Podcast
Questions and Answers
The surface area of a cube is 726 cm², find its volume?
The surface area of a cube is 726 cm², find its volume?
What is the sum of least and maximum value of x so that 5 digit Number 427x5 is divisible by 9?
What is the sum of least and maximum value of x so that 5 digit Number 427x5 is divisible by 9?
If tan θ = x²y², find sin θ + tan θ?
If tan θ = x²y², find sin θ + tan θ?
Find the value of 8 + 9 × 8 + 9 + 8 + 8 + 8 / 5 + 5 × 5 + 5 + 7 + 7 + 7?
Find the value of 8 + 9 × 8 + 9 + 8 + 8 + 8 / 5 + 5 × 5 + 5 + 7 + 7 + 7?
Signup and view all the answers
A right angled triangle of area 364.5 unit is to be made such that hypotenuse is √2 times the base, then for triangle hypotenuse is _______ more than the base.
A right angled triangle of area 364.5 unit is to be made such that hypotenuse is √2 times the base, then for triangle hypotenuse is _______ more than the base.
Signup and view all the answers
On selling an item at a certain price after a discount of 25%, there is a profit of 25%. Find the ratio of cost price to marked price?
On selling an item at a certain price after a discount of 25%, there is a profit of 25%. Find the ratio of cost price to marked price?
Signup and view all the answers
Ram travelled an equal distance with the speed of 45 km/hr, 50 km/hr and 60 km/hr. What is the average speed of Ram during the whole journey?
Ram travelled an equal distance with the speed of 45 km/hr, 50 km/hr and 60 km/hr. What is the average speed of Ram during the whole journey?
Signup and view all the answers
The price of an electric bike was 130,000 last year. This year price got increased by 20%. What is the price of the bike this year?
The price of an electric bike was 130,000 last year. This year price got increased by 20%. What is the price of the bike this year?
Signup and view all the answers
Evaluate (255) + 5 + (279) × 5 - (366) × (3 + 2) = ?
Evaluate (255) + 5 + (279) × 5 - (366) × (3 + 2) = ?
Signup and view all the answers
The price of an item increased from 2700 to 3600. Find the increase percentage?
The price of an item increased from 2700 to 3600. Find the increase percentage?
Signup and view all the answers
A man can row 9 km/h in still water. It takes him 3 times as long to row upstream as to row downstream. What is the rate of the stream?
A man can row 9 km/h in still water. It takes him 3 times as long to row upstream as to row downstream. What is the rate of the stream?
Signup and view all the answers
Three quantities costing Rs. 25/kg, Rs 35/kg & Rs. 45/kg are mixed in ratio 2:3:x. Find the value of x such that the mixture is worth Rs. 40.
Three quantities costing Rs. 25/kg, Rs 35/kg & Rs. 45/kg are mixed in ratio 2:3:x. Find the value of x such that the mixture is worth Rs. 40.
Signup and view all the answers
If the radii of two circles are 5 cm & 4 cm and the length of common external tangent is 6 cm, find the distance between the two circle's centers?
If the radii of two circles are 5 cm & 4 cm and the length of common external tangent is 6 cm, find the distance between the two circle's centers?
Signup and view all the answers
If A: B = 4: 5 and B: C = 6: 7, find A: C: B?
If A: B = 4: 5 and B: C = 6: 7, find A: C: B?
Signup and view all the answers
A can do work in 10 days, B can do double the work in 15/2 days, if they work together, then how long will the work be completed?
A can do work in 10 days, B can do double the work in 15/2 days, if they work together, then how long will the work be completed?
Signup and view all the answers
The simple interest on a sum for 12 years is 4/5 of the sum. Find the rate of interest per annum?
The simple interest on a sum for 12 years is 4/5 of the sum. Find the rate of interest per annum?
Signup and view all the answers
Find the mode of the following data: 12, 14, 15, 04, 9, 9, 7, 7, 7, 2, 5, 7, 9, 12, 13.
Find the mode of the following data: 12, 14, 15, 04, 9, 9, 7, 7, 7, 2, 5, 7, 9, 12, 13.
Signup and view all the answers
Which of the following statements are true?
Which of the following statements are true?
Signup and view all the answers
If P(A) = 9/13 & P(B) = 7/13 & P(AB) = 4/13, find P(A|B)?
If P(A) = 9/13 & P(B) = 7/13 & P(AB) = 4/13, find P(A|B)?
Signup and view all the answers
M varies inversely as (N² + 4).
M varies inversely as (N² + 4).
Signup and view all the answers
Study Notes
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.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
