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Lesson Posted on 13 Jun Learn Mathematics +2

Savita

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which one is greater ? 6√10 , 3√4, 2√5 Hind to solve this question : take LCM and then find greater Question ask in CGL , CHSL , Bank PO . Question type : Surds and its type . read more

which one is greater ?

6√10 ,  3√4,  2√5

Hind to solve this question : take LCM and then find greater

Question ask in CGL , CHSL , Bank PO .

Question type : Surds and its type .

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Answered on 18 Apr Learn Sphere

Nazia Khanum

Problem Analysis: Given Parameters: Length of road roller: 120 cm Diameter of road roller: 84 cm Number of complete revolutions to level the playground: 500 Cost per square meter: Rs. 2 Solution: Determine Area Covered by Each Revolution: The road roller covers a circular area with each revolution. Formula... read more

Problem Analysis:

• Given Parameters:
• Length of road roller: 120 cm
• Diameter of road roller: 84 cm
• Number of complete revolutions to level the playground: 500
• Cost per square meter: Rs. 2

Solution:

1. Determine Area Covered by Each Revolution:

• The road roller covers a circular area with each revolution.
• Formula for area of a circle: A=πr2A=πr2, where rr is the radius.
• Given diameter, D=84D=84 cm, so radius r=D/2=42r=D/2=42 cm.
• Calculate area covered by each revolution: Arev=π×(42)2Arev=π×(42)2 sq.cm.
2. Calculate Total Area Covered:

• Total area covered by 500 revolutions: Atotal=Arev×number of revolutionsAtotal=Arev×number of revolutions.
3. Convert Area to Square Meters:

• Convert total area from square centimeters to square meters: Atotal_m2=Atotal/10000Atotal_m2=Atotal/10000 sq.m.
4. Determine Cost of Levelling:

• Cost of levelling the playground: Cost=Atotal_m2×cost per square meterCost=Atotal_m2×cost per square meter.
5. Final Calculation:

• Substitute values and calculate the cost.

Detailed Calculation:

1. r=842=42r=284=42 cm
2. Arev=π×(42)2Arev=π×(42)2 sq.cm.
3. Atotal=Arev×500Atotal=Arev×500 sq.cm.
4. Atotal_m2=Atotal10000Atotal_m2=10000Atotal sq.m.
5. Cost=Atotal_m2×2Cost=Atotal_m2×2 Rs.

The cost of levelling the playground at Rs. 2 per square meter is Rs. [insert calculated value].

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Answered on 18 Apr Learn Sphere

Nazia Khanum

To find the cost of the cloth required to make a conical tent, we'll need to: Calculate the slant height of the conical tent. Find the total surface area of the tent. Determine the length of cloth required. Calculate the cost of the cloth. Solution: Step 1: Calculate Slant Height (l) Given: Radius... read more

To find the cost of the cloth required to make a conical tent, we'll need to:

1. Calculate the slant height of the conical tent.
2. Find the total surface area of the tent.
3. Determine the length of cloth required.
4. Calculate the cost of the cloth.

Solution:

Step 1: Calculate Slant Height (l)

Given:

• Radius (r) = 7 m
• Height (h) = 24 m

Using Pythagoras theorem, we can find the slant height (l) of the cone: l=r2+h2l=r2+h2

l=72+242l=72+242

l=49+576l=49+576 l=625l=625

l=25 ml=25m

Step 2: Find Total Surface Area of the Tent

Total surface area (A) of a cone is given by: A=πr(r+l)A=πr(r+l)

A=π×7×(7+25)A=π×7×(7+25) A=π×7×32A=π×7×32 A≈704 m2A≈704m2

Step 3: Determine Length of Cloth Required

Given:

• Width of cloth (w) = 5 m

The length of cloth required will be equal to the circumference of the base of the cone, which is: C=2πrC=2πr

C=2π×7C=2π×7 C≈44 mC≈44m

Step 4: Calculate Cost of Cloth

Given:

• Rate of cloth (R) = Rs. 50 per meter

The cost of cloth required will be: Cost=Length of cloth required×Rate of clothCost=Length of cloth required×Rate of cloth

Cost=44×50Cost=44×50 Cost=Rs.2200Cost=Rs.2200

Conclusion:

The cost of the 5 m wide cloth required at the rate of Rs. 50 per metre is Rs. 2200.

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Answered on 18 Apr Learn Sphere

Nazia Khanum

Calculate the number of lead balls that can be made from a sphere of radius 8 cm, with each ball having a radius of 1 cm. Solution: Step 1: Calculate Volume of Sphere The volume of the original sphere can be calculated using the formula: V=43πr3V=34πr3, where rr is the radius of the sphere. Substituting... read more

Calculate the number of lead balls that can be made from a sphere of radius 8 cm, with each ball having a radius of 1 cm.

