SSC CGL Tier 1 2024 Quantitative Aptitude Questions with Solutions

Explanation

The price a customer pays, also known as the Selling Price (SP), is determined by subtracting the discount from the Marked Price (MP). The formula for Selling Price is: SP = MP × (100 - Discount%)/100

Substituting the given values into the formula: SP = 15,620 × (100 - 27)/100 SP = 15,620 × 73/100 SP = 15,620 × 0.73

Performing the multiplication: 15,620 × 0.73 = 11,402.6

The question asks for the price rounded to the nearest tens. Since the last digit before the decimal is 2 (which is less than 5), we round down to the nearest multiple of ten, resulting in 11,400.

Explanation

To determine the value of (sin A)/(1 - cos A) in terms of p, we examine the product of the given expression for p and the expression we wish to evaluate:

p · (sin A)/(1 - cos A) = (sin A)/(1 + cos A) · (sin A)/(1 - cos A)

Multiplying the numerators and denominators gives:

(sin² A)/((1 + cos A)(1 - cos A)) = (sin² A)/(1 - cos² A)

Using the fundamental trigonometric identity sin² A + cos² A = 1, we know that 1 - cos² A = sin² A. Substituting this into the equation:

(sin² A)/(sin² A) = 1

Since the product of p and (sin A)/(1 - cos A) is 1, it follows that:

(sin A)/(1 - cos A) = ¹⁄ₚ

Alternatively, using half-angle identities, p = (2 sin(A/2) cos(A/2))/(2 cos²(A/2)) = tan(A/2). The expression to be evaluated is (2 sin(A/2) cos(A/2))/(2 sin²(A/2)) = cot(A/2). Since cot(A/2) = (1)/(tan(A/2)), the result is ¹⁄ₚ.

Explanation

The initial volume of the solution is 500 ml (half a litre). Since it contains 15% alcohol, the volume of alcohol is 0.15 × 500 = 75 ml, and the volume of water is 500 - 75 = 425 ml.

To reach a 50% alcohol concentration, the volume of alcohol and water in the final mixture must be equal. As only alcohol is being added, the volume of water remains constant at 425 ml. Therefore, the final total volume of alcohol must also be 425 ml.

The quantity of alcohol to be mixed is the difference between the final alcohol volume and the initial alcohol volume: 425 ml - 75 ml = 350 ml

Explanation

In an isosceles triangle, angles opposite to equal sides are equal. Since AC = BC, the angle opposite side AC is equal to the angle opposite side BC, meaning ∠ ABC = ∠ BAC. Given that ∠ BAC = 70°, it follows that ∠ ABC = 70°.

By the Exterior Angle Theorem, the exterior angle ∠ ACD formed by producing side BC to D is equal to the sum of the two interior opposite angles: ∠ ACD = ∠ BAC + ∠ ABC = 70° + 70° = 140°

Substituting these values into the given expression: 2.5∠ ACD - 1.5∠ ABC = 2.5(140°) - 1.5(70°) 2.5 × 140 = 350 1.5 × 70 = 105 350° - 105° = 245°

Explanation

Given 8cot θ = 7, we have cot θ = ⅞. Since tan θ = (1)/(cot θ), then tan θ = ⁸⁄₇.

The expression (1 + sinθ)/(cosθ) is equivalent to (1)/(cosθ) + (sinθ)/(cosθ) = secθ + tanθ.

Using the trigonometric identity sec² θ = 1 + tan² θ: sec² θ = 1 + (⁸⁄₇)² = 1 + ⁶⁴⁄₄₉ = (49 + 64)/(49) = ¹¹³⁄₄₉ sec θ = √(¹¹³⁄₄₉) = √(113){7}

Substituting the values of sec θ and tan θ into the simplified expression: sec θ + tan θ = √(113){7} + ⁸⁄₇ = 8 + √(113){7}

Explanation

To find the value of ((x)/(y) + ʸ⁄ₓ)² - 3, first simplify the expression inside the parentheses. The sum of the fractions is given by (x² + y²)/(xy).

Using the given values x = √6 + 2 and y = √6 - 2, the product xy is calculated using the difference of squares identity (a + b)(a - b) = a² - b²: xy = (√6 + 2)(√6 - 2) = (√6)² - 2² = 6 - 4 = 2

Next, calculate x² + y² using the algebraic identity (a+b)² + (a-b)² = 2(a² + b²): x² + y² = (√6 + 2)² + (√6 - 2)² = 2((√6)² + 2²) = 2(6 + 4) = 2(10) = 20

Now, substitute these values back into the expression for the sum: (x)/(y) + ʸ⁄ₓ = (x² + y²)/(xy) = ²⁰⁄₂ = 10

Finally, calculate the required value: ((x)/(y) + ʸ⁄ₓ)² - 3 = 10² - 3 = 100 - 3 = 97

Explanation

In Δ ABC, since DE ∥ BC, the Basic Proportionality Theorem (Thales's Theorem) states that the line divides the other two sides in the same ratio. Therefore, (AD)/(DB) = (AE)/(EC).

