SSC CGL Tier 1 2017 Quantitative Aptitude Questions with Solutions

Explanation

The sum of cubes of two numbers can be expressed using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Let x represent the value of a + b. Substituting the given values a³ + b³ = 19 and ab = -6 into this formula results in the following equation:

19 = x³ - 3(-6)(x) 19 = x³ + 18x

Rearranging the equation into standard cubic form gives x³ + 18x - 19 = 0. By testing the values for x, we find that when x = 1:

1³ + 18(1) - 19 = 1 + 18 - 19 = 0

Alternatively, looking for two numbers a and b such that a + b = 1 and ab = -6, we can use the factors of -6. For a = 3 and b = -2, the product is 3 × (-2) = -6 and the sum of cubes is 3³ + (-2)³ = 27 - 8 = 19. The sum a + b is 3 + (-2) = 1.

Explanation

Let the required fraction be x. The problem states that the sum of four times the fraction and six times its reciprocal is 11, which can be expressed as: 4x + ⁶⁄ₓ = 11

To find the value of x, we can substitute the value from the options. Using x = ¾: 4(¾) + 6(⁴⁄₃) = 3 + (2 × 4) = 3 + 8 = 11

This confirms that ¾ satisfies the condition. Alternatively, the equation can be solved by multiplying through by x to form a quadratic equation: 4x² - 11x + 6 = 0 Factoring the quadratic equation: 4x² - 8x - 3x + 6 = 0 4x(x - 2) - 3(x - 2) = 0 (4x - 3)(x - 2) = 0

Solving for x gives x = ¾ or x = 2. Among the given options, ¾ is the correct value.

Explanation

The n-th term of an arithmetic progression is given by aₙ = a + (n-1)d, where a is the first term and d is the common difference. Given the 3rd term a₃ = a + 2d = -13 and the 6th term a₆ = a + 5d = -4, we can find d by subtracting the first equation from the second: (a + 5d) - (a + 2d) = -4 - (-13), which simplifies to 3d = 9, so d = 3.

Substituting d = 3 back into the equation for the 3rd term gives a + 2(3) = -13, resulting in a = -19. The sum of the first n terms is calculated using the formula Sₙ = ⁿ⁄₂ [2a + (n-1)d]. For n = 12:

S₁₂ = ¹²⁄₂ [2(-19) + (12-1)3] S₁₂ = 6 [-38 + 11(3)] S₁₂ = 6 [-38 + 33] S₁₂ = 6 [-5] = -30

Explanation

When a point (x, y) is reflected in a vertical line x = k, the y-coordinate remains the same, while the new x-coordinate x' is determined by the formula x' = 2k - x. This is because the line x = k is the perpendicular bisector of the segment joining the original point and its image, meaning the midpoint of the horizontal distance must lie on x = k.

Given the point (5, 2) and the line of reflection x = -3, we substitute x = 5 and k = -3 into the formula. The calculation for the new x-coordinate is x' = 2(-3) - 5 = -6 - 5 = -11. Since the reflection is across a vertical line, the y-coordinate does not change, so y' = 2. Therefore, the coordinates of the reflected point are (-11, 2).

Explanation

The centroid G(x, y) of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) is the arithmetic mean of the coordinates of its vertices, given by the formula G = ((x₁ + x₂ + x₃)/(3), (y₁ + y₂ + y₃)/(3)).

Substituting the given vertices A(1, -5), B(4, 0), and C(-2, 2) into the formula:

The x-coordinate is x = (1 + 4 + (-2))/(3) = ³⁄₃ = 1.

The y-coordinate is y = (-5 + 0 + 2)/(3) = ⁻³⁄₃ = -1.

This results in the centroid coordinates (1, -1).

