SSC CGL Tier 1 2022 Quantitative Aptitude Questions with Solutions

Explanation

The volume V of a frustum of a cone with height h and radii of the ends R and r is given by the formula V = ⅓ π h (R² + r² + Rr). Substituting the given values h = 7 cm, R = 5 cm, and r = 3 cm, along with π = ²²⁄₇, we get:

V = ⅓ × ²²⁄₇ × 7 × (5² + 3² + 5 × 3)

V = ⅓ × 22 × (25 + 9 + 15)

V = ⅓ × 22 × 49

V = ¹⁰⁷⁸⁄₃ = 359.333...

Rounding to one decimal place, the volume is 359.3 cm³.

Explanation

Let u be the speed of the boat in still water and v be the speed of the stream. The time taken to cover the distance against the stream (upstream) is tᵤ = 9 hours 36 minutes, which is 9 + ³⁶⁄₆₀ = 9 + ⅗ = ⁴⁸⁄₅ hours. The time taken to cover the same distance along the stream (downstream) is t_d = 6 hours.

Since distance is constant, the ratio of speeds is inversely proportional to the ratio of time taken. (u + v)/(u - v) = (tᵤ)/(t_d) = (48/5)/(6) = ⁸⁄₅

Applying the componendo and dividendo rule: ((u + v) + (u - v))/((u + v) - (u - v)) = (8 + 5)/(8 - 5) ²ᵘ⁄₂ᵥ = ¹³⁄₃ ᵘ⁄ᵥ = ¹³⁄₃

The ratio of the speed of the boat in still water to that of the stream is 13:3.

Explanation

Let the total work be the Least Common Multiple (LCM) of 15, 20, and 25. Total work = LCM(15, 20, 25) = 300 units.

The efficiency of each person is calculated as: Efficiency of Raju = 300 / 15 = 20 units/day Efficiency of Shobh = 300 / 20 = 15 units/day Efficiency of Mohan = 300 / 25 = 12 units/day

In one cycle of three days (where they work on alternate days), the work completed is: Work in 3 days = 20 + 15 + 12 = 47 units.

To find the number of full cycles, we divide the total work by the work done per cycle: 300 / 47 = 6 full cycles with a remainder. Work done in 6 cycles (6 × 3 = 18 days) = 47 × 6 = 282 units.

Remaining work = 300 - 282 = 18 units.

On the 19th day, it is Raju's turn to work. Raju's efficiency is 20 units per day. Time taken by Raju to finish the remaining 18 units = 18 / 20 = 9/10 day.

Total time to finish the work = 18 + 9/10 = 18 ⁹⁄₁₀ days.

Explanation

The algebraic identity for the square of a trinomial is (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx. To find the value of the expression a² + b² + c² - 2ab - 2bc + 2ca, we compare it with the standard identity. The squared terms a², b², and c² correspond to x², y², and z². Looking at the cross-product terms, -2ab and -2bc are negative, while +2ca is positive. This indicates that the variable b must have an opposite sign compared to a and c. By setting x = a, y = -b, and z = c, the identity expands as (a - b + c)² = a² + (-b)² + c² + 2(a)(-b) + 2(-b)(c) + 2(c)(a) = a² + b² + c² - 2ab - 2bc + 2ca. This matches the given expression exactly.

Explanation

The average speed for a journey where equal distances are covered at two different speeds, v₁ and v₂, is determined by the harmonic mean of the speeds:

Average Speed = 2v₁v₂{v₁ + v₂}

Given that the jeep travels to a destination at v₁ = 60 km/h and returns the same distance at v₂ = 30 km/h, the calculation is as follows:

Average Speed = (2 × 60 × 30)/(60 + 30)

Average Speed = ³⁶⁰⁰⁄₉₀

Average Speed = 40 km/h

The result can be verified using the total distance and total time method. The time taken for the forward journey is 300 / 60 = 5 hours, and the time for the return journey is 300 / 30 = 10 hours. The total distance for the round trip is 300 + 300 = 600 km, and the total time taken is 5 + 10 = 15 hours. Dividing the total distance by the total time gives 600 / 15 = 40 km/h.

Explanation

The average is calculated by dividing the total sum of the values by the number of observations. Using the formula Average = (Sum of all values)/(Total number of values), we first find the sum of the people in all five states.

