 = 90 + 2(ab + bc + ca). To isolate the term containing the required expression, subtract 90 from 256, which gives 166 = 2(ab + bc + ca). Dividing both sides by 2 results in (ab + bc + ca) = 83.

Explanation

Given the trigonometric equation (3sinθ - cosθ)/(cosθ + sinθ) = 1, we begin by cross-multiplying the denominator to the right side:

3sinθ - cosθ = cosθ + sinθ

To solve for the relationship between sine and cosine, we rearrange the terms by adding cosθ to both sides and subtracting sinθ from both sides:

3sinθ - sinθ = cosθ + cosθ 2sinθ = 2cosθ

Dividing both sides by 2 gives the identity sinθ = cosθ. The cotangent function is defined as the ratio of the cosine of an angle to its sine:

cotθ = (cosθ)/(sinθ)

By substituting cosθ = sinθ into this ratio, we obtain:

cotθ = (sinθ)/(sinθ) = 1

Explanation

To find when two runners meet again at the starting point, calculate the time each takes to complete one full lap and then find the Least Common Multiple (LCM) of those times.

First, convert the speeds from km/h to m/s using the factor ⁵⁄₁₈: Sony speed v₁ = 18 × ⁵⁄₁₈ = 5 m/s Mony speed v₂ = 24 × ⁵⁄₁₈ = ²⁰⁄₃ m/s

Next, calculate the time taken by each runner to complete one lap of the 200 m track: Time for Sony T₁ = ²⁰⁰⁄₅ = 40 seconds Time for Mony T₂ = (200)/(20/3) = 200 × ³⁄₂₀ = 30 seconds

The time they meet at the starting point is the LCM(T₁, T₂): LCM(40, 30) = 120 seconds

Explanation

Using modular arithmetic properties, we evaluate the expression (265)⁴⁰⁸¹ + 9 266. Since 265 = 266 - 1, it follows that 265 ≡ -1 266. Substituting this into the term with the exponent gives (-1)⁴⁰⁸¹ + 9 266. Because 4081 is an odd integer, (-1)⁴⁰⁸¹ = -1. Thus, the expression becomes -1 + 9, which results in a remainder of 8.

Explanation

The given equation a² + b² + c² = ab + bc + ac can be rearranged by multiplying both sides by 2 to obtain 2a² + 2b² + 2c² - 2ab - 2bc - 2ac = 0. This expression is equivalent to the algebraic identity (a - b)² + (b - c)² + (c - a)² = 0. For the sum of squares of real numbers to be zero, each individual squared term must be zero, which leads to a - b = 0, b - c = 0, and c - a = 0. Consequently, the condition implies that a = b = c.

By substituting b = a and c = a into the target expression, we simplify the numerator and the denominator separately. The numerator becomes 11a⁴ + 13a⁴ + 17a⁴ = 41a⁴. The denominator becomes 17a²(a²) + 9a²(a²) + 15a²(a²) = 17a⁴ + 9a⁴ + 15a⁴ = 41a⁴. Dividing the numerator by the denominator, we get (41a⁴)/(41a⁴) = 1.

Explanation

In △ ABC, since DE ∥ BC, △ ADE ∼ △ ABC by AA similarity. The ratio of their corresponding sides is DE/BC = AD/AB. Given AD:DB = 7:9, we have AB = AD + DB = 7 + 9 = 16 units. Therefore, the ratio DE/BC = 7/16.

Since DE ∥ BC, the triangles △ DEF and △ CBF are similar by AA similarity because ∠ FED = ∠ FBC and ∠ FDE = ∠ FCB (alternate interior angles). For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Using the formula for the ratio of areas: Area(△ DEF) / Area(△ CBF) = (DE/BC)² Substituting the side ratio: Area ratio = (7/16)² = 49/256

Explanation

Since A and B can individually complete the work in 30 days, we can assume the total work is 30 units, and each person’s efficiency is 1 unit per day.

A takes a day off after every four days of work, meaning A works for 4 days and rests on the 5th day. Therefore, A rests on days 5, 10, 15, 20, and so on. B takes a day off after every five days of work, meaning B works for 5 days and rests on the 6th day. Therefore, B rests on days 6, 12, 18, 24, and so on.

By the end of the 18th day: Total days off for A = 3 days (Days 5, 10, 15) Total days worked by A = 18 - 3 = 15 days Work done by A = 15 × 1 = 15 units

Total days off for B = 3 days (Days 6, 12, 18) Total days worked by B = 18 - 3 = 15 days Work done by B = 15 × 1 = 15 units

Total work done by A and B together = 15 + 15 = 30 units.