Explanation

The Selling Price (SP) can be calculated using the successive percentage change formula or by calculating the Marked Price (MP) first. Given the Cost Price (CP) is Rs 1600 and the mark-up is 60 percent, the Marked Price is:

MP = CP × (1 + ⁶⁰⁄₁₀₀) = 1600 × 1.6 = 2560

A discount of 25 percent is then applied to this Marked Price to find the Selling Price:

SP = MP × (1 - ²⁵⁄₁₀₀) = 2560 × 0.75 = 2560 × ¾ = 1920

Alternatively, the net profit percentage can be found using the formula for successive changes: Net Percentage = m - d - (m × d)/(100), where m is the mark-up and d is the discount. Substituting the values gives 60 - 25 - (60 × 25)/(100) = 35 - 15 = 20 percent profit. The Selling Price is therefore 120 percent of the Cost Price:

SP = 1600 × 1.2 = 1920

Explanation

In Δ ABC, since DE ∥ BC, the triangles Δ ADE and Δ ABC are similar by AA (Angle-Angle) similarity. The ratio of the sides AD : DB is given as 2 : 5. The total length of side AB is the sum of segments AD and DB: AB = AD + DB = 2 + 5 = 7 units.

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Therefore: (Area(Δ ADE))/(Area(Δ ABC)) = ((AD)/(AB))² = (²⁄₇)² = ⁴⁄₄₉

Let the area of Δ ADE be 4k and the area of Δ ABC be 49k. The area of the quadrilateral BDEC is the difference between the area of the larger triangle and the smaller triangle: Area(BDEC) = Area(Δ ABC) - Area(Δ ADE) = 49k - 4k = 45k

The required ratio of the area of Δ ADE to the area of the quadrilateral BDEC is: Area(Δ ADE) : Area(BDEC) = 4k : 45k = 4 : 45

Given that the area of Δ ADE is 8 sq cm, we can see that 4k = 8, which means k = 2. Thus, the area of BDEC is 45 × 2 = 90 sq cm. The ratio 8 : 90 simplifies to 4 : 45.

Explanation

The trigonometric values for the specific angles are sec 45° = √2 and tan 30° = (1)/(√(3)). Substituting these values into the given expression √2 sec 45° - tan 30° gives:

√2 · √2 - (1)/(√(3))

Since √2 · √2 = 2, the expression simplifies to:

2 - (1)/(√(3))

To combine the terms under a single denominator, we multiply the integer 2 by √3{√3}:

2√3{√3} - (1)/(√(3)) = 2√3 - 1{√3}

Explanation

In any triangle, the sum of the interior angles is 180°. Since △ ABC is right-angled at B (m∠B = 90°) and m∠A = 60°, the measure of angle C is found by subtracting the known angles from the total sum:

∠C = 180° - (90° + 60°) = 30°

The problem asks for the value of (1/√3) cosec C. Substituting the value of angle C and using the trigonometric identity cosec 30° = 1/sin 30°:

(1/√3) cosec 30° = (1/√3) × 2 = 2/√3

Explanation

The relationship between cosecant and cotangent is defined by the trigonometric identity csc² θ - cot² θ = 1. Rearranging this formula to find cotangent gives cot θ = √(csc² θ - 1). Substituting the given value csc θ = 25/7 into the equation, we get cot θ = √((25/7)² - 1) = √(625/49 - 1). Simplifying the expression inside the square root leads to √((625 - 49)/49) = √(576/49) = 24/7.

This can also be solved using the properties of a right-angled triangle where csc θ represents the ratio of the hypotenuse to the perpendicular. Taking the hypotenuse as 25 and the perpendicular as 7, the base is calculated using the Pythagorean theorem as √(25² - 7²) = √(625 - 49) = √(576) = 24. Since cot θ is the ratio of the base to the perpendicular, the value is 24/7.

Explanation

The inflation rates for country D in 2015 and 2016, according to the provided bar graph data, were 2% and 5% respectively.

To find the percentage by which the inflation in 2016 was greater than that in 2015, we use the percentage change formula:

Percentage Increase = Inflation₂₀₁₆ - Inflation₂₀₁₅{Inflation₂₀₁₅} × 100

Substituting the values: Percentage Increase = (5 - 2)/(2) × 100 = ³⁄₂ × 100 = 1.5 × 100 = 150%

Explanation

Based on the bar graph provided in the exam, the annual inflation rate for country C in the year 2015 was 8%, while the annual inflation rate for country A in the same year was 4%. To find the ratio of inflation in country C to country A:

Ratio = (Inflation in Country C (2015))/(Inflation in Country A (2015)) = ⁸⁄₄ = ²⁄₁

Thus, the ratio of inflation in country C to country A for the year 2015 is 2 : 1.

