SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

Let the Selling Price (SP) of one pineapple be Rs 1. The Selling Price of 175 pineapples is Rs 175. The problem states that the gain is equal to the SP of 50 pineapples, so the total Gain is Rs 50.

The relationship between Selling Price (SP), Cost Price (CP), and Gain is given by the formula: Gain = SP - CP

Substituting the values for 175 pineapples: 50 = 175 - CP of 175 pineapples CP of 175 pineapples = 175 - 50 = 125

The gain percentage is calculated based on the Cost Price: Gain % = ( (Gain)/(CP) ) × 100 Gain % = ( ⁵⁰⁄₁₂₅ ) × 100

Simplifying the fraction ⁵⁰⁄₁₂₅ gives ⅖. Gain % = ⅖ × 100 = 40%

Explanation

Let the marks of B be 100. Since A has 20% more marks than B, the marks of A are 100 + (20% of 100) = 120.

To calculate how much B's marks are less than A's marks as a percentage, we use A's marks as the base value in the denominator. The difference between their marks is 120 - 100 = 20.

Percentage less = ( (Difference)/(Value of A) ) × 100

Percentage less = ( ²⁰⁄₁₂₀ ) × 100 = ⅙ × 100 = 16.66%

Alternatively, for any problem where one quantity is x% more than another, the second is less than the first by ( (x)/(100 + x) ) × 100%. Substituting x = 20 gives ( ²⁰⁄₁₂₀ ) × 100, which results in 16.66%.

Explanation

Let the distance between Delhi and Jaipur be D. The first train takes 6 hours (10 a.m. to 4 p.m.), so its speed v₁ is D/6. The second train takes 5 hours (12 p.m. to 5 p.m.), so its speed v₂ is D/5.

By 12 p.m., the first train has already traveled for 2 hours, covering a distance of v₁ × 2 = 2D/6 = D/3. The remaining distance between the two trains at 12 p.m. is D - D/3 = 2D/3. Since they are moving toward each other, their relative speed is v₁ + v₂ = D/6 + D/5 = 11D/30.

The time t taken to meet after 12 p.m. is the remaining distance divided by the relative speed:

t = (2D/3)/(11D/30) = ⅔ × ³⁰⁄₁₁ = ²⁰⁄₁₁ hours

Converting this to hours and minutes, t = 1 hour and 9/11 of an hour. Since 9/11 × 60 ≈ 49 minutes, the trains meet approximately 1 hour and 49 minutes after 12 p.m., which is 1:49 p.m.

Explanation

For simple interest (SI) over 1 year at 8% per annum, the interest is SI = (P × 8 × 1)/(100) = 0.08P. When compound interest (CI) is compounded half-yearly, the annual rate is divided by 2 and the number of periods is doubled. Thus, the rate per period is r = 4% and the number of periods is n = 2.

The compound interest is calculated as CI = P (1 + ⁴⁄₁₀₀)² - P = P(1.04)² - P = 1.0816P - P = 0.0816P. The difference between the compound interest and simple interest is 0.0816P - 0.08P = 0.0016P.

Given that this difference is Rs 64, we set up the equation 0.0016P = 64. Solving for the principal P, we get P = (64)/(0.0016) = (64 × 10000)/(16). Simplifying the fraction gives 4 × 10000 = 40000. Alternatively, for two periods, the difference between CI and SI is given by the formula D = P (ʳ⁄₁₀₀)². Substituting the values, 64 = P (⁴⁄₁₀₀)², which leads to 64 = P (¹⁶⁄₁₀₀₀₀), resulting in P = 40000.

Explanation

According to the Factor Theorem, if (x - a) and (x - b) are factors of a quadratic equation, the equation can be represented as (x - a)(x - b) = 0. Given the factors (x - 2) and (x + 3), the equation is: (x - 2)(x + 3) = 0

Expanding the product: x(x + 3) - 2(x + 3) = 0 x² + 3x - 2x - 6 = 0 x² + x - 6 = 0

Comparing this result with the given equation x² + k₁x + k₂ = 0, we equate the coefficients of corresponding terms: Coefficient of x: k₁ = 1 Constant term: k₂ = -6

Alternatively, using Vieta's formulas where the roots are x = 2 and x = -3: Sum of roots: 2 + (-3) = -k₁ ⟹ -1 = -k₁ ⟹ k₁ = 1 Product of roots: 2 × (-3) = k₂ ⟹ k₂ = -6

Explanation

Given the equation x - y = 7, we can rearrange the terms to match the components of the expression (x - 15)³ - (y - 8)³. By subtracting 15 from both sides of the equation, we get x - 15 - y = 7 - 15, which simplifies to x - 15 - y = -8. Adding 8 to both sides results in (x - 15) - y + 8 = 0, which can be rewritten as (x - 15) - (y - 8) = 0.