The total work of 30 units is exactly completed on the 18th day. Since the work started on 01 August 2022, the 18th day is 18 August 2022.

Explanation

The amount of canvas required to make a conical tent corresponds to the curved surface area of the cone, as the base is the floor and does not require canvas. The formula for the curved surface area is A = π r l, where r is the radius and l is the slant height. First, the slant height is determined using the relation l = √(r²⁾⁺ʰ^^^². Given the radius r = 6 units and height h = 8 units, the slant height is l = √(6²⁾⁺⁸^^^² = √(36 + 64) = √(100) = 10 units. Substituting these values into the area formula using π ≈ 22/7 gives A = (22/7) × 6 × 10 = 1320 / 7. Dividing 1320 by 7 results in approximately 188.5714, which rounded to two decimal places is 188.57 units.

Explanation

To determine the smallest number that must be added to 888 so that the sum is exactly divisible by 35, we first divide 888 by 35 to find the remainder.

Performing the division: 888 ÷ 35 35 × 2 = 70 88 - 70 = 18 Bringing down the last digit, we have 188. 35 × 5 = 175 188 - 175 = 13

The remainder is 13. For a number to be perfectly divisible by a divisor after adding a value, the value to be added is the difference between the divisor and the remainder.

Required number = Divisor - Remainder Required number = 35 - 13 = 22

When 22 is added to 888, the result is 910, which is exactly 35 × 26.

Explanation

Let the two natural numbers be x and y. According to the problem, the square of the difference is (x - y)² = 324, which gives the difference x - y = √(324) = 18. The product of the numbers is xy = 144.

To find the difference between the squares, x² - y², we use the identity x² - y² = (x - y)(x + y). We already have x - y = 18, so we need to find x + y. Using the algebraic identity (x + y)² = (x - y)² + 4xy, we substitute the known values:

(x + y)² = 324 + 4(144) (x + y)² = 324 + 576 (x + y)² = 900

Taking the square root, we get x + y = √(900) = 30. Finally, the positive difference between the squares of the numbers is:

x² - y² = (18)(30) = 540

Explanation

Let the total work be 15 units. Since A can finish the work in 15 days, the efficiency of A is 15 / 15 = 1 unit per day.

A works for the first two days and the last two days, making a total of 4 days. The work completed by A is 4 × 1 = 4 units. The remaining work is 15 - 4 = 11 units. This remaining work is completed by B. Since the total time taken to complete the work is 20 days and A worked for 4 of those days, B worked for the remaining 20 - 4 = 16 days.

The efficiency of B is the work done divided by the time taken, which is 11 / 16 units per day. To find the time taken by A and B together, we first find their combined efficiency: 1 + ¹¹⁄₁₆ = (16 + 11)/(16) = ²⁷⁄₁₆ units per day.

The time taken by A and B together to finish the 15 units of work is: Time = (Total Work)/(Combined Efficiency) = (15)/(27/16) = (15 × 16)/(27) = (5 × 16)/(9) = ⁸⁰⁄₉ days.

Explanation

The expression is evaluated following the order of operations dictated by the VBODMAS rule, where operations inside brackets and of are prioritized.

First, calculate the terms within the parentheses inside the curly braces. In the first set of parentheses, 25 of 12 ÷ 30 = (25 × 12) ÷ 30 = 300 ÷ 30 = 10. In the second set, 5⁴ ÷ 5 of 125 = 625 ÷ (5 × 125) = 625 ÷ 625 = 1.

Substituting these values back into the curly braces, the expression becomes 51 ÷ {25 + 10 - 1}. Simplifying the addition and subtraction inside the braces gives 25 + 10 - 1 = 34.

The final step is to divide 51 by 34. Both numbers are divisible by their greatest common divisor, 17. Thus, the calculation is ⁵¹⁄₃₄ = (17 × 3)/(17 × 2) = ³⁄₂.