Sum = 474 + 500 + 444 + 495 + 580 = 2493

There are 5 states in total (F, G, H, I, and J).

Average = ²⁴⁹³⁄₅ = 498.6

Explanation

The sum of the ages of the man and his son is calculated by multiplying their average age by the number of individuals: 55 × 2 = 110 years. Given the current ratio of their ages is 7:4, the sum of the ratio units is 7 + 4 = 11 units.

To find the value of one ratio unit, divide the total sum by the total units: 110 / 11 = 10 years. Therefore, the man's current age is 7 × 10 = 70 years, and the son's current age is 4 × 10 = 40 years.

After 6 years, the man's age will be 70 + 6 = 76 years and the son's age will be 40 + 6 = 46 years. The required ratio of their ages after 6 years is 76:46. Simplifying this by dividing both terms by their greatest common divisor, 2, gives the final ratio of 38:23.

Explanation

The profit percentage in cases where a dishonest dealer sells goods at cost price using a false weight is given by the formula:

Profit % = (Claimed Weight - Actual Weight)/(Actual Weight) × 100

Given that the profit is 15% and the claimed weight is 1 kg or 1000 gm, let the actual weight used be x gm. Substituting the values into the formula:

15 = (1000 - x)/(x) × 100

Dividing both sides by 5:

3 = (1000 - x)/(x) × 20 3x = 20000 - 20x 23x = 20000 x = ²⁰⁰⁰⁰⁄₂₃

Performing the division:

x ≈ 869.565...

Rounding off to one digit after the decimal, we get 869.6 gm. Thus, the dealer uses a weight of 869.6 gm for every kilogram.

Explanation

The movement of the coconut tree trunk describes an arc of a circle where the length of the tree corresponds to the radius r and the distance covered by the top of the tree is the arc length s. The relationship between these variables is defined by the formula s = rθ, where θ is the angle in radians.

First, convert the angle from degrees to radians. Given that 180° = π radians: θ = 18° × (π)/(180°) = (π)/(10) radians

Substitute the arc length s = 44 metres and the angle θ = (π)/(10) into the formula: 44 = r × (π)/(10)

Rearranging to solve for r: r = (44 × 10)/(π)

Using the value of π ≈ ²²⁄₇: r = (440)/(22){7} = (440 × 7)/(22) r = 20 × 7 = 140 metres

Explanation

To find the year with the highest total exports, calculate the sum of the exports for companies P, Q, and R for each year provided in the table:

Total exports in 2015: 2000 + 4000 + 3000 = 9000 Total exports in 2016: 1000 + 5000 + 2000 = 8000 Total exports in 2017: 2000 + 4000 + 4000 = 10000 Total exports in 2018: 3000 + 5000 + 4000 = 12000 Total exports in 2019: 4000 + 3000 + 5000 = 12000

The highest total export value is 12000. Among the years listed in the choices, the year 2019 yields this maximum total.

Explanation

When the distance between the centres of two circles d = C₁C₂ is exactly equal to the sum of their radii r₁ + r₂, the two circles touch each other externally at a single point. In this geometric configuration, there are exactly three common tangents. Two of these are direct common tangents that lie on the outer sides of the circles. The third is a transverse common tangent that passes through the point of external contact, perpendicular to the line joining the centres C₁C₂. Thus, the total number of common tangents is three.

Explanation

To simplify the expression (1 + sin θ)/(cos θ), multiply both the numerator and the denominator by the conjugate of the numerator, which is 1 - sin θ. This yields:

(1 + sin θ)/(cos θ) · (1 - sin θ)/(1 - sin θ) = (1 - sin² θ)/(cos θ(1 - sin θ))

According to the Pythagorean identity sin² θ + cos² θ = 1, the term 1 - sin² θ is equal to cos² θ. Substituting this into the expression gives:

(cos² θ)/(cos θ(1 - sin θ))

Given that θ ≠ (π)/(2), the value of cos θ is not zero, allowing for the cancellation of one factor of cos θ from both the numerator and the denominator. The expression simplifies to:

(cos θ)/(1 - sin θ)

Explanation

Savings represent the portion of income remaining after all expenditures are deducted. Given that Pankaj spends 65% of his total earnings, his savings percentage is the complement of his spending, which is 100% - 65% = 35%.