The problem provides the relationship 5AE = 3EC, which can be rewritten as the ratio (AE)/(EC) = ⅗. Substituting this into the proportionality equation gives (AD)/(DB) = ⅗.

We know that the total length AB = AD + DB. Using the ratio (AD)/(DB) = ⅗, we can express AD as ⅗ DB. Substituting this into the equation for the total length of the side, we get 6.4 = ⅗ DB + DB.

Combining the terms on the right side results in 6.4 = ⁸⁄₅ DB. To find the value of DB, we calculate DB = (6.4 × 5)/(8). Since 6.4 ÷ 8 = 0.8, then DB = 0.8 × 5 = 4 units.

Explanation

The algebraic identity for the difference of cubes is x³ - y³ = (x - y)³ + 3xy(x - y). Let x = 2a and y = b. Substituting the given values 2a - b = 3 and 8a³ - b³ = 999:

999 = 3³ + 3(2a)(b)(3) 999 = 27 + 18ab 972 = 18ab ab = 54

To find the value of 4a² - b², use the identity 4a² - b² = (2a - b)(2a + b). The value of 2a + b is determined using the relation (2a + b)² = (2a - b)² + 8ab:

(2a + b)² = 3² + 8(54) (2a + b)² = 9 + 432 = 441 2a + b = √(441) = 21

Substituting these values into the expression for the difference of squares: 4a² - b² = 3 × 21 = 63

Explanation

The total surface area (TSA) of a right circular cylinder is the sum of the curved surface area and the area of its two circular bases. The formula is given by:

TSA = 2 π r (h + r)

Given the radius r = 21 cm and the height h = 5 cm, we substitute these values into the formula using π = ²²⁄₇:

TSA = 2 × ²²⁄₇ × 21 × (5 + 21)

Simplifying the expression by dividing 21 by 7 gives:

TSA = 2 × 22 × 3 × 26 TSA = 132 × 26

Multiplying 132 by 26:

132 × 20 = 2640 132 × 6 = 792 2640 + 792 = 3432 cm²

Explanation

According to the circle theorem, the angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Let D be a point on the major arc of the circle. Since the arc ABC subtends an angle of 138° at the center O, the angle subtended by this arc at point D on the circumference is: ∠ ADC = ½ × ∠ AOC = (138°)/(2) = 69°

The points A, B, C, and D lie on the circle, making ABCD a cyclic quadrilateral. A key property of a cyclic quadrilateral is that the exterior angle formed by producing one of its sides is equal to the interior opposite angle.

When the chord AB is produced to a point P, ∠ CBP becomes the exterior angle at vertex B. The interior opposite angle to ∠ ABC is ∠ ADC. Therefore: ∠ CBP = ∠ ADC = 69°

Explanation

The net profit or loss percentage when a value is increased by x% and then decreased by y% can be calculated using the successive percentage formula:

Net Percentage Change = x + y + ˣʸ⁄₁₀₀

In this problem, Raman increases the cost price by 16% to fix the sale price (x = +16) and then offers a discount of 12% on that fixed price (y = -12). Substituting these values into the formula:

Profit Percentage = 16 - 12 + (16 × (-12))/(100)

Profit Percentage = 4 - ¹⁹²⁄₁₀₀

Profit Percentage = 4 - 1.92 = 2.08%

Alternatively, if the cost price is assumed to be 100, the fixed price becomes 116. The selling price is 12% less than the fixed price, calculated as 116 × (1 - 0.12) = 116 × 0.88 = 102.08. The profit is the difference between the selling price and the cost price, which is 102.08 - 100 = 2.08. Since the base is 100, the profit percentage is 2.08%.