Explanation

The trigonometric identity sin A = cos B implies that the angles A and B are complementary when they are acute, meaning A + B = 90°. Given the equation sin (5x - 25°) = cos (5y + 25°), we can set the sum of the angles to 90°:

(5x - 25°) + (5y + 25°) = 90°

Simplifying the expression by combining like terms, the constants -25° and +25° cancel each other out:

5x + 5y = 90°

Factoring out the coefficient 5:

5(x + y) = 90°

Dividing both sides by 5 to find the value of (x + y):

x + y = (90°)/(5) = 18°

Explanation

Let the total work be the LCM of 10, 12, and 6, which is 60 units. The daily efficiencies of the pairs are calculated as Work / Time. Efficiency of (A + B) is 60 / 10 = 6 units/day, Efficiency of (B + C) is 60 / 12 = 5 units/day, and Efficiency of (C + A) is 60 / 6 = 10 units/day.

Summing these efficiencies gives 2(A + B + C) = 6 + 5 + 10 = 21 units/day. Therefore, the combined efficiency of A, B, and C is A + B + C = 21 / 2 = 10.5 units/day. To find the efficiency of A alone, subtract the efficiency of (B + C) from the total: A = 10.5 - 5 = 5.5 units/day, which is equivalent to 11 / 2 units/day. The time taken by A alone to finish the work is Total Work / Efficiency of A, which is 60 / (11/2) = 120 / 11 days.

Explanation

To find the average of the probable weights, we first identify the range of weights that satisfies all given conditions simultaneously. According to Raghav, the weight W is such that 64 < W < 74. His sister states 60 < W < 69. His mother states W ≤ 68, and his father states W ≤ 67.

By intersecting these inequalities: From Raghav and his sister, we get 64 < W < 69. Including the mother's and father's constraints, the upper limit is further restricted to W ≤ 67. Therefore, the common range for the weight is 64 < W ≤ 67.

The probable integer weights in this range are 65, 66, and 67. The average of these weights is calculated as: Average = (65 + 66 + 67)/(3) Average = ¹⁹⁸⁄₃ = 66

Explanation

The relationship between Cost Price (CP) and Marked Price (MP) when both the discount percentage (D%) and profit percentage (P%) are known is given by the formula: (CP)/(MP) = (100 - D%)/(100 + P%)

Given that the marked price is ₹4,200, the discount is 5%, and the profit is 14%, substitute these values into the formula: CP = 4200 × (100 - 5)/(100 + 14) CP = 4200 × ⁹⁵⁄₁₁₄

To simplify the calculation, reduce the fraction ⁹⁵⁄₁₁₄ by dividing both the numerator and the denominator by their greatest common divisor, 19: 95 ÷ 19 = 5 114 ÷ 19 = 6

Thus, the equation becomes: CP = 4200 × ⅚ CP = 700 × 5 CP = 3500

Explanation

The fundamental trigonometric identity states that 1 + tan² A = sec² A. By rearranging this formula to isolate tan² A, we subtract 1 from both sides to obtain tan² A = sec² A - 1. The expression tan² A = 1 - sec² A is the negative of this correct identity, meaning it would imply tan² A = -tan² A, which is mathematically false for all values where the tangent is non-zero.

Explanation

According to the Internal Angle Bisector Theorem, the angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle. In △ XYZ, since XQ is the internal bisector of ∠ X that intersects the side YZ at point Q, the ratio of the segments YQ and QZ is equal to the ratio of the sides XY and XZ.

The relationship is expressed as: (YQ)/(QZ) = (XY)/(XZ)

By substituting the given lengths XY = 4 cm and XZ = 5 cm into the formula, we get: (YQ)/(QZ) = ⅘

This results in the ratio YQ:QZ = 4:5.

Explanation

When two trains travel in the same direction, their relative speed is the difference between their individual speeds. Here, the relative speed is 70 - 34 = 36 km/h. To convert this speed from km/h to m/s, we multiply by ⁵⁄₁₈.

Relative speed in m/s = 36 × ⁵⁄₁₈ = 10 m/s.

The distance covered when one train crosses another is equal to the sum of the lengths of both trains. Using the formula Distance = Speed × Time, the combined length of the trains is 10 m/s × 25 seconds = 250 metres.

Explanation

The total number of valid votes is calculated by first finding the total votes cast and then excluding the invalid portion.

Total votes cast = 85% of 3,00,000 = 0.85 × 3,00,000 = 2,55,000.

Since 10% of these votes were invalid, the remaining 90% are valid. Total valid votes = 90% of 2,55,000 = 0.90 × 2,55,000 = 2,29,500.