This shows that (x - 15) = (y - 8). Let a = x - 15 and b = y - 8. Since a = b, it follows that a³ = b³. Therefore, the value of the expression a³ - b³ is 0. Alternatively, using the method of substitution, we can pick values that satisfy the condition x - y = 7. If we let x = 15, then y must be 8. Substituting these values into the target expression gives (15 - 15)³ - (8 - 8)³ = 0³ - 0³ = 0.

Explanation

Given the equations x - y - √(18) = -1 and x + y - 3√2 = 1. Since √(18) = 3√2, these simplify to: x - y = 3√2 - 1 x + y = 3√2 + 1

The term (x² - y²) is calculated using the identity (a + b)(a - b) = a² - b²: x² - y² = (x + y)(x - y) = (3√2 + 1)(3√2 - 1) = (3√2)² - (1)² = 18 - 1 = 17

To find xy, we use the identity 4xy = (x + y)² - (x - y)²: 4xy = (3√2 + 1)² - (3√2 - 1)² Applying (a + b)² - (a - b)² = 4ab: 4xy = 4(3√2)(1) = 12√2 Dividing by 4, xy = 3√2.

Substituting these values into the required expression 12xy(x² - y²): 12 × (3√2) × 17 = 36 × 17 × √2 36 × 17 = 612 Result = 612√2

Explanation

From the given ratios, we have p/q = r/s = t/u = √5. Squaring these terms, we obtain p²/q² = r²/s² = t²/u² = 5. This allows us to express the squared variables in the numerator in terms of those in the denominator: p² = 5q², r² = 5s², and t² = 5u².

Substituting these values into the given expression:

(3p² + 4r² + 5t²)/(3q² + 4s² + 5u²) = (3(5q²) + 4(5s²) + 5(5u²))/(3q² + 4s² + 5u²)

Factoring out the common multiplier 5 from the numerator:

(5(3q² + 4s² + 5u²))/(3q² + 4s² + 5u²) = 5

This result follows the general property of ratios which states that if a/b = c/d = e/f = k, then the value of (maⁿ + ncⁿ + peⁿ)/(mbⁿ + ndⁿ + pfⁿ) is equal to kⁿ. In this problem, k = √5 and n = 2, leading directly to (√5)² = 5.

Explanation

For triangles sharing the same vertex and having their bases on the same straight line, the ratio of their areas is equal to the ratio of the lengths of their bases. Both triangle ABD and triangle ADC share the vertex A and have their bases BD and DC on the straight line BC. Consequently, they share the same perpendicular height h dropped from vertex A to the base line BC.

The area of a triangle is calculated using the formula Area = ½ × base × height. Given that BC = 9 cm and DC = 3 cm, the length of the base BD is BD = BC - DC = 9 - 3 = 6 cm.

The ratio of the areas is:

(Area(△ ABD))/(Area(△ ADC)) = (½ × BD × h)/(½ × DC × h) = (BD)/(DC)

Substituting the values:

Ratio = 6/3 = 2/1

This simplifies to 2 : 1.

Explanation

In a right-angled triangle, the altitude RS drawn to the hypotenuse PQ can be found using the area of the triangle. First, determine the length of the hypotenuse PQ using the Pythagorean theorem: PQ = √(PR² + RQ²) = √(3² + 4²) = √25 = 5 cm.

The area of triangle PQR can be expressed in two ways: Area = ½ × PR × RQ Area = ½ × PQ × RS

Equating these two expressions: PR × RQ = PQ × RS 3 × 4 = 5 × RS 12 = 5 × RS RS = 12/5 cm.

Explanation

The maximum number of points of intersection for n distinct lines in a two-dimensional plane is given by the formula ⁿC₂ = ⁿ⁽ⁿ⁻¹⁾⁄₂. This mathematical limit occurs when no two lines are parallel and no more than two lines intersect at any single point. Substituting n = 4 into the formula, the maximum number of intersection points is ⁴C₂ = (4 × 3)/(2) = 6. Any configuration of four lines, whether they are parallel, concurrent, or in general position, will result in a number of intersection points between 0 and 6 inclusive. Consequently, it is geometrically impossible for four lines to produce 7 points of intersection.

Explanation

The expression csc 2A + cot 2A can be simplified by expressing each term in terms of sine and cosine:

csc 2A + cot 2A = (1)/(sin 2A) + (cos 2A)/(sin 2A) = (1 + cos 2A)/(sin 2A)

Using the trigonometric double angle identities 1 + cos 2A = 2cos² A and sin 2A = 2sin A cos A, the expression is transformed as follows:

(1 + cos 2A)/(sin 2A) = 2cos² A{2sin A cos A}

After cancelling the common factor 2cos A from both the numerator and the denominator, the expression simplifies to:

(cos A)/(sin A) = cot A

Explanation

The expression sin³ A + cos³ B + tan³ C - 3sin A cos B tan C is in the algebraic form x³ + y³ + z³ - 3xyz. This identity is equivalent to (x + y + z)(x² + y² + z² - xy - yz - zx).

