SSC CGL Tier 1 2014 Quantitative Aptitude Questions with Solutions

Explanation

When two circles touch each other externally, the distance between their centers is exactly equal to the sum of their radii, denoted as d = r₁ + r₂. In this specific geometric configuration, there are exactly three common tangents that can be drawn. Two of these are external common tangents, which lie on opposite sides of the circles and do not intersect the line segment connecting the centers between the circles. The third is a single internal common tangent that passes through the unique point of contact where the two circles meet. This internal tangent is perpendicular to the line joining the centers of the circles at the point of tangency. Thus, the total number of common tangents is three.

Explanation

The average age of the group is determined by calculating the weighted mean of the ages. The formula for the weighted average is x = (Σ (fᵢ × xᵢ))/(Σ fᵢ), where fᵢ represents the frequency of each age and xᵢ represents the age itself.

Sum of the ages of all students = (3 × 20) + (4 × 21) + (3 × 22) Sum of the ages = 60 + 84 + 66 = 210 years

The total number of students in the group is 3 + 4 + 3 = 10.

Average age = ²¹⁰⁄₁₀ = 21 years

This result can also be observed through the symmetry of the data. Since there are an equal number of students (3 each) at the same distance (1 year) from the central age of 21, the deviations from the mean cancel each other out, leaving 21 as the arithmetic mean.

Explanation

The filling rate of the tap is 1/10 of the cistern per hour. With the leak present, the time taken to fill the cistern increases to 10 + 2 = 12 hours, which means the net filling rate is 1/12 of the cistern per hour. The rate at which the leak empties the cistern is the difference between the tap's filling rate and the net filling rate:

Rateₗₑₐₖ = ⅒ - ¹⁄₁₂

To solve this, use the common denominator of 60:

Rateₗₑₐₖ = ⁶⁄₆₀ - ⁵⁄₆₀ = ¹⁄₆₀

The time required for the leak to empty the full cistern is the reciprocal of its emptying rate. Thus, the leak will empty the full cistern in 60 hours.

Explanation

Let the capacity of the tank be C litres. The inlet pipe A fills the tank at a rate of 8000 litres per hour. The outlet pipe B empties the full tank in 3 hours, meaning its rate of emptying is C/3 litres per hour. In problems of this type where the net time to fill the tank matches the emptying time of the outlet pipe, the net filling rate when both pipes are open is 8000 - C/3 litres per hour.

Applying the work-rate formula where the total capacity is the product of the net rate and the time taken: C = (8000 - C/3) × 3 C = 24000 - C 2C = 24000 C = 12000

Thus, the capacity of the tank is 12000 litres. This confirms that the inlet pipe A fills 24000 litres in 3 hours, while the outlet pipe B removes 12000 litres in the same duration, resulting in a filled tank of 12000 litres.

Explanation

Let the distance to the school be d km. When walking at 4 km/h, the time taken is d/4 hours. When walking at 5 km/h, the time taken is d/5 hours. The difference in arrival times is 5 minutes late versus 10 minutes early, which is a total time difference of 15 minutes. Converting this to hours, we get 15/60 = 1/4 hour.

The distance can be calculated using the formula for speed and time: d/4 - d/5 = 1/4

To solve for d, find a common denominator for the fractions: (5d - 4d)/(20) = 1/4 d/20 = 1/4 d = 20/4 d = 5 km

Alternatively, using the direct formula d = (s₁ × s₂)/(s₂ - s₁) × Δ t: d = (4 × 5)/(5 - 4) × ¹⁵⁄₆₀ d = 20 × ¼ = 5 km

Explanation

In an arithmetic progression with an odd number of terms, the average is always equal to the middle term. For the five consecutive odd numbers a, b, c, d, e, the middle term is c. Since consecutive odd numbers have a common difference of 2, the terms can be expressed in terms of the first number a as b = a + 2, c = a + 4, d = a + 6, and e = a + 8. Thus, the average is a + 4.

This can also be calculated using the standard formula for average. The sum of the five numbers is a + (a + 2) + (a + 4) + (a + 6) + (a + 8) = 5a + 20. Dividing this sum by the total count of 5 gives the average: (5a + 20)/(5) = a + 4.