Dharam secured 1,37,700 valid votes. To find the percentage of total valid votes secured by Dharam, we use the formula: Percentage = (Dharam's valid votes)/(Total valid votes) × 100 Percentage = (1,37,700)/(2,29,500) × 100

Simplifying the fraction by dividing both numbers by 100 gives 1377 / 2295. Dividing both by their common factor 459 (or step-by-step through 9 and 51): 1377 ÷ 459 = 3 2295 ÷ 459 = 5 Percentage = ⅗ × 100 = 60%

Explanation

For constant trigonometric expressions, the fastest approach is to substitute a specific value for θ. Let θ = 0°. Since sin 0° = 0$ and cos 0° = 1, the expression 12(sin⁴ θ + cos⁴ θ) + 18(sin⁶ θ + cos⁶ θ) + 78 sin² θ cos² θ simplifies to: 12(0⁴ + 1⁴) + 18(0⁶ + 1⁶) + 78(0²)(1²) 12(1) + 18(1) + 0 = 30

Alternatively, using standard trigonometric identities, we know that sin⁴ θ + cos⁴ θ = 1 - 2 sin² θ cos² θ and sin⁶ θ + cos⁶ θ = 1 - 3 sin² θ cos² θ. Substituting these into the original expression gives: 12(1 - 2 sin² θ cos² θ) + 18(1 - 3 sin² θ cos² θ) + 78 sin² θ cos² θ 12 - 24 sin² θ cos² θ + 18 - 54 sin² θ cos² θ + 78 sin² θ cos² θ Combining the constants yields 12 + 18 = 30, and combining the coefficients of sin² θ cos² θ yields -24 - 54 + 78 = 0. Thus, the value of the expression is 30.

Explanation

In an equilateral triangle △ PQR, all internal angles are equal to 60°, which means ∠ PRQ = 60°. When a triangle is inscribed in a circle, its vertices lie on the circumference. According to the property of circles, angles subtended by the same arc at any point on the remaining part of the circumference are equal. In this geometry, both ∠ PRQ and ∠ PSQ are subtended by the same arc PQ. Therefore, the measure of ∠ PSQ must be equal to the measure of ∠ PRQ. Given that ∠ PRQ = 60°, it follows that ∠ PSQ = 60°.

Explanation

The overall percentage is calculated using the weighted average of the individual subject percentages, where the maximum marks of each subject serve as the weights: Overall Percentage = (Σ (Percentage obtained × Maximum marks))/(Σ Maximum marks)

Substituting Hitansh's values into the equation: Overall Percentage = ((67 × 100) + (70 × 150) + (64 × 100) + (55 × 100) + (75 × 100))/(100 + 150 + 100 + 100 + 100) Overall Percentage = (6700 + 10500 + 6400 + 5500 + 7500)/(550) Overall Percentage = ³⁶⁶⁰⁰⁄₅₅₀ = ⁷³²⁄₁₁ ≈ 66.5454%

Rounding the result to two decimal places gives 66.55%.

Explanation

In a 1500 m race, when X reaches the finish line, Y has covered 1500 - 100 = 1400 m and Z has covered 1500 - 240 = 1260 m. The ratio of the speeds of Y and Z is constant and equal to the ratio of the distances they cover in the same time: V_Y / V_Z = 1400 / 1260 = 10 / 9.

To find the distance Z covers when Y completes the 1500 m race, we use the ratio: D_Z = (V_Z / V_Y) × 1500 D_Z = (9 / 10) × 1500 = 1350 m

The distance by which Y beats Z is the difference between the total race distance and the distance Z covers in the time Y takes to finish: Beating Distance = 1500 - 1350 = 150 m

Explanation

The cost of cloth is directly proportional to its length. First, calculate the cost of one meter of cloth using the unitary method: Cost per meter = (Total Cost)/(Total Length) = ⁹⁶⁰⁰⁄₁₂₀ = ₹80

Next, find the cost for the required length of 147 meters by multiplying the unit rate by the total length: Cost of 147m = 147 × 80 = ₹11,760

Explanation

The expression x⁸ - (1)/(x⁸) is simplified using the difference of squares identity as (x⁴ + (1)/(x⁴))(x⁴ - (1)/(x⁴)), which further expands to (x⁴ + (1)/(x⁴))(x² + (1)/(x²))(x + ¹⁄ₓ)(x - ¹⁄ₓ). Starting with the given x + ¹⁄ₓ = 5, we square both sides to find x² + (1)/(x²) = (x + ¹⁄ₓ)² - 2 = 5² - 2 = 23. Squaring this result gives x⁴ + (1)/(x⁴) = (x² + (1)/(x²))² - 2 = 23² - 2 = 529 - 2 = 527.