Solution:

Step 1: Calculate Volume of Sphere

• The volume of the original sphere can be calculated using the formula: V=43πr3V=34πr3, where rr is the radius of the sphere.
• Substituting the given value of r=8r=8 cm into the formula: V=43π(8)3=43π×512=20483πV=34π(8)3=34π×512=32048π cubic cm.

Step 2: Calculate Volume of Each Lead Ball

• The volume of each lead ball can be calculated using the formula: Vball=43πr3Vball=34πr3, where rr is the radius of the lead ball.
• Substituting the given value of r=1r=1 cm into the formula: Vball=43π(1)3=43πVball=34π(1)3=34π cubic cm.

Step 3: Determine Number of Lead Balls

• To find the number of lead balls that can be made, divide the volume of the original sphere by the volume of each lead ball.
• Number of balls=Volume of original sphereVolume of each lead ballNumber of balls=Volume of each lead ballVolume of original sphere
• Number of balls=20483π43πNumber of balls=34π32048π
• Number of balls=20484=512Number of balls=42048=512

Step 4: Conclusion

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Answered on 18 Apr Learn Real Numbers

Nazia Khanum

Visualizing 3.765 on the Number Line Introduction Visualizing numbers on a number line can be a helpful technique to understand their placement and relationship to other numbers. Let's explore how we can visualize the number 3.765 using successive magnification. Steps to Visualize 3.765 Identify... read more

Visualizing 3.765 on the Number Line

Introduction

Visualizing numbers on a number line can be a helpful technique to understand their placement and relationship to other numbers. Let's explore how we can visualize the number 3.765 using successive magnification.

Steps to Visualize 3.765

1. Identify the Initial Position:

2. First Magnification:

• Zoom in on the integer part, 3, of the number.
• Place 3 on the number line and divide the interval between 3 and 4 into ten equal parts.
• Locate the position of 0.765 within this interval. Since 0.765 lies between 0 and 1, it would be helpful to break down the interval further.
3. Second Magnification:

• Zoom in on the interval between 3 and 4.
• Divide this interval into ten equal parts again.
• Now, locate the position of 0.765 within this smaller interval.
• Continue this process of successive magnification until you reach a level of detail that allows you to pinpoint the position of 0.765 accurately.
4. Final Visualization:

• After several magnifications, you'll notice that 0.765 falls between two consecutive integers on the number line.
• Approximate the position of 0.765 relative to the nearest integers, 3 and 4, based on the magnification level.

Conclusion

Visualizing numbers on the number line using successive magnification helps in understanding their precise location and relationship to other numbers. By breaking down the intervals into smaller parts, we can accurately locate decimal numbers like 3.765 on the number line.

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Answered on 18 Apr Learn Real Numbers

Nazia Khanum

Introduction: In mathematics, adding radical expressions involves combining like terms to simplify the expression. Radical expressions contain radicals, which are expressions that include square roots, cube roots, etc.

+53 and 2−332−33

.

1. Identify Like Terms:

• 2222

and 22

• are like terms.
• 5353

and −33−33

• are like terms.
• Combine Like Terms:

• Add the coefficients of like terms:
• For 22
• : 2+1=32+1=3
• For 33
• : 5−3=25−3=2
• Write the Result:

• The sum of 22+5322

+53 and 2−332−33 is:

• 32+2332

+23

• .

+53 and 2−332−33 simplifies to 32+2332+23

.

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Nazia Khanum

Graphing the Equation x + 2y = 6 To graph the equation x+2y=6x+2y=6, we'll first rewrite it in slope-intercept form (y=mx+by=mx+b): x+2y=6x+2y=6 2y=−x+62y=−x+6 y=−12x+3y=−21x+3 Plotting the Graph To plot the graph, we'll identify two points and draw a line through them: Intercept... read more

Graphing the Equation x + 2y = 6

To graph the equation x+2y=6x+2y=6, we'll first rewrite it in slope-intercept form (y=mx+by=mx+b):

x+2y=6x+2y=6 2y=−x+62y=−x+6 y=−12x+3y=−21x+3

Plotting the Graph

To plot the graph, we'll identify two points and draw a line through them:

1. Intercept Method:

• y-intercept (when x = 0): y=−12(0)+3=3y=−21(0)+3=3 Therefore, the y-intercept is (0, 3).
• x-intercept (when y = 0): 0=−12x+30=−21x+3 −12x=3−21x=3 x=−6x=−6 Therefore, the x-intercept is (-6, 0).
2. Slope Method: From the slope-intercept form y=−12x+3y=−21x+3, the slope is -1/2, meaning the line decreases by 1 unit in the y-direction for every 2 units in the x-direction.

