SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

The expression a³ + b³ + c³ - 3abc follows the algebraic identity ½(a + b + c)((a - b)² + (b - c)² + (c - a)²). When the terms a, b, and c are in an arithmetic progression with a common difference d, this identity simplifies directly to 9 × middle term × d².

Given the values a = 73, b = 74, and c = 75, they form an arithmetic progression with the middle term b = 74 and a common difference d = 1. Substituting these into the simplified formula:

Value = 9 × 74 × 1² Value = 666

This formula is derived from the fact that the sum (a + b + c) is 3b for terms in arithmetic progression, and the sum of the squared differences (a - b)² + (b - c)² + (c - a)² equals (-d)² + (-d)² + (2d)², which is 6d². Combining these with the ½ factor from the original identity yields 9bd².

Explanation

Given x² + (1)/(x²⁾ = ³¹⁄₉, we can find the value of x + ¹⁄ₓ using the identity (x + ¹⁄ₓ)² = x² + (1)/(x²⁾ + 2. Substituting the given value results in (x + ¹⁄ₓ)² = ³¹⁄₉ + 2 = (31 + 18)/(9) = ⁴⁹⁄₉. Since x > 0, we take the positive square root to find x + ¹⁄ₓ = ⁷⁄₃.

To find the value of x³ + (1)/(x³⁾, we use the algebraic identity x³ + (1)/(x³⁾ = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ). Substituting x + ¹⁄ₓ = ⁷⁄₃ into the formula:

x³ + (1)/(x³⁾ = (⁷⁄₃)³ - 3(⁷⁄₃) x³ + (1)/(x³⁾ = ³⁴³⁄₂₇ - 7 x³ + (1)/(x³⁾ = (343 - 189)/(27) x³ + (1)/(x³⁾ = ¹⁵⁴⁄₂₇

Explanation

The expression can be simplified by factorizing each quadratic polynomial using the splitting the middle term method. The first fraction consists of x² - 5x + 6 = (x - 2)(x - 3) and x² - 3x + 2 = (x - 1)(x - 2). The second fraction consists of x² - 7x + 12 = (x - 3)(x - 4) and x² - 5x + 4 = (x - 1)(x - 4). Substituting these factorized forms into the original expression gives:

((x - 2)(x - 3))/((x - 1)(x - 2)) ÷ ((x - 3)(x - 4))/((x - 1)(x - 4))

Simplifying the individual fractions by canceling common terms (x - 2) in the first and (x - 4) in the second results in:

(x - 3)/(x - 1) ÷ (x - 3)/(x - 1)

Since any non-zero expression divided by itself equals 1, the division simplifies directly. Alternatively, multiplying by the reciprocal of the second term gives:

(x - 3)/(x - 1) × (x - 1)/(x - 3) = 1

Explanation

Given the equation x - ¹⁄ₓ = 3, squaring both sides yields x² - 2 + (1)/(x²⁾ = 9, which simplifies to x² + (1)/(x²⁾ = 11. To evaluate the fraction 2x⁴ + 3x³ + 13x² - 3x + 2{3x⁴ + 3}, divide both the numerator and the denominator by x². This transforms the expression into:

2x² + 3x + 13 - ³⁄ₓ + (2)/(x²⁾{3x² + (3)/(x²⁾}

Grouping the terms with identical coefficients allows for the substitution of the known identities:

2(x² + (1)/(x²⁾) + 3(x - ¹⁄ₓ) + 13{3(x² + (1)/(x²⁾)}

Now, substitute x - ¹⁄ₓ = 3 and x² + (1)/(x²⁾ = 11 into the rearranged expression:

(2(11) + 3(3) + 13)/(3(11))

(22 + 9 + 13)/(33) = ⁴⁴⁄₃₃ = ⁴⁄₃

Explanation

For a triangle with sides a, b, and c to exist, the triangle inequality theorem states that the length of the third side must be greater than the absolute difference of the other two sides and less than their sum. If PQ = 983 and PR = 893, and the third side QR = x is an integer, then the range for x is defined as |983 - 893| < x < 983 + 893. This simplifies to 90 < x < 1876.

The number of possible integer values for x in the open interval (90, 1876) is calculated as 1876 - 90 - 1 = 1785. Alternatively, a shortcut formula to find the number of possible integer values for the third side when two sides a and b are known is 2 × min(a, b) - 1. Substituting the given values, we get 2 × 893 - 1, which equals 1786 - 1 = 1785.

Explanation

The expression sec⁶ A - tan⁶ A - 3 sec² A tan² A can be simplified using the algebraic identity x³ - y³ - 3xy(x - y) = (x - y)³. Let x = sec² A and y = tan² A. Substituting these into the identity, we get (sec² A)³ - (tan² A)³ - 3 sec² A tan² A (sec² A - tan² A) = (sec² A - tan² A)³.

