SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The overall gain or loss percentage for two successive transactions can be calculated using the net change formula x + y + ˣʸ⁄₁₀₀, where x and y represent the percentage changes. In this case, the first transaction is a 20% loss (x = -20) and the second transaction is a 30% gain (y = +30).

Net percentage change = -20 + 30 + (-20 × 30)/(100) = 10 - 6 = 4%.

Since the result is positive, there is an overall gain of 4% on the initial investment. The total gain in rupees is: Overall Gain = 42000 × ⁴⁄₁₀₀ = 1680.

Alternatively, the first selling price is 42000 × 0.80 = 33600. This amount is used to purchase the second item, which is then sold for 33600 × 1.30 = 43680. The overall gain is the final amount minus the initial cost: 43680 - 42000 = 1680.

Explanation

The volume V of a cylinder is calculated using the formula V = π r² h, where r is the radius and h is the height. Given the radius r = 8 cm and the height h = 14 cm, and using the value π = ²²⁄₇, the volume is determined as:

V = ²²⁄₇ × 8² × 14

V = ²²⁄₇ × 64 × 14

By simplifying the fraction with the height value:

V = 22 × 64 × 2

V = 44 × 64

Performing the multiplication results in V = 2816 cm³.

Explanation

Let the current ages of Prachi and her daughter be P and D respectively.

According to the first condition, seven years ago: P - 7 = 4(D - 7) P - 4D = -21 (Equation 1)

According to the second condition, four years from now: P + 4 = 2.5(D + 4) P + 4 = 2.5D + 10 P - 2.5D = 6 (Equation 2)

To find the value of D, subtract Equation 2 from Equation 1: (P - 4D) - (P - 2.5D) = -21 - 6 -1.5D = -27 D = 18

Substituting D = 18 into Equation 2: P - 2.5(18) = 6 P - 45 = 6 P = 51

The sum of their present ages is P + D = 51 + 18 = 69 years.

Explanation

The area of a sector of a circle can be calculated directly using the relationship between the arc length and the radius. The formula is Area A = ½ × r × l, where r is the radius and l is the arc length.

Given the radius r = 15.75 cm and the arc length l = 11 cm, substituting these values into the formula gives:

A = ½ × 15.75 × 11

A = (173.25)/(2)

A = 86.625 cm²

Explanation

The expression is a direct application of the cubic algebraic identity a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca). Let the variables be defined as a = 8.3, b = 9.2, and c = 6.1. By substituting these values into the given fraction, the expression takes the form:

a³ + b³ + c³ - 3abc{a² + b² + c² - ab - bc - ca}

Using the identity to expand the numerator, we can rewrite the fraction to show the common terms:

(a + b + c)(a² + b² + c² - ab - bc - ca){a² + b² + c² - ab - bc - ca}

The quadratic factor (a² + b² + c² - ab - bc - ca) appearing in both the numerator and the denominator cancels out, assuming it is non-zero. This leaves the simplified result as the sum of the three variables:

a + b + c

By substituting the numerical values back into this simplified expression, we perform the following addition:

8.3 + 9.2 + 6.1 = 17.5 + 6.1 = 23.6

The sum of 8.3 and 9.2 results in 17.5, and adding 6.1 to that total yields the final value of 23.6.

Explanation

The trigonometric identity relating tangent and secant is sec²(t) = 1 + tan²(t). Substituting the given value tan(t) = ⅓ into the formula, we get sec²(t) = 1 + (⅓)². This simplifies to sec²(t) = 1 + ⅑, which equals ¹⁰⁄₉. Taking the square root of both sides results in sec(t) = ± √(¹⁰⁄₉), which simplifies to ± √(10){3}. Based on the options provided, the positive value √(10){3} is the intended solution.

Explanation

In the triangles Δ PQR and Δ SRQ, we are given that PQ = SR and ∠ PQR = ∠ SRQ = 90°. Additionally, the side QR is common to both triangles, which can be written as QR = RQ to maintain the correct vertex correspondence.