To determine the value of x - ¹⁄ₓ, we use the algebraic identity (x - ¹⁄ₓ)² = (x + ¹⁄ₓ)² - 4. Substituting the given value, we get (x - ¹⁄ₓ)² = 5² - 4 = 21. Since x > 1, x - ¹⁄ₓ must be positive, so x - ¹⁄ₓ = √(21). Combining all calculated components, the final product is 527 × 23 × 5 × √(21). Calculating the numerical part, 527 × 23 = 12121, and 12121 × 5 = 60605. Thus, the value is 60605√(21).

Explanation

The Simple Interest (SI) earned is calculated as the difference between the Amount (A) and the Principal (P). Thus, SI = 11,780 - 9,500 = 2,280. Using the standard formula for simple interest SI = (P × R × T)/(100), where P is the principal, R is the rate per annum, and T is the time in years, the time can be found using T = (SI × 100)/(P × R). Substituting the known values into the equation gives T = (2,280 × 100)/(9,500 × 8). Simplifying this expression, we have T = (228,000)/(76,000), which equals 3 years.

Explanation

The Cost Price (CP) of the watch is calculated by dividing the Selling Price (SP) by the remaining percentage after the loss. Since the shopkeeper loses 20%, the Selling Price of ₹2,000 represents 80% of the Cost Price. CP = (2000)/(0.80) = 2500

With the Cost Price established at ₹2,500, we determine the profit or loss for the new Selling Price of ₹3,000. Because the Selling Price is higher than the Cost Price, there is a gain. The gain is calculated as: Gain = 3000 - 2500 = 500

The gain percentage is then found by comparing the gain to the Cost Price: Gain % = ⁵⁰⁰⁄₂₅₀₀ × 100 = ⅕ × 100 = 20%

Explanation

The expression is simplified by applying the VBODMAS rule, which dictates the order of operations as Viniculum, Brackets, Of, Division, Multiplication, Addition, and Subtraction.

First, solve the square root and the basic division outside the curly brackets. √(25) simplifies to 5, and 12 ÷ 3 simplifies to 4.

Next, evaluate the terms inside the parentheses within the curly brackets. The rule specifies that the of operator is performed before division. For the first term, 16 of 3 equals 48, which is then divided by 6 to get 8. For the second term, 18 of ½ equals 9, and 54 divided by 9 results in 6.

Now, substitute these results back into the curly bracket expression: {20 + 8 - 6}. Performing the addition and subtraction gives 28 - 6, which equals 22.

Finally, combine all values into the main square bracket: [5 + 4 - 22]. This simplifies to 9 - 22, resulting in the final value of -13.

Explanation

Using the algebraic identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), we substitute the given values a + b + c = 6 and a² + b² + c² = 14: 6² = 14 + 2(ab + bc + ca) 36 = 14 + 2(ab + bc + ca) 22 = 2(ab + bc + ca)

The expression to be evaluated is E = (a - b)² + (b - c)² + (c - a)². Expanding this expression, we get: E = a² - 2ab + b² + b² - 2bc + c² + c² - 2ca + a² E = 2(a² + b² + c²) - 2(ab + bc + ca)

Now, substituting the values a² + b² + c² = 14 and 2(ab + bc + ca) = 22 into the expanded form: E = 2(14) - 22 E = 28 - 22 = 6

Explanation

In simple interest, the interest earned is directly proportional to the time duration for a fixed principal and rate. When an amount doubles in 8 years, the simple interest (SI) earned is equal to the principal (P). Using the simple interest formula SI = (P · r · t)/(100), we can establish the relationship P = (P · r · 8)/(100), which simplifies to r = 12.5%.

To find the time required for Rs. 25,000 to become Rs. 1,00,000, we first calculate the required interest: 1,00,000 - 25,000 = 75,000. Since the principal is Rs. 25,000, the required interest is exactly three times the principal (3P). Because the interest earned is linear, if it takes 8 years to earn interest equal to the principal (1P), it will take 3 × 8 = 24 years to earn interest equal to 3P.

Mathematically, substituting the values into the formula: 75,000 = (25,000 · 12.5 · t)/(100). Dividing both sides by 25,000 yields 3 = (12.5 · t)/(100). Multiplying both sides by 100 gives 300 = 12.5 · t, and solving for t results in t = (300)/(12.5) = 24 years.