Untitled Quiz

Surface Area and Volume of a Cube

• The surface area of a cube is 726 cm²
• To find the volume of a cube, you first need to find the length of each side
• The formula for the surface area of a cube is 6a², where 'a' is the length of a side
• Therefore, 6a² = 726 cm²
• a² = 121 cm²
• a = 11 cm
• The volume of a cube is calculated by a³, therefore the volume of the cube is 11 cm³ = 1331 cm³

Divisibility by 9

• A number is divisible by 9 if the sum of its digits is divisible by 9.
• The 5-digit number 427x5 is divisible by 9
• The sum of the digits of 427x5 is 18 + x
• To find the smallest value of x, x = 0 (18 + 0 = 18 is divisible by 9)
• To find the largest value of x, x = 9 (18 + 9 = 27 is divisible by 9)
• The sum of the smallest and largest values of x is 0 + 9 = 9

Trigonometry

• Given that tan θ = x²y²
• The formula for tan θ = sin θ / cos θ
• The formula for sin θ = √(1 - cos²θ)
• To find sin θ + tan θ, we need to find sin θ and cos θ
• We know that sin θ = √(1 - cos²θ) and tan θ = sin θ / cos θ, therefore cos θ = sin θ / tan θ
• Substituting the value of tan θ in the equation for cos θ, we get cos θ = sin θ / x²y²
• Squaring both sides, we get cos²θ = sin²θ / (x⁴y⁴)
• Using the identity sin²θ + cos²θ = 1, we can substitute the value of cos²θ to get sin²θ + (sin²θ / (x⁴y⁴)) = 1
• Solving for sin²θ, we get sin²θ (1 + 1 / (x⁴y⁴)) = 1
• sin²θ = (x⁴y⁴) / (x⁴y⁴ + 1)
• sin θ = √((x⁴y⁴) / (x⁴y⁴ + 1))
• Therefore, sin θ + tan θ = √((x⁴y⁴) / (x⁴y⁴ + 1)) + x²y²

Order of Operations

• 8 + 9 × 8 + 9 + 8 + 8 + 8 / 5 + 5 × 5 + 5 + 7 + 7 + 7
• Following the order of operations (PEMDAS/BODMAS)
• First, perform multiplication: 9 × 8 = 72, 5 × 5 = 25
• Then, perform division: 8 / 5 = 1.6
• Now the expression becomes: 8 + 72 + 9 + 8 + 8 + 1.6 + 5 + 25 + 5 + 7 + 7 + 7
• Finally, perform addition: 183.6
• Simplification: 183.6 / 27 = 6.8, rounded to the nearest whole number is 7

Right Triangle

• The area of a right-angled triangle is 364.5 units.
• The hypotenuse is √2 times the base.
• We want to find how much longer the hypotenuse is than the base.
• The area of a right-angled triangle = (1/2) * base * height.
• Here the height is equal to the base, because the hypotenuse is √2 times the base (it's a 45-45-90 triangle).
• Area = (1/2) * base * base = 364.5 units
• Base² = 364.5 * 2 = 729
• Base = √729 = 27 units
• Hypotenuse = √2 * base = √2 * 27 = 27√2 units
• Hypotenuse - Base = 27√2 - 27 = 27(√2 - 1) ≈ 11.5 units (rounded to the nearest tenth)

Profit and Discount

• A seller sells an item at a price that results in a 25% profit after a 25% discount.
• We need to find the ratio of the cost price to the marked price.
• Let the cost price be CP and the marked price be MP.
• The selling price (SP) is equal to MP - (25/100)MP = (75/100)MP
• After the discount, the seller makes a profit of 25%, meaning SP = (125/100)CP
• Therefore, (75/100)MP = (125/100)CP
• Simplifying: 3MP = 5CP
• The ratio of CP: MP = 3:5

Average Speed

• Ram travels an equal distance at speeds of 45 km/hr, 50 km/hr, and 60 km/hr.
• To find the average speed, we need to consider the total distance traveled and the total time taken.
• Let the distance travelled in each leg be 'd'.
• Time taken for the first leg = d/45 hours
• Time taken for the second leg = d/50 hours
• Time taken for the third leg = d/60 hours
• Total distance = d + d + d = 3d
• Total time = (d/45) + (d/50) + (d/60) = (20d + 18d + 15d)/900 = 53d/900 hours
• Average speed = Total distance / Total time = 3d / (53d/900) = 2700/53 ≈ 50.9 km/hr (rounded off)

Percentage Increase

• The price of an item increases from 2700 to 3600.
• To find the percentage increase, use the following formula:
• Percentage increase = ( (New Price - Old Price) / Old Price ) * 100
• Percentage increase = ((3600 - 2700) / 2700) * 100
• Percentage increase = (900/2700) * 100 = 33.33% ≈ 33% (rounded to the nearest whole number)