Explanation

To find the value of the given expression, divide both the numerator and the denominator by x:

Numerator: (x⁴ - 1)/(x²){x} = x³ - (1)/(x³)

Denominator: (3x² + 5x - 3)/(x) = 3x + 5 - ³⁄ₓ = 3(x - ¹⁄ₓ) + 5

Given that x - ¹⁄ₓ = 1, we can calculate the value of x³ - (1)/(x³) using the algebraic identity: (x - ¹⁄ₓ)³ = x³ - (1)/(x³) - 3(x)(¹⁄ₓ)(x - ¹⁄ₓ)

Substituting x - ¹⁄ₓ = 1 into the identity: (1)³ = x³ - (1)/(x³) - 3(1) 1 = x³ - (1)/(x³) - 3 x³ - (1)/(x³) = 4

Now, substitute the values into the simplified expression: (x³ - 1)/(x³){3(x - ¹⁄ₓ) + 5} = (4)/(3(1) + 5) = ⁴⁄₈ = ½

Explanation

The given equation is sin(π x / 2) = x² - 2x + 2. We can rewrite the quadratic expression on the right-hand side by completing the square: x² - 2x + 2 = (x - 1)² + 1. Since (x - 1)² ≥ 0 for all real x, the minimum value of the right-hand side is 1, occurring at x = 1.

On the left-hand side, the sine function has a maximum value of 1, meaning sin(π x / 2) ≤ 1. For the equation to hold, both sides must be equal to 1 simultaneously. Solving (x - 1)² + 1 = 1 yields x = 1. Substituting x = 1 into the sine function gives sin(π(1) / 2) = sin(π / 2) = 1, which satisfies the condition.

Explanation

The expression is evaluated using complementary angle identities and standard trigonometric values. According to the identity sin θ = cos(90° - θ), since 43° + 47° = 90°, the term sin 43° is equal to cos 47°, making the ratio sin 43° / cos 47° = 1. Similarly, since 19° + 71° = 90°, the identity cos θ = sin(90° - θ) implies that cos 19° is equal to sin 71°, making the ratio cos 19° / sin 71° = 1.

For the final term, the standard value of cos 60° is 1/2. Substituting this into the term 8cos ²⁶⁰° yields 8 × (1/2)² = 8 × 1/4 = 2. Combining these results, the simplified expression becomes 1 + 1 - 2 = 0.

Explanation

To determine the largest fraction, we compare their decimal values or use the cross-multiplication method. The decimal equivalent of 8 / 11 is approximately 0.727, which is higher than the values of the other fractions in the group: 0.4, 0.6, and approximately 0.647. Using the cross-multiplication rule, where a fraction a / b is greater than c / d if the product ad is greater than bc, we can verify the standing of 8 / 11. When comparing 8 / 11 with 11 / 17, the product 8 × 17 = 136 exceeds 11 × 11 = 121. Similarly, when comparing 8 / 11 with 3 / 5, the product 8 × 5 = 40 exceeds 11 × 3 = 33. Because 8 / 11 consistently produces the higher product and the largest decimal value, it is the greatest fraction among those listed.

Explanation

Let the Cost Price (CP) be x. The shopkeeper marks the goods 40% above the CP, so the Marked Price (MP) is 1.40x. A discount of 5% is then applied to this Marked Price to find the Selling Price (SP).

Using the relation SP = MP × (1 - Discount Rate): 1064 = 1.40x × (1 - 0.05) 1064 = 1.40x × 0.95 1064 = 1.33x

Solving for x: x = (1064)/(1.33) = 800

The Cost Price of the goods is 800. Profit is the difference between the Selling Price and the Cost Price: Profit = SP - CP Profit = 1064 - 800 = 264

Explanation

Consider a circle with center O and radius r. Let AB be the chord that subtends an angle of 60° at the center, such that ∠ AOB = 60°. In △ OAB, the segments OA and OB are both radii of the circle, therefore OA = OB = r. Since two sides are equal, △ OAB is an isosceles triangle, which implies that the base angles opposite those sides are equal: ∠ OAB = ∠ OBA. Using the property that the sum of angles in a triangle is 180°, we calculate 60° + ∠ OAB + ∠ OBA = 180°. This simplifies to 2 × ∠ OAB = 120°, resulting in ∠ OAB = 60° and ∠ OBA = 60°. Because all three interior angles are 60°, △ OAB is an equilateral triangle. In an equilateral triangle, all sides are of equal length, so the chord AB is equal to the radius r. The ratio between the chord and the radius is r : r, which simplifies to 1 : 1.