To determine the absolute value of his savings, apply this percentage to his total earnings of Rs. 720: Savings = 35% × 720 Savings = ³⁵⁄₁₀₀ × 720

This can be simplified by reducing the fraction or using decimal multiplication: Savings = ⁷⁄₂₀ × 720 Savings = 7 × 36 Savings = 252

Explanation

When two circles of radii r₁ and r₂ touch each other externally, the length of their direct common tangent is given by the standard formula L = 2√(r₁ r₂). This formula is derived from the general expression for a direct common tangent, L = √(d² - (r₁ - r₂)²), where the distance between centers d equals r₁ + r₂ for externally touching circles. Substituting the given values r₁ = 18 cm and r₂ = 8 cm into the formula yields L = 2√(18 × 8) = 2√(144). Calculating the square root results in 2 × 12 = 24 cm.

Explanation

The calculation is divided into two 2-year periods since the interest is added to the principal every second year. For the first two years, the principal P₁ = 15,600, the rate R = 10%, and time T = 2 years. The interest I₁ is calculated using the formula I = (P × R × T)/(100):

I₁ = (15,600 × 10 × 2)/(100) = 3,120

According to the condition, this interest is added to the principal for the next period. The new principal P₂ for the third and fourth years becomes:

P₂ = 15,600 + 3,120 = 18,720

The interest for the next two years I₂ is calculated on this new principal:

I₂ = (18,720 × 10 × 2)/(100) = 3,744

The total interest earned at the end of 4 years is the sum of the interest from both periods:

Total Interest = I₁ + I₂ = 3,120 + 3,744 = 6,864

Explanation

In a cubical room, the floor is a square with side length a. The area of the floor is given as a² = 64 m², which means the side length of the cube is a = √(64) = 8 m. The longest rod that can be placed inside a cubical room corresponds to the space diagonal of the cube. The formula for the length of the space diagonal of a cube with side a is d = a√3. Substituting the value of the side length, the length of the longest rod is 8√3 m.

Explanation

In a proportion a : b :: c : d, the ratio of the first two terms is equal to the ratio of the last two terms, which is mathematically represented as a/b = c/d. This relationship can also be expressed as the product of the extremes being equal to the product of the means, ad = bc.

Given the terms a = 51, b = 68, and d = 108, we substitute these into the proportion formula to find the third term c:

51/68 = c/108

To simplify the calculation, notice that both 51 and 68 are multiples of 17. Reducing the fraction 51/68 gives 3/4. The equation then becomes:

3/4 = c/108

Solving for c by multiplying both sides by 108:

c = (3/4) × 108

Dividing 108 by 4 yields 27. Therefore:

c = 3 × 27 = 81

Explanation

The Angle-Side-Angle (ASA) congruence rule states that two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle. In geometry, the included side is the side that is common to both angles, effectively connecting the two vertices where the angles are located. For example, in △ ABC and △ DEF, if ∠ B = ∠ E, ∠ C = ∠ F, and the side connecting these angles is equal (BC = EF), then the triangles are congruent by the ASA criterion. This rule ensures that once the positions of two angles and the length of the side between them are fixed, the entire geometry of the triangle is uniquely determined.

Explanation

The given equation is 2 sin θ + 2 sin² θ = 2. Dividing both sides by 2, we obtain sin θ + sin² θ = 1. By rearranging the terms, we get sin θ = 1 - sin² θ.

Using the fundamental trigonometric identity cos² θ = 1 - sin² θ, we can substitute this into our simplified equation to find that sin θ = cos² θ. Squaring both sides of this relation yields sin² θ = cos⁴ θ.

The expression to be evaluated is 2 cos⁴ θ + 2 cos² θ. Substituting sin² θ for cos⁴ θ and sin θ for cos² θ, the expression becomes 2 sin² θ + 2 sin θ. Since the original equation states that 2 sin θ + 2 sin² θ = 2, the value of the required expression is 2.