Evaluating the individual trigonometric terms for the given angles: x = sin A = sin 30° = ½ y = cos B = cos 60° = ½ z = tan C = tan 135° = tan(180° - 45°) = -1

Calculating the sum x + y + z: x + y + z = ½ + ½ + (-1) = 1 - 1 = 0

Since the sum x + y + z = 0, the entire expression (x + y + z)(x² + y² + z² - xy - yz - zx) simplifies to 0. Thus, the value of sin³ A + cos³ B + tan³ C - 3sin A cos B tan C is 0.

Explanation

The expression f(θ) = tan² θ + cot² θ + sin² θ + cos² θ + sec² θ + csc² θ can be simplified using standard trigonometric identities. We know that sin² θ + cos² θ = 1, sec² θ = 1 + tan² θ, and csc² θ = 1 + cot² θ. Substituting these into the original expression:

f(θ) = tan² θ + cot² θ + 1 + (1 + tan² θ) + (1 + cot² θ) f(θ) = 3 + 2tan² θ + 2cot² θ f(θ) = 3 + 2(tan² θ + cot² θ)

To find the minimum value of tan² θ + cot² θ, we use the Arithmetic Mean-Geometric Mean (AM-GM) inequality. For positive real numbers a and b, the inequality states that (a + b)/2 ≥ √(ab). Applying this to a = tan² θ and b = cot² θ:

(tan² θ + cot² θ)/2 ≥ √(tan² θ · cot² θ)

Since tan θ · cot θ = 1, the product tan² θ · cot² θ is also 1. This simplifies to:

tan² θ + cot² θ ≥ 2

Substituting this minimum value back into the simplified expression for f(θ):

f(θ) ≥ 3 + 2(2) = 7

This minimum value is attained when tan² θ = cot² θ, which occurs when θ = 45°.

Explanation

The distribution of teaching hours for Mr. Raghav indicates that Algebra accounts for 13 percent of the total time, while Modern Maths accounts for 8 percent. To find the difference in the number of hours, first calculate the difference in their percentage shares:

Percentage difference = 13% - 8% = 5%

Now, apply this percentage difference to the total teaching duration of 500 hours:

Difference in hours = ⁵⁄₁₀₀ × 500 = 25

This calculation shows that Mr. Raghav taught Algebra for 25 more hours than Modern Maths.

Explanation

In the teaching schedule for this exam shift, the time allocated for Geometry is 6% of the total duration and the time for Arithmetic is 46%.

Given that the teaching time for Geometry is 36 hours, the value of 1% of the total time is calculated as: 1% = (36 hours)/(6) = 6 hours

Using this, the total time spent on the Arithmetic section is: Arithmetic time = 46 × 6 = 276 hours

The problem specifies that the time taken for teaching the topic of Ratio is one-fourth of the total time spent on Arithmetic: Ratio time = ²⁷⁶⁄₄ = 69 hours

Explanation

The percentage distribution for the teaching hours of the subjects is as follows: Data Interpretation is 15%, Modern Maths is 9%, Number System is 28%, and Geometry is 30%. The combined percentage for Data Interpretation and Modern Maths is 15% + 9% = 24%. Given that this 24% represents 96 hours of teaching time, we can determine the time for 1% by dividing the total hours by the percentage: 96 / 24 = 4 hours per 1%. The combined percentage for Number System and Geometry is 28% + 30% = 58%. To find the total hours for these two subjects, we multiply their combined percentage by the value of 1%: 58 × 4 = 232.

Explanation

The total percentage of time for all topics in the original pie chart is 100%. Based on the data, the time allotted to Arithmetic is 46%. Therefore, the total time allotted to all topics except Arithmetic is: 100% - 46% = 54%

The time devoted to the new topic, Problem Solving, is 10% of the time of all topics except Arithmetic: 10% of 54% = 0.10 × 54 = 5.4%

To find the central angle made by Problem Solving in the new pie chart, we multiply its percentage share by 360 degrees (the total angle of a circle): Central Angle = (5.4)/(100) × 360° Central Angle = 0.054 × 360° = 19.44°

Explanation

To find the count of numbers from 700 to 950 that are neither divisible by 3 nor by 7, we use the principle of inclusion-exclusion. First, calculate the total number of terms in the range: Total numbers N = (950 - 700) + 1 = 251