Explanation

Let a = x - 10 and b = y - 5. The expression to be evaluated is a³ + b³. First, find the sum of a and b:

a + b = (x - 10) + (y - 5) a + b = x + y - 15

The problem states that x + y = 15. Substituting this into the equation for the sum:

a + b = 15 - 15 = 0

According to the algebraic identity, if a + b = 0, then a = -b. Cubing both sides gives a³ = (-b)³, which simplifies to a³ = -b³. Therefore, a³ + b³ = 0. Alternatively, using the expansion formula a³ + b³ = (a + b)(a² - ab + b²), since the factor (a + b) is 0, the entire product and the resulting expression equal 0.

Explanation

To find the required gain percent on the remaining goods, we use the weighted average method. Let the total quantity of goods be represented as 3 units. The target is a 20% gain on the whole transaction, meaning the total gain should be 3 × 20 = 60 units of profit.

One-third of the goods (1 unit) is sold at a 10% loss, which contributes 1 × (-10) = -10 to the total profit. Let x be the gain percent required on the remaining two-thirds (2 units). The sum of the gains and losses on the individual parts must equal the total desired gain:

-10 + 2x = 60 2x = 70 x = 35%

Alternatively, using the total purchase price of 450, the desired total selling price for a 20% gain is 450 × 1.20 = 540. The cost price of the first one-third is 450 / 3 = 150, and it is sold at a 10% loss, yielding a selling price of 150 × 0.90 = 135. The selling price required for the remaining part is 540 - 135 = 405. Since the cost price of the remaining part is 450 - 150 = 300, the required gain is 405 - 300 = 105. The gain percentage on the remainder is calculated as (105 / 300) × 100 = 35%.

Explanation

The total volume of water wasted is calculated by determining the total number of drops over the given period and converting that volume into litres. First, calculate the total number of seconds in 300 days: Total Seconds = 300 × 24 × 60 × 60 = 25,920,000 seconds

Since the tap drips at a rate of one drop per second, the total number of drops is 25,920,000. The problem states that 600 drops make 100 ml, which means the volume of one drop is 100 / 600 = 1/6 ml.

Total volume in millilitres is: Vₘₗ = 25,920,000 × ⅙ = 4,320,000 ml

To convert this volume into litres, divide by 1000: Vₗᵢₜᵣₑₛ = (4,320,000)/(1000) = 4320 litres

Using a simplified unit conversion approach: V = (300 days × 24 hr/day × 3600 sec/hr × 1 drop/sec × 100 ml)/(600 drops × 1000 ml/litre) V = (300 × 24 × 36 × 100)/(6 × 1000) = (50 × 24 × 36)/(10) = 5 × 24 × 36 = 120 × 36 = 4320

Explanation

According to the Midpoint Theorem, the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and equal to half of its length. In triangle ABC, where P, Q, and R are the midpoints of sides AB, BC, and CA respectively, the side lengths of triangle PQR are determined by the lengths of the sides of triangle ABC. Specifically, PQ = ½ AC, QR = ½ AB, and PR = ½ BC. Since ABC is an equilateral triangle, its sides are all equal, such that AB = BC = CA. Substituting this into the midpoint relations gives PQ = QR = PR = ½ AB. Since all three sides of triangle PQR are equal, it is by definition an equilateral triangle.

Explanation

The savings of 25 represent the absolute value of the 20% discount allowed on the article's marked price. Let the marked price be M. The relationship is given by the formula Discount = Discount Rate × M. Substituting the known values, we get 25 = 0.20 × M, which simplifies to M = (25)/(0.20) = 125. The amount paid by the man is the marked price minus the discount, calculated as 125 - 25 = 100. Alternatively, since the discount is 20%, the man pays 80% of the marked price. If 20% corresponds to 25, then 80% corresponds to 25 × 4 = 100.

Explanation

In a cubical room with side length a, the floor is a square with area A = a². The problem states that the area of the floor is 48 m², so a² = 48. The longest rod that can be placed in a cubical room is equal to the length of its space diagonal. The formula for the space diagonal d of a cube is d = √(a²⁾⁺ᵃ^^^² ⁺ ᵃ^² = √(3a²⁾. By substituting the known value of a² into the formula, we get d = √(3 × 48) = √(144) = 12 metres.