Explanation

The trigonometric identity relating secant and tangent is sec² θ - tan² θ = 1. This can be expressed as a difference of squares:

(sec θ + tan θ)(sec θ - tan θ) = 1

Given that sec θ + tan θ = √5, substitute this value into the equation:

√5 (sec θ - tan θ) = 1

Solving for sec θ - tan θ:

sec θ - tan θ = (1)/(√(5))

To match the format of the options, rationalize the denominator by multiplying both the numerator and the denominator by √5:

sec θ - tan θ = 1 × √5{√5 × √5} = √5{5}

Explanation

Let the efficiency of B be 5 units per day. Since A is 40% more efficient than B, the efficiency of A is 5 + (40% of 5) = 7 units per day. The ratio of efficiencies of A and B is 7 : 5. The total work is the product of B's efficiency and the time B takes alone: Total Work = 5 × 36 = 180 units. When working together, the combined efficiency of A and B is 7 + 5 = 12 units per day. Therefore, the time taken by both to finish the work is (Total Work)/(Combined Efficiency) = ¹⁸⁰⁄₁₂ = 15 days.

Explanation

The value of the algebraic expression p³ + q³ + r³ - 3pqr is governed by the identity p³ + q³ + r³ - 3pqr = (p + q + r)(p² + q² + r² - pq - qr - rp). In this specific case, the sum of the variables is p + q + r = 8.15 + 9.06 + (-17.21). Adding the positive terms gives 8.15 + 9.06 = 17.21, and subtracting the third term results in 17.21 - 17.21 = 0. Since the term (p + q + r) is equal to zero, the product of zero and the remaining quadratic factor in the identity must also be zero.

Explanation

To find the Highest Common Factor (HCF) of 98, 175, and 210, we first determine the prime factorization of each number: 98 = 2 × 7 × 7 = 2 × 7² 175 = 5 × 5 × 7 = 5² × 7 210 = 2 × 3 × 5 × 7

The HCF is calculated by taking the product of the lowest powers of all common prime factors. In this set of numbers, the only prime factor present in all three factorizations is 7. Comparing the powers of 7 (7², 7¹, and 7¹), the lowest power is 7¹. Thus, the HCF is 7. Alternatively, testing for divisibility shows that 7 divides 98 (14 × 7), 175 (25 × 7), and 210 (30 × 7), and it is the greatest such integer that divides all three.

Explanation

For the seven-digit number 52A6B7C to be divisible by 33, it must be divisible by both 3 and 11.

For divisibility by 11, the difference between the sum of the digits at odd positions and the sum of the digits at even positions must be a multiple of 11: (5 + A + B + C) - (2 + 6 + 7) = A + B + C - 10 = 11k Since A, B, and C are prime digits, they must be from the set {2, 3, 5, 7}. The minimum possible sum A+B+C is 2+2+2=6 and the maximum is 7+7+7=21. Within this range, A + B + C - 10 can only be 0 (resulting in A + B + C = 10) or 11 (resulting in A + B + C = 21).

For divisibility by 3, the total sum of the digits must be a multiple of 3: 5 + 2 + A + 6 + B + 7 + C = 20 + (A + B + C) = 3m If A + B + C = 21, the sum is 20 + 21 = 41, which is not divisible by 3. If A + B + C = 10, the sum is 20 + 10 = 30, which is divisible by 3. Thus, A + B + C = 10.

The only set of three prime digits that sums to 10 is {2, 3, 5}. To maximize the value of 2A + 3B + C, we assign the largest weight (3) to the largest digit (5) and the next largest weight (2) to the next largest digit (3). Setting B = 5, A = 3, C = 2: 2(3) + 3(5) + 2 = 6 + 15 + 2 = 23

Explanation

The total number of tractor T5 sold by both companies P and Q is the sum of their individual sales, which is 110 + 150 = 260. Similarly, the total number of tractor T2 sold by both companies is 40 + 50 = 90.

To find how much percent more the sales of T5 are compared to T2, use the percentage increase formula:

Percentage Increase = (Total T5 - Total T2)/(Total T2) × 100

Substituting the calculated values:

Percentage Increase = (260 - 90)/(90) × 100

Percentage Increase = ¹⁷⁰⁄₉₀ × 100 = ¹⁷⁰⁰⁄₉

Dividing 1700 by 9 results in approximately 188.88%.