Next, find the count of numbers divisible by 3 (n(3)), divisible by 7 (n(7)), and divisible by both 3 and 7 (i.e., divisible by LCM(3, 7) = 21). n(3) = ⌊ 950/3 ⌋ - ⌊ 699/3 ⌋ = 316 - 233 = 83 n(7) = ⌊ 950/7 ⌋ - ⌊ 699/7 ⌋ = 135 - 99 = 36 n(21) = ⌊ 950/21 ⌋ - ⌊ 699/21 ⌋ = 45 - 33 = 12

The number of values divisible by either 3 or 7 is given by the union formula: n(3 ∪ 7) = n(3) + n(7) - n(21) n(3 ∪ 7) = 83 + 36 - 12 = 107

The number of values divisible by neither 3 nor 7 is the total count minus those divisible by 3 or 7: Result = N - n(3 ∪ 7) Result = 251 - 107 = 144

Explanation

Let the total work be the Least Common Multiple (LCM) of 20 and 25, which is 100 units. Since A completes the work in 20 days, A's efficiency is 100 / 20 = 5 units per day. Since B completes the same work in 25 days, B's efficiency is 100 / 25 = 4 units per day. When working together, their combined daily efficiency is 5 + 4 = 9 units per day. In 3 days, the total work completed by both is 3 × 9 = 27 units. To find the percentage of the total work completed, we use the formula (Work done / Total work) × 100. Substituting the values gives (27 / 100) × 100 = 27%.

Explanation

The area of a trapezium is calculated using the formula Area = ½ × (a + b) × h, where a and b are the lengths of the two parallel sides and h is the perpendicular distance (height) between them.

Given the lengths of the parallel sides as 18 m and 24 m, and the height as 12 m, the area is:

Area = ½ × (18 + 24) × 12

Area = ½ × 42 × 12

Area = 21 × 12 = 252 m²

Explanation

The net discount for two successive discounts of x% and y% is calculated using the formula: Net Discount = (x + y - ˣʸ⁄₁₀₀)%

Given the successive discounts are 50% and 10%, let x = 50 and y = 10. Substituting these values into the formula: Net Discount = (50 + 10 - (50 × 10)/(100))% Net Discount = (60 - ⁵⁰⁰⁄₁₀₀)% Net Discount = (60 - 5)% Net Discount = 55%

This can also be verified by assuming an initial price of 100. After the first discount of 50%, the price becomes 50. Applying a second discount of 10% on the remaining 50 results in a further reduction of 5. The final price is 50 - 5 = 45. The total reduction from the original price is 100 - 45 = 55, which represents a 55% net discount.

Explanation

To find the final ratio, we first ensure the capacity of each bottle is equal by finding the Least Common Multiple (LCM) of the total parts in each mixture. The total parts for the three mixtures are 2 + 5 = 7, 3 + 4 = 7, and 4 + 5 = 9. The LCM(7, 7, 9) = 63. We assume the capacity of each bottle is 63 units.

In the first bottle, the quantity of milk is (2 / 7) × 63 = 18 units and water is (5 / 7) × 63 = 45 units. In the second bottle, the quantity of milk is (3 / 7) × 63 = 27 units and water is (4 / 7) × 63 = 36 units. In the third bottle, the quantity of milk is (4 / 9) × 63 = 28 units and water is (5 / 9) × 63 = 35 units.

When emptied into a large bottle, the total quantity of milk is 18 + 27 + 28 = 73 units. The total quantity of water is 45 + 36 + 35 = 116 units. The resulting ratio of milk to water in the large bottle is 73 : 116.

Explanation

The total age of the 6 family members is found by multiplying the count by the average: 6 × 20 = 120 years. When the servant joins, the total number of people becomes 7. The average age increases by 25%, so the new average is 20 × 1.25 = 25 years. The total age of the 7 people is 7 × 25 = 175 years. The age of the servant is the difference between the two totals: 175 - 120 = 55 years.

Alternatively, using the deviation method, the servant's age must cover the new average for himself plus the total increase he brings to the existing 6 members. The increase in average is 25 - 20 = 5 years. Thus, Servant age = New Average + (Original count × Increase) = 25 + (6 × 5) = 25 + 30 = 55 years.

Explanation

Let the initial cost price (CP₁) be 100. Since the profit is 190% of the cost price, the initial profit is 190. The selling price (SP) is determined by the sum of the cost price and the profit: SP = CP₁ + Profit = 100 + 190 = 290

According to the problem, the cost price increases by 10%, while the selling price remains unchanged. The new cost price (CP₂) is: CP₂ = 100 × (1 + 0.10) = 110

The new profit earned under these conditions is: New Profit = SP - CP₂ = 290 - 110 = 180

To find what percentage the new profit is of the selling price, we use the following ratio: Percentage = ( (New Profit)/(SP) ) × 100 Percentage = ( ¹⁸⁰⁄₂₉₀ ) × 100 = ¹⁸⁰⁰⁄₂₉ ≈ 62.06%

Rounding to the nearest whole number as per the options, the profit is approximately 62% of the selling price.