Explanation

The price index at the end of 2016 is determined by applying the annual inflation rates for Country D consecutively to the starting index value. For Country D, the inflation rate was 5% in 2015 and 2% in 2016.

The calculation for the index at the end of 2015 is: 200 × (1 + ⁵⁄₁₀₀) = 200 × 1.05 = 210

The index at the end of 2016 is then calculated by applying the 2016 inflation rate to the value reached at the end of 2015: 210 × (1 + ²⁄₁₀₀) = 210 × 1.02 = 214.2

This can also be expressed using the compound growth formula: Final Index = Initial Index × (1 + r₁) × (1 + r₂) Final Index = 200 × 1.05 × 1.02 = 214.2

The result represents the cumulative increase in the price level over the two-year period.

Explanation

For the numbers 55+x, 100+x, 65+x, and 116+x to be in proportion, the product of the extremes must equal the product of the means. This is represented by the equation ⁵⁵⁺ˣ⁄₁₀₀₊ₓ = ⁶⁵⁺ˣ⁄₁₁₆₊ₓ.

A direct formula to find the value x to be added to four numbers a, b, c, d to make them proportional is x = (ad - bc)/((b+c) - (a+d)).

Substituting the given values a = 55, b = 100, c = 65, and d = 116 into the formula: x = ((55 × 116) - (100 × 65))/((100 + 65) - (55 + 116)) x = (6380 - 6500)/(165 - 171) x = ⁻¹²⁰⁄₋₆ x = 20

By adding 20 to each number, we get 75, 120, 85, and 136. We can verify the proportion by calculating the ratios: ⁷⁵⁄₁₂₀ = (5 × 15)/(8 × 15) = ⅝ ⁸⁵⁄₁₃₆ = (5 × 17)/(8 × 17) = ⅝

Since the ratios are equal, adding 20 results in numbers that are in proportion.

Explanation

Let the average after 10 matches be A. Since the average increased by 5 runs after the 10th match, the average after 9 matches was A - 5. The total runs after 10 matches is the sum of the total runs after 9 matches and the runs scored in the 10th match. This relationship is expressed as 10A = 9(A - 5) + 111. Expanding the equation gives 10A = 9A - 45 + 111. Subtracting 9A from both sides results in A = 111 - 45, which yields A = 66.

Alternatively, using the shortcut for average problems: the new average is found by subtracting the total increase distributed among the previous matches from the latest score. The 111 runs scored in the 10th match provided an increase of 5 runs for each of the 9 preceding matches. Therefore, the average after the 10th match is 111 - (9 × 5) = 111 - 45 = 66. This shortcut effectively accounts for the runs required to raise the previous matches to the new average level.

Explanation

To find the profit percentage, equate the number of items for both cost and selling prices. The Cost Price (CP) of 6 bananas is given as Rs 25. The vendor sells 3 bananas for Rs 20, so the Selling Price (SP) for 6 bananas is 20 × 2 = Rs 40.

The profit is calculated as: Profit = SP - CP = 40 - 25 = Rs 15

The profit percentage is determined by the formula: Profit Percentage = (Profit)/(CP) × 100 Profit Percentage = ¹⁵⁄₂₅ × 100 Profit Percentage = ⅗ × 100 = 60%

Explanation

The payment distribution between labourer A and labourer B is based on a percentage relationship that can be simplified into a ratio. Given that A is paid 160% of what B is paid, if B's pay is represented as 100%, then A's pay is 160%. This provides a ratio of A to B as 160:100, which simplifies to 8:5 after dividing both parts by 20. The total combined daily wage of Rs 650 represents the sum of these ratio parts. The total number of units in the ratio is calculated as: 8 + 5 = 13

To determine the monetary value of a single unit, the total daily pay is divided by the total number of units: ⁶⁵⁰⁄₁₃ = 50

Since labourer B's wage is equivalent to 5 units in the simplified ratio, the total amount paid to B is found by multiplying the value of one unit by the number of parts assigned to B: 5 × 50 = 250

Explanation

The total travel time is the sum of the time spent on foot and the time spent on the bicycle. Let x represent the distance travelled on foot. The distance covered by bicycle is therefore 60 - x. Using the fundamental relation Time = (Distance)/(Speed), the equation for the total time is:

ˣ⁄₆ + (60 - x)/(12) = 7

To solve for x, multiply the entire equation by the least common multiple of the speeds, which is 12:

2x + (60 - x) = 84

Simplifying the left side:

x + 60 = 84 x = 24

The distance travelled on foot is 24 km. This result is consistent with a total time of 7 hours, as 4 hours are spent walking (²⁴⁄₆) and 3 hours are spent cycling (³⁶⁄₁₂).