Explanation

From the given table, the number of scissors sold by shopkeeper R is 45 and the number of scissors sold by shopkeeper S is 25. To determine the ratio of the number of scissors sold by R to the number of scissors sold by S, we set up the following relation: Ratio = (R)/(S) = ⁴⁵⁄₂₅ To simplify this fraction, we divide both the numerator and the denominator by their greatest common divisor, which is 5: Ratio = (45 ÷ 5)/(25 ÷ 5) = ⁹⁄₅ Expressing this in ratio form gives 9:5.

Explanation

The Marked Price (MP) is calculated by applying a 40% markup to the Cost Price (CP) of Rs. 1,500. Using the formula MP = CP × (1 + (Markup%)/(100)), we get MP = 1500 × (1 + 0.40) = 1500 × 1.4 = 2100. To realize a profit of Rs. 75, the Selling Price (SP) must be the sum of the Cost Price and the desired profit, which is SP = CP + Profit = 1500 + 75 = 1575.

The discount amount is the reduction from the Marked Price to the Selling Price, calculated as Discount = MP - SP = 2100 - 1575 = 525. The discount percentage is then found by dividing the discount amount by the Marked Price and multiplying by 100. Applying the formula Discount% = ((Discount)/(MP)) × 100, we have ⁵²⁵⁄₂₁₀₀ × 100 = ⁵²⁵⁄₂₁ = 25%.

Explanation

The trigonometric identity tan A · tan B = 1 implies that A + B = 90°, because tan A = (1)/(tan B) = cot B = tan(90° - B). Given the equation tan 3θ · tan 7θ = 1, we can set up the linear equation:

3θ + 7θ = 90° 10θ = 90° θ = 9°

We verify the condition that 7θ is an acute angle: 7 × 9° = 63°, which is indeed acute. Now, we substitute θ = 9° into the expression cot 15θ:

cot(15 × 9°) = cot 135°

To evaluate cot 135°, we use the quadrant rule cot(180° - x) = -cot x:

cot 135° = cot(180° - 45°) = -cot 45°

Since the value of cot 45° is 1, we get:

-cot 45° = -1

Explanation

To find x + 2/x, we first determine x² + 4/x² by adding 2(x²)(4/x²) = 8 to both sides of the given equation, resulting in (x² + 4/x²)² = 15617 + 8 = 15625, which gives x² + 4/x² = 125. Next, we find (x + 2/x)² by adding 2(x)(2/x) = 4 to 125, resulting in (x + 2/x)² = 129. Therefore, x + 2/x = √129.

Explanation

The number of common tangents to two circles is determined by comparing the distance between their centres d with the sum and difference of their radii r₁ and r₂. Given the diameters are 12 cm and 20 cm, the radii are r₁ = ¹²⁄₂ = 6 cm and r₂ = ²⁰⁄₂ = 10 cm. The sum of the radii is r₁ + r₂ = 6 + 10 = 16 cm. The distance between the centres is given as d = 16 cm. Since d = r₁ + r₂, the two circles touch each other externally at a single point. For two circles touching externally, there are exactly three common tangents: two direct common tangents and one transverse common tangent that passes through the point of contact.

Explanation

The growth percentage for each product is calculated using the formula: Growth % = (Value in 2022 - Value in 2021)/(Value in 2021) × 100

Applying this to Product E, the production increased from 30 in 2021 to 50 in 2022. The growth is (50 - 30)/(30) × 100 = ²⁰⁄₃₀ × 100 = 66.67%.

For comparison, the growth percentages for the other products are as follows: Product A is ³⁰⁄₁₂₀ × 100 = 25%; Product B is ⁴⁰⁄₁₆₀ × 100 = 25%; Product C is ⁵⁰⁄₂₅₀ × 100 = 20%; Product D is ³⁰⁄₆₀ × 100 = 50%; and Product F is ²²⁄₁₁₀ × 100 = 20%. Product E shows the highest growth rate among all the products listed.

Explanation

The area A of an equilateral triangle with side length a is determined by the formula A = √3{4} a². Given that the side length a is 16 cm, substituting this value into the formula gives A = √3{4} × (16)². Since 16² = 256, the expression simplifies to A = √3{4} × 256. Dividing 256 by 4 equals 64, resulting in an area of 64√3 cm².

Explanation

The expression involves subtracting a number from itself (1.6 cubed minus 1.6 cubed). Any number subtracted from itself results in zero.