Explanation

The cost price of the house is X lakhs. The owner sold the house for 25% more than the purchase price, which means the gain is 25% of the initial cost. Using the percentage formula, the gain is calculated as: Gain = ²⁵⁄₁₀₀ × X = (X)/(4) lakhs

The tax is specified as 50% of this gain. To find the tax amount, we apply the tax rate to the profit: Tax = ⁵⁰⁄₁₀₀ × Gain Tax = ½ × (X)/(4) Tax = (X)/(8) lakhs

This result is obtained by first determining the absolute profit from the percentage increase and then taking half of that profit as the tax liability.

Explanation

According to the Midpoint Theorem, the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is half its length. In triangle ABC, the points B', C', and A' are given as the midpoints of sides AB, BC, and AC respectively. Consequently, the sides of the medial triangle A'B'C' satisfy the relationships A'B' = ½ BC, B'C' = ½ AC, and A'C' = ½ AB.

Because each side of Δ A'B'C' is exactly half the length of the corresponding side in Δ ABC, the two triangles are similar with a side ratio of 1:2. A fundamental property of geometry states that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore, the ratio of the area of Δ A'B'C' to the area of Δ ABC is (1/2)², which simplifies to 1/4. This implies that joining the midpoints of the sides of any triangle divides it into four congruent triangles of equal area, making the area of the central triangle exactly one-fourth of the original triangle's area.

Explanation

When numbers are expressed in a ratio a:b:c, they can be represented as ax, bx, and cx, where x is their Highest Common Factor (HCF), provided that a, b, and c have no common factor other than 1. In this case, the ratio is 1:2:3. Since the HCF of 1, 2, and 3 is 1, the given HCF of 12 must be the common multiplier x. By multiplying each term of the ratio by the HCF, we obtain the actual numbers: 1 × 12 = 12, 2 × 12 = 24, and 3 × 12 = 36.

Explanation

To find the value of x, simplify the nested expressions starting from the innermost parentheses.

In the first term, calculate the value inside the brackets: -4(-a) = 4a -2(4a) = -8a So, the first term becomes x(-8a) = -8ax.

In the second term, calculate the value inside the brackets: -2(-a) = 2a -2(2a) = -4a So, the second term becomes 5(-4a) = -20a.

Substitute these values back into the original equation: -8ax - 20a = 4a

Dividing the entire equation by a: -8x - 20 = 4 -8x = 24 x = ²⁴⁄₋₈ x = -3

Explanation

The time taken by two individuals working together can be calculated using their rates of work. If A can complete the work in m days, A's rate of work is 1/m of the task per day. Similarly, if B can complete the work in n days, B's rate of work is 1/n of the task per day.

When both work together, their combined rate of work per day is the sum of their individual rates: Combined Rate = ¹⁄ₘ + ¹⁄ₙ

To add these fractions, we find a common denominator, which is mn: Combined Rate = (n + m)/(mn)

The total time taken to complete the work is the reciprocal of the combined rate: Total Days = (1)/(Combined Rate) = (1)/((m + n) / mn) = (mn)/(m + n)

Explanation

Let the distance between point P and the base of the building be x and the length of the flagstaff be L. In the right-angled triangle formed by point P and the building, the relationship is given by tan(30°) = ¹⁰⁄ₓ. Since tan(30°) = (1)/(√(3)), we find x = 10√3 meters.

In the right-angled triangle formed by point P and the top of the flagstaff, the total height is 10 + L. The relationship is given by tan(45°) = (10 + L)/(x). Since tan(45°) = 1, the equation becomes 1 = (10 + L)/(10√(3)), which leads to 10√3 = 10 + L.

Solving for L, we get L = 10√3 - 10 = 10(√3 - 1). Substituting the given value √3 = 1.732, the length of the flagstaff is L = 10(1.732 - 1) = 10(0.732) = 7.32 meters.

Explanation

To find the value of the given expression, substitute the standard trigonometric values: sin (π)/(6) = ½, cos (π)/(4) = (1)/(√(2)), cot (π)/(3) = (1)/(√(3)), sec (π)/(6) = (2)/(√(3)), tan (π)/(4) = 1, and sin (π)/(2) = 1.