From the fundamental trigonometric identity, we know that sec² A - tan² A = 1. Substituting this value into the expanded form, the term (sec² A - tan² A) becomes 1, and the right side of the equation becomes 1³. Therefore, sec⁶ A - tan⁶ A - 3 sec² A tan² A (1) = 1. The simplified value of the given expression is 1.

Explanation

The expression can be simplified by converting all trigonometric ratios into sine and cosine using basic identities.

The first part of the expression is (csc A - sin A) = (1)/(sin A) - sin A = (1 - sin² A)/(sin A). Using the identity sin² A + cos² A = 1, this becomes (cos² A)/(sin A).

The second part is (sec A - cos A) = (1)/(cos A) - cos A = (1 - cos² A)/(cos A), which simplifies to (sin² A)/(cos A).

The third part is (tan A + cot A) = (sin A)/(cos A) + (cos A)/(sin A) = (sin² A + cos² A)/(sin A cos A). Since sin² A + cos² A = 1, this becomes (1)/(sin A cos A).

Multiplying these three simplified components gives: ((cos² A)/(sin A)) × ((sin² A)/(cos A)) × ((1)/(sin A cos A))

Combining the terms in the numerator and denominator: (cos² A sin² A)/(sin² A cos² A) = 1

Alternatively, by substituting a specific value such as A = 45°, where sin A = cos A = (1)/(√(2)), csc A = sec A = √2, and tan A = cot A = 1: (√2 - (1)/(√(2)))(√2 - (1)/(√(2)))(1 + 1) = ((2-1)/(√(2)))((2-1)/(√(2)))(2) = (1)/(√(2)) × (1)/(√(2)) × 2 = ½ × 2 = 1

Explanation

The expression cos⁴ A - sin⁴ A can be factored as a difference of squares using the algebraic identity a² - b² = (a - b)(a + b). By letting a = cos² A and b = sin² A, the expression becomes (cos² A - sin² A)(cos² A + sin² A). According to the fundamental trigonometric identity, cos² A + sin² A = 1. Substituting this value into the equation, we are left with cos² A - sin² A. This is the standard double-angle formula for cosine, which is equal to cos 2A.

Explanation

Prime numbers between 60 and 80 are integers that have no positive divisors other than 1 and themselves. By checking the odd numbers in this range (excluding even numbers as they are divisible by 2), we identify the following primes: 61, 67, 71, 73, and 79. Numbers such as 63 (divisible by 3 and 7), 65 (divisible by 5), 69 (divisible by 3 and 23), 75 (divisible by 3 and 5), and 77 (divisible by 7 and 11) are excluded.

The sum of these prime numbers is calculated as follows: 61 + 67 + 71 + 73 + 79 = 351

Explanation

First, determine the typing speed of both individuals. A types 36 pages in 9 hours, so the rate of A is R_A = ³⁶⁄₉ = 4 pages per hour. B types 40 pages in 5 hours, so the rate of B is R_B = ⁴⁰⁄₅ = 8 pages per hour.

The total work is divided into two parts of 60 pages each. For the first 60 pages, A works alone. The time taken is T₁ = (60)/(R_A) = ⁶⁰⁄₄ = 15 hours. For the last 60 pages, A and B work together. Their combined rate is R_A + R_B = 4 + 8 = 12 pages per hour. The time taken for this part is T₂ = ⁶⁰⁄₁₂ = 5 hours.

The total time taken to complete the 120-page book is the sum of the time for both parts: Tₜₒₜₐₗ = T₁ + T₂ = 15 + 5 = 20 hours.

Explanation

In a right-angled triangle rotated about its hypotenuse, the resulting solid is a double cone. The hypotenuse BC of the triangle serves as the combined height of the two cones, and the altitude AD from the right angle to the hypotenuse serves as the common radius r.

First, calculate the hypotenuse BC using the Pythagorean theorem: BC = √(AB² + AC²) = √(5² + 12²) = √(25 + 144) = 13 cm

The radius r is the altitude to the hypotenuse. Using the area of the triangle relation ½ × AB × AC = ½ × BC × r: r = (AB × AC)/(BC) = (5 × 12)/(13) = ⁶⁰⁄₁₃ cm

The total volume V of the double cone is the sum of the volumes of the two individual cones: V = ⅓ π r² h₁ + ⅓ π r² h₂ = ⅓ π r² (h₁ + h₂) = ⅓ π r² (BC)

Substituting the values: V = ⅓ × ²²⁄₇ × (⁶⁰⁄₁₃)² × 13 V = ⅓ × ²²⁄₇ × ³⁶⁰⁰⁄₁₆₉ × 13 = (22 × 1200)/(7 × 13) = ²⁶⁴⁰⁰⁄₉₁ ≈ 290.11 cm³

Rounding to the nearest whole number, the approximate volume is 290 cm³.

