SSC CGL Tier 1 2020 Quantitative Aptitude Questions with Solutions

Explanation

Let the fifth number be x. The question states that the average of the first four numbers is three times the fifth number, so the average of the first four numbers is 3x. The sum of these first four numbers is calculated by multiplying their average by the count, which gives 4 × 3x = 12x.

The sum of all five numbers is the sum of the first four numbers plus the fifth number, which is 12x + x = 13x. The average of all five numbers is given as 85.8. We can set up the equation using the relationship:

Average of five numbers = (Sum of five numbers)/(5)

85.8 = ¹³ˣ⁄₅

Multiplying both sides by 5 gives:

13x = 429

Dividing by 13 to solve for x:

x = ⁴²⁹⁄₁₃ = 33

Explanation

The expression is solved using the BODMAS rule, which dictates the order of operations as Brackets, Orders, Division, Multiplication, Addition, and Subtraction.

Following this order, first perform the division operation: 11 ÷ 11 = 1. The expression is now 11 + 11 × 11 - 1.

Next, perform the multiplication operation: 11 × 11 = 121. The expression becomes 11 + 121 - 1.

Then, perform the addition operation: 11 + 121 = 132. The expression is simplified to 132 - 1.

Finally, perform the subtraction operation: 132 - 1 = 131.

Explanation

The expression can be simplified using trigonometric complementary angle identities, where sin(90° - θ) = cos θ and tan(90° - θ) = cot θ.

First, simplify the ratio of sine and cosine terms: (sin 22°)/(cos 68°) = (sin 22°)/(sin(90° - 68°)) = (sin 22°)/(sin 22°) = 1

Next, simplify the product of tangent terms. Note that tan θ · tan(90° - θ) = tan θ · cot θ = 1: tan 20° · tan 40° · tan 45° · tan 50° · tan 70° = (tan 20° · tan 70°) · (tan 40° · tan 50°) · tan 45° = (tan 20° · cot 20°) · (tan 40° · cot 40°) · (1) = 1 · 1 · 1 = 1

Then, simplify the ratio involving tan 15° and cot 75°: (tan 15°)/(cot 75°) = (tan 15°)/(tan(90° - 75°)) = (tan 15°)/(tan 15°) = 1

Substituting these simplified values into the expression using the given coefficients (2, 8, 5, 5): 2 ((sin 22°)/(cos 68°)) + 8(tan 20° … tan 70°) - 5 ((tan 15°)/(cot 75°)) - 5 = 2(1) + 8(1) - 5(1) - 5 = 2 + 8 - 5 - 5 = 10 - 10 = 0

Explanation

The given expression can be rearranged as ((csc A + 1) + cot A)((csc A + 1) - cot A) - 2csc A. By applying the algebraic identity (x + y)(x - y) = x² - y², where x = csc A + 1 and y = cot A, the expression simplifies to (csc A + 1)² - cot² A - 2csc A. Expanding the squared term results in csc² A + 2csc A + 1 - cot² A - 2csc A. Using the fundamental trigonometric identity csc² A - cot² A = 1, the expression reduces to 1 + 1 + 2csc A - 2csc A, which equals 2. This result can also be quickly verified by substituting a specific angle such as A = 90°, where csc 90° = 1 and cot 90° = 0, leading to (1 + 0 + 1)(1 - 0 + 1) - 2(1) = 4 - 2 = 2.

Explanation

According to the property of remainders in modular arithmetic, the remainder of a product is the product of the remainders of the individual terms. The question states that when the integer n is divided by 5, the remainder is 2. This can be written as n ≡ 2 5.

To find the remainder of 7n when divided by 5, we multiply the remainder of 7 by the remainder of n. Dividing 7 by 5 gives a remainder of 2, because 7 = 5 × 1 + 2. Applying the product rule for remainders, the remainder for 7n is 2 × 2 = 4. Since 4 is less than the divisor 5, the final remainder is 4. Alternatively, substituting n = 5k + 2 into the expression 7n yields 7(5k + 2) = 35k + 14. When 35k + 14 is divided by 5, 35k leaves no remainder, and 14 leaves a remainder of 4.