Explanation

The time taken by Ram to travel from point A to point B is 6 hours (from 8 a.m. to 2 p.m.). The time taken by Raju to travel from point B to point A is 10 hours (from 8 a.m. to 6 p.m.). Since both start at the same time (8 a.m.) and move towards each other on the same track, the time taken for them to meet can be calculated using the relative speed formula or the shortcut for simultaneous starts.

Let the distance between A and B be D. The speed of Ram is S₁ = D/6 and the speed of Raju is S₂ = D/10. The time t after 8 a.m. when they meet is given by the distance divided by their relative speed:

t = (D)/(S₁ + S₂) = (D)/(D){6 + (D)/(10)}

t = (1)/(5+3){30} = ³⁰⁄₈ = 3.75 hours

Alternatively, using the shortcut formula for meeting time t = (T₁ × T₂)/(T₁ + T₂), where T₁ and T₂ are the individual times:

t = (6 × 10)/(6 + 10) = ⁶⁰⁄₁₆ = 3.75 hours

Converting 3.75 hours into hours and minutes: 3 hours + 0.75 × 60 minutes = 3 hours 45 minutes.

Adding 3 hours and 45 minutes to the starting time of 8 a.m. results in 11:45 a.m.

Explanation

The volume of water remains constant when transferred from the conical vessel to the cylindrical vessel. The volume of a cone is given by V_cₒₙₑ = ⅓ π r² h and the volume of a cylinder is V_c_yₗᵢₙ_dₑᵣ = π R² H.

Given for the cone: internal radius r = 20 cm and height h = 27 cm. Given for the cylinder: internal radius R = 15 cm.

Equating the volumes: ⅓ π (20)² (27) = π (15)² H

The constant π cancels out from both sides: ⅓ × 400 × 27 = 225 × H 400 × 9 = 225 × H 3600 = 225 × H

To find H, divide 3600 by 225: H = ³⁶⁰⁰⁄₂₂₅ = 16 cm

The water will rise to a height of 16 cm in the cylindrical vessel.

Explanation

The relationship is expressed as X = kY², where k is the constant of proportionality. Substituting the initial values X = 12 and Y = 2 into the equation gives 12 = k(2²), which simplifies to 12 = 4k, hence k = 3. To find the value of X when Y = 3, substitute k = 3 and Y = 3 into the original formula: X = 3(3²). This calculation yields X = 3 × 9, which equals 27.

Explanation

The Highest Common Factor (HCF) is the largest positive integer that divides each of the given numbers without leaving a remainder. To find the HCF of 60, 148, and 382, we determine the prime factorization of each number:

60 = 2² × 3 × 5 148 = 2² × 37 382 = 2 × 191

The HCF is calculated by taking the product of the lowest power of each common prime factor. In this case, the only common prime factor present in all three numbers is 2. The lowest power of 2 across the three factorizations is 2¹. Therefore, the HCF is 2.

Alternatively, we can use the divisibility rule for the provided options. While 60 and 148 are divisible by 4, 382 is not because its last two digits (82) are not divisible by 4. Furthermore, 382 is not divisible by 3 because the sum of its digits (3 + 8 + 2 = 13) is not a multiple of 3, which eliminates 6 and 24 as potential common factors. Since all three numbers are even, they are all divisible by 2.

Explanation

The relative speed of the policeman with respect to the thief is the difference in their speeds since they are running in the same direction. Relative speed vᵣₑₗ = vₚₒₗᵢ_cₑₘₐₙ - vₜₕᵢₑ_f = 12 - 10 = 2 m/sec.

The time t required to close the initial gap of 40 metres is calculated as: t = (Initial Distance)/(Relative Speed) = ⁴⁰⁄₂ = 20 seconds.

The total distance covered by the policeman to catch the thief is the product of his speed and the time taken: Distance = vₚₒₗᵢ_cₑₘₐₙ × t = 12 × 20 = 240 metres.

Alternatively, since time is constant for both, the ratio of distances covered is equal to the ratio of their speeds. The ratio of policeman's speed to thief's speed is 12:10 or 6:5. The difference of 1 unit in the ratio represents the 40-metre lead. Therefore, the distance covered by the policeman is 6 × 40 = 240 metres.