Explanation

The marks obtained by student A are 18% more than student B. Given B got 220 marks, A's marks are calculated as 220 × (1 + ¹⁸⁄₁₀₀) = 220 × 1.18 = 259.6.

Student A's marks are also 12% less than student C's marks. Let C represent the marks obtained by student C. This relationship is expressed as A = C × (1 - ¹²⁄₁₀₀), which simplifies to 259.6 = 0.88 × C.

To find C, divide A's marks by 0.88: C = (259.6)/(0.88) = ²⁵⁹⁶⁰⁄₈₈. Dividing both the numerator and denominator by 8 gives ³²⁴⁵⁄₁₁, which results in 295.

Explanation

Person A travels alone for 2 hours from 9:00 a.m. to 11:00 a.m. The distance covered by A in this duration is 25 km/h × 2 h = 50 km. At 11:00 a.m., when Person B starts moving, the distance remaining between A and B is 260 km - 50 km = 210 km. Since both are moving toward each other, their relative speed is the sum of their individual speeds: Relative Speed = 25 km/h + 10 km/h = 35 km/h. The time taken for them to meet after 11:00 a.m. is calculated as Time = (Remaining Distance)/(Relative Speed) = ²¹⁰⁄₃₅ = 6 hours. Adding 6 hours to the 11:00 a.m. start time results in 5:00 p.m.

Explanation

The relationship between the principal P, the amount A, the annual rate of interest r, and the time n in years is given by the formula A = P(1 + ʳ⁄₁₀₀)ⁿ. Substituting the given values P = 2500, A = 2970.25, and n = 2 into the formula:

2970.25 = 2500(1 + ʳ⁄₁₀₀)²

Rearranging the equation to isolate the rate term:

(1 + ʳ⁄₁₀₀)² = (2970.25)/(2500)

(1 + ʳ⁄₁₀₀)² = ²⁹⁷⁰²⁵⁄₂₅₀₀₀₀

Taking the square root of both sides:

1 + ʳ⁄₁₀₀ = √(²⁹⁷⁰²⁵⁄₂₅₀₀₀₀)

1 + ʳ⁄₁₀₀ = ⁵⁴⁵⁄₅₀₀

1 + ʳ⁄₁₀₀ = 1.09

ʳ⁄₁₀₀ = 0.09

r = 9

Alternatively, the total interest earned is 2970.25 - 2500 = 470.25. The total percentage increase over 2 years is (470.25 / 2500) × 100 = 18.81%. For a compound interest rate r, the effective rate over 2 years is r + r + (r² / 100). Setting r = 9, we get 9 + 9 + 0.81 = 18.81%, which confirms the result.

Explanation

The algebraic identity for the given expression is x³ + y³ + z³ - 3xyz = (x + y + z)((x + y + z)² - 3(xy + yz + zx)). From the first given equation, (1/x) + (1/y) + (1/z) = 0, we find the common denominator xyz to get (yz + xz + xy) / xyz = 0. This implies that xy + yz + zx = 0.

Substituting the known values x + y + z = 9 and xy + yz + zx = 0 into the identity, the calculation becomes 9 × (9² - 3 × 0). Simplifying this results in 9 × 81, which equals 729.

Explanation

To find the value of x³ - (1)/(x³), we first determine the value of x - ¹⁄ₓ. Starting with the given equation x⁴ + (1)/(x⁴) = 34, we add 2 to both sides to complete the square: x⁴ + (1)/(x⁴) + 2 = 36 (x² + (1)/(x²))² = 36 x² + (1)/(x²) = 6

Next, we subtract 2 from both sides of this result to find the square of x - ¹⁄ₓ: x² + (1)/(x²) - 2 = 6 - 2 (x - ¹⁄ₓ)² = 4 x - ¹⁄ₓ = 2

Using the algebraic identity for the difference of cubes, a³ - b³ = (a - b)³ + 3ab(a - b), we substitute a = x and b = ¹⁄ₓ: x³ - (1)/(x³) = (x - ¹⁄ₓ)³ + 3(x)(¹⁄ₓ)(x - ¹⁄ₓ) x³ - (1)/(x³) = (x - ¹⁄ₓ)³ + 3(x - ¹⁄ₓ)

Substituting the previously derived value x - ¹⁄ₓ = 2: x³ - (1)/(x³) = (2)³ + 3(2) x³ - (1)/(x³) = 8 + 6 = 14

Explanation

The given equations can be rewritten as x + y = 1 and x² + y² = 2. Using the standard algebraic identity (x + y)² = x² + y² + 2xy, we substitute the values from the equations directly into the formula. This gives (1)² = 2 + 2xy. Simplifying the equation results in 1 = 2 + 2xy, which leads to 2xy = 1 - 2. Therefore, 2xy = -1, and solving for the product gives xy = -1/2.