Explanation

The relationship between the Cost Price (CP) and the List Price (MP) can be established using the discount and profit percentages. The Selling Price (SP) is obtained by subtracting the discount from the List Price: SP = MP × (1 - (D)/(100)). Simultaneously, the Selling Price is also defined by the profit earned on the Cost Price: SP = CP × (1 + (P)/(100)).

Given the List Price (MP) is Rs 1360 and the discount is 15%, the Selling Price is: SP = 1360 × (1 - ¹⁵⁄₁₀₀) = 1360 × 0.85 = 1156

Now, using the profit percentage of 15.6%, we can determine the Cost Price: CP = (SP)/(1 + P){100} = (1156)/(1 + 15.6){100} = (1156)/(1.156)

By shifting the decimal point, the calculation becomes: CP = (1156 × 1000)/(1156) = 1000

Thus, the cost price of the trolley bag is Rs 1000.

Explanation

Let the weights of the three pieces be 3x, 5x, and 7x. The total weight of the diamond before breaking was W = 3x + 5x + 7x = 15x. Given that the price P is directly proportional to the square of the weight W², we have P = kW², where k is a constant.

The original price of the diamond was Pₒᵣᵢ_gᵢₙₐₗ = k(15x)² = 225kx². After the diamond broke, the sum of the prices of the three pieces was Pₚᵢₑ_cₑₛ = k(3x)² + k(5x)² + k(7x)² = (9 + 25 + 49)kx² = 83kx².

The loss in value is Pₒᵣᵢ_gᵢₙₐₗ - Pₚᵢₑ_cₑₛ = 225kx² - 83kx² = 142kx². We are given that this loss equals Rs 42600. Therefore, 142kx² = 42600, which gives kx² = ⁴²⁶⁰⁰⁄₁₄₂ = 300. To find the original price, we calculate 225kx² = 225 × 300 = 67500.

Explanation

The total weight of all 100 students is calculated by multiplying the number of students by their average weight, which is 100 × 32 = 3200 kg. The sum of the weights of the first 49 students is 49 × 30 = 1470 kg, and the sum of the weights of the last 50 students is 50 × 34 = 1700 kg. The 50th student is the only one not included in these two groups. Therefore, the weight of the 50th student is the difference between the total weight of the 100 students and the combined weight of the other 99 students.

Weight of 50th student = 3200 - (1470 + 1700) = 3200 - 3170 = 30 kg.

Explanation

The selling price for a fixed quantity of items is proportional to the percentage value relative to the cost price. To find the new selling price SP₂ for 90 pens from the initial selling price SP₁ given a loss percentage L and a target profit percentage P, use the relation SP₂ = SP₁ × (100 + P)/(100 - L). Substituting the given values where SP₁ = 80, L = 20, and P = 20: SP₂ = 80 × (100 + 20)/(100 - 20) SP₂ = 80 × ¹²⁰⁄₈₀ SP₂ = 120

Explanation

Let the initial income of Raman be I = 100. Since he spends 80% of his income, his initial expenditure is E = 80. His initial savings are calculated as S = I - E = 100 - 80 = 20.

According to the question, the income increases by 25%, making the new income Iₙₑ_w = 100 × 1.25 = 125. The expenditure increases by 10%, making the new expenditure Eₙₑ_w = 80 × 1.10 = 88.

The new savings are Sₙₑ_w = Iₙₑ_w - Eₙₑ_w = 125 - 88 = 37. To find the percentage increase in savings, we use the formula:

Percentage Increase = Sₙₑ_w - S{S} × 100

Substituting the values, we get:

Percentage Increase = (37 - 20)/(20) × 100 = ¹⁷⁄₂₀ × 100 = 17 × 5 = 85%

Explanation

When a journey consists of equal distances covered at different speeds, the average speed is the harmonic mean of those speeds. The speeds for the four trips are v₁ = 60 km/hr, v₂ = 30 km/hr, v₃ = 15 km/hr, and v₄ = 7.5 km/hr.