Explanation

The effective compound interest rate for two years at a rate of r% per annum is calculated using the successive percentage formula R = r + r + (r × r)/(100). Substituting the given rate of 12%:

Effective Rate = 12 + 12 + (12 × 12)/(100) = 24 + 1.44 = 25.44%

The compound interest earned is given as Rs 10176. We can set up the equation for the principal P as follows:

25.44% of P = 10176

(25.44)/(100) × P = 10176

P = (10176 × 100)/(25.44)

P = (10176 × 10000)/(2544)

Dividing 10176 by 2544 yields exactly 4, as 2500 × 4 = 10000 and 44 × 4 = 176.

P = 4 × 10000 = 40000

Explanation

To find the value of x, we start with the given equation: (2(4x)/(5) - ¾){3} - ⁵⁄₃ = -⅙

First, we move the constant term -⁵⁄₃ to the right side of the equation: (2(4x)/(5) - ¾){3} = ⁵⁄₃ - ⅙

To subtract the fractions on the right, we find a common denominator, which is 6: ⁵⁄₃ - ⅙ = ¹⁰⁄₆ - ⅙ = ⁹⁄₆ = ³⁄₂

Now the equation is: (2(4x)/(5) - ¾){3} = ³⁄₂

Multiply both sides by 3 and divide by 2 to isolate the expression within the parentheses: ⁴ˣ⁄₅ - ¾ = ³⁄₂ · ³⁄₂ = ⁹⁄₄

Next, add ¾ to both sides: ⁴ˣ⁄₅ = ⁹⁄₄ + ¾ = ¹²⁄₄ = 3

Finally, solve for x by multiplying both sides by ⁵⁄₄: x = 3 · ⁵⁄₄ = ¹⁵⁄₄

Explanation

We use the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). Let s = a + b. Substituting the given values a³ + b³ = 35 and ab = 6 into the identity, we get 35 = s³ - 3(6)(s). This simplifies to the cubic equation s³ - 18s - 35 = 0. By testing small integer values or factors of 35, we find that if s = 5, then 5³ - 18(5) - 35 = 125 - 90 - 35 = 0.

Alternatively, we can find the values of a and b by inspection. Since ab = 6 and a³ + b³ = 35, we look for factors of 6 such as 2 and 3. Testing these values, we see that 2³ + 3³ = 8 + 27 = 35, which satisfies both given conditions. Therefore, the sum a + b = 2 + 3 = 5.

Explanation

The largest 4-digit number is 9999. To find the largest 4-digit number exactly divisible by 93, we divide 9999 by 93 to identify the remainder.

9999 ÷ 93 = 107 with a remainder of 48. This can be expressed using the division algorithm as 9999 = (93 × 107) + 48.

By subtracting the remainder from the dividend, we obtain the largest multiple of 93 that does not exceed 9999: 9999 - 48 = 9951

Alternatively, multiplying the largest possible integer quotient 107 by the divisor 93 yields the same result: 93 × 107 = 9951

Explanation

To find the time taken to complete the job, let the total work be the Least Common Multiple (LCM) of the individual days taken by A, B, and C. The LCM of 20, 30, and 60 is 60 units.

The efficiency (work done per day) of each person is: Efficiency of A = 60 / 20 = 3 units/day Efficiency of B = 60 / 30 = 2 units/day Efficiency of C = 60 / 60 = 1 unit/day

When A, B, and C work together for 1 day, the total work completed is the sum of their efficiencies multiplied by the time: Work done in 1 day = (3 + 2 + 1) × 1 = 6 units

The remaining work to be completed by C alone is: Remaining work = Total work - Work completed = 60 - 6 = 54 units

The number of days C takes to finish the remaining part is: Time = Remaining work / Efficiency of C = 54 / 1 = 54 days

Explanation

The perimeter P of a rectangle is given by the formula P = 2(l + b), where l represents the length and b represents the breadth. Given the perimeter is 90 cm and the breadth is 20 cm, the equation is: 90 = 2(l + 20) 45 = l + 20 l = 25 cm The area of a rectangle is calculated using the formula Area = l × b. Substituting the values for length and breadth: Area = 25 × 20 = 500 sq cm

Explanation

The marked price is the initial price of an item before any deductions. A discount of 25 percent implies that the item is sold at 75 percent of its marked price, since 100% - 25% = 75%.