Substituting these values into the expression: (1)/(√(2)) · ½ · (1)/(√(2)) - (1)/(√(3)) · (2)/(√(3)) + (5 · 1)/(12 · 1)

Simplifying each term individually: The first term is (1)/(√(2) · 2) · ½ = ½ · ½ = ¼. The second term is (1 · 2)/(√(3) · 3) = ⅔. The third term is ⁵⁄₁₂.

Now, combine the simplified terms: ¼ - ⅔ + ⁵⁄₁₂

To perform the subtraction and addition, convert the fractions to a common denominator of 12: ³⁄₁₂ - ⁸⁄₁₂ + ⁵⁄₁₂ = (3 - 8 + 5)/(12) = ⁰⁄₁₂ = 0

Explanation

Given sin Θ = 3/5, we identify the opposite side as 3 and the hypotenuse as 5. Using the Pythagorean identity a² + b² = c², the adjacent side is √(5² - 3²) = 4. From these values, we determine the required trigonometric ratios: cos Θ = 4/5, tan Θ = 3/4, cot Θ = 4/3, and csc Θ = 5/3.

Substituting these into the expression (tan Θ + cos Θ)/(cot Θ + csc Θ):

The numerator is ¾ + ⅘ = (15 + 16)/(20) = ³¹⁄₂₀. The denominator is ⁴⁄₃ + ⁵⁄₃ = ⁹⁄₃ = 3.

The final value is (31/20)/(3) = ³¹⁄₆₀.

Explanation

To determine the relation between a, b, p, and q, we square both given equations and add them together to eliminate the trigonometric terms. Squaring the first equation a cos Θ + b sin Θ = p yields a² cos² Θ + b² sin² Θ + 2ab sin Θ cos Θ = p². Squaring the second equation a sin Θ - b cos Θ = q yields a² sin² Θ + b² cos² Θ - 2ab sin Θ cos Θ = q².

When these two squared equations are added, the term +2ab sin Θ cos Θ from the first equation and the term -2ab sin Θ cos Θ from the second equation cancel each other out. This addition results in the expression a² (cos² Θ + sin² Θ) + b² (sin² Θ + cos² Θ) = p² + q².

By applying the fundamental trigonometric identity sin² Θ + cos² Θ = 1, the terms within the parentheses simplify to 1. Substituting this value into the equation gives a²(1) + b²(1) = p² + q², which establishes the relationship a² + b² = p² + q².

Explanation

In a pie chart, the central angle for a specific category is calculated using the formula: Central Angle = (Frequency of the Category)/(Total Frequency) × 360°

The total number of students is given as 60. The number of students who chose football as their favourite sport is 15. Substituting these values into the formula: Angle for Football = ¹⁵⁄₆₀ × 360°

Since ¹⁵⁄₆₀ simplifies to ¼, the calculation becomes: Angle for Football = ¼ × 360° = 90°

Explanation

In △ ABC, since AB = AC, the angles opposite these sides are equal, meaning ∠ ABC = ∠ ACB. The exterior angle ∠ ACD and the interior angle ∠ ACB form a linear pair on the line BD, so ∠ ACB = 180° - ∠ ACD = 180° - 120° = 60°.

Given the equality of the base angles in an isosceles triangle, ∠ ABC = ∠ ACB = 60°. The sum of the interior angles in any triangle is 180°, expressed as ∠ A + ∠ ABC + ∠ ACB = 180°. Substituting the known values, we get ∠ A + 60° + 60° = 180°, which simplifies to ∠ A = 180° - 120° = 60°. Alternatively, using the exterior angle theorem, ∠ ACD = ∠ A + ∠ ABC. Substituting 120° = ∠ A + 60° also yields ∠ A = 60°.

Explanation

The total remuneration is distributed in the ratio of the work done by each person. Since A, B, and C worked together for the same duration of 6 days, their share is directly proportional to their efficiencies.

Let the total work be the least common multiple of the time taken by each, which is LCM(16, 24, 6) = 48 units. The efficiency of A is 48 / 16 = 3 units per day. The efficiency of B is 48 / 24 = 2 units per day. The combined efficiency of A, B, and C is 48 / 6 = 8 units per day. Therefore, the efficiency of C is 8 - (3 + 2) = 3 units per day.

The ratio of efficiencies for A, B, and C is 3 : 2 : 3. The sum of the ratio parts is 3 + 2 + 3 = 8. Dividing the total remuneration by the total ratio parts gives 400 / 8 = 50 rupees per part. Consequently, the share of A is 3 × 50 = 150 rupees, the share of B is 2 × 50 = 100 rupees, and the share of C is 3 × 50 = 150 rupees. This distribution corresponds to the values A:150, B:100, and C:150.