Using the harmonic mean formula for four equal distances: Average Speed = (4)/(1){v₁ + (1)/(v₂) + (1)/(v₃) + (1)/(v₄)}

To simplify the calculation, assume each trip distance is 60 km (the LCM of the speeds). Time for trip 1 = 60 / 60 = 1 hour Time for trip 2 = 60 / 30 = 2 hours Time for trip 3 = 60 / 15 = 4 hours Time for trip 4 = 60 / 7.5 = 8 hours

Total distance covered = 4 × 60 = 240 km Total time taken = 1 + 2 + 4 + 8 = 15 hours

Average speed = (Total Distance)/(Total Time) = ²⁴⁰⁄₁₅ = 16 km/hr

Explanation

The total simple interest paid is the sum of interest accrued over three distinct periods. The formula for simple interest is I = (P × R × T)/(100). When the interest rate changes over different time intervals, the total interest is calculated as I = (P)/(100) × (r₁ t₁ + r₂ t₂ + r₃ t₃), where P is the principal amount.

Given the values: Interest for the first 3 years at 3% = 3 × 3 = 9% of P Interest for the next 5 years at 4% = 5 × 4 = 20% of P Interest for the next 7 years at 6% = 7 × 6 = 42% of P

Total interest percentage = 9 + 20 + 42 = 71% of P

According to the question, the total interest paid is Rs 2059. Therefore: ⁷¹⁄₁₀₀ × P = 2059 P = (2059 × 100)/(71)

Dividing 2059 by 71: 2059 ÷ 71 = 29 P = 29 × 100 = 2900

Explanation

Using the algebraic identity x³ + y³ = (x + y)((x + y)² - 3xy), we substitute the given values x + y = 5 and x³ + y³ = 35:

35 = 5(5² - 3xy) 7 = 25 - 3xy 3xy = 18 xy = 6

Now, to find the positive difference |x - y|, we use the relationship between the sum and difference of two numbers:

(x - y)² = (x + y)² - 4xy (x - y)² = 5² - 4(6) (x - y)² = 25 - 24 = 1 |x - y| = √1 = 1

Alternatively, by inspecting the system of equations x + y = 5 and xy = 6, we can identify the integers 2 and 3. Their sum is 5 and their product is 6. The positive difference between these two numbers is 3 - 2 = 1.

Explanation

To simplify the expression for x, we rationalize the denominator inside the square root by multiplying the numerator and denominator by the conjugate of the denominator, which is 2 + √3:

x = √(2+{3){2-√3} × 2+√3{2+√3}}

x = √((2+{3))²{2² - (√3)²}}

x = √((2+{3))²{4-3}} = √((2+3))² = 2 + √3

Now, we calculate x²:

x² = (2 + √3)² = 4 + 3 + 4√3 = 7 + 4√3

Substitute the values of x² and x into the given expression x² + x - 9:

x² + x - 9 = (7 + 4√3) + (2 + √3) - 9

Grouping the rational and irrational terms:

x² + x - 9 = (7 + 2 - 9) + (4√3 + √3)

x² + x - 9 = 0 + 5√3 = 5√3

Explanation

To find the value of the expression (x²)/(3z) + (y³)/(3xz) + (z²)/(3x), we first combine the fractions by finding a common denominator. The least common multiple of the denominators 3z, 3xz, and 3x is 3xz.

Multiplying the numerator and denominator of each term to match the common denominator, the expression becomes: (x² · x)/(3xz) + (y³)/(3xz) + (z² · z)/(3xz) = (x³ + y³ + z³)/(3xz)

Using the standard algebraic identity, if x + y + z = 0, then x³ + y³ + z³ = 3xyz. Substituting this identity into the numerator of our expression: (3xyz)/(3xz) = y

The common factors 3, x, and z in the numerator and denominator cancel out, leaving the result as y.

Explanation

Given the equation x - ¹⁄ₓ = 1, we can find the value of x + ¹⁄ₓ by using the identity: (x + ¹⁄ₓ)² = (x - ¹⁄ₓ)² + 4 (x + ¹⁄ₓ)² = 1² + 4 = 5 x + ¹⁄ₓ = ± √5

Now, simplify the expression within the parentheses: E = ¹⁄ₓ₋₁ - ¹⁄ₓ₊₁ + (1)/(x²+1) - (1)/(x²-1) E = ( ¹⁄ₓ₋₁ - ¹⁄ₓ₊₁ ) - ( (1)/(x²-1) - (1)/(x²+1) )

Using the common denominator for each pair: ¹⁄ₓ₋₁ - ¹⁄ₓ₊₁ = ((x+1) - (x-1))/(x²-1) = (2)/(x²-1) (1)/(x²-1) - (1)/(x²+1) = ((x²+1) - (x²-1))/((x²-1)(x²+1)) = (2)/(x⁴-1)

Subtracting these results: E = (2)/(x²-1) - (2)/(x⁴-1) = (2(x²+1) - 2)/(x⁴-1) = (2x²)/(x⁴-1)

From the given x - ¹⁄ₓ = 1, we have x² - 1 = x. Substituting this into the denominator x⁴ - 1 = (x²-1)(x²+1): x⁴ - 1 = x(x²+1)