Explanation

The total number of shares issued by the company is 10 lakh, which is 1,000,000. Meera currently holds 45% of these shares. The number of shares Meera holds initially is calculated as:

Initial shares of Meera = 1,000,000 × ⁴⁵⁄₁₀₀ = 450,000

Jeevan decides to sell 20,000 of his shares to Meera. To find the total number of shares Meera will have after the purchase, we add the acquired shares to her initial holding:

Total shares of Meera = 450,000 + 20,000 = 470,000

Explanation

The total number of employees in company D is calculated by multiplying the number of zones by the average number of employees per zone. Using the values from the table:

Total employees = 25 × 450 = 11,250

The distribution of male and female employees in company D is given by the ratio 5:4. The total number of ratio units is 5 + 4 = 9. To find the number of male employees, we use the proportion of the male ratio to the total ratio:

Number of male employees = ⁵⁄₉ × 11,250

First, divide 11,250 by 9: 11,250 ÷ 9 = 1,250

Then, multiply by the male ratio component: 1,250 × 5 = 6,250

Explanation

The relationship between the Amount A, Principal P, Rate R, and Time T in simple interest is given by the formula A = P (1 + (RT)/(100)). Substituting the given values A = 21420, R = 9.5%, and T = 2 years into the formula:

21420 = P (1 + (9.5 × 2)/(100))

21420 = P (1 + ¹⁹⁄₁₀₀)

21420 = P (1 + 0.19)

21420 = 1.19P

To find P, divide the amount by 1.19:

P = (21420)/(1.19) = ²¹⁴²⁰⁰⁰⁄₁₁₉

By division, 2142 ÷ 119 = 18. Thus, P = 18000.

Explanation

Let the smaller even number be x. Since the numbers are consecutive even integers, the next number is x + 2. Their sum is given as 174:

x + (x + 2) = 174 2x + 2 = 174 2x = 172 x = 86

Alternatively, the average of two consecutive even numbers is the odd number lying between them. Dividing the total sum by the count of numbers gives the midpoint: 174 / 2 = 87. The two consecutive even numbers surrounding the midpoint 87 are 86 and 88. The smaller of these two values is 86.

Explanation

Since the police car and the thief start their movement simultaneously and the chase ends when the thief is caught, the time t for both is constant. When time is constant, the ratio of speeds is directly proportional to the ratio of distances covered: vₚ / vₜ = dₚ / dₜ.

The thief covers a distance dₜ = 1.5 km. The police car starts 250 m (or 0.25 km) behind the thief, so the total distance covered by the police car is dₚ = 1.5 km + 0.25 km = 1.75 km.

Substituting the known values into the ratio:

70 / vₜ = 1.75 / 1.5

vₜ = (70 × 1.5) / 1.75

vₜ = 105 / 1.75 = 60 km/h

Explanation

Based on the relationship Income = Expenditure + Savings , let Suresh's initial expenditure be 300 and his initial savings be 100 , following the ratio 3 : 1 . Therefore, his initial income is 300 + 100 = 400 .

After a 25% increase, the new income becomes 400 × 1.25 = 500 . With a 20% increase in savings, the new savings become 100 × 1.20 = 120 . The new expenditure is calculated by subtracting the new savings from the new income: 500 - 120 = 380 .

The increase in expenditure is 380 - 300 = 80 . To find the percentage increase, we use the formula (Increase / Original Expenditure) × 100 . This gives (80 / 300) × 100 = 80 / 3 , which simplifies to 26 ⅔% .

Explanation

The area of a triangle can be calculated using the formula Area = ½ × base × height. Let the sides of the triangle be a, b, c and the corresponding altitudes to these sides be hₐ, h_b, h_c. Given that the altitudes from two vertices are equal, let hₐ = h_b. Since the area of the triangle is constant regardless of the base used for the calculation, we have the relation ½ × a × hₐ = ½ × b × h_b. Substituting hₐ for h_b in the equation leads to ½ a hₐ = ½ b hₐ. By canceling the common terms, we find that a = b. A triangle that possesses two sides of equal length is defined as an isosceles triangle.

Explanation

According to the Power of a Point theorem, for a point P located outside a circle with center O and radius r, the product of the lengths of the segments of any secant line passing through P is constant and equal to the square of the distance from P to the center minus the square of the radius. This is expressed by the formula:

PA · PB = OP² - r²

Substituting the given values into the formula:

12 · 22.5 = 17² - r²

270 = 289 - r²

Rearranging the equation to solve for r:

r² = 289 - 270

r² = 19

r = √(19) cm

Explanation

The cost price (CP) of the article is ₹1,000 and the marked price (MP) is ₹1,500. The salesman applies two successive discounts: a trade discount of 20% and a cash discount of 10%. The final selling price (SP) is calculated using the formula SP = MP × (1 - d₁{100}) × (1 - d₂{100}).