Explanation

Let cost price of first book = Rs.x. Cost price of second book = 300 - x. For no profit no loss: x*(120/100) + (300-x)*(90/100) = 300. Solving: 3x = 300, x = 100. Selling price = 100 * 120/100 = Rs.120

Explanation

Let the sum of the n numbers be S. Based on the given information, we can set up two linear equations: S - 2n = 102 S - 5n = 12

By subtracting the second equation from the first, we eliminate S to find the value of n: (S - 2n) - (S - 5n) = 102 - 12 3n = 90 n = 30

Next, we substitute n = 30 into the first equation to determine the total sum S: S - 2(30) = 102 S - 60 = 102 S = 162

The average of the n numbers is the total sum divided by the number of elements: Average = (S)/(n) = ¹⁶²⁄₃₀ = 5.4

Explanation

The total sum of the four terms is determined by multiplying the average by the number of terms: 30 × 4 = 120. Let the first term be represented by x and the sum of the remaining three terms be S. Based on the given condition, the first term is one-third of the sum of the remaining terms, expressed as x = ⅓ S, which implies S = 3x. The total sum of all four terms is the sum of the first term and the remaining terms, so x + S = 120. By substituting 3x for S, the equation becomes x + 3x = 120, which simplifies to 4x = 120. Dividing both sides by 4 gives x = 30.

Explanation

Let t be the time in hours spent travelling on the bicycle. Since the total duration of the journey is 9 hours, the time spent travelling on foot is 9 - t hours. Using the relation Distance = Speed × Time, the total distance covered is the sum of the distances covered by both modes: 5(9 - t) + 10t = 50 45 - 5t + 10t = 50 5t = 50 - 45 5t = 5 t = 1 The distance travelled on the bicycle is obtained by multiplying the speed of the bicycle by this time duration: 10 km/h × 1 hour = 10 km

Alternatively, this can be solved using the weighted average method. The average speed for the entire journey is 50/9 km/h. The ratio of time spent on foot to time spent on the bicycle is given by the ratio of the differences between the speeds and the average speed: (10 - 50/9) : (50/9 - 5) (90/9 - 50/9) : (50/9 - 45/9) 40/9 : 5/9 = 8 : 1 The total time of 9 hours is divided into 9 parts (8 + 1). Thus, the time spent on the bicycle is 1/9 × 9 = 1 hour. The distance covered on the bicycle is 10 km/h × 1 hour = 10 km.

Explanation

For a tangential quadrilateral, sums of lengths of opposite sides are equal: AB + CD = BC + AD. Substituting 8 + 6 = 7 + AD, so AD = 14 - 7 = 7 cm.

Explanation

From the intersecting chords property: RQ * RP = SR * RT. 14.4 * (14.4 + 11.2) = 12.8 * RT. 14.4 * 25.6 = 12.8 * RT. RT = 28.8 cm. TS = RT - SR = 28.8 - 12.8 = 16 cm

Explanation

From the given relation a = 2b = 8c, we can express a and b in terms of c as a = 8c and b = 4c. Using the equation a + b + c = 13, we substitute these terms to find the value of c:

8c + 4c + c = 13 13c = 13 c = 1

Since c = 1, it follows that a = 8 and b = 4. Now, substitute these values into the expression √(a² + b² + c²){2c}:

√(8² + 4² + 1²){2(1)} = √(64 + 16 + 1){2}

√(81){2} = ⁹⁄₂

Alternatively, the expression can be simplified in terms of c directly without solving for the specific value of c, as long as c ≠ 0:

√((8c)² + (4c)² + c²){2c} = √(64c² + 16c² + c²){2c} = √(81c²){2c} = (9c)/(2c) = ⁹⁄₂

Explanation

The reduction in profit is equal to the difference in the discount amounts offered on the market value of the bat. Initially, the man planned a discount of 8%, but he increased it to 10% for his brother. This change represents an additional discount of 10% - 8% = 2% of the market value. Since the cost price remains constant, this increase in discount directly results in a corresponding reduction in profit.