Explanation

Given the equation x + (1)/(x + 7) = 0, add 7 to both sides to simplify the expression: (x + 7) + (1)/(x + 7) = 7

Let k = x + 7, which implies x = k - 7. Substituting these into the required expression x - (1)/(x + 7) gives: (k - 7) - ¹⁄ₖ = (k - ¹⁄ₖ) - 7

To find the value of k - ¹⁄ₖ, use the algebraic identity (k - ¹⁄ₖ)² = (k + ¹⁄ₖ)² - 4. Substituting k + ¹⁄ₖ = 7: (k - ¹⁄ₖ)² = 7² - 4 = 49 - 4 = 45 k - ¹⁄ₖ = √(45) = 3√5

Substituting the value of k - ¹⁄ₖ back into the expression (k - ¹⁄ₖ) - 7: 3√5 - 7

Explanation

Since PQ is the diameter of the circle, the angle subtended by it at any point on the circumference is a right angle. Therefore, in triangle PQR, ∠ PRQ = 90°. The sum of angles in a triangle is 180°, so ∠ PQR = 180° - (∠ PRQ + ∠ RPQ) = 180° - (90° + 38°) = 52°. In a cyclic quadrilateral, the sum of opposite angles is 180°. For the cyclic quadrilateral PQRS, the opposite angles are ∠ PSR and ∠ PQR. Thus, ∠ PSR = 180° - ∠ PQR = 180° - 52° = 128°.

Explanation

The side of the rhombus is s = 6 cm and the smaller diagonal is d₁ = 6 cm. Since the side length and the diagonal are equal, the rhombus is composed of two equilateral triangles joined at their bases. The diagonals of a rhombus bisect each other at right angles, allowing the use of the Pythagorean theorem in the right-angled triangle formed by the half-diagonals and the side:

(d₁{2})² + (d₂{2})² = s²

Substituting the known values s = 6 and d₁ = 6:

3² + (d₂{2})² = 6² 9 + (d₂{2})² = 36 (d₂{2})² = 27 d₂{2} = √(27) = 3√3 d₂ = 6√3 cm

The side of the required equilateral triangle is equal to the bigger diagonal of the rhombus, so a = d₂ = 6√3 cm. The area of an equilateral triangle is calculated as:

Area = √3{4} a² Area = √3{4} × (6√3)² Area = √3{4} × (36 × 3) Area = √3{4} × 108 Area = 27√3 cm²

Explanation

The expression (sec A + cos A)(sec A - cos A) follows the algebraic identity (x + y)(x - y) = x² - y². By applying this, the expression simplifies to sec² A - cos² A.

Using the fundamental trigonometric identities sec² A = 1 + tan² A and cos² A = 1 - sin² A, we substitute these into the simplified expression: (1 + tan² A) - (1 - sin² A)

Expanding the brackets, we get: 1 + tan² A - 1 + sin² A = tan² A + sin² A

Explanation

To simplify the expression, we convert the trigonometric functions into their sine and cosine equivalents. The denominator cot A + tan A is simplified as follows:

cot A + tan A = (cos A)/(sin A) + (sin A)/(cos A)

To add these fractions, we find a common denominator:

(cos² A + sin² A)/(sin A cos A)

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

(1)/(sin A cos A)

Now, we substitute this back into the original fraction (cot A)/(cot A + tan A):

(cot A)/(1){sin A cos A} = cot A × (sin A cos A)

Substituting cot A = (cos A)/(sin A) into the equation:

(cos A)/(sin A) × sin A cos A = cos² A

Using the identity cos² A = 1 - sin² A, we arrive at the simplified value 1 - sin² A.

Explanation

The expression (tan A)/(1 - cot A) + (cot A)/(1 - tan A) can be simplified by substituting cot A = (1)/(tan A). This gives:

(tan A)/(1 - 1){tan A} + (1)/(tan A){1 - tan A} = (tan² A)/(tan A - 1) + (1)/(tan A(1 - tan A))

Taking a common denominator of tan A(tan A - 1), the expression becomes:

(tan³ A - 1)/(tan A(tan A - 1)) = ((tan A - 1)(tan² A + tan A + 1))/(tan A(tan A - 1)) = (tan² A + tan A + 1)/(tan A)

Simplifying further, we get tan A + 1 + cot A. Using the identity tan A + cot A = (sin A)/(cos A) + (cos A)/(sin A) = (sin² A + cos² A)/(sin A cos A) = (1)/(sin A cos A), and multiplying the numerator and denominator by 2, we have:

tan A + cot A = (2)/(2 sin A cos A) = (2)/(sin 2A)

Substituting this back into our simplified expression, we find that (tan A)/(1 - cot A) + (cot A)/(1 - tan A) = (2)/(sin 2A) + 1. Therefore, the value of the original equation ((tan A)/(1 - cot A) + (cot A)/(1 - tan A)) - (2)/(sin 2A) is 1.