The formula relating the selling price SP, marked price MP, and discount percentage d is: SP = MP × (1 - ᵈ⁄₁₀₀)

By rearranging the formula to solve for the marked price, we get: MP = (SP)/(1 - d){100}

Substituting the given values SP = 9750 and d = 25: MP = (9750)/(1 - 0.25) MP = (9750)/(0.75)

To simplify the division, we can express 0.75 as the fraction 3/4: MP = 9750 × ⁴⁄₃ MP = 3250 × 4 MP = 13000

Explanation

To find the ratio A : B : C from the equation 3A = 2B = 4C, we equate all terms to the Least Common Multiple (LCM) of the coefficients 3, 2, and 4. The LCM of these values is 12. By setting 3A = 12, we obtain A = 12/3 = 4. Setting 2B = 12 yields B = 12/2 = 6, and setting 4C = 12 results in C = 12/4 = 3. Alternatively, the ratio is proportional to the reciprocals of the coefficients, which is ⅓ : ½ : ¼. Multiplying each part of this ratio by the common denominator 12 converts the fractions into the integer ratio 4 : 6 : 3.

Explanation

The total cost of the first 5 items is found by multiplying the average by the number of items: 5 × 650 = 3250. When a 6th item costing 1400 is added, the total cost for the 6 items becomes 3250 + 1400 = 4650. The new average is calculated by dividing this total sum by the new number of items: 4650 / 6 = 775.

Alternatively, using the deviation method, the 6th item exceeds the current average by 1400 - 650 = 750. This surplus is shared equally among all 6 items, increasing the average by 750 / 6 = 125. Adding this to the original average gives 650 + 125 = 775.

Explanation

The cost price (CP) of the apples is calculated using the selling price (SP) and the loss percentage. When the vendor sells at Rs 170 with a 15% loss, the relationship is CP = (SP × 100)/(100 - Loss%). Substituting the values, we get CP = (170 × 100)/(100 - 15) = (170 × 100)/(85) = Rs 200.

To find the profit percentage when selling at Rs 230, we first determine the profit amount using Profit = New SP - CP = 230 - 200 = Rs 30. The profit percentage is then calculated as Profit% = (Profit)/(CP) × 100. Substituting the calculated values gives ³⁰⁄₂₀₀ × 100 = 15%.

Explanation

Let the fraction be x. According to the problem, the sum of the fraction and thrice its reciprocal is x + 3/x = 73/20. By substituting x = 5/4, the reciprocal becomes 4/5. The calculation is as follows:

5/4 + 3(4/5) = 5/4 + 12/5

To add these terms, we use the common denominator of 20:

(25 + 48) / 20 = 73/20

The value satisfies the given condition perfectly.

Explanation

The value of x can be determined using the percentage formula: Value = (Rate)/(100) × x. Given that 40 is 0.05% of x, the equation is written as 40 = (0.05)/(100) × x. Rearranging to solve for x gives x = (40 × 100)/(0.05). This simplifies to x = (4000)/(0.05). To eliminate the decimal in the denominator, multiply both the numerator and the denominator by 100, resulting in x = ⁴⁰⁰⁰⁰⁰⁄₅. Dividing 400000 by 5 yields x = 80000.

Explanation

Since the distance traveled is the same in both cases, speed is inversely proportional to time (S ∝ 1/T). This relationship is expressed by the formula S₁ × T₁ = S₂ × T₂, where S is speed and T is time.