Explanation

To find the minimum price, we must compare the final costs under both coupon options. The prices of the three shirts are Rs. 1,250, Rs. 1,540, and Rs. 1,375.

Using the first coupon, a 30% discount is applied to the total sum of the prices. The total cost before discount is 1250 + 1540 + 1375 = 4165. The discount amount is 30% of 4165 = 0.30 × 4165 = 1249.5. Thus, the final price using the first coupon is 4165 - 1249.5 = 2915.5.

Using the second coupon, an 80% discount is applied only to the costliest shirt, which is Rs. 1,540. The discount on this shirt is 80% of 1540 = 0.80 × 1540 = 1232. The reduced price of the costliest shirt becomes 1540 - 1232 = 308. The total price using the second coupon is the sum of the two regular-priced shirts and the discounted shirt: 1250 + 1375 + 308 = 2933.

Comparing the two final prices, Rs. 2,915.50 and Rs. 2,933, the minimum price at which Raghav can purchase the three shirts is Rs. 2,915.50.

Explanation

In simple interest, the interest earned is directly proportional to time for a constant principal and interest rate. When a sum of money doubles itself, the interest earned is equal to the principal P (since Amount = P + Interest, and 2P = P + P). This interest P is accumulated over a period of 7 years.

To determine the time required for the sum to become 5 times of itself, the total interest needed is 5P - P = 4P. Using the standard relation for simple interest timing: T₁{T₂} = n₁ - 1{n₂ - 1}

Where T₁ = 7 years, n₁ = 2 (doubles), and n₂ = 5 (five times). Substituting these values into the formula: (7)/(T₂₎ = (2 - 1)/(5 - 1) (7)/(T₂₎ = ¼ T₂ = 7 × 4 = 28

Thus, it will take 28 years for the sum to become 5 times of itself.

Explanation

To find the required difference, we calculate the sum of the joined employees for both categories across all years from 2015 to 2019.

Total number of Business Analysts who joined is 80 + 63 + 78 + 42 + 67 = 330.

Total number of Sales Representatives who joined is 50 + 46 + 38 + 34 + 22 = 190.

The difference between the total number of Business Analysts joined and the total number of Sales Representatives joined is 330 - 190 = 140.

Explanation

The algebraic identity relating the sum of three variables, their squares, and their pairwise products is (x + y + z)² = x² + y² + z² + 2(xy + yz + zx). By substituting the given values x + y + z = 0 and x² + y² + z² = 40 into this formula, we get:

0² = 40 + 2(xy + yz + zx)

0 = 40 + 2(xy + yz + zx)

-40 = 2(xy + yz + zx)

xy + yz + zx = ⁻⁴⁰⁄₂

xy + yz + zx = -20

Explanation

A number is divisible by 11 if the difference between the sum of the digits at odd positions and the sum of the digits at even positions is 0 or a multiple of 11. For the number 54p3987, we calculate these sums starting from the left. The digits at odd positions are 5, p, 9, and 7, so their sum is 5 + p + 9 + 7 = 21 + p. The digits at even positions are 4, 3, and 8, so their sum is 4 + 3 + 8 = 15.

The difference between these sums is (21 + p) - 15 = 6 + p. For the number to be divisible by 11, this difference 6 + p must be a multiple of 11. Since p is a single-digit integer ranging from 0 to 9, the possible values for 6 + p range from 6 to 15. Within this range, the only multiple of 11 is 11 itself. Setting 6 + p = 11 gives p = 11 - 6 = 5.

Explanation

To find the profit percentage, consider the cost price (CP) and selling price (SP) relative to a standard weight of 1000 units. Let the cost price be ₹1 per unit, so 1000 units cost ₹1000. The dealer sells at a 7% loss on the cost price, making the selling price SP = 1000 × (1 - 0.07) = ₹930. Since the dealer uses 18% less weight, the actual quantity delivered is 1000 × (1 - 0.18) = 820 units. The actual cost price to the dealer for this quantity is ₹820.

The profit percentage is calculated using the formula (SP - CP)/(CP) × 100. Substituting the actual values gives (930 - 820)/(820) × 100 = ¹¹⁰⁄₈₂₀ × 100. This simplifies to ¹¹⁰⁰⁄₈₂, which equals approximately 13.4146%. Rounded to two decimal places, the profit percentage is 13.41%.