Explanation

To find the number of literate people in City 4, we use the formula provided in the question:

Literate people = (Percentage of literate people × Total population)/(100)

From the table, the percentage of literate people in City 4 is 63%. The total population of City 4 is given as 600,000.

Substituting these values into the formula:

Literate people in City 4 = ⁶³⁄₁₀₀ × 600,000

Literate people in City 4 = 63 × 6,000 = 378,000

Explanation

For City 2, the total population is 220,000 and the literacy rate is 85%. The number of literate people is 220,000 × ⁸⁵⁄₁₀₀ = 187,000. The ratio of literate males to females is 7 : 4. Thus, the number of literate males in City 2 is: Literate Males (City 2) = 187,000 × ⁷⁄₇₊₄ = 187,000 × ⁷⁄₁₁ = 17,000 × 7 = 119,000

For City 6, the total population is 200,000 and the literacy rate is 58%. The number of literate people is 200,000 × ⁵⁸⁄₁₀₀ = 116,000. The ratio of literate males to females is 2 : 3. Thus, the number of literate females in City 6 is: Literate Females (City 6) = 116,000 × ³⁄₂₊₃ = 116,000 × ⅗ = 23,200 × 3 = 69,600

The required ratio of literate males of City 2 to literate females of City 6 is: Ratio = 119,000 : 69,600 Ratio = 1,190 : 696 Dividing both sides by 2: Ratio = 595 : 348

Explanation

The ratio of literate males to females in City 5 is 9 : 7. Given that the number of literate females is 259,210, we can find the total number of literate people (L) in City 5 by using the proportion of females in the literate population:

L × (7)/(9 + 7) = 259,210 L × ⁷⁄₁₆ = 259,210 L = (259,210 × 16)/(7) = 37,030 × 16 = 592,480

The literacy rate of City 5 is 92% of the total population (P). We use the provided formula to find the total population:

(L)/(P) × 100 = 92 (592,480)/(P) = 0.92 P = (592,480)/(0.92) = 644,000

Explanation

To find the number of literate people in each city, we multiply the total population by the percentage of literacy:

Number of literate people in City 1 = 250,000 × 0.80 = 200,000 Number of literate people in City 2 = 200,000 × 0.85 = 170,000 Number of literate people in City 3 = 220,000 × 0.78 = 171,600 Number of literate people in City 4 = 300,000 × 0.63 = 189,000 Number of literate people in City 5 = 150,000 × 0.92 = 138,000 Number of literate people in City 6 = 400,000 × 0.58 = 232,000

Comparing these values, the descending order of the number of literate people is: 232,000 > 200,000 > 189,000 > 171,600 > 170,000 > 138,000

This corresponds to the sequence: City 6 > City 1 > City 4 > City 3 > City 2 > City 5.

Explanation

The total number of key presses required to type numbers from 121 to 1346 is calculated by determining the quantity of 3-digit and 4-digit numbers within the specified range.

The range of 3-digit numbers starts from 121 and ends at 999. The total count of numbers in this segment is 999 - 121 + 1 = 879. Since each of these numbers requires 3 key presses, the total presses for this segment are 879 × 3 = 2637.

The range of 4-digit numbers starts from 1000 and ends at 1346. The total count of numbers in this segment is 1346 - 1000 + 1 = 347. Each of these numbers requires 4 key presses, resulting in 347 × 4 = 1388 presses.

By summing the key presses from both segments, we get 2637 + 1388 = 4025.

Explanation

In labor-related payment problems, wages are distributed in the ratio of the work done by each individual. The total work is represented as 1 unit. Since Sandy and Mandy complete ⁸⁄₁₃ of the work, the remaining share done by Andy is calculated by subtracting their combined work from the total:

Work done by Andy = 1 - ⁸⁄₁₃ = ⁵⁄₁₃

The total payment for the job is Rs 2600. Andy's share is determined by the fraction of the work he contributed relative to the whole:

Share of Andy = ⁵⁄₁₃ × 2600

Dividing 2600 by 13 results in 200. Multiplying this value by 5 yields the final share:

5 × 200 = 1000

Explanation

The volume of the initial solid cone is calculated using the formula V_cₒₙₑ = ⅓ π r₁² h₁. Substituting the given radius of 8 cm and height of 24 cm, the volume is V_cₒₙₑ = ⅓ × π × 8² × 24 = 512π cubic cm.