According to the Side-Angle-Side (SAS) congruence criterion, two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle. Here, the sides PQ and QR of Δ PQR and the corresponding sides SR and RQ of Δ SRQ are equal, and the included angles ∠ PQR and ∠ SRQ are both 90°.

The vertex correspondence is P ↔ S, Q ↔ R, and R ↔ Q. This mapping ensures that PQ corresponds to SR, QR corresponds to RQ, and PR corresponds to SQ. Thus, Δ PQR ≅ Δ SRQ by the SAS congruence rule.

Explanation

Let the cost price (CP) of the article be 100 units. According to the question, the marked price (MP) is 26% more than the cost price, which gives MP = 100 + 0.26 × 100 = 126 units. A discount of 32% is then applied to the marked price to find the selling price (SP).

SP = MP × (1 - (Discount)/(100)) = 126 × (1 - 0.32) = 126 × 0.68 = 85.68 units.

The loss percentage can be calculated using the formula: Loss% = (CP - SP)/(CP) × 100 Loss% = (100 - 85.68)/(100) × 100 = 14.32%

Alternatively, using the successive percentage change formula x + y + ˣʸ⁄₁₀₀, where x = 26 (markup) and y = -32 (discount): Net Change = 26 - 32 + (26 × (-32))/(100) = -6 - ⁸³²⁄₁₀₀ = -6 - 8.32 = -14.32%

The negative sign indicates a loss of 14.32%.

Explanation

In a right-angled isosceles triangle, the two legs are equal in length. Let the length of each leg be a. The area A of the triangle is given by the formula A = ½ × base × height = ½ a². Given the area is 50 square units: ½ a² = 50 a² = 100 a = 10 units.

According to the Pythagorean theorem, the hypotenuse h is calculated as: h² = a² + a² = 2a² h = √(2a²) = a√2 Substituting a = 10: h = 10√2 units.

Alternatively, the area of a right-angled isosceles triangle can be expressed directly in terms of its hypotenuse using the formula A = (h²)/(4). Substituting the given values: 50 = (h²)/(4) h² = 200 h = √(200) = 10√2 units.

Explanation

Using the balance equation Income = Expenditure + Savings, let the initial income be 100 units. Since the person spends 80%, the initial expenditure is 80 units and the initial savings are 20 units.

The new income after a 20% increase is 100 × 1.20 = 120 units. The new expenditure after a 10% increase is 80 × 1.10 = 88 units. The new savings are determined by the difference between the new income and new expenditure: S' = 120 - 88 = 32 units.

The percentage increase in savings is calculated using the formula: Percentage Increase = (New Savings - Initial Savings)/(Initial Savings) × 100

Substituting the values: Percentage Increase = (32 - 20)/(20) × 100 = ¹²⁄₂₀ × 100 = 60%

The provided initial income of ₹16,000 confirms this proportion but is not required for solving percentage-based change problems when the initial ratios are already established.

Explanation

According to the divisibility rule for the number 9, the remainder obtained when an integer is divided by 9 is identical to the remainder obtained when the sum of its digits is divided by 9. This principle stems from the property that 10ⁿ ≡ 1 9 for any non-negative integer n, which allows any number to be expressed as the sum of its digits in modular arithmetic base 9.

To find the remainder for 28735429, we first calculate the sum of its digits: 2 + 8 + 7 + 3 + 5 + 4 + 2 + 9 = 40

Next, we determine the remainder when this sum is divided by 9: 40 = (9 × 4) + 4

Since the sum of the digits 40 leaves a remainder of 4 when divided by 9, the original number 28735429 also leaves a remainder of 4 when divided by 9. This can also be verified by finding the digital root of the sum, where 4 + 0 = 4.

Explanation

For any triangle, the sum of the interior angles must be π radians. If one angle is the average of the other two, let the angles be A, B, and C such that B = (A + C)/(2). This condition implies that 2B = A + C. Substituting this into the triangle sum property A + B + C = π gives 3B = π, which means one of the angles must be (π)/(3).