Substituting the given values: 90 × 35 = S₂ × 21

Solving for the new speed S₂: S₂ = (90 × 35)/(21) S₂ = (90 × 5)/(3) S₂ = 30 × 5 = 150 km/hr

The required increase in speed is the difference between the new speed and the original speed: Increase in speed = S₂ - S₁ Increase in speed = 150 - 90 = 60 km/hr

Explanation

In compound interest, the interest earned in any specific year grows at the given rate r relative to the interest earned in the preceding year. The interest for the nᵗʰ year (Iₙ) can be calculated from the interest of the mᵗʰ year (Iₘ) using the formula Iₙ = Iₘ (1 + ʳ⁄₁₀₀)ⁿ⁻ᵐ.

Given that the interest for the 2ⁿᵈ year (I₂) is Rs 1200 and the rate (r) is 10%, we calculate the interest for the 4ᵗʰ year (I₄) by substituting the values into the formula:

I₄ = 1200 × (1 + ¹⁰⁄₁₀₀)⁴⁻² I₄ = 1200 × (1.1)² I₄ = 1200 × 1.21 I₄ = 1452

Explanation

To find the value of x, simplify the equation 17/3 + [3(2x - 5/3)]/2 = 1/6 by multiplying all terms by 6 to clear the fractions: 2(17) + 3[3(2x - 5/3)] = 1 34 + 9(2x - 5/3) = 1 Distributing the 9 through the parentheses results in: 34 + 18x - 15 = 1 Combining the constant terms 34 - 15 gives: 19 + 18x = 1 Subtract 19 from both sides: 18x = -18 Divide by 18: x = -1

Explanation

The value of a³ + b³ is determined using the algebraic identity a³ + b³ = (a + b)³ - 3ab(a + b). By substituting the given values a + b = 5 and ab = 6 into this formula, the expression becomes 5³ - 3(6)(5). Performing the arithmetic gives 125 - 90 = 35.

Alternatively, the integers a and b that satisfy the equations a + b = 5 and ab = 6 can be identified as 2 and 3. Calculating the sum of their cubes yields 2³ + 3³ = 8 + 27 = 35.

Explanation

Let the fraction be x. According to the problem, the sum of the fraction and three times its reciprocal is 37/10. This relation can be expressed as:

x + ³⁄ₓ = ³⁷⁄₁₀

To solve for x, multiply the entire equation by 10x to eliminate the denominators:

10x² + 30 = 37x

Rearranging this into a standard quadratic equation ax² + bx + c = 0:

10x² - 37x + 30 = 0

Using the quadratic formula x = -b ± √(b² - 4ac){2a}:

x = 37 ± √((-37)² - 4 × 10 × 30){2 × 10}

x = 37 ± √(1369 - 1200){20}

x = 37 ± √(169){20} = (37 ± 13)/(20)

This results in two possible values for x:

x₁ = (37 + 13)/(20) = ⁵⁰⁄₂₀ = ⁵⁄₂

x₂ = (37 - 13)/(20) = ²⁴⁄₂₀ = ⁶⁄₅

The value 5/2 fulfills the given condition. Substituting 5/2 back into the original expression:

⁵⁄₂ + 3 × ⅖ = ⁵⁄₂ + ⁶⁄₅ = (25 + 12)/(10) = ³⁷⁄₁₀

Explanation

The n-th term of an arithmetic progression is determined by the formula aₙ = a + (n - 1)d, where a is the first term and d is the common difference. Based on the values provided, we can establish the following system of linear equations: a + 3d = 11 and a + 6d = -4. Subtracting the first equation from the second gives (a + 6d) - (a + 3d) = -4 - 11, which simplifies to 3d = -15, resulting in d = -5. Substituting the value of d back into the first equation gives a + 3(-5) = 11, so a - 15 = 11 and a = 26. To find the 15th term, we apply the formula a₁₅ = a + 14d = 26 + 14(-5) = 26 - 70 = -44.

Explanation

When a point (x, y) is reflected across a horizontal line y = k, the x-coordinate of the image remains identical to the x-coordinate of the original point, while the y-coordinate changes such that the line y = k is the midpoint between the original y and the reflected y'. The formula for the reflected y-coordinate is y' = 2k - y.

In this problem, the point is (5, -1) and the line of reflection is y = 2. Substituting the values x = 5, y = -1, and k = 2 into the reflection transformation:

x' = 5 y' = 2(2) - (-1) = 4 + 1 = 5

The resulting coordinates for the reflected point are (5, 5).