Thus, the expression becomes: E = (2x²)/(x(x²+1)) = (2x)/(x²+1)

Dividing the numerator and denominator by x: E = (2)/(x + 1){x}

Substituting x + ¹⁄ₓ = ± √5: E = (2)/(± √(5)) = ± (2)/(√(5))

Explanation

In a circle with centre O and diameter PQ, the central angle ∠ POQ is 180°. Given that the chord RS is parallel to PQ and ∠ ROS = 40°, the symmetry of the circle ensures that the arcs PR and QS are equal. The sum of the central angles along the diameter is ∠ POR + ∠ ROS + ∠ SOQ = 180°. Substituting the given values, 2 × ∠ POR + 40° = 180°, which leads to ∠ POR = ∠ SOQ = 70°.

The angle ∠ TPQ in △ PTQ is the same as the inscribed angle ∠ RPQ. This angle subtends the arc RSQ. The measure of arc RSQ is the sum of the central angles ∠ ROS and ∠ SOQ, which is 40° + 70° = 110°. Since an inscribed angle is half the measure of its intercepted arc, ∠ RPQ = ½ × 110° = 55°. By symmetry, the inscribed angle ∠ TQP (or ∠ SQP) subtends arc RSP, which is also 70° + 40° = 110°, making ∠ TQP = 55°.

In △ PTQ, the sum of the interior angles is 180°. Therefore, the measure of ∠ PTQ is calculated as 180° - (∠ TPQ + ∠ TQP) = 180° - (55° + 55°) = 70°.

Explanation

Given the numerical relationship involving 576 (24²) and 625 (25²), we utilize the Pythagorean triplet (7, 24, 25). To evaluate the expression, we identify the trigonometric ratios as sin θ = 24/25 and cos θ = 7/25.

Substituting these values into the given expression: (sin θ - cos θ)/(sin θ + cos θ) = (24)/(25) - ⁷⁄₂₅{²⁴⁄₂₅ + ⁷⁄₂₅}

Perform the subtraction in the numerator and addition in the denominator: (17/25)/(31/25)

The common denominator 25 cancels out, leaving: ¹⁷⁄₃₁

Explanation

Using the complementary angle relationship sin θ = cos(90° - θ), the term sin 51° can be rewritten as: sin 51° = cos(90° - 51°) = cos 39°

Substituting this into the first part of the expression: (cos 39°)/(sin 51°) = (cos 39°)/(cos 39°) = 1

Next, applying the trigonometric identity sin² θ + cos² θ = 1 to the terms under the square root: sin² 39° + sin² 51° = sin² 39° + cos² 39° = 1

The given expression simplifies to ³⁄₂(1) - √1. Since the square root of 1 can be considered as both 1 and -1 in this context:

Using the positive root (+1): ³⁄₂ - 1 = (3 - 2)/(2) = ½

Using the negative root (-1): ³⁄₂ - (-1) = ³⁄₂ + 1 = (3 + 2)/(2) = ⁵⁄₂

Thus, both 1/2 and 5/2 are obtained as values for the expression.

Explanation

The given expression for cot A can be simplified using the half-angle trigonometric identities sin B = 2 sin(B/2) cos(B/2) and 1 - cos B = 2 sin²(B/2). Substituting these into the equation:

cot A = (2 sin(B/2) cos(B/2))/(2 sin²(B/2))

cot A = (cos(B/2))/(sin(B/2)) = cot(B/2)

From this relation, we can determine that A = B/2 (or more generally A = nπ + B/2). Multiplying both sides by 2 gives 2A = B + 2nπ.

Applying the cotangent function to both sides:

cot 2A = cot(B + 2nπ) = cot B

Explanation

To convert a binary number to its decimal equivalent, the positional notation method is used where each digit is multiplied by 2ⁿ, where n is the position of the digit starting from zero at the least significant bit (rightmost).

For the binary value 1101111₂, the calculation is: (1 × 2⁶) + (1 × 2⁵) + (0 × 2⁴) + (1 × 2³) + (1 × 2²) + (1 × 2¹) + (1 × 2⁰) = 64 + 32 + 0 + 8 + 4 + 2 + 1 = 111

By summing these values, the binary string 1101111 is converted to the decimal integer 111.

Explanation

Let the total work be 1 unit. The efficiency of A is 1/18 work per day and the efficiency of B is 1/36 work per day. Their combined efficiency when working together is:

Eₜₒₜₐₗ = ¹⁄₁₈ + ¹⁄₃₆ = (2 + 1)/(36) = ³⁄₃₆ = ¹⁄₁₂ unit per day

In 9 days, the amount of work completed by A and B together is calculated by multiplying their combined efficiency by the number of days:

Work done = 9 × ¹⁄₁₂ = ¾

The remaining work is the difference between the total work and the work completed:

Work left = 1 - ¾ = ¼

To convert the fraction of remaining work into a percentage, multiply by 100:

Percentage of work left = ¼ × 100 = 25

After working together for 9 days, 25 percent of the work remains to be finished.