In the first set of angles (π)/(6), (π)/(3), and (π)/(2), the sum is (π)/(6) + (2π)/(6) + (3π)/(6) = (6π)/(6) = π, which satisfies the triangle property. To verify the average condition, we calculate the average of the smallest and largest angles: (π)/(6) + (π)/(2){2} = (π + 3π)/(6){2} = (4π)/(12) = (π)/(3). Since the third angle is exactly (π)/(3), the condition is satisfied. This set corresponds to the first numerical label, which is mapped to option d.

Explanation

The line segment joining the center of a circle to an external point bisects the angle between the two tangents drawn from that point to the circle. Given that the tangents PA and PB are inclined to each other at ∠ APB = 60°, the angle ∠ APO is half of this value: ∠ APO = ½ ∠ APB = (60°)/(2) = 30°

A tangent to a circle is perpendicular to the radius at the point of contact. Therefore, the radius OA is perpendicular to the tangent PA, making ∠ OAP = 90°. In the right-angled triangle △ OAP, the sum of the interior angles is 180°: ∠ POA + ∠ OAP + ∠ APO = 180° ∠ POA + 90° + 30° = 180° ∠ POA = 180° - 120° = 60°

Explanation

Let the current age of B be x years. According to the problem, the age of A is 3x and the age of C is x + 5. The sum of their ages is given as 75 years:

3x + x + (x + 5) = 75 5x + 5 = 75 5x = 70 x = 14

The current age of B is 14 years, and the current age of A is 3 × 14 = 42 years.

To find the required ratio: Age of B five years ago = 14 - 5 = 9 years Age of A three years from now = 42 + 3 = 45 years

The ratio of B's age five years ago to A's age three years from now is 9 : 45. Dividing both terms by their greatest common divisor, 9, we get the simplified ratio 1 : 5.

Explanation

When two variables x and y vary inversely, their product remains constant. This relationship is expressed by the mathematical formula x · y = k, where k is the constant of variation. Using the initial values provided, x = 2 and y = 6, we can determine the value of the constant:

k = x₁ · y₁ = 2 · 6 = 12

To find the value of y when x = 3, we substitute these values into the inverse variation equation x₂ · y₂ = k:

3 · y = 12

Solving for y:

y = ¹²⁄₃ = 4

Alternatively, the relationship can be calculated using the ratio x₁ y₁ = x₂ y₂. Substituting the given values yields 2 · 6 = 3 · y, which simplifies to 12 = 3y, resulting in y = 4.

Explanation

Simple Interest (SI) is calculated using the formula SI = (P × R × T)/(100), where P is the principal amount, R is the annual interest rate, and T is the time in years.

For the first sum of ₹5,000 at a rate of 15% for 3 years: SI₁ = (5000 × 15 × 3)/(100) = 50 × 45 = 2,250

For the second sum of ₹8,000 at a rate of 12% for 4 years: SI₂ = (8000 × 12 × 4)/(100) = 80 × 48 = 3,840

The positive difference between the interests paid is: Difference = |SI₂ - SI₁| = 3,840 - 2,250 = 1,590

Explanation

To find the time when two people starting from the same point on a circular track meet for the first time while moving in the same direction, we use the formula t = (L)/(v₁ - v₂), where L is the length of the track and v₁ and v₂ are their respective speeds.

First, convert the speeds from km/h to m/s by multiplying by ⁵⁄₁₈: Varun speed = 90 × ⁵⁄₁₈ = 25 m/s Sandeep speed = 108 × ⁵⁄₁₈ = 30 m/s

The relative speed of Sandeep with respect to Varun since they are moving in the same direction is: Relative Speed = 30 - 25 = 5 m/s

The time taken to meet again for the first time is the time taken to cover the track length of 1635 m at this relative speed: t = ¹⁶³⁵⁄₅ = 327 s

Explanation

Let the capacity of the tank be the Least Common Multiple (LCM) of the filling times 25, 50, and 100, which is 100 units. The filling efficiencies (units per minute) for the pipes are calculated as follows: Efficiency of Pipe M (E_M) = 100 / 25 = 4 Efficiency of Pipe N (E_N) = 100 / 50 = 2 Efficiency of Pipe S (E_S) = 100 / 100 = 1

Let the total time taken to fill the tank be T minutes. Based on the operational pattern described: Pipe N works only for the first 10 minutes. Pipe M starts after 10 minutes and works until the tank is full, so it works for T - 10 minutes. Pipe S starts at the beginning but is closed 15 minutes before the tank is full, so it works for T - 15 minutes.