Let M be the market value of the bat. The problem states that this 2% difference is equivalent to Rs. 70. Using the percentage formula: 2% of M = 70 ²⁄₁₀₀ × M = 70 M = (70 × 100)/(2) M = 35 × 100 M = 3500

By calculating the difference in discount rates and equating it to the loss in profit, the market value is determined to be Rs. 3,500.

Explanation

The trade balance for a country in a given year is defined by the formula: Trade Balance = Exports - Imports. To find the average trade balance over the four-year period, we first calculate the trade balance for each specific year:

For 2016: 130 - 125 = 5 crore For 2017: 150 - 145 = 5 crore For 2018: 175 - 165 = 10 crore For 2019: 200 - 188 = 12 crore

Next, we calculate the sum of these trade balances: 5 + 5 + 10 + 12 = 32 crore. The average trade balance is determined by dividing this total by the number of years (4):

Average Trade Balance = (Σ Trade Balances)/(n) = ³²⁄₄ = 8 crore.

Explanation

Volume of cuboid = Base area x height. 330 = 15 x height. Height = 330/15 = 22 cm

Explanation

The volume of a sphere is given by the formula V = ⁴⁄₃ π r³. Since the volume is proportional to the cube of the radius, the ratio of the volumes of spheres A and B is (V_A)/(V_B) = (r_A³)/(r_B³) = ¹²⁵⁄₆₄. Taking the cube root of this ratio, we find the ratio of their radii (r_A)/(r_B) = ³√(125){³√(64)} = ⁵⁄₄.

The sum of the radii is given as r_A + r_B = 36 cm. Using the ratio 5:4, we can determine the radius of sphere A as r_A = (5)/(5 + 4) × 36 = ⁵⁄₉ × 36 = 20 cm.

The surface area of a sphere is calculated using the formula A = 4 π r². For sphere A, substituting the radius r_A = 20 cm gives A = 4 π (20)² = 4 π × 400 = 1600π cm².

Explanation

To simplify the expression (a + b - c)³ + (a - b + c)³ - 8a³, we can use the algebraic identity: If x + y + z = 0, then x³ + y³ + z³ = 3xyz.

Let the three terms be: x = a + b - c y = a - b + c z = -2a (Note that -8a³ = (-2a)³)

First, check the sum of x, y, and z: x + y + z = (a + b - c) + (a - b + c) + (-2a) x + y + z = (a + a - 2a) + (b - b) + (-c + c) x + y + z = 2a - 2a + 0 + 0 = 0

Since the sum is zero, we apply the identity: (a + b - c)³ + (a - b + c)³ + (-2a)³ = 3(a + b - c)(a - b + c)(-2a) = -6a(a + b - c)(a - b + c)

To match the structure of the provided options, we can distribute the negative sign into one of the factors: -6a(a + b - c)(a - b + c) = 6a[-(a + b - c)](a - b + c) = 6a(c - a - b)(a - b + c)

This expression matches the factors and the coefficient in option a).

Explanation

Total expenditure in 2014 is the sum of all expenditure items for that year: Total Expenditure (2014) = Salary + Fuel and Transport + Bonus + Interest on Loans Total Expenditure (2014) = 284 + 98 + 3.00 + 23.4 = 408.4

Total expenditure in 2018 is the sum of all expenditure items for that year: Total Expenditure (2018) = Salary + Fuel and Transport + Bonus + Interest on Loans Total Expenditure (2018) = 425 + 141 + 3.96 + 49.4 = 619.36

To find what percent the total expenditure in 2014 was of the total expenditure in 2018, we use the formula: Percentage = (Total Expenditure 2014)/(Total Expenditure 2018) × 100

Substituting the calculated totals: Percentage = (408.4)/(619.36) × 100 ≈ 66.32%

Explanation

To determine the company with a continuous decrease in production, the monthly data for Company I is analyzed as follows: production in January was 327 lakh tonne, which decreased to 326 lakh tonne in February. This downward trend continued in March with a production of 320 lakh tonne, followed by 318 lakh tonne in April, and finally reaching 310 lakh tonne in May. By comparing the values chronologically, we observe that 327 > 326 > 320 > 318 > 310. Because each successive month shows a lower production value than the preceding month without any instance of stagnation or increase, Company I is the only company that satisfies the condition of continuous decrease throughout the entire five-month period.