Explanation

In a square, the relationship between the diagonal d and the side s is governed by the formula d = s√2, derived from the Pythagorean theorem where s² + s² = d². Given that the diagonal is 12 cm, we substitute this value into the equation:

12 = s√2

To solve for the side s, we rearrange the formula:

s = (12)/(√(2))

To simplify this expression, we rationalize the denominator by multiplying both the numerator and the denominator by √2:

s = 12 × √2{√2 × √2}

s = 12√2{2}

s = 6√2

Thus, the length of the side of the square is 6√2 cm.

Explanation

The marked price of the item is Rs 2500. After the initial discount of 30%, the price becomes 2500 × (1 - ³⁰⁄₁₀₀) = 2500 × 0.7 = 1750.

The item is finally sold for Rs 1400. The additional discount is calculated on the price after the first discount, which is Rs 1750. The amount of additional discount provided is 1750 - 1400 = 350.

To find the additional discount percentage, we use the formula: Additional Discount % = (Additional Discount Amount)/(Price after first discount) × 100

Substituting the values, we get: ³⁵⁰⁄₁₇₅₀ × 100 = ⅕ × 100 = 20%

Explanation

The third proportional c of two numbers a and b is defined by the continued proportion a:b = b:c. This relationship can be expressed as (a)/(b) = (b)/(c), which leads to the formula c = b²{a}. Given the values a = 10 and b = 25, we substitute them into the formula: c = 25²{10} = ⁶²⁵⁄₁₀ = 62.5.

Explanation

The total weight of the class is calculated by multiplying the total number of students by the class average weight: 50 × 55.44 = 2772 kg. Since there are 27 boys in a class of 50, the number of girls is 50 - 27 = 23. The total weight of the boys is the product of the number of boys and their average weight: 27 × 72 = 1944 kg. Subtracting the weight of the boys from the total class weight gives the total weight of the girls: 2772 - 1944 = 828 kg. The average weight of the girls is then found by dividing this total weight by the number of girls: ⁸²⁸⁄₂₃ = 36 kg.

Explanation

The relationship between selling price (SP), cost price (CP), and profit or loss percentage is given by the formula SP = CP × (100 ± percentage)/(100). In the first scenario, a selling price of Rs 1900 results in a 5% loss, meaning that Rs 1900 represents 95% of the cost price. This is calculated as CP = (1900)/(0.95) = 2000.

To realize a 15% profit, the saree must be sold at 115% of its cost price. Using the previously determined cost price of Rs 2000, the new selling price is calculated as SPₙₑ_w = 2000 × ¹¹⁵⁄₁₀₀ = 20 × 115 = 2300.

A faster approach involves using the direct proportion between the selling prices and their respective percentage values. The required price can be found using the expression 1900 × ¹¹⁵⁄₉₅. Simplifying the fraction by dividing both 115 and 95 by 5 results in ²³⁄₁₉. Thus, the calculation becomes 1900 × ²³⁄₁₉ = 100 × 23 = 2300.

Explanation

Let M and S represent the marks in Math and Science respectively. The problem states that Math marks are 60% of the total marks (M + S). This implies that Science marks must represent the remaining 40% of the total marks, as 100% - 60% = 40%.

We can express the ratio of Math marks to Science marks as: M : S = 60 : 40 = 3 : 2

Let M = 3x and S = 2x. The difference between the marks is given as M - S = 30. Substituting the ratio values: 3x - 2x = 30 x = 30

The Science marks are calculated as: S = 2x = 2 × 30 = 60

Explanation

Let Tₑ be the time taken by the Express train and T_d be the time taken by the Duronto train. According to the first condition, Tₑ = T_d + 1. When the speed of the Express train increases by 50%, its speed becomes 1.5 times the original, so the new time taken is Tₑ / 1.5 = ⅔ Tₑ. The second condition states this new time is 2 hours less than the Duronto train, so ⅔ Tₑ = T_d - 2.

The total difference in time for the Express train between the two scenarios is the difference between being 1 hour behind and 2 hours ahead of the Duronto train, which is 1 + 2 = 3 hours. We can solve for Tₑ using the relation: Tₑ - ⅔ Tₑ = 3 ⅓ Tₑ = 3 Tₑ = 9 hours

Since the Express train takes 9 hours and this is 1 hour more than the Duronto train, the time taken by the Duronto train is T_d = 9 - 1 = 8 hours. The speed of the Duronto train is: Speed = (Distance)/(Time) = ⁴³²⁄₈ = 54 km/hr

Explanation

When interest is compounded yearly for one year at 12%, the interest is 4000 × ¹²⁄₁₀₀ = 480.