The sum of the water filled by each pipe equals the total capacity of 100 units: (E_N × 10) + (E_M × (T - 10)) + (E_S × (T - 15)) = 100 (2 × 10) + 4(T - 10) + 1(T - 15) = 100 20 + 4T - 40 + T - 15 = 100 5T - 35 = 100 5T = 135 T = 27

The total time taken to fill the tank is 27 minutes.

Explanation

To simplify the expression 21 - 4.9 ÷ 7 + 3.9 × 0.4 + 0.9, the order of operations defined by the BODMAS rule is followed, prioritizing division and multiplication before addition and subtraction. First, perform the division 4.9 ÷ 7, which equals 0.7. Next, calculate the multiplication 3.9 × 0.4, which results in 1.56. The expression is now simplified to 21 - 0.7 + 1.56 + 0.9. Proceeding with subtraction and addition from left to right, subtract 0.7 from 21 to get 20.3. Then, add 1.56 to 20.3, resulting in 21.86. Finally, adding 0.9 to 21.86 yields the final result of 22.76.

Explanation

The temperature on the 16th day is determined by calculating the overlap between the two given periods. January contains 31 days. The total sum of temperatures for the entire month is the product of the monthly average and the number of days: Total Sum = 31 × 24 = 744°C.

The sum of temperatures for the first sixteen days is 16 × 22 = 352°C. The sum of temperatures for the last sixteen days is 16 × 26 = 416°C.

When the sum of the first sixteen days and the last sixteen days are added together, the 16th day is counted twice, while all other days are counted once. Therefore, the temperature of the 16th day can be expressed as: Temperature of 16th day = (Sum of first 16 days + Sum of last 16 days) - Total monthly sum Temperature of 16th day = (352 + 416) - 744 Temperature of 16th day = 768 - 744 = 24°C.

Explanation

The simplification uses the algebraic identity for the difference of cubes, which is a³ - b³ = (a - b)(a² + ab + b²). By substituting a = sin A and b = cos A, the numerator sin³ A - cos³ A can be rewritten as (sin A - cos A)(sin² A + sin A cos A + cos² A).

When this factored form is placed back into the original fraction, the common term (sin A - cos A) in both the numerator and the denominator cancels out. This leaves the expression sin² A + sin A cos A + cos² A. Using the trigonometric identity sin² A + cos² A = 1, the terms are further reduced to 1 + sin A cos A.

Explanation

In △ AFC, the sum of interior angles is 180°. Given ∠ ACF = 50° and ∠ AFC = 30°, we find ∠ FAC = 180° - (50° + 30°) = 100°. Since F, A, B are collinear, the angle ∠ CAB is supplementary to ∠ FAC, which gives ∠ CAB = 180° - 100° = 80°.

By the Alternate Segment Theorem, the angle between the tangent FC and the chord AC is equal to the angle subtended by that chord in the alternate segment. Therefore, ∠ ABC = ∠ ACF = 50°. In △ ABC, the sum of the angles is 180°, so ∠ ACB = 180° - (∠ CAB + ∠ ABC) = 180° - (80° + 50°) = 50°.

The central angle theorem states that the angle subtended by a chord at the center is double the angle subtended by it at any point on the circumference. Consequently, ∠ AOB = 2 × ∠ ACB = 2 × 50° = 100°.

Explanation

The effective percentage for two successive discounts of d₁% and d₂% is given by the formula D = d₁ + d₂ - (d₁ × d₂)/(100). Substituting the values 15% and 20%, we get:

Effective Discount = 15 + 20 - (15 × 20)/(100) Effective Discount = 35 - 3 = 32%

The difference between the two discount rates is 32% - 25% = 7%. Given that this difference in value is ₹28, let the total bill amount be x.