Explanation

The distance covered by a wheel in one complete revolution is equivalent to its circumference. The formula for the circumference C of a circle in terms of its diameter d is C = π d.

Substituting the given distance of 1,100 cm for the circumference and using the standard approximation π = ²²⁄₇:

1,100 = ²²⁄₇ × d

To find the diameter d, rearrange the equation:

d = (1,100 × 7)/(22)

By dividing 1,100 by 22, we get 50:

d = 50 × 7 d = 350 cm

Explanation

Let the total work be the Least Common Multiple (LCM) of 24, 16, and 32. Total Work = LCM(24, 16, 32) = 96 units

The efficiency (work done per day) of each individual is: Efficiency of A = ⁹⁶⁄₂₄ = 4 units/day Efficiency of B = ⁹⁶⁄₁₆ = 6 units/day Efficiency of C = ⁹⁶⁄₃₂ = 3 units/day

A and C work together for 6 days. Their combined efficiency is 4 + 3 = 7 units/day. Work done by A and C in 6 days = 7 × 6 = 42 units

The remaining work to be completed by B is: Remaining Work = 96 - 42 = 54 units

The number of days required by B to complete this remaining task is: Time = (Remaining Work)/(Efficiency of B) = ⁵⁴⁄₆ = 9 days

Explanation

The total distance to be covered in both cases is the sum of the lengths of the two trains, L_X + L_Y. Using the relative speed formula, distance D = Relative Speed × Time. When trains move in the same direction, relative speed is V_X - V_Y. When they move in opposite directions, relative speed is V_X + V_Y.

Given V_X = 60 km/h and L_Y = 225 m, we can set up the equality because the distance L_X + L_Y is the same: (V_X - V_Y) × 72 = (V_X + V_Y) × 18 4(60 - V_Y) = 60 + V_Y 240 - 4V_Y = 60 + V_Y 180 = 5V_Y V_Y = 36 km/h

To find L_X, substitute V_Y into the opposite direction equation, converting speeds to m/s by multiplying by ⁵⁄₁₈: L_X + 225 = (60 + 36) × ⁵⁄₁₈ × 18 L_X + 225 = 96 × 5 L_X + 225 = 480 L_X = 255 m

Explanation

Using the algebraic identity x⁴ + x² y² + y⁴ = (x² + xy + y²)(x² - xy + y²), we can find the value of the second factor by substituting the given values. Given x⁴ + x² y² + y⁴ = 21 and x² + xy + y² = 7, we have 21 = 7 × (x² - xy + y²), which simplifies to x² - xy + y² = 3.

To find the components for the required expression, we perform operations on the two equations: x² + xy + y² = 7 x² - xy + y² = 3

Adding the two equations yields 2(x² + y²) = 10, so x² + y² = 5. Subtracting the second equation from the first yields 2xy = 4, so xy = 2, which implies x² y² = 4.

The expression to be evaluated is (1)/(x²) + (1)/(y²), which can be rewritten as (x² + y²)/(x² y²). Substituting the values x² + y² = 5 and x² y² = 4 into this formula gives ⁵⁄₄.

Explanation

The trigonometric identities for complementary angles state that csc A = sec(90° - A) and tan A = cot(90° - A). In the given expression, the term csc(85° + θ) and the term sec(5° - θ) are complementary because their angles sum to 90°: (85° + θ) + (5° - θ) = 90°. This implies csc(85° + θ) = sec(5° - θ), so the subtraction of these two terms results in zero.