When interest is compounded half-yearly, the annual rate is divided by 2 and the number of periods in a year is 2. Thus, the effective rate per period is r = 6% and the number of periods is n = 2. Using the formula for compound interest CI = P[(1 + ʳ⁄₁₀₀)ⁿ - 1]: CI = 4000[(1 + ⁶⁄₁₀₀)² - 1] = 4000[(1.06)² - 1] = 4000[1.1236 - 1] = 4000 × 0.1236 = 494.4.

The difference between the two interests is 494.4 - 480 = 14.4.

A faster approach for a one-year period is to recognize that the difference between yearly and half-yearly compounding is the interest earned on the first half-year's interest during the second half-year. Interest for the first half-year = 4000 × ⁶⁄₁₀₀ = 240. Difference = Interest on 240 at 6% for the remaining half-year = 240 × ⁶⁄₁₀₀ = 14.4.

Explanation

The given linear equation is 7x - (3(2x - 3))/(2) = ½. To simplify the expression and solve for x, multiply the entire equation by 2 to eliminate the denominator:

2(7x) - 3(2x - 3) = 1

Now, expand the bracket on the left side:

14x - 6x + 9 = 1

Group the terms containing x and simplify:

8x + 9 = 1

Subtract 9 from both sides of the equation to isolate the term with x:

8x = 1 - 9

8x = -8

Divide both sides by 8:

x = ⁻⁸⁄₈ = -1

Explanation

The value of the expression a³ + b³ is determined using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). By substituting the given values a + b = 4 and ab = 3 into this formula, the calculation becomes 4³ - 3(3)(4). Solving this gives 64 - 36, which results in 28.

Alternatively, for a faster approach, one can identify two numbers whose sum is 4 and product is 3. The numbers 1 and 3 satisfy these conditions since 1 + 3 = 4 and 1 × 3 = 3. Calculating the sum of their cubes gives 1³ + 3³ = 1 + 27 = 28.

Explanation

Let the required fraction be x. According to the problem statement, the sum of this fraction and 7 times its reciprocal is equal to 11/2. This can be expressed as the equation x + 7 · (1/x) = 11/2.

To solve for x, multiply the entire equation by 2x to clear the denominators: 2x² + 14 = 11x

Rearranging the terms into a standard quadratic equation ax² + bx + c = 0 gives: 2x² - 11x + 14 = 0

We solve this quadratic equation by splitting the middle term: 2x² - 7x - 4x + 14 = 0 x(2x - 7) - 2(2x - 7) = 0 (x - 2)(2x - 7) = 0

The roots of the equation are x = 2 or x = 7/2. Between the given options, the value 7/2 is present. We can verify this by substituting 7/2 into the original expression: 7/2 + 7 · (2/7) = 7/2 + 2 = 7/2 + 4/2 = 11/2

Explanation

The common difference d of an arithmetic progression is determined by the formula d = (aₘ - aₙ)/(m - n). Substituting the values for the 9th term a₉ = 10 and the 3rd term a₃ = -8, we calculate d = (10 - (-8))/(9 - 3) = ¹⁸⁄₆ = 3. The 16th term a₁₆ can be found using the relation aₖ = aₘ + (k - m)d. By substituting the known 9th term and the common difference, we get a₁₆ = 10 + (16 - 9) × 3 = 10 + 21 = 31.

Explanation

The reflection of a point (x, y) in a horizontal line y = k results in an image (x', y') where the x-coordinate remains unchanged (x' = x) because the line is parallel to the x-axis. The y-coordinate of the image is determined by the fact that the line of reflection is the perpendicular bisector of the segment joining the point and its image. This implies that the midpoint of the y-coordinates must lie on the line y = k.

Using the midpoint formula for the y-coordinate: (y + y')/(2) = k y' = 2k - y

Substituting the given values x = 1, y = 2, and k = 3: x' = 1 y' = 2(3) - 2 = 6 - 2 = 4

The reflected point is (1, 4).

Explanation

According to the section formula, if a point (x, y) divides a line segment joining B(x₁, y₁) and C(x₂, y₂) in the ratio m:n, the y-coordinate of the dividing point is given by y = (m y₂ + n y₁)/(m+n). In this problem, point A(2, 1) divides the segment BC where B = (1, -3) and C = (4, y) in the ratio 2:3.