7% of x = 28 ⁷⁄₁₀₀ × x = 28 x = (28 × 100)/(7) x = 4 × 100 = 400

The total amount of the bill is ₹400.

Explanation

To be divisible by 2, 5, 10, and 11, a number must be divisible by their least common multiple. Since 10 is already a multiple of 2 and 5, we find LCM(10, 11) = 110. A number is divisible by 110 if it ends in 0 and satisfies the divisibility rule for 11. The number 589270 ends in 0, fulfilling the first requirement. For divisibility by 11, the difference between the sum of digits in the odd positions and the sum of digits in the even positions must be 0 or a multiple of 11. In 589270, the sum of digits in the odd positions from the right is 0 + 2 + 8 = 10, and the sum of digits in the even positions is 7 + 9 + 5 = 21. The difference is |21 - 10| = 11, which is a multiple of 11, confirming the number is divisible by 110.

Explanation

For a system of linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to have infinitely many solutions, the ratio of their coefficients and constants must be equal: (a₁)/(a₂) = (b₁)/(b₂) = (c₁)/(c₂).

Given the equations 2x + y = a and 8x + by = 12, we identify the coefficients: a₁ = 2, b₁ = 1, c₁ = a a₂ = 8, b₂ = b, c₂ = 12

Substituting these into the condition for infinite solutions: ²⁄₈ = (1)/(b) = ᵃ⁄₁₂

The ratio ²⁄₈ simplifies to ¼. By setting the other ratios equal to ¼, we solve for a and b: For b: (1)/(b) = ¼ ⟹ b = 4 For a: ᵃ⁄₁₂ = ¼ ⟹ a = ¹²⁄₄ = 3

The sum of the values of a and b is: a + b = 3 + 4 = 7

Explanation

To calculate the simple interest, use the formula SI = (P × R × T)/(100), where P is the principal amount, R is the annual interest rate, and T is the time period in years. In this case, the principal P is ₹6,700, the rate R is 12%, and the time T is 13 months. Since the interest rate is per annum, the time must be converted into years by dividing by 12, giving T = ¹³⁄₁₂ years.

Substituting these values into the formula: SI = (6700 × 12 × 13)/(12){100}

The 12 in the numerator and denominator cancel each other out, simplifying the expression to: SI = (6700 × 13)/(100) SI = 67 × 13

Multiplying 67 by 13 gives 871. Thus, the simple interest earned over 13 months is ₹871.

Explanation

The trigonometric identity states that if tan A · tan B = 1, then A + B = 90°. This relation arises because tan A = (1)/(tan B), which is equivalent to tan A = cot B. Since cot B = tan(90° - B), it follows that A = 90° - B, or A + B = 90°.

Given the equation tan 4θ · tan 6θ = 1, we can set the sum of the angles to 90°: 4θ + 6θ = 90° 10θ = 90° θ = 9°

The problem requires finding the value of cot 5θ. Substituting the calculated value of θ into the expression: cot 5θ = cot(5 × 9°) cot 5θ = cot 45°

According to standard trigonometric values, cot 45° = 1.

Explanation

A number is divisible by 8 if the number formed by its last three digits is divisible by 8. For 91876a2, the last three digits are 6a2. This can be expressed as 600 + 10a + 2. Since 600 is already divisible by 8 (8 × 75 = 600), we only need the remaining part 10a + 2 to be divisible by 8. We test the smallest possible digits for a. If a = 0, 10(0) + 2 = 2. If a = 1, 10(1) + 2 = 12. If a = 2, 10(2) + 2 = 22. None of these are divisible by 8. When a = 3, the expression becomes 10(3) + 2 = 32, which is divisible by 8 (8 × 4 = 32). Therefore, 3 is the smallest value that makes the entire number divisible by 8.

Explanation

The time taken by Pipe A to fill the tank is 1 ⅓ hours, which equals ⁴⁄₃ hours. The time taken by Pipe B is 2 hours. The combined rate of both pipes filling the tank is the sum of their individual rates.