Substituting the values, we get: SP = 1500 × (1 - ²⁰⁄₁₀₀) × (1 - ¹⁰⁄₁₀₀) SP = 1500 × 0.8 × 0.9 SP = 1500 × 0.72 = 1080

The gain made by the salesman is the difference between the selling price and the cost price: Gain = SP - CP = 1080 - 1000 = 80

The gain percentage is calculated as: Gain % = ((Gain)/(CP)) × 100 = (⁸⁰⁄₁₀₀₀) × 100 = 8%

Explanation

To simplify the expression 626 × 2912 ÷ 8138, we look for common factors between the numbers. First, observe that 626 and 8138 share a common divisor. By testing for factors, we find that 626 = 2 × 313. Checking the denominator, we find that 8138 is also divisible by 313:

8138 ÷ 313 = 26

Now, we can rewrite the entire expression as:

(626 × 2912)/(8138) = ((2 × 313) × 2912)/(26 × 313)

The term 313 cancels out from the numerator and the denominator:

(2 × 2912)/(26) = ²⁹¹²⁄₁₃

Next, we perform the division of 2912 by 13: 29 ÷ 13 = 2 with a remainder of 3. 31 ÷ 13 = 2 with a remainder of 5. 52 ÷ 13 = 4.

The resulting quotient is 224.

Explanation

Let the monthly incomes of A, B, and C be 2x, 9x, and 3x respectively. Let their monthly expenses be 3y, 9y, and 5y respectively. Savings are calculated as Income minus Expenses.

According to the given condition, A's saving is half of his total income: S_A = 2x - 3y = ½(2x) 2x - 3y = x x = 3y

Now, substitute x = 3y into the expressions for the savings of A, B, and C: S_A = 2(3y) - 3y = 6y - 3y = 3y S_B = 9(3y) - 9y = 27y - 9y = 18y S_C = 3(3y) - 5y = 9y - 5y = 4y

The ratio of their savings is: S_A : S_B : S_C = 3y : 18y : 4y = 3 : 18 : 4

Explanation

Let the efficiency of pipe A be E_A units/minute. Since pipe B fills the tank twice as fast as A, its efficiency E_B is 2E_A units/minute. The combined efficiency of both pipes working together is E_A + E_B = E_A + 2E_A = 3E_A units/minute.

The total work (capacity of the tank) is the product of the combined efficiency and the time taken to fill it: Work = 3E_A × 45 = 135E_A units.

The time taken by pipe A alone to fill the tank is the total work divided by pipe A's efficiency: Time_A = (Total Work)/(E_A₎ = 135E_A{E_A} = 135 minutes.

Explanation

To find the meeting times, first convert the speeds from km/h to m/s by multiplying by ⁵⁄₁₈. The speed of Bhaskar is 27 × ⁵⁄₁₈ = 7.5 m/s, and the speed of Vinay is 45 × ⁵⁄₁₈ = 12.5 m/s.

In a circular race, the time taken for two people to meet for the first time anywhere on the track is given by the formula T = (L)/(vᵣₑₗ₎, where L is the length of the track and vᵣₑₗ is the relative speed.

When they run in opposite directions, their relative speed is the sum of their individual speeds: vᵣₑₗ = 12.5 + 7.5 = 20 m/s. The time taken to meet is ¹⁶⁰⁰⁄₂₀ = 80 seconds. Converting this to minutes and seconds, 80 seconds is 1 minute 20 seconds.

When they run in the same direction, their relative speed is the difference between their individual speeds: vᵣₑₗ = 12.5 - 7.5 = 5 m/s. The time taken to meet is ¹⁶⁰⁰⁄₅ = 320 seconds. Converting this to minutes and seconds, 320 seconds is 5 minutes 20 seconds.

Explanation

The expression sin⁴ θ + cos⁴ θ can be simplified using the identity a² + b² = (a+b)² - 2ab. By substituting a = sin² θ and b = cos² θ, we get:

sin⁴ θ + cos⁴ θ = (sin² θ + cos² θ)² - 2 sin² θ cos² θ

Since sin² θ + cos² θ = 1, the expression becomes:

1 - 2 sin² θ cos² θ

This can be further rewritten using the double-angle formula sin 2θ = 2 sin θ cos θ, which implies sin² 2θ = 4 sin² θ cos² θ. Substituting this back into the expression:

1 - ½ (4 sin² θ cos² θ) = 1 - ½ sin² 2θ

The value of sin² 2θ ranges from 0 to 1. To find the greatest value of the entire expression, we subtract the smallest possible value of ½ sin² 2θ. The minimum value of sin² 2θ is 0, occurring when 2θ = nπ. Substituting 0 into the simplified expression gives the maximum value:

1 - ½(0) = 1

Explanation

To find the percentage of the total expenditure in 2006 relative to the total expenditure in 2009, first calculate the sum of all expenditure categories for each of those years.