Similarly, the term tan(55° + θ) and the term cot(35° - θ) are complementary because (55° + θ) + (35° - θ) = 90°. This implies tan(55° + θ) = cot(35° - θ). In the expression, these terms are given as -tan(55° + θ) + cot(35° - θ), which also results in zero. Combining these two parts, the total value is 0 + 0 = 0.

Explanation

When 200 is divided by a positive integer x with a remainder of 8, the relationship can be expressed as 200 = qx + 8, where q is an integer and x > 8. This simplifies to qx = 192, meaning x must be a factor of 192 that is strictly greater than the remainder 8.

To find the number of factors of 192, we first determine its prime factorization: 192 = 2⁶ × 3¹. The total number of factors is calculated using the formula (a+1)(b+1), where a and b are the exponents of the prime factors. Thus, total factors = (6 + 1) × (1 + 1) = 7 × 2 = 14.

The factors of 192 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, and 192. Since the divisor x must be greater than the remainder 8, we exclude the factors 1, 2, 3, 4, 6, and 8. There are 6 such factors to exclude. Subtracting these from the total, we get 14 - 6 = 8 valid values for x.

Explanation

The given equation x² - 3x + 1 = 0 can be rewritten by dividing all terms by x, which gives x - 3 + ¹⁄ₓ = 0, or x + ¹⁄ₓ = 3. To find the value of x⁶ + (1)/(x⁶⁾, first determine x² + (1)/(x²⁾ using the relation x² + (1)/(x²⁾ = (x + ¹⁄ₓ)² - 2. Substituting the value results in 3² - 2 = 7.

Next, use the cubic identity a³ + b³ = (a + b)³ - 3ab(a + b) by setting a = x² and b = (1)/(x²⁾. This yields the expression x⁶ + (1)/(x⁶⁾ = (x² + (1)/(x²⁾)³ - 3(x² · (1)/(x²⁾)(x² + (1)/(x²⁾). Substituting the previously calculated value of 7 into this formula gives 7³ - 3(7) = 343 - 21 = 322.

Explanation

Let the initial cost of production be 100, making the initial selling price 125. After the changes, the new cost of production is 120 and the new selling price is 137.5 (125 increased by 10%). The new profit is 17.5, and the profit percentage is (17.5 / 120) * 100, which is approximately 14.6%.

Explanation

The relationship between the speeds of two objects starting from opposite ends and the time taken to reach their destinations after meeting is given by the ratio Sa/Sb = √(Tb/Ta). Given that Sa is 20% more than Sb, Sa/Sb = 1.2/1 = 6/5. Substituting the values into the formula: 6/5 = √(x / 2.5). Squaring both sides gives 36/25 = x / 2.5, which simplifies to x = (36 · 2.5) / 25 = 3.6, or 3 3/5.

Explanation

Successive discounts of 20% and 25% result in a single equivalent discount calculated as d = 20 + 25 - (20 × 25)/(100) = 45 - 5 = 40%. Let the cost price of the article be 100x. The dealer marks the article 40% above the cost price, making the marked price MP = 100x × 1.40 = 140x. The selling price is then determined by applying the 40% discount to the marked price: SP = 140x × (1 - 0.40) = 140x × 0.60 = 84x.

The loss incurred is the difference between the cost price and the selling price, which is 100x - 84x = 16x. The problem states that the dealer suffers a loss of ₹ 140, so 16x = 140. Solving for x gives x = ¹⁴⁰⁄₁₆ = 8.75. The cost price of the article is 100x = 100 × 8.75 = 875.

Explanation

The expression 515 × 485 can be solved efficiently using the algebraic identity for the difference of squares, (a + b)(a - b) = a² - b². By identifying the midpoint between the two numbers, we can rewrite them as follows: 515 = 500 + 15 485 = 500 - 15 Substituting these values into the identity where a = 500 and b = 15: 515 × 485 = (500 + 15)(500 - 15) = 500² - 15² Calculating the squares: 500² = 250,000 15² = 225 Performing the final subtraction: 250,000 - 225 = 249,775

Explanation

To find the required ratio, first identify the number of girls enrolled in the Arts stream and then calculate the total number of girls in the remaining streams. According to the table, the number of girls in the Arts stream is 45.