Explanation

The Y-axis is defined by the equation x = 0. When a line segment joining two points (x₁, y₁) and (x₂, y₂) is divided by the Y-axis, the x-coordinate of the point of intersection must be zero. According to the section formula, the x-coordinate of a point dividing the segment joining (x₁, y₁) and (x₂, y₂) in the ratio m : n is given by x = (m x₂ + n x₁)/(m + n).

Given the points (2, 3) and (-2, 1), we substitute x₁ = 2, x₂ = -2, and x = 0 into the formula. This gives 0 = (m(-2) + n(2))/(m + n). Multiplying both sides by m + n results in 0 = -2m + 2n, which simplifies to 2m = 2n. Dividing both sides by 2n yields ᵐ⁄ₙ = ²⁄₂ = ¹⁄₁. Thus, the ratio m : n is 1 : 1.

A faster method to find the ratio in which the Y-axis divides the segment joining (x₁, y₁) and (x₂, y₂) is using the ratio formula -x₁ : x₂. Substituting the given x-coordinates, we get -2 : -2, which simplifies directly to 1 : 1.

Explanation

To find the slope of the line given by the equation 2x - 5y = 12, we can convert it into the slope-intercept form, which is expressed as y = mx + c. In this form, m represents the slope of the line and c represents the y-intercept.

Starting with the given equation: 2x - 5y = 12

Isolate the term containing y by subtracting 2x from both sides of the equation: -5y = -2x + 12

To solve for y, divide every term in the equation by -5: y = ⁻²⁄₋₅x + ¹²⁄₋₅ y = ⅖x - ¹²⁄₅

By comparing this result to the slope-intercept form y = mx + c, we can see that the coefficient of x is ⅖. Therefore, the slope m is ⅖.

Alternatively, for any linear equation in the standard form Ax + By = C, the slope can be calculated using the direct formula m = -(A)/(B). Substituting the coefficients from the given equation where A = 2 and B = -5: m = -²⁄₋₅ = ⅖

Explanation

In an arithmetic progression, the sum of n terms is given by Sₙ = ⁿ⁄₂ [2a + (n-1)d], where a is the first term and d is the common difference. For an odd number of terms, the sum can also be calculated as Sₙ = n × aₘᵢ_d, where aₘᵢ_d is the middle term of the series.

The common difference d is calculated using the given 4th and 7th terms: d = (a₇ - a₄)/(7 - 4) = (-4 - 11)/(3) = -5

For the first 11 terms, the middle term is the 6th term (a₆). Using the 4th term: a₆ = a₄ + 2d a₆ = 11 + 2(-5) = 1

Applying the sum formula using the middle term: S₁₁ = 11 × a₆ = 11 × 1 = 11

Explanation

According to the theorem on the areas of similar triangles, the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. For Δ XYZ ∼ Δ PQR, this relationship is expressed as:

(Area(Δ XYZ))/(Area(Δ PQR)) = ((XY)/(PQ))²

Given that the ratio of the corresponding sides XY:PQ is 5:1 and the area of Δ PQR is 5 sq cm, we substitute these values into the formula:

(Area(Δ XYZ))/(5) = (⁵⁄₁)²

(Area(Δ XYZ))/(5) = 25

Area(Δ XYZ) = 5 × 25 = 125 sq cm

Explanation

The trigonometric values for the specified angles are cot 60° = (1)/(√(3)) and sec 30° = (2)/(√(3)). Substituting these values into the given expression yields:

cot 60° - sec 30° = (1)/(√(3)) - (2)/(√(3))

Since the denominators are identical, the numerators are subtracted directly:

(1 - 2)/(√(3)) = -(1)/(√(3))

Explanation

In the right-angled triangle ABC, where the right angle is at B, the side AB is the base (adjacent side) relative to angle A, and AC is the hypotenuse. The relationship between the base and the hypotenuse is given by the cosine trigonometric ratio:

cos(A) = (AB)/(AC)

Substituting the given values m∠ A = 30° and AC = 8 cm into the formula:

cos(30°) = (AB)/(8)

Since the value of cos(30°) is √3{2}, the equation becomes:

√3{2} = (AB)/(8)

Solving for AB:

AB = 8 · √3{2}

AB = 4√3 cm

Explanation

In a right-angled triangle, cot θ is defined as the ratio of the base to the perpendicular. Given cot θ = ²⁴⁄₇, we can let the base b = 24 and the perpendicular p = 7.