The total expenditure in 2006 is the sum of Salary, Fuel and Transport, Bonus, Interest on Loan, and Taxes: Total₂₀₀₆ = 225 + 98 + 4.25 + 29.7 + 110 = 466.95 lakh rupees.

The total expenditure in 2009 is the sum of the same categories for that year: Total₂₀₀₉ = 295 + 145 + 6.25 + 49.6 + 118 = 613.85 lakh rupees.

The required percentage is calculated using the formula: Percentage = Total₂₀₀₆{Total₂₀₀₉} × 100

Substituting the calculated values: Percentage = (466.95)/(613.85) × 100 ≈ 0.7606907 × 100 ≈ 76.069%

Rounding off to two places of decimals, the value is 76.07%.

Explanation

The numerator of the fraction, 4x² - 49y², is a difference of squares that can be factored using the algebraic identity a² - b² = (a + b)(a - b). In this case, a = 2x and b = 7y, which gives:

4x² - 49y² = (2x + 7y)(2x - 7y)

Substituting this factored form back into the original expression:

((2x + 7y)(2x - 7y))/(2x + 7y) + 7y - 2x

The common factor (2x + 7y) in the numerator and denominator cancels out, simplifying the fraction to (2x - 7y). The expression then becomes:

2x - 7y + 7y - 2x

By grouping and simplifying the like terms, the 2x and -2x terms cancel each other out, and the -7y and +7y terms also cancel each other out. This results in a final value of 0.

Explanation

The angle sum property of a triangle states that the sum of all interior angles is always 180°. For triangle PQR, this is expressed as: ∠ P + ∠ Q + ∠ R = 180°

The problem provides the sum of two angles, ∠ P + ∠ Q = 84°. By substituting this given value into the angle sum formula, we can solve for the third angle: 84° + ∠ R = 180°

Subtracting 84° from both sides gives: ∠ R = 180° - 84° ∠ R = 96°

Explanation

The volume V of a cone is determined by the formula V = ⅓ π r² h. Given the base radius r = 3.5 cm (which is equivalent to ⁷⁄₂ cm) and the height h = 18 cm, the volume is calculated by substituting these values into the formula:

V = ⅓ × ²²⁄₇ × (⁷⁄₂)² × 18

V = ⅓ × ²²⁄₇ × ⁴⁹⁄₄ × 18

Simplifying the expression, the 49 in the numerator and 7 in the denominator reduce to 7. Similarly, dividing the height 18 by the 3 in the denominator leaves 6:

V = (22 × 7 × 6)/(4)

Further reduction is achieved by dividing 22 by 2 and 6 by 2, which cancels the 4 in the denominator:

V = 11 × 7 × 3 = 231 cm³

Explanation

To find the percentage by which Shyam's income is less than Ram's income, we use the standard percentage comparison formula. If one value is x% more than another, the second value is less than the first by (x)/(100 + x) × 100%.

Given that Ram's income is 2.5% more than Shyam's income, we substitute x = 2.5 into the formula:

Percentage Less = (2.5)/(100 + 2.5) × 100

Percentage Less = (2.5)/(102.5) × 100

Multiplying both the numerator and the denominator by 10 to remove the decimals gives:

Percentage Less = ²⁵⁄₁₀₂₅ × 100

Simplifying the fraction by dividing both 25 and 1025 by 25:

Percentage Less = ¹⁄₄₁ × 100 = ¹⁰⁰⁄₄₁

Performing the division:

100 ÷ 41 ≈ 2.439...%

Rounding to two decimal places, we get approximately 2.43%.

Explanation

Using the rule of alligation, the ratio of the quantities of the two types of rice is determined by the differences between their individual prices and the target mean price of the mixture. Let Q₁ be the quantity of rice priced at ₹4 per kg and Q₂ be the quantity of rice priced at ₹10 per kg.

The ratio of the quantities is calculated as follows: Q₁{Q₂} = (Price of dearer rice - Mean price)/(Mean price - Price of cheaper rice) Q₁{15} = (10 - 6.50)/(6.50 - 4) Q₁{15} = (3.50)/(2.50) Q₁{15} = ⁷⁄₅

Solving for Q₁: Q₁ = 15 × ⁷⁄₅ Q₁ = 3 × 7 Q₁ = 21 kg

Explanation

The given equation is (x)/(x² - 2x + 1) = ⅓. Taking the reciprocal of both sides gives (x² - 2x + 1)/(x) = 3. By dividing each term in the numerator by x, we obtain x - 2 + ¹⁄ₓ = 3, which simplifies to x + ¹⁄ₓ = 5.