The number of girls in the other streams is the sum of girls in Science, Commerce, and Vocational streams. Total girls in other streams = 18 (Science) + 42 (Commerce) + 30 (Vocational) = 90.

The ratio of the number of girls in Arts to the number of girls in all other streams is calculated as follows: Ratio = 45 : 90

Dividing both sides by the greatest common divisor, which is 45, we get: Ratio = 1 : 2

Explanation

The total cost price (TCP) of the car is the sum of the purchase price and the maintenance expenses. Given a selling price (SP) of Rs. 1,56,250 and a profit of 25%, the total cost price can be calculated using the formula:

TCP = (SP)/(1 + Profit %){100}

Substituting the given values:

TCP = (1,56,250)/(1 + 25){100} = (1,56,250)/(1.25)

TCP = 1,25,000

The total cost price is Rs. 1,25,000. Since the dealer purchased the car for Rs. 1,08,500, the amount spent on maintenance is the difference between the total cost price and the purchase price:

Maintenance = TCP - Purchase Price

Maintenance = 1,25,000 - 1,08,500 = 16,500

Therefore, the dealer spent Rs. 16,500 on the maintenance of the car.

Explanation

To identify which vowels occur less than 80 times, we examine the frequency table provided for each vowel and compare the values to the threshold of 80. The vowel a has a frequency of 75, which satisfies the condition 75 < 80. The vowel o has a frequency of 70, which also satisfies the condition 70 < 80. The vowel u has a frequency of 75, which satisfies the condition 75 < 80. In contrast, the vowel i has a frequency of exactly 80, which is equal to but not less than 80, and the vowel e has a frequency of 125, which is greater than 80. Based on these comparisons, the vowels that occur with a frequency lower than 80 are a, o, and u.

Explanation

The total work is measured in man-days. Initially, 8 persons can finish the work in 20 days, so the total work is 8 × 20 = 160 man-days. After 5 days, the work completed is 8 × 5 = 40 man-days. The remaining work is 160 - 40 = 120 man-days.

To complete this remaining 120 man-days of work in the next 8 days, we use the formula Work = Persons × Days. Let the total number of persons required be M. 120 = M × 8 M = ¹²⁰⁄₈ = 15

Since there are already 8 persons in the group, the number of additional persons required is 15 - 8 = 7. Alternatively, using the chain rule for the remaining work: M₁ D₁ = M₂ D₂, we have 8 × 15 = (8 + x) × 8, which simplifies to 15 = 8 + x, giving x = 7.

Explanation

To find the percentage of income derived from overtime for employee D, the specific values for overtime and total income must be extracted from the table. For employee D, the income from overtime is 2,500 and the total monthly income is 40,000.

The required percentage is calculated by dividing the overtime income by the total income and multiplying by 100:

Percentage = ²⁵⁰⁰⁄₄₀₀₀₀ × 100

Simplifying the fraction by canceling the zeros:

Percentage = ²⁵⁰⁰⁄₄₀₀

Percentage = ²⁵⁄₄

Percentage = 6.25%

Explanation

To find the actual average, we calculate the net change in the total sum of scores caused by the error. The score was recorded as 88 instead of the correct value of 58, which means the initial sum was higher than it should have been.

The difference between the correct score and the incorrect score is 58 - 88 = -30. This indicates that the total sum must be decreased by 30 to reflect the actual scores. To find the effect on the average, we divide this total change by the number of students in the class:

Change in average = (Total change)/(Number of students) = ⁻³⁰⁄₃₀ = -1

By subtracting this value from the original average, we obtain the actual average: 69 - 1 = 68.