Explanation

Let the number be x. The problem states that adding 75 to 75% of x results in x itself. Mathematically, this is expressed as: 0.75x + 75 = x

By rearranging the equation to isolate x, we see that 75 represents the remaining 25% of the number: x - 0.75x = 75 0.25x = 75 x = (75)/(0.25) = 300

To find 40% of this number, multiply the result by 0.40: 40% of 300 = 0.40 × 300 = 120

Explanation

When a train passes a platform, the total distance covered is the sum of the length of the train and the length of the platform. The formula for the total distance is D = Lₜᵣₐᵢₙ + Lₚₗₐₜ_fₒᵣₘ. Given that the train length is 50 metres and the platform length is 100 metres, the total distance D is 50 + 100 = 150 metres. The speed v is determined by the formula v = (D)/(t), where t is the time taken. Substituting the values into the formula gives v = ¹⁵⁰⁄₁₀ = 15 m/s.

Explanation

The series follows the mathematical pattern Tₙ = Tₙ₋₁ × k + k, where k starts from 2 and increases by 1 for each subsequent term.

Starting with the first term: 3 × 2 + 2 = 8 8 × 3 + 3 = 27 27 × 4 + 4 = 112

To find the missing term in the sequence, we apply the next multiplier and increment in the pattern, which is 5: 112 × 5 + 5 = 560 + 5 = 565

This logic is confirmed by calculating the final term of the series using the next increment, 6: 565 × 6 + 6 = 3390 + 6 = 3396

Explanation

The difference between compound interest and simple interest for a period of 2 years is given by the formula D = P(r/100)², where D is the difference, P is the principal sum, and r is the annual rate of interest.

Given that the difference D = 300 and the rate r = 10%, we substitute these values into the formula:

300 = P(10/100)²

300 = P(1/10)²

300 = P(1/100)

P = 300 × 100

P = 30,000

Explanation

Let x be the number to be subtracted from both terms of the ratio 11:15 to obtain the ratio 2:3. This can be represented by the following equation:

(11 - x)/(15 - x) = ⅔

To solve for x, cross-multiply the terms: 3(11 - x) = 2(15 - x) 33 - 3x = 30 - 2x

Rearranging the equation to isolate x gives: 33 - 30 = 3x - 2x x = 3

Alternatively, observe that the difference between the terms in the ratio 11:15 is 4, while the difference in 2:3 is 1. Multiplying the target ratio 2:3 by 4 to equate the differences gives 8:12. Comparing the original ratio 11:15 with the adjusted ratio 8:12 shows that 11 - 8 = 3 and 15 - 12 = 3 must be subtracted from each term.

Explanation

The expression simplifies using the complementary angle identities sin(90° - θ) = cos θ and tan(90° - θ) = cot θ. Since 7.5° and 82.5° add up to 90°, we can replace sin² 82.5° with cos² 7.5°. This allows the first part of the expression to be evaluated as sin² 7.5° + cos² 7.5° = 1, following the standard Pythagorean identity sin² θ + cos² θ = 1. Similarly, 2° and 88° are complementary, so tan 88° = cot 2°. The term tan² 2° tan² 88° becomes tan² 2° cot² 2°. Using the reciprocal identity tan θ · cot θ = 1, this part simplifies to 1. Adding the two simplified components results in 1 + 1 = 2.

Explanation

The relationship between the Cost Price (CP) and the Marked Price (MP) when a discount is given and a profit is earned is expressed by the formula (CP)/(MP) = (100 - Discount%)/(100 + Profit%).

Substituting the given values into the formula where the Marked Price (MP) is 850, the discount is 4%, and the profit is 20%: (CP)/(850) = (100 - 4)/(100 + 20) (CP)/(850) = ⁹⁶⁄₁₂₀

Reducing the fraction ⁹⁶⁄₁₂₀ by dividing both the numerator and denominator by their greatest common divisor, 24, we get: (CP)/(850) = ⅘

To find the Cost Price, multiply both sides by 850: CP = ⅘ × 850 CP = 4 × 170 CP = 680

Explanation

The Cost Price (CP) of the item is determined by subtracting the gain from the selling price: CP = 1800 - 200 = 1600.