To find the value of the cubic expression x³ + (1)/(x³), we use the algebraic identity x³ + (1)/(x³) = (x + ¹⁄ₓ)³ - 3(x + ¹⁄ₓ). This formula is derived from the expansion of (x + ¹⁄ₓ)³.

Substituting the value x + ¹⁄ₓ = 5 into the identity: x³ + (1)/(x³) = 5³ - 3(5) x³ + (1)/(x³) = 125 - 15 = 110

Explanation

Using the trigonometric identity sec θ = (1)/(cos θ), the given values for x and y can be simplified as follows:

x = sec θ cos θ = (1)/(cos θ) · cos θ = 1

y = sec θ sin θ = (sin θ)/(cos θ) = tan θ

The value of z is given as z = tan θ. Substituting these simplified forms into the expression (x²)/(a²) + (y²)/(b²) - (z²)/(c²) (where a, b, c typically represent the coefficients in the definitions of x, y, z, such as x = a sec θ cos θ, etc.):

ˣ⁄ₐ = 1 ⟹ (x²)/(a²) = 1

(y)/(b) = tan θ ⟹ (y²)/(b²) = tan² θ

(z)/(c) = tan θ ⟹ (z²)/(c²) = tan² θ

Substituting these into the expression:

1 + tan² θ - tan² θ = 1

Explanation

The shopkeeper uses a faulty weight of 3 kg that has an error of 100 g. In the context of a seller's profit, this implies the weight used measures 3 kg on the scale but only delivers 3000 g - 100 g = 2900 g of potatoes. Therefore, the actual quantity sold to Charu is 2.9 kg.

The cost price for the seller is 12 per kg. The total cost price (CP) for the actual weight delivered is: CP = 2.9 × 12 = 34.8

The selling price (SP) for the transaction is given as 46. The profit earned by the shopkeeper is the difference between the selling price and the total cost price of the potatoes actually provided: Profit = SP - CP = 46 - 34.8 = 11.2

Explanation

In △ XAY, the condition XA = YA implies that it is an isosceles triangle where the angles opposite the equal sides are equal. If we consider the base angles ∠ AXY and ∠ AYX to be 50° each, then the vertex angle at A is calculated as: ∠ XAY = 180° - (50° + 50°) = 80°

Since point A lies on the line segment YZ, the angles ∠ XAY and ∠ XAZ form a linear pair and are supplementary: ∠ XAZ = 180° - ∠ XAY = 180° - 80° = 100°

In △ XAZ, the sum of the interior angles must be 180°. Using the given value ∠ AZX = 19°, we can find the measure of the remaining angle ∠ AXZ: ∠ AXZ = 180° - (∠ XAZ + ∠ AZX) ∠ AXZ = 180° - (100° + 19°) ∠ AXZ = 180° - 119° = 61°

Explanation

Substituting the values x = 2, y = -3, and z = 4 into the system of linear equations is the fastest way to verify the solution. For the first equation, x + 1.5y + 2z = 2 + 1.5(-3) + 2(4) = 2 - 4.5 + 8 = 5.5. For the second equation, x + 5y + 7z = 2 + 5(-3) + 7(4) = 2 - 15 + 28 = 15. For the third equation, 3x + 11y + 13z = 3(2) + 11(-3) + 13(4) = 6 - 33 + 52 = 25.

Explanation

The sum of any three consecutive integers is always divisible by 3. This property can be expressed using the formula for the sum of three consecutive numbers n, n+1, and n+2: n + (n+1) + (n+2) = 3n + 3 = 3(n+1)

For the given numbers, the sum is 715 + 716 + 717. Applying the formula with n = 715: 3(715 + 1) = 3 × 716 = 2148

Since the sum results in a product of 3 and an integer, it is divisible by 3. This can also be verified using the divisibility rule for 3, where the sum of the digits of the result (2 + 1 + 4 + 8 = 15) is divisible by 3, confirming that the total sum is divisible by 3.

Explanation

Given cot θ = √(11), it follows that cot² θ = 11 and tan² θ = ¹⁄₁₁. Using the fundamental trigonometric identities csc² θ = 1 + cot² θ and sec² θ = 1 + tan² θ, we find the required values: csc² θ = 1 + 11 = 12 and sec² θ = 1 + ¹⁄₁₁ = ¹²⁄₁₁. Substituting these into the expression csc² θ - sec² θ{csc² θ + sec² θ}, we obtain:

(12 - 12)/(11){12 + ¹²⁄₁₁}

By factoring out 12 from the numerator and the denominator, the expression simplifies to:

(12(1 - 1)/(11)){12(1 + ¹⁄₁₁)} = (1 - 1)/(11){1 + ¹⁄₁₁} = (11 - 1)/(11){(11 + 1)/(11)} = ¹⁰⁄₁₂ = ⅚

Explanation

To find the values of x, y, and z, we solve the given system of linear equations using the method of elimination. The equations are:

1. x + y + z = 12
2. x + y - z = 6
3. x - y + z = 4

Subtracting Eq 2 from Eq 1 eliminates the variables x and y: (x + y + z) - (x + y - z) = 12 - 6 2z = 6 ⟹ z = 3

Subtracting Eq 3 from Eq 1 eliminates the variables x and z: (x + y + z) - (x - y + z) = 12 - 4 2y = 8 ⟹ y = 4

Adding Eq 2 and Eq 3 eliminates the variables y and z: (x + y - z) + (x - y + z) = 6 + 4 2x = 10 ⟹ x = 5

The values obtained are x = 5, y = 4, and z = 3. Substituting these values back into the first equation confirms the solution: 5 + 4 + 3 = 12, which is correct. Similarly, for the second equation: 5 + 4 - 3 = 6, and for the third equation: 5 - 4 + 3 = 4. All conditions are satisfied.

Explanation

The distance between the policeman and the thief is 500 metres, which is equal to 0.5 km. The time taken to catch the thief is 15 minutes, which is equivalent to 15/60 = 0.25 hours. Let the speed of the policeman be v km/h. Since both are running in the same direction, the relative speed is (v - 15) km/h.

Using the formula for relative speed, Relative Speed = Distance / Time, we substitute the known values: v - 15 = 0.5 / 0.25 v - 15 = 2 v = 17 km/h

Alternatively, the relative speed in metres per minute is 500 / 15 = 100/3 m/min. Converting this to km/h, we multiply by 60/1000, which gives (100/3) × (60/1000) = 2 km/h. The policeman's speed is the sum of the thief's speed and this relative speed, resulting in 15 + 2 = 17 km/h.

Explanation

To find the combined ratio A:B:C, the value of the common variable B must be the same in both given ratios. We are given A:B = 6:8 and B:C = 5:12. First, we simplify the ratio A:B = 6:8 to 3:4 by dividing both parts by 2. Now, we have A:B = 3:4 and B:C = 5:12.

The value of B in the first ratio is 4 and in the second ratio is 5. The Least Common Multiple (LCM) of 4 and 5 is 20. To make the value of B equal to 20 in both ratios, we multiply the terms of the first ratio by 5 and the terms of the second ratio by 4.

Multiplying A:B = 3:4 by 5 gives A:B = (3 × 5) : (4 × 5) = 15:20. Multiplying B:C = 5:12 by 4 gives B:C = (5 × 4) : (12 × 4) = 20:48.

Since the value of B is now 20 in both expressions, we can combine them to find the ratio A:B:C = 15:20:48.

Explanation

Given 2 tan θ = 3, we find that tan θ = ³⁄₂.

To evaluate the expression (3sinθ - 2cosθ)/(3sinθ + 2cosθ), we can divide both the numerator and the denominator by cosθ. This converts the sine and cosine terms into tangent terms:

(3sinθ)/(cosθ) - (2cosθ)/(cosθ){(3sinθ)/(cosθ) + (2cosθ)/(cosθ)} = (3tanθ - 2)/(3tanθ + 2)

Substituting the value tan θ = ³⁄₂ into the equation:

(3(3)/(2)) - 2{3(³⁄₂) + 2} = (9)/(2) - 2{⁹⁄₂ + 2}

To simplify the fractions, we perform the arithmetic in the numerator and denominator:

(9 - 4)/(2){(9 + 4)/(2)} = (5)/(2){¹³⁄₂}

The common denominator 2 cancels out, resulting in ⁵⁄₁₃.

Explanation

From the first equation p cos A = 2q sin A, we can write (q)/(cos A) = (p)/(2 sin A). Substituting this into the second equation 2p csc A - q sec A = 3, which is equivalent to (2p)/(sin A) - (q)/(cos A) = 3, we get:

(2p)/(sin A) - (p)/(2 sin A) = 3

Multiplying the entire equation by 2 sin A yields:

4p - p = 6 sin A 3p = 6 sin A p = 2 sin A

Now, substitute p = 2 sin A back into the first equation:

(2 sin A) cos A = 2q sin A 2q = 2 cos A q = cos A

The value of the expression p² + 4q² is:

p² + 4q² = (2 sin A)² + 4(cos A)² p² + 4q² = 4 sin² A + 4 cos² A p² + 4q² = 4(sin² A + cos² A)