Explanation

In a circle with centre O, the radius OA is perpendicular to the tangent PAQ at the point of contact A, which means ∠ OAQ = 90°. In triangle OAB, the sides OA and OB are radii, making it an isosceles triangle where ∠ OAB = ∠ OBA. Given ∠ OBA = 32°, we have ∠ OAB = 32°. The angle x is defined as ∠ BAQ, which can be calculated as ∠ OAQ - ∠ OAB = 90° - 32° = 58°.

By the Alternate Segment Theorem, the angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment. Here, the angle between tangent AQ and chord AB is ∠ BAQ = x, and the angle in the alternate segment is ∠ ACB = y. Therefore, y = x = 58°. The required sum is x + y = 58 + 58 = 116.

Explanation

According to the divisibility rule for 9, a number is divisible by 9 if the sum of its digits is divisible by 9. For the number 1190*6, let the missing digit be x. The sum of the digits is calculated as 1 + 1 + 9 + 0 + x + 6 = 17 + x. To make the number divisible by 9, the sum 17 + x must be a multiple of 9. The smallest multiple of 9 that is greater than or equal to 17 is 18. By setting 17 + x = 18, we find that x = 1. Therefore, replacing the asterisk with 1 results in a sum of 18, which fulfills the condition for divisibility.

Explanation

Following the BODMAS rule, the operations within the parentheses are performed first. Inside the bracket (26 - 13 × 2), multiplication takes precedence over subtraction, so 13 × 2 = 26. The expression inside the bracket then becomes 26 - 26 = 0. Substituting this value back into the original equation results in 0 ÷ 2 + 1. According to the order of operations, division is performed before addition, yielding 0 ÷ 2 = 0. Finally, the addition of 0 + 1 results in a value of 1.

Explanation

The given expression is the product cos 10° cos 30° cos 50° cos 70° cos 90°. In trigonometry, the value of cos 90° is exactly 0. Since zero multiplied by any finite real number results in zero, the presence of cos 90° as a factor in the product makes the entire expression equal to 0.

Explanation

The net profit percentage resulting from a markup followed by a discount can be calculated using the successive percentage change formula: Net Change = x + y + ˣʸ⁄₁₀₀.

In this problem, the markup x is +20% and the discount y is -10%. Substituting these values into the formula: Net Change = 20 - 10 + (20 × (-10))/(100) Net Change = 10 - 2 = 8%

Because the result is positive, it indicates a profit of 8%. Alternatively, if the cost price is assumed to be 100, the marked price at 20% above cost is 120. A 10% discount on 120 reduces the price by 12 (10% of 120), resulting in a selling price of 108. The difference between the selling price of 108 and the cost price of 100 confirms a profit of 8%.

Explanation

This problem utilizes the standard algebraic identity: if a + b + c = 0, then a³ + b³ + c³ - 3abc = 0. Given the equation x + 3y + 2 = 0, we can assign a = x, b = 3y, and c = 2. The sum of these three terms is zero as per the given condition. Substituting these values into the cubic identity, we have x³ + (3y)³ + (2)³ - 3(x)(3y)(2) = 0. Expanding the powers and the product results in x³ + 27y³ + 8 - 18xy = 0.

Explanation

The total number of students in the class is the sum of all frequencies provided in the table: Total students = 5 + 11 + 18 + 38 + 39 + 24 + 15 = 150

The number of students who obtained less than 50 marks is the sum of the frequencies for the intervals 10-20, 20-30, 30-40, and 40-50: Students with less than 50 marks = 5 + 11 + 18 + 38 = 72

To find the percentage, use the formula: Percentage = (Number of students with less than 50 marks / Total number of students) × 100 Percentage = (72 / 150) × 100 Percentage = (24 / 50) × 100 Percentage = 48%

Explanation

In an isosceles triangle, the angles opposite to the equal sides are equal. Given AB = AC, the angles ∠ ABC and ∠ ACB are equal. Since ∠ ABC = 72°, it follows that ∠ ACB = 72°. The sum of the internal angles of any triangle is 180°. Using the relation ∠ BAC + ∠ ABC + ∠ ACB = 180°, the calculation is ∠ BAC + 72° + 72° = 180°. Adding the base angles gives ∠ BAC + 144° = 180°. Subtracting 144° from 180° results in ∠ BAC = 36°.