The selling price of 1800 was reached after a 10% discount on the Marked Price (MP). Therefore, 1800 = MP × (1 - 0.10), which simplifies to 1800 = 0.9 × MP. Solving for the Marked Price gives MP = (1800)/(0.9) = 2000.

If the item is sold without a discount, the selling price becomes the Marked Price. In this scenario, the profit would be 2000 - 1600 = 400. The profit percentage is calculated on the Cost Price: ⁴⁰⁰⁄₁₆₀₀ × 100 = ¼ × 100 = 25%.

Explanation

The rate of discount is calculated using the formula: Rate of Discount = (Marked Price - Selling Price)/(Marked Price) × 100

Given the marked price is 1200 and the selling price is 1100, the absolute discount offered is: 1200 - 1100 = 100

Substituting the values into the formula: Rate of Discount = ¹⁰⁰⁄₁₂₀₀ × 100 Rate of Discount = ¹⁰⁰⁄₁₂ Rate of Discount = ²⁵⁄₃ Rate of Discount = 8 ⅓%

This value is represented in the options as 81 / 3%.

Explanation

Let the radius of both circles be r. Since the circle with centre A passes through B, the distance AB = r. Similarly, since the circle with centre B passes through A, BA = r. The points of intersection C and D lie on both circles, which means AC = BC = r and AD = BD = r.

In △ ABC, all three sides AB, BC, and AC are equal to r, making it an equilateral triangle. Therefore, ∠ ABC = 60°. In △ ABD, all three sides AB, BD, and AD are also equal to r, making it an equilateral triangle as well. Therefore, ∠ ABD = 60°. The angle ∠ DBC is the sum of these two angles, ∠ ABC + ∠ ABD = 60° + 60° = 120°.

Explanation

The total work rate is the sum of the individual work rates of the man, the woman, and the boy. Let Tₘ, T_w, and T_b be the time taken by the man, woman, and boy respectively, and Tₜ be the time taken together. The relationship is defined by:

1/Tₘ + 1/T_w + 1/T_b = 1/Tₜ

Given Tₜ = 6, Tₘ = 10, and T_w = 24, we substitute these values into the formula:

1/10 + 1/24 + 1/T_b = 1/6

To find the rate of the boy, subtract the rates of the man and woman from the total rate:

1/T_b = 1/6 - 1/10 - 1/24

The least common multiple of 6, 10, and 24 is 120. Converting the fractions:

1/T_b = 20/120 - 12/120 - 5/120 1/T_b = (20 - 12 - 5) / 120 1/T_b = 3/120 1/T_b = 1/40

Taking the reciprocal, T_b = 40. The boy takes 40 days to finish the work alone.

Explanation

The 6th number is common to both the first group of six numbers and the last group of six numbers. Therefore, the sum of the first six numbers and the last six numbers will exceed the total sum of the 11 numbers by the value of this 6th number.

Total sum of 11 numbers = 11 × 63 = 693 Sum of the first 6 numbers = 6 × 60 = 360 Sum of the last 6 numbers = 6 × 65 = 390

To find the 6th number, subtract the total sum of the 11 numbers from the combined sum of the two overlapping groups: 6th number = (360 + 390) - 693 = 750 - 693 = 57

Alternatively, using the deviation method from the overall average of 63: Deviation of the first 6 numbers = 6 × (60 - 63) = 6 × (-3) = -18 Deviation of the last 6 numbers = 6 × (65 - 63) = 6 × (+2) = +12 Net deviation = -18 + 12 = -6 The 6th number = 63 + (-6) = 57

Explanation

Given the equation x² + (1)/(x²) = 66, we can find the value of x - ¹⁄ₓ using the algebraic identity (x - ¹⁄ₓ)² = x² + (1)/(x²) - 2. Substituting the given value, we have (x - ¹⁄ₓ)² = 66 - 2 = 64, which leads to x - ¹⁄ₓ = ± √(64) = ± 8.

The expression to be evaluated, (x² - 1 + 2x)/(x), can be simplified by dividing each term in the numerator by x. This yields (x²)/(x) - ¹⁄ₓ + ²ˣ⁄ₓ = x - ¹⁄ₓ + 2. Substituting the two possible values for x - ¹⁄ₓ into this expression: If x - ¹⁄ₓ = 8, then 8 + 2 = 10. If x - ¹⁄ₓ = -8, then -8 + 2 = -6. Thus, the possible values are 10 and -6.