Substituting the given values into the y-coordinate formula: 1 = (2(y) + 3(-3))/(2 + 3)

Multiplying both sides by the denominator: 1 · 5 = 2y - 9 5 = 2y - 9

Solving for y: 2y = 5 + 9 2y = 14 y = 7

Explanation

A line intersects the X-axis at the point where its y-coordinate is equal to zero. To find the coordinates of this point for the equation 2x + 5y = -6, we substitute y = 0 into the given equation:

2x + 5(0) = -6

2x = -6

Solving for x by dividing both sides by 2:

x = -6 / 2 = -3

Since the x-coordinate is -3 and the y-coordinate is 0, the line cuts the X-axis at the point (-3, 0).

Explanation

In a right-angled triangle, the altitude BD drawn from the vertex of the right angle B to the hypotenuse AC creates two smaller triangles that are similar to each other and to the original triangle. According to the Right Triangle Altitude Theorem, also known as the Geometric Mean Theorem, the length of this altitude is the geometric mean of the lengths of the two segments it creates on the hypotenuse.

The mathematical relationship is expressed as BD² = AD × DC. By substituting the given values AD = 9 cm and DC = 16 cm into this formula, we get BD² = 9 × 16. Multiplying these values yields BD² = 144. To find the length of the altitude BD, we take the square root of both sides, resulting in BD = √(144) = 12 cm. This geometric property is derived from the similarity of △ ADB and △ BDC, where the ratio of corresponding sides is constant, specifically AD/BD = BD/DC.

Explanation

The trigonometric values for the given angles are tan 45° = 1 and sec 60° = 2 (since cos 60° = 1/2). Substituting these values into the expression:

tan 45° + (4)/(√(3)) sec 60° = 1 + (4)/(√(3))(2)

= 1 + (8)/(√(3))

To combine these terms into a single fraction, find the common denominator, which is √3:

= √3 · 1 + 8{√3}

= √3 + 8{√3}

Explanation

In right-angled triangle Δ DEF, where ∠ E = 90°, the length of the side DE can be determined using the trigonometric ratio for tangent. For the given angle ∠ D = 30°, the side EF is the opposite side and DE is the adjacent side. The relationship is defined as:

tan(D) = (EF)/(DE)

Substituting the known values m ∠ D = 30° and EF = 2√3 cm:

tan(30°) = 2√3{DE}

Using the standard trigonometric value tan(30°) = (1)/(√(3)):

(1)/(√(3)) = 2√3{DE}

Cross-multiplying to solve for DE:

DE = 2√3 × √3 DE = 2 × 3 DE = 6 cm.

Explanation

The relationship between sine and cosine is given by the trigonometric identity sin² θ + cos² θ = 1. Substituting the given value of sin θ = 20/29:

(20/29)² + cos² θ = 1

400/841 + cos² θ = 1

cos² θ = 1 - 400/841

cos² θ = 441/841

Taking the square root of both sides, we get:

cos θ = √(441/841) = 21/29

Alternatively, using the Pythagorean triplet (20, 21, 29), where the opposite side is 20 and the hypotenuse is 29, the adjacent side is calculated as √(29²⁾⁻²⁰^^^² = √(841 - 400) = √(441) = 21. Since cos θ is the ratio of the adjacent side to the hypotenuse, the value is 21/29.

Explanation

If A completes 1/3 of the work in 3 days, the time taken by A to complete the entire work is 3 × 3 = 9 days. Similarly, if B completes 1/2 of the work in 9 days, the time taken by B to complete the entire work is 9 × 2 = 18 days.

Using the rate of work formula, the combined rate of A and B per day is: Rate_A₊_B = ⅑ + ¹⁄₁₈ = (2 + 1)/(18) = ³⁄₁₈ = ⅙ job per day

To calculate the time required to finish half of the job (Work = 1/2): Time = (Work)/(Rate) = (1/2)/(1/6) = ½ × 6 = 3 days

Explanation

The area of a rhombus is calculated using the formula Area = ½ × d₁ × d₂, where d₁ and d₂ represent the lengths of the diagonals. Given the diagonal lengths of 25 cm and 20 cm, the area is:

Area = ½ × 25 × 20 Area = 25 × 10 = 250 sq cm.

Explanation

The discount offered is the difference between the Marked Price (MP) and the Selling Price (SP). Given that MP = Rs 4800 and SP = Rs 4200, the absolute discount is calculated as:

Discount = MP - SP = 4800 - 4200 = 600

The discount percentage is always calculated based on the Marked Price using the following formula:

Discount % = (Discount)/(MP) × 100

Substituting the known values into the formula:

Discount % = ⁶⁰⁰⁄₄₈₀₀ × 100

By simplifying the fraction, ⁶⁰⁰⁄₄₈₀₀ reduces to ⅛. Converting this fraction into a percentage:

Discount % = ⅛ × 100 = 12.5%