Rate of Pipe A = (1)/(4/3) = ¾ tank per hour Rate of Pipe B = ½ tank per hour

Combined Rate = ¾ + ½ = ³⁺²⁄₄ = ⁵⁄₄ tanks per hour

The total time T required to fill the tank is the reciprocal of the combined rate: T = (1)/(5/4) = ⅘ hours

To convert this into minutes, multiply by 60: T = ⅘ × 60 = 4 × 12 = 48 minutes

Explanation

The given equation a² + b² + c² = 2(a + c - 1) can be rearranged by bringing all terms to one side: a² + b² + c² - 2a - 2c + 2 = 0. By grouping the terms related to a, b, and c, we can form perfect squares:

(a² - 2a + 1) + b² + (c² - 2c + 1) = 0

This simplifies to (a - 1)² + b² + (c - 1)² = 0. For the sum of squares of real numbers to be zero, each individual term must independently equal zero. Therefore:

a - 1 = 0 ⟹ a = 1 b = 0 c - 1 = 0 ⟹ c = 1

Substituting these values into the required expression: a³ + b³ + c³ = 1³ + 0³ + 1³ = 1 + 0 + 1 = 2

Explanation

First, calculate the upstream and downstream speeds. The upstream speed u is given as (2 km)/(1 hr) = 2 km/hr. The downstream speed v is (1 km)/(10/60 hr) = 6 km/hr.

The speed of the boat in stationary water x is the average of the downstream and upstream speeds: x = (u + v)/(2) = (2 + 6)/(2) = 4 km/hr

To find the time taken to travel 6 km in stationary water, use the formula t = (Distance)/(Speed): t = (6 km)/(4 km/hr) = 1.5 hours

Converting 1.5 hours into hours and minutes: 1.5 hours = 1 hour + (0.5 × 60) minutes = 1 hr 30 min

Explanation

The values of the variables can be verified by substituting x = -5, y = 3, and z = 8 into the given system of linear equations.

For the first equation: 5(-5) - 3(3) + 7(8) = -25 - 9 + 56 = -34 + 56 = 22

For the second equation: 3(-5) - 5(3) - 2(8) = -15 - 15 - 16 = -30 - 16 = -46

For the third equation: 2(-5) - 2(3) + 5(8) = -10 - 6 + 40 = -16 + 40 = 24

Since the values satisfy all three equations simultaneously, x = -5, y = 3, and z = 8 is the unique solution to the system.

Explanation

To find the curved surface area of a conical tent, we use the formula CSA = π r l, where r is the radius of the base and l is the slant height.

Given the height h = 8 m and the intended dimensions for this problem, the radius r is 8√3 m. The slant height l is calculated using the Pythagorean relation:

l = √(r² + h²)

Substituting the values: l = √((83))² + 8² l = √(192 + 64) l = √(256) l = 16 m

Now, calculate the curved surface area: CSA = π × r × l CSA = π × 8√3 × 16 CSA = 128√3π m²

Explanation

The value of sin 150° can be determined using the trigonometric reduction formula sin(180° - θ) = sin θ. Since 150° is in the second quadrant where the sine function is positive, we substitute θ = 30° into the formula: sin 150° = sin(180° - 30°) = sin 30°. The exact value of sin 30° is 1/2, which equals 0.5. Alternatively, using the identity sin(90° + θ) = cos θ, we have sin(90° + 60°) = cos 60° = 0.5.

Explanation

The mean proportional between two numbers a and b is defined by the formula x = √(ab). This value x represents the geometric mean of the two numbers, where the ratio a:x is equal to the ratio x:b. Substituting the given values a = 2 and b = 32 into the formula, we get x = √(2 × 32). Performing the multiplication inside the radical results in √(64), which simplifies to 8.