Explanation

The sum of the interior angles of a triangle is 180°. In Δ ABC, the measure of ∠ A is calculated as ∠ A = 180° - (∠ B + ∠ C) = 180° - (70° + 50°) = 60°. The incentre I is the point where the internal angle bisectors of the triangle intersect. For any triangle, the angle at the incentre subtended by a side is given by the formula ∠ BIC = 90° + (∠ A)/(2). By substituting the calculated value of ∠ A into this expression, we get ∠ BIC = 90° + (60°)/(2) = 90° + 30° = 120°.

Alternatively, since I is the incentre, BI and CI are the bisectors of ∠ B and ∠ C respectively. In Δ BIC, the angles at the base are ∠ IBC = (70°)/(2) = 35° and ∠ ICB = (50°)/(2) = 25°. Using the angle sum property for Δ BIC, the required angle is ∠ BIC = 180° - (35° + 25°) = 180° - 60° = 120°.

Explanation

According to the midpoint theorem in geometry, joining the midpoints of the three sides of a triangle divides the original triangle into four smaller triangles that are congruent to each other. Consequently, each of these four triangles has an area equal to exactly one-fourth of the area of the original triangle. In Δ ABC, if D, E, and F are the midpoints of the sides, the area of the medial triangle Δ DEF is given by the relation Area(Δ DEF) = ¼ × Area(Δ ABC). Substituting the given value of 24 sq. units for the area of Δ ABC, the calculation becomes ¼ × 24 = 6 sq. units.

Explanation

The given equation is a² + a + 1 = 0. Multiplying both sides of the equation by (a - 1), we obtain (a - 1)(a² + a + 1) = 0. Using the algebraic identity for the difference of cubes, where x³ - y³ = (x - y)(x² + xy + y²), the expression simplifies to a³ - 1³ = 0. This leads to a³ = 1. To find the value of a⁹, we use the laws of exponents to write it as (a³)³. Substituting the value of a³ into this expression gives 1³ = 1.

Explanation

In a right-angled triangle, the circumcentre is located at the midpoint of the hypotenuse. This is a direct consequence of Thales's Theorem, which states that the angle subtended by a diameter at any point on the circle's circumference is a right angle (90°). In this geometry, the hypotenuse of the triangle acts as the diameter of its circumcircle. Since the circumcentre of any triangle is the center of its circumcircle, and the center of a diameter lies on the diameter itself, the circumcentre must lie on the side of the triangle that forms the hypotenuse. This characteristic is unique to right-angled triangles, as the circumcentre lies inside acute triangles and outside obtuse triangles.

Explanation

From the given equation x + 2/x = 1, we multiply by x to get x² + 2 = x, or x² - x + 2 = 0. Substituting x² + 2 = x into the numerator of the expression (x² + x + 2), we get x + x = 2x. From x² - x + 2 = 0, we can derive x(1 - x) = 2, so the denominator x²(1 - x) becomes x * [x(1 - x)] = x * 2 = 2x. Thus, the value is 2x / 2x = 1.

Explanation

In △ PQR, the vertex P is common to both △ PQS and △ PSR. Since the bases QS and SR of these triangles lie on the same straight line QR, both triangles share the same perpendicular height h from vertex P to the base QR.

The area of a triangle is given by the formula Area = ½ × base × height.

The length of QS is calculated by subtracting SR from the total length QR: QS = QR - SR = 4.5 cm - 1.5 cm = 3.0 cm

The ratio of the areas of △ PQS and △ PSR is: (Area(△ PQS))/(Area(△ PSR)) = (1)/(2) × QS × h{½ × SR × h} = (QS)/(SR)

Substituting the values: Ratio = (3.0)/(1.5) = ²⁄₁ = 2:1

Explanation

According to the Inscribed Angle Theorem, the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the circumference. In the case of a semi-circle, the arc is a diameter, which subtends a straight angle of 180° at the center of the circle. Therefore, the angle subtended by the semi-circle at any point on the circle's boundary is half of the central angle: (180°)/(2) = 90°. This specific geometric property is known as Thales' Theorem, which establishes that any triangle inscribed in a semi-circle with the diameter as one side must be a right-angled triangle, making the inscribed angle a right angle.