Boat and Stream

• A man can row a boat at a speed of 9 km/hr in still water.
• He takes three times longer to row upstream than downstream.
• We need to find the speed of the stream.
• Let the speed of the stream be 'x' km/hr.
• The speed of the boat upstream is (9 - x) km/hr.
• The speed of the boat downstream is (9 + x) km/hr.
• Let the distance traveled be 'd' km.
• Time taken upstream = Distance / Speed = d / (9 - x) hours
• Time taken downstream = Distance / Speed = d / (9 + x) hours
• We are given that the time taken upstream is three times the time taken downstream.
• So, d / (9 - x) = 3 * (d / (9 + x))
• Simplifying: 9 + x = 27 - 3x
• 4x = 18
• x = 4.5 km/hr

Mixture

• Three quantities, costing Rs. 25/kg, Rs. 35/kg, and Rs. 45/kg, are mixed in the ratio 2:3:x to create a mixture worth Rs. 40/kg.
• We need to find the value of x.
• Let's assume we have 2 units of the Rs. 25/kg quantity, 3 units of the Rs. 35/kg quantity, and x units of the Rs. 45/kg quantity.
• The total weight of the mixture = 2 + 3 + x units
• The total cost of the mixture = (2 * 25) + (3 * 35) + (x * 45) = 50 + 105 + 45x = 155 + 45x
• The average cost of the mixture = Total cost / Total weight = (155 + 45x) / (5 + x)
• We are given that the average cost of the mixture is Rs. 40/kg.
• So, (155 + 45x) / (5 + x) = 40
• 155 + 45x = 200 + 40x
• 5x = 45
• x = 9

External Tangent and Circle Centers

• The radii of two circles are 5 cm and 4 cm.
• The length of the common external tangent is 6 cm.
• We need to find the distance between the centers of the two circles.
• Draw a perpendicular from the center of each circle to the common external tangent.
• This creates two right triangles.
• The sides of the larger triangle are 5 cm (radius), 6 cm (external tangent), and the distance between the centers plus 4 cm (radius of the smaller circle).
• The sides of the smaller triangle are 4 cm (radius), 6 cm (external tangent), and the distance between the centers minus 5 cm (radius of the larger circle).
• Using the Pythagorean theorem on the larger triangle, we get: 5² + 6² = (Distance + 4)²
• Solving for the distance: 25 + 36 = Distance² + 8Distance + 16
• Distance² + 8Distance - 45 = 0
• Factoring the quadratic equation, we get (Distance + 15)(Distance - 3) = 0
• Therefore, the distance between the centers = 3 cm

Ratio

• Given that A:B = 4:5 and B:C = 6:7, we need to find A:C:B.
• To find the ratio of A:C, we need to find a common factor for B in both ratios.
• This can be done by multiplying the first ratio by 6 and the second ratio by 5.
• This gives us A:B = 24:30 and B:C = 30:35.
• Now, we can see that B is represented by the same value in both ratios.
• Therefore, the ratio of A:C:B is 24:35:30.

Work and Time

• A can complete a work in 10 days, while B can complete twice the work in 15/2 days.
• We need to find how long it takes them to complete the work together.
• First, we find B's efficiency compared to A.
• B takes 15/2 days to complete twice the work, meaning B takes 15/4 days to complete one work (15/2 * 1/2 = 15/4).
• A takes 10 days to complete the work, and B takes 15/4 days, so B is 10 / (15/4) = 8/3 times more efficient than A.
• Let A's work rate be 1 unit per day, then B's work rate is 8/3 units per day.
• Combined work rate = 1 + 8/3 = 11/3 units per day
• Time = Total work/Work rate = 1 / (11/3) = 3/11 days

Simple Interest

• The simple interest on a sum of money in 12 years is 4/5 of the sum.
• We need to find the rate of interest per annum.
• Let the principal sum be 'P'.
• SI = (4/5)P
• The formula for Simple Interest = (P * R * T) / 100
• Substituting the known values: (4/5)P = (P * R * 12) / 100
• Simplifying: 4/5 = 12R/100
• R = 33.33% ≈ 25/3% (rounded to the nearest third)

Mode

• The mode of a data set is the value that appears most frequently.
• The data set given is: 12, 14, 15, 04, 9, 9, 7, 7, 7, 2, 5, 7, 9, 12, 13.
• The number 7 appears most frequently (4 times).
• Therefore, the mode of the data set is 7.

Formulas for Mean, Median, and Mode

• The relationship between the mean, median, and mode is:
• Mean - Mode = 3(Mean - Median)
• This formula holds true for unimodal distributions that are slightly skewed.
• The other provided formulas are not accurate representations of the relationship between mean, median, and mode.

Conditional Probability

• Given that P(A) = 9/13, P(B) = 7/13, and P(A∩B) = 4/13, we need to find P(A|B).
• The formula for conditional probability is:
• P(A|B) = P(A∩B) / P(B)
• Substituting the given values: P(A|B) = (4/13) / (7/13)
• P(A|B) = 4/7

Variation

• M varies inversely as (N² + 4)
• This means: M = k / (N² + 4), where 'k' is a constant of proportionality.
• This relationship shows that as (N² + 4) increases, M decreases, and vice versa. The product of M and (N² + 4) will always be a constant value.