Explanation

Let the initial cost price CP be 100. Given that the profit is 320% of the cost, the profit is 320. The selling price SP is the sum of the cost price and the profit: SP = 100 + 320 = 420

If the cost increases by 10%, the new cost price CPₙₑ_w becomes: CPₙₑ_w = 100 + 10 = 110

The selling price remains constant at 420. The new profit is calculated by subtracting the new cost price from the selling price: Profitₙₑ_w = 420 - 110 = 310

To find what percentage of the selling price this profit represents, use the formula: Percentage = Profitₙₑ_w{SP} × 100 Percentage = ³¹⁰⁄₄₂₀ × 100 = ³¹⁄₄₂ × 100 Percentage = ¹⁵⁵⁰⁄₂₁ ≈ 73.809%

Rounding to one decimal place, the profit is 73.8% of the selling price.

Explanation

The perimeter of a sector is defined as the sum of its arc length and the two radii that bound the sector. The formula for the arc length l of a sector with radius r and central angle θ in degrees is l = (θ)/(360) × 2π r.

Given the radius r = 12 cm, the central angle θ = 30°, and π = 3.14, the arc length is calculated as: l = ³⁰⁄₃₆₀ × 2 × 3.14 × 12 l = ¹⁄₁₂ × 24 × 3.14 l = 2 × 3.14 = 6.28 cm

The total perimeter P is the sum of the arc length and twice the radius: P = l + 2r P = 6.28 + 2(12) P = 6.28 + 24 = 30.28 cm

Explanation

The cost price of the mixture is determined by the weighted average of the two varieties. The formula for the weighted mean is:

CP = w₁c₁ + w₂c₂{w₁ + w₂}

Here, w₁ = 5 kg and c₁ = 4 per kg, while w₂ = 3 kg and c₂ = 6 per kg. Substituting these values into the expression:

CP = ((5 × 4) + (3 × 6))/(5 + 3)

CP = (20 + 18)/(8)

CP = ³⁸⁄₈ = 4.75

This calculation provides the direct average cost per kilogram of the total 8 kg mixture based on the individual purchase prices and quantities.

Explanation

Let Suresh's initial spending be E = 300 and his initial saving be S = 100, consistent with the 3:1 ratio. Since total income I = E + S, his initial salary is I = 300 + 100 = 400. When the salary increases by 25%, the new salary becomes 400 + (0.25 × 400) = 500. The saving grows by 10%, so the new saving becomes 100 + (0.10 × 100) = 110.

To find the new expenditure, we subtract the new saving from the new salary: 500 - 110 = 390. The increase in expenditure is the difference between the new and old spending values, which is 390 - 300 = 90. The percentage increase in expenditure is calculated as ⁹⁰⁄₃₀₀ × 100 = 30%.

Explanation

From the given equation 7 cos A = 6, we find that cos A = ⁶⁄₇.

To find the numerical value of the expression (csc A + cot A)/(csc A - cot A), we rewrite the trigonometric terms in terms of sin A and cos A: csc A = (1)/(sin A) and cot A = (cos A)/(sin A)

Substituting these into the expression: (1)/(sin A) + (cos A)/(sin A){(1)/(sin A) - (cos A)/(sin A)} = (1 + cos A)/(1 - cos A)

Now, substitute the value of cos A = ⁶⁄₇ into the simplified expression: (1 + 6)/(7){1 - ⁶⁄₇} = (7 + 6)/(7){(7 - 6)/(7)} = (13)/(7){⅐}

By canceling out the common denominator of 7, we get: ¹³⁄₁ = 13

Explanation

The length of an arc L is calculated using the formula L = (θ)/(360°) × 2π r, where θ is the sectorial angle and r is the radius. Substituting the given values r = 9.6 units and θ = 75°:

L = ⁷⁵⁄₃₆₀ × 2π × 9.6

Simplifying the fraction ⁷⁵⁄₃₆₀ by dividing both the numerator and the denominator by 15 gives ⁵⁄₂₄. Substituting this back into the equation:

L = ⁵⁄₂₄ × 19.2π

Dividing 19.2 by 24 yields 0.8. Finally, multiplying 5 by 0.8π results in:

L = 4π

Explanation

The given equation is p(√3 + cot 30°) = tan³ 60° - 2sin 60°. Using standard trigonometric values, we have cot 30° = √3, tan 60° = √3, and sin 60° = √3{2}.