Explanation

According to the Power of a Point theorem, for an external point P from which two secants PCD and PBA are drawn to a circle, the relation PA × PB = PC × PD holds true. Given PD = 15.3 cm and CD = 11.9 cm, the total length of the secant segment PC = PD + CD = 15.3 + 11.9 = 27.2 cm. Substituting the given values into the equation, we have 30.6 × PB = 27.2 × 15.3. Solving for PB gives PB = (27.2 × 15.3)/(30.6) = (27.2)/(2) = 13.6 cm. Since AB is the diameter lying on the secant line from P, its length is the distance between the two intersection points A and B, calculated as AB = PA - PB = 30.6 - 13.6 = 17 cm. The radius of the circle is half the length of the diameter, which is ¹⁷⁄₂ = 8.5 cm.

Explanation

In a circle, the tangent at any point is perpendicular to the radius through the point of contact. Consequently, the radius OQ is perpendicular to the tangent PQ, forming a right-angled triangle OQP where ∠ OQP = 90° and the distance from the center OP is the hypotenuse.

Using the Pythagorean theorem in △ OQP: OQ² + PQ² = OP² r² + 12² = 13² r² + 144 = 169 r² = 169 - 144 = 25 r = √(25) = 5 cm

The radius of the circle is determined by the (5, 12, 13) Pythagorean triplet, where 13 is the hypotenuse and 12 is one of the sides.

Explanation

The sum of cubes of two variables can be calculated using the algebraic identity p³+q³ = (p+q)³ - 3pq(p+q). By substituting the given values p+q=7 and pq=5 into the formula, we obtain p³+q³ = (7)³ - 3(5)(7). Evaluating the expression gives 343 - 105, which results in 238.

Explanation

To find the share of a specific person in a ratio, we divide the total amount by the sum of the ratio parts and multiply by the specific person's ratio value. The sum of the ratios for A, B, and C is 3 + 4 + 2 = 9 parts. The total amount to be divided is Rs. 990. Therefore, the value of one part is calculated as 990 / 9 = 110. Since B's portion of the ratio corresponds to 4 parts, B's share is calculated by multiplying the value of one part by 4, resulting in 4 × 110 = 440. This can also be represented by the formula B = (4)/(3 + 4 + 2) × 990, which simplifies to ⁴⁄₉ × 990 = 4 × 110 = 440.

Explanation

The expenditure on an item is the product of its price and the quantity consumed. When the expenditure is to remain constant, the price and consumption are inversely proportional. The percentage reduction in consumption required to keep the total budget unchanged when the price increases by r% is determined by the formula Reduction% = (r)/(100 + r) × 100%. Substituting the given price increase of r = 25% into the formula, we get (25)/(100 + 25) × 100 = ²⁵⁄₁₂₅ × 100 = ⅕ × 100 = 20%.

Explanation

Train 1 starts at 3:00 p.m. and Train 2 starts at 3:30 p.m. In the 30-minute interval before Train 2 begins its journey, Train 1 covers a distance of 50 km/h × 0.5 h = 25 km. The remaining distance between the two stations at 3:30 p.m. is 575 km - 25 km = 550 km.

Since the trains are moving toward each other, their relative speed is the sum of their individual speeds: 50 km/h + 60 km/h = 110 km/h. The time required for the trains to meet after 3:30 p.m. is calculated using the formula Time = (Remaining Distance)/(Relative Speed) = (550 km)/(110 km/h) = 5 hours.

The meeting point distance from station A is the total distance covered by Train 1 from 3:00 p.m. until the meeting time. This is the sum of the initial distance covered in the first half-hour and the distance covered during the 5 hours of travel: 25 km + (50 km/h × 5 h) = 25 km + 250 km = 275 km.