The volume of the final solid cylinder is determined by the formula V_c_yₗᵢₙ_dₑᵣ = π r₂² h₂. Given a radius of 6 cm and height of 6 cm, the volume is V_c_yₗᵢₙ_dₑᵣ = π × 6² × 6 = 216π cubic cm.

The material wasted in the process is the difference between the volume of the cone and the volume of the cylinder: Wasted Volume = 512π - 216π = 296π cubic cm. To find the waste percentage relative to the original material, the formula is Waste % = (Wasted Volume)/(Initial Volume) × 100. Substituting the values gives (296π)/(512π) × 100 = ³⁷⁄₆₄ × 100 = 57.8125%, which rounds to 57.8%.

Explanation

The net discount for two successive discounts of x% and y% is calculated using the formula x + y - ˣʸ⁄₁₀₀. Given the discounts are 20% and 30%, substituting these values into the formula gives 20 + 30 - (20 × 30)/(100). This simplifies to 50 - ⁶⁰⁰⁄₁₀₀, resulting in 50 - 6 = 44%.

Alternatively, using the 100-base method, if the initial price is 100, a 20% discount reduces the price to 80. A subsequent 30% discount on 80 is calculated as 0.30 × 80 = 24. Subtracting this from 80 gives a final price of 56. The total reduction from the original price of 100 to 56 is 44, which corresponds to a net discount of 44%.

Explanation

To find the mixing ratio, first determine the cost price (CP) of the mixture using the selling price (SP) and the profit percentage. Given SP = 42 and a profit of 20%, the cost price is calculated as: CP = (SP)/(1 + Profit%){100} = (42)/(1.2) = 35 Rs/kg.

Using the rule of alligation, the ratio in which the two types of sugar (Rs 30/kg and Rs 45/kg) must be mixed to achieve a mean price of Rs 35/kg is: Ratio = (Price of Dearer - Mean Price) : (Mean Price - Price of Cheaper) Ratio = (45 - 35) : (35 - 30) Ratio = 10 : 5 = 2 : 1

Explanation

The arithmetic mean follows the property of homogeneity under linear transformations. If every value in a data set is multiplied by a constant k, the average of the data set is also multiplied by that same constant k. This is because the average is defined as A = (Σ xᵢ)/(n). When every individual number xᵢ is doubled, the new sum of the numbers becomes Σ (2xᵢ) = 2 Σ xᵢ. Dividing this new sum by the total count n results in the new average Aₙₑ_w = (2 Σ xᵢ)/(n) = 2 × Aₒₗ_d. Substituting the given original average of 7, the new average is 2 × 7 = 14.

Explanation

Let the initial cost price (CP) be 100u. Since the trader makes a 20% profit, the initial selling price (SP) is 120u.

The problem states that if the trader had bought the article at 60% less, the new cost price would be 100u - 60% of 100u = 40u. On this new cost price, the trader makes a 50% profit. The new selling price is calculated as: New SP = 40u × (1 + 0.50) = 40u × 1.5 = 60u

The difference between the original SP and the new SP is given as Rs 90. Setting up the equation based on the units: 120u - 60u = 90 60u = 90 u = ⁹⁰⁄₆₀ = 1.5

The value of the original cost price is 100u: Value = 100 × 1.5 = 150

Explanation

Let the total income of Amit be x. After donating 20 percent of his income to the school, the remaining amount is 80 percent of x. He then deposits 20 percent of this remaining balance into his bank, which leaves him with 80 percent of that remainder.

Using the successive percentage formula for the final value: Final Amount = x × (1 - ²⁰⁄₁₀₀) × (1 - ²⁰⁄₁₀₀)

Substituting the given final amount: 12800 = x × ⁸⁰⁄₁₀₀ × ⁸⁰⁄₁₀₀ 12800 = x × ⅘ × ⅘ 12800 = x × ¹⁶⁄₂₅

Solving for x: x = (12800 × 25)/(16) x = 800 × 25 x = 20000

The initial income of Amit is Rs 20000.

Explanation

When two objects move in opposite directions, their relative speed is the sum of their individual speeds. In this scenario, the relative speed V is: V = 43 + 51 = 94 km/h

To convert the speed from km/h to m/s, the value is multiplied by ⁵⁄₁₈: V = 94 × ⁵⁄₁₈ m/s

The distance covered by the slower train to cross a man sitting in the faster train is exactly equal to the length of the slower train itself. Using the standard formula Distance = Speed × Time, where the time taken is given as 9 seconds: Length = (94 × ⁵⁄₁₈) × 9

By simplifying the calculation, we divide 9 into 18: Length = 94 × ⁵⁄₂ Length = 47 × 5 = 235 metres