Explanation

The original average age of the six family members is 40 years. When a guest is included, the total number of people becomes 7, and the average age increases by 12.5%.

The increase in the average age is calculated as 12.5% of 40: Increase = 40 × (12.5)/(100) = 40 × ⅛ = 5 years.

The new average age for the 7 people is 40 + 5 = 45 years.

To find the age of the guest, we use the property that the new member must account for the old average plus the total increase they bring to the entire group: Age of guest = Old Average + (New Total Members × Increase in Average) Age of guest = 40 + (7 × 5) Age of guest = 40 + 35 = 75 years.

Alternatively, using the sum method: Sum of ages of 6 members = 6 × 40 = 240 Sum of ages of 7 people = 7 × 45 = 315 Age of guest = 315 - 240 = 75 years.

Explanation

The expression involves the simplification of trigonometric terms using standard identities. We know that 1 + tan² A = sec² A. Substituting this into the first part of the expression:

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

By expressing tan A and sec A in terms of sine and cosine:

(sin A / cos A)/(1 / cos² A) = (sin A)/(cos A) × cos² A = sin A cos A

Now, simplifying the second part of the expression:

(sin A)/(sec A) = sin A × (1)/(sec A) = sin A cos A

Since both terms simplify to sin A cos A, their difference is:

sin A cos A - sin A cos A = 0

Explanation

To obtain an equation with roots α² and β² from an equation with roots α and β, replace x with √x in the original equation. Substituting into 6x² + 13x + 7 = 0 gives 6(√x)² + 13√x + 7 = 0, which simplifies to 6x + 7 = -13√x. Squaring both sides to remove the radical yields (6x + 7)² = (-13√x)². Expanding the terms results in 36x² + 84x + 49 = 169x. Collecting all terms on one side of the equation produces 36x² - 85x + 49 = 0.

Explanation

When a solid metallic sphere is melted and recast into a cone, the volume remains constant. Let the radius of the sphere be R = 12 cm. The base diameter of the cone is 12 cm, so its radius r = 6 cm. Let the height of the cone be h.

The volume of the sphere is given by Vₛ = ⁴⁄₃ π R³ and the volume of the cone is given by V_c = ⅓ π r² h. Equating the two volumes:

⁴⁄₃ π R³ = ⅓ π r² h

Canceling ⅓ π from both sides, we get:

4 R³ = r² h

Substituting the given values:

4 × (12)³ = (6)² × h

4 × 1728 = 36 × h

h = (4 × 1728)/(36)

h = ⁶⁹¹²⁄₃₆ = 192 cm

Explanation

Let the initial salary be 100. Initial cost of living expenditure is 60% of 100, which is 60. Initial saving is the difference between salary and expenditure: 100 - 60 = 40.

The salary receives two successive increments of 30% and 60%. New Salary = 100 × (1 + ³⁰⁄₁₀₀) × (1 + ⁶⁰⁄₁₀₀) New Salary = 100 × 1.3 × 1.6 = 208.

The saving increases by 250% of the initial saving. Increase in saving = ²⁵⁰⁄₁₀₀ × 40 = 100. New saving = Initial saving + Increase = 40 + 100 = 140.

New cost of living expenditure is the difference between the new salary and new saving: New Expenditure = 208 - 140 = 68.

To find what percentage the new expenditure is of the increased salary: Percentage = ((New Expenditure)/(New Salary)) × 100 Percentage = (⁶⁸⁄₂₀₈) × 100 = ¹⁷⁄₅₂ × 100 ≈ 32.692%.

Rounding to the nearest decimal place as indicated by the options, the value is 32.7%.

Explanation

For any two similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding linear dimensions, such as altitudes, medians, or sides. Given two triangles with areas A₁ and A₂ and corresponding heights h₁ and h₂, the relationship is expressed as (A₁)/(A₂) = ((h₁)/(h₂))².

Substituting the given ratio of the areas x²: y² into the formula, we have (x²)/(y²) = ((h₁)/(h₂))². To find the ratio of the heights, we take the square root of both sides of the equation: √((x²)/(y²)) = (h₁)/(h₂), which simplifies to (x)/(y) = (h₁)/(h₂). Therefore, the ratio of the corresponding heights is x: y.