Since sin² A + cos² A = 1, we have:

p² + 4q² = 4(1) = 4

Explanation

Since Δ EFG ∼ Δ HIJ with ∠ E = ∠ H and ∠ F = ∠ I, the triangles are similar by AA (Angle-Angle) criterion, and the corresponding vertices are E ↔ H, F ↔ I, and G ↔ J. For similar triangles, the ratios of the lengths of corresponding sides are equal:

(EF)/(HI) = (FG)/(IJ) = (EG)/(HJ)

The problem states 3EF = HI, which implies the ratio of the sides of Δ HIJ to Δ EFG is:

(HI)/(EF) = 3

Given that FG = 9 cm, we substitute the known values into the side ratio equation:

(IJ)/(FG) = (HI)/(EF) = 3

IJ = 3 × FG

IJ = 3 × 9 = 27 cm

Explanation

Let u and v be the speeds of the two cars in km/h, where u > v. When moving in the same direction, the relative speed is u - v. Since they cover the 20 km gap in 1 hour, the relative speed is u - v = 20 / 1 = 20. When moving towards each other, the relative speed is u + v. They meet in 12 minutes, which is 12/60 = 1/5 hour, so the relative speed is u + v = 20 / (1/5) = 100. Solving the system of linear equations u - v = 20 and u + v = 100 by adding them gives 2u = 120, hence u = 60. Substituting u = 60 into the equation u + v = 100 gives 60 + v = 100, hence v = 40. The speeds of the two cars are 60 km/h and 40 km/h.

Explanation

According to the Exterior Angle Theorem of a triangle, the measure of an exterior angle at any vertex is equal to the sum of the measures of its two opposite (remote) interior angles. Given that two interior angles of the triangle are 45° and 65°, the exterior angle associated with the third (remaining) interior angle is simply the sum of these two opposite angles.

Exterior Angle = 45° + 65° = 110°

Alternatively, by the Angle Sum Property, the sum of all interior angles in a triangle is 180°. The third interior angle is 180° - (45° + 65°) = 180° - 110° = 70°. Since an interior angle and its corresponding exterior angle form a linear pair, they are supplementary. Thus, the exterior angle is 180° - 70° = 110°.

Explanation

When a right triangle with sides 3 cm, 4 cm, and 5 cm is rotated about the side of 3 cm, that side becomes the axis of the cone, representing its height h = 3 cm. The other leg of the triangle, which is 4 cm, becomes the radius of the base r = 4 cm. The hypotenuse of 5 cm represents the slant height of the cone.

The volume V of a cone is calculated using the formula: V = ⅓ π r² h

Substituting the dimensions of the formed cone: V = ⅓ × π × 4² × 3 V = ⅓ × π × 16 × 3 V = 16π cm³

Explanation

The time taken by two pipes to fill a pool together is calculated using the formula T = (xy)/(x + y), where x and y represent the time taken by each pipe individually. Given x = 30 hours and y = 45 hours, the calculation is:

T = (30 × 45)/(30 + 45) T = ¹³⁵⁰⁄₇₅ T = 18 hours

Alternatively, using the LCM method, let the total capacity of the pool be the LCM of 30 and 45, which is 90 units. The filling rate of pipe L is 90 / 30 = 3 units per hour, and the filling rate of pipe M is 90 / 45 = 2 units per hour. The combined filling rate when both pipes are open is 3 + 2 = 5 units per hour. The time required to fill the 90-unit pool is 90 / 5 = 18 hours.

Explanation

Using the algebraic relation where if z + (1)/(z) = k, then z² + (1)/(z²) = k² - 2, let z = √x and k = 7. Substituting the given value √x + (1)/(√(x)) = 7 into this identity, the expression for x + ¹⁄ₓ becomes 7² - 2. Calculating the square and subtracting the constant yields 49 - 2, which equals 47.

Explanation

The rule of alligation is the most efficient way to solve problems involving the weighted average of two parts. Let the quantity sold at 10% profit be Q₁ and the quantity sold at 40% profit be Q₂. The ratio of these quantities is determined by the difference between the individual profit percentages and the mean profit percentage.

Ratio Q₁ : Q₂ = (Profit₂ - Mean Profit) : (Mean Profit - Profit₁)

Substituting the given values: Ratio Q₁ : Q₂ = (40 - 20) : (20 - 10) = 20 : 10 = 2 : 1

The total weight of 1000 kg is divided into 2 + 1 = 3 parts. The quantity sold at 40% profit corresponds to 1 part of this ratio.

Quantity sold at 40% profit = (1)/(2 + 1) × 1000 = ¹⁰⁰⁰⁄₃ = 333⅓ kg