Substituting these values into the equation: p(√3 + √3) = (√3)³ - 2(√3{2})

Simplifying both sides: p(2√3) = 3√3 - √3 2√3 p = 2√3

Dividing both sides by 2√3: p = 2√3{2√3} = 1

Explanation

To find the percentage, first calculate the total marks obtained by adding 93, 90, 90, 95, and 78, which equals 446. Since there are 5 subjects each out of 100, the maximum possible marks are 500. The percentage is (446/500) * 100, which results in 89.2%.

Explanation

The perimeter of the remaining three-quarters of the pizza is the sum of the arc length and the two straight radial edges. The arc length is (3/4) × 2 × (22/7) × 21 = 99 cm, and the two radii are 2 × 21 = 42 cm, totaling 141 cm.

Explanation

The cost of the first type of rice is 250/5 = ₹50 per kg, and the second type is 280/5 = ₹56 per kg. Using the weighted average formula for the 7:8 ratio, the net cost is [(7 * 50) + (8 * 56)] / (7 + 8) = (350 + 448) / 15 = 798 / 15 = ₹53.20 per kg.

Explanation

Let the number of votes polled by S be x. According to the problem, R gets 110% of the votes of S, which can be expressed as 1.1x. The difference between the votes of R and S is given as 770.

We set up the equation based on the difference: 1.1x - x = 770 0.1x = 770 x = (770)/(0.1) = 7700

The total number of votes polled is the sum of the votes obtained by R and S. Since there are no invalid votes, the total is: Total Votes = R + S = 1.1x + x = 2.1x Substituting the value of x: Total Votes = 2.1 × 7700 = 16170

Explanation

A number is a multiple of 6 if it is divisible by both 2 and 3. The given number 321y72 ends with the digit 2, which is even, satisfying the condition for divisibility by 2 regardless of the value of y. For divisibility by 3, the sum of the digits of the number must be a multiple of 3.

The sum of the digits is 3 + 2 + 1 + y + 7 + 2 = 15 + y. For the total sum to be divisible by 3, the expression 15 + y must result in a multiple of 3. Since 15 is already a multiple of 3, the digit y must itself be a multiple of 3. The possible values for the digit y are 0, 3, 6, and 9. Comparing these values, the least value that y can take is 0.

Explanation

Let the Cost Price (CP) be 100 units. Given that the article is sold at a profit of 250%, the profit earned is 250% of 100, which equals 250 units. The Selling Price (SP) is calculated using the formula SP = CP + Profit. Substituting the values, we get SP = 100 + 250 = 350 units. The ratio of the Cost Price to the Selling Price is CP : SP = 100 : 350. Dividing both sides of the ratio by their greatest common divisor, 50, simplifies the expression to 2 : 7.

Explanation

For a system of linear equations, the most efficient method to determine the correct solution from the given options is direct substitution. Substituting the values x = 1, y = 1, and z = 1 into each of the three equations confirms their validity.

For the first equation: 5(1) + 4(1) - 8(1) = 5 + 4 - 8 = 1

For the second equation: 7(1) - 9(1) + 1 = 7 - 9 + 1 = - 1

For the third equation: 2(1) + 3(1) - 4(1) = 2 + 3 - 4 = 1

Since the values satisfy all three equations simultaneously, the solution to the system is x = 1, y = 1, and z = 1.

Explanation

The simple interest for the first year is calculated on the initial principal of ₹6,23,305. Using the formula I = P × r × t, the interest for the first year I₁ is: I₁ = 6,23,305 × 10% × 1 = 62,330.5

At the end of the first year, Pranav repays ₹16,390 towards the principal. Therefore, the remaining principal P₂ for the second year is: P₂ = 6,23,305 - 16,390 = 6,06,915

The interest for the second year I₂ is calculated on this reduced principal: I₂ = 6,06,915 × 10% × 1 = 60,691.5

To clear all dues at the end of the second year, Pranav must pay the remaining principal and the total interest accrued over both years. The total payment A is: A = P₂ + I₁ + I₂ A = 6,06,915 + 62,330.5 + 60,691.5 A = 6,06,915 + 1,23,022 A = 7,29,937