According to the Pythagorean theorem, the hypotenuse h is calculated as h = √(p² + b²). Substituting the given values, h = √(7² + 24²) = √(49 + 576) = √(625) = 25. This corresponds to the well-known Pythagorean triple (7, 24, 25).

The secant function sec θ is defined as the ratio of the hypotenuse to the base. Thus, sec θ = (h)/(b) = ²⁵⁄₂₄.

Explanation

When a point (x, y) is reflected in a vertical line x = k, the y-coordinate of the image remains the same as that of the original point, while the x-coordinate changes. The line x = k acts as the perpendicular bisector of the line segment joining the point and its image. The x-coordinate of the image x' is determined by the formula x' = 2k - x.

For the point (-1, 3) and the reflection line x = -4, we substitute x = -1 and k = -4 into the formula. This gives x' = 2(-4) - (-1) = -8 + 1 = -7. Since the y-coordinate remains constant at 3, the coordinates of the reflected point are (-7, 3).

Explanation

The coordinates of a triangle vertex (x₃, y₃) are derived from the centroid formula as x₃ = 3x_G - x₁ - x₂ and y₃ = 3y_G - y₁ - y₂, where (x_G, y_G) represents the centroid and (x₁, y₁), (x₂, y₂) are the other two vertices. Using the values provided for the centroid (1, -4) and vertices A(3, -4) and B(0, 5):

x₃ = 3(1) - 3 - 0 = 0 y₃ = 3(-4) - (-4) - 5 = -12 + 4 - 5 = -13

Explanation

For a linear equation in the general form Ax + By = C, the slope m is given by the formula m = -A/B. In the given equation ax + 5y = 8, the coefficients are A = a and B = 5. Therefore, the slope of the line is m = -a/5.

The problem states that the slope of the line is -4/3. By equating these two expressions for the slope, we get:

-a/5 = -4/3

Multiplying both sides by -1 gives:

a/5 = 4/3

Multiplying both sides by 5 to solve for a:

a = (4 × 5) / 3 = 20/3

Explanation

In Δ ABC, since DE ∥ BC, Δ ADE is similar to Δ ABC by the AA similarity criterion. The ratio of the side lengths of these triangles is determined by AD and AB. Given AD:DB = 2:5, the length AB can be expressed as AD + DB = 2 + 5 = 7 units. Thus, the ratio of the sides AD/AB is 2/7.

The ratio of the areas of similar triangles is equal to the square of the ratio of their corresponding sides. This relationship is expressed as Area(Δ ADE) / Area(Δ ABC) = (AD/AB)². Substituting the known values, we get Area(Δ ADE) / 98 = (2/7)² = 4/49. Solving for the area of the smaller triangle, Area(Δ ADE) = (4/49) × 98 = 4 × 2 = 8 sq cm.

The area of the quadrilateral BDEC is found by subtracting the area of the smaller triangle Δ ADE from the total area of Δ ABC. Therefore, Area(BDEC) = Area(Δ ABC) - Area(Δ ADE) = 98 - 8 = 90 sq cm.

Explanation

The values of the trigonometric functions are cot 45° = 1 and csc 60° = (1)/(sin 60°) = (2)/(√(3)). Substituting these values into the given expression:

cot 45° - (1)/(√(3)) csc 60° = 1 - (1)/(√(3)) ( (2)/(√(3)) )

Multiplying the terms in the second part of the expression:

1 - ⅔

To subtract, use a common denominator:

³⁄₃ - ⅔ = ⅓

Explanation

In the right-angled triangle Δ DEF, the value of the angle F is given as 45°. The expression to be evaluated is 2 sin F × cot F. By applying the trigonometric identity cot F = (cos F)/(sin F), the expression can be simplified directly:

2 sin F × (cos F)/(sin F) = 2 cos F

Substitute the standard trigonometric value cos 45° = (1)/(√(2)) into the simplified expression:

2 × cos 45° = 2 × (1)/(√(2)) = √2