Plotting the Points and Drawing the Line

Using the intercepts and the slope, we plot the points (0, 3) and (-6, 0), then draw a line through them.

Finding the Value of x when y = -3

Given y=−3y=−3, we substitute this value into the equation y=−12x+3y=−21x+3 and solve for x:

−3=−12x+3−3=−21x+3 −12x=−3−3−21x=−3−3 −12x=−6−21x=−6 x=−6×(−2)x=−6×(−2) x=12x=12

Conclusion

• The graph of the equation x+2y=6x+2y=6 is a straight line passing through points (0, 3) and (-6, 0).
• The value of xx when y=−3y=−3 is x=12x=12.
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Nazia Khanum

Understanding Linear Equations: Linear equations are fundamental in mathematics, representing straight lines on a coordinate plane. They're expressed in the form of ax+b=0ax+b=0, where aa and bb are constants. Identifying Axis: In the context of linear equations, the term "axis" typically refers to... read more

Understanding Linear Equations: Linear equations are fundamental in mathematics, representing straight lines on a coordinate plane. They're expressed in the form of ax+b=0ax+b=0, where aa and bb are constants.

Identifying Axis: In the context of linear equations, the term "axis" typically refers to either the x-axis or the y-axis on a Cartesian plane.

Analyzing the Equation: The linear equation provided is x−2=0x−2=0.

Finding the Axis: To determine which axis the given linear equation is parallel to, let's analyze the equation:

1. Equation Form:

• x−2=0x−2=0
2. Solving for x:

• x=2x=2
3. Interpretation:

• This equation indicates that no matter what value y takes, x will always be 2. This implies that the line represented by this equation is parallel to the y-axis.

Conclusion: The linear equation x−2=0x−2=0 is parallel to the y-axis.

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Nazia Khanum

Problem Statement: Find the value of x2+y2x2+y2, given x+y=12x+y=12 and xy=32xy=32. Solution: Step 1: Understanding the problem We have two equations: x+y=12x+y=12 xy=32xy=32 We need to find the value of x2+y2x2+y2. Step 2: Solving the equations We'll use the method of substitution to solve... read more

Problem Statement: Find the value of x2+y2x2+y2, given x+y=12x+y=12 and xy=32xy=32.

Solution:

Step 1: Understanding the problem

• We have two equations:
1. x+y=12x+y=12
2. xy=32xy=32
• We need to find the value of x2+y2x2+y2.

Step 2: Solving the equations

• We'll use the method of substitution to solve for xx and yy.
• From x+y=12x+y=12, we can express yy in terms of xx as y=12−xy=12−x.
• Substitute this expression for yy into equation 2: xy=32xy=32.
• We get x(12−x)=32x(12−x)=32.

Step 3: Finding the values of xx and yy

• Expanding the equation, we have 12x−x2=3212x−x2=32.
• Rearranging terms, we get x2−12x+32=0x2−12x+32=0.
• Now, we solve this quadratic equation for xx.
• We can use factoring or the quadratic formula to find the values of xx.
• Upon solving, we find two solutions for xx, let's call them x1x1 and x2x2.

Step 4: Finding corresponding values of yy

• Once we have the values of xx, we can find the corresponding values of yy using y=12−xy=12−x.

Step 5: Calculating x2+y2x2+y2

• For each pair of xx and yy, calculate x2+y2x2+y2.
• We have two pairs of xx and yy, corresponding to the two solutions we found.
• So, we calculate x12+y12x12+y12 and x22+y22x22+y22.

Step 6: Presenting the solution

• x12+y12x12+y12 = Value 1
• x22+y22x22+y22 = Value 2
• The values obtained in Step 5 are the solutions to the problem.

• x2+y2=x2+y2= The sum of Value 1 and Value 2.

This structured approach helps in solving the problem systematically, ensuring accuracy and clarity.

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Nazia Khanum

Given: x2+y2+z2=83x2+y2+z2=83 x+y+z=15x+y+z=15 To Find: x3+y3+z3−3xyzx3+y3+z3−3xyz Approach: Use the identity (x+y+z)3=x3+y3+z3+3(x+y)(y+z)(z+x)(x+y+z)3=x3+y3+z3+3(x+y)(y+z)(z+x). Given x+y+z=15x+y+z=15, find x3+y3+z3x3+y3+z3. Also, find (x+y)(y+z)(z+x)(x+y)(y+z)(z+x). Substitute... read more

Given:

• x2+y2+z2=83x2+y2+z2=83
• x+y+z=15x+y+z=15

To Find:

• x3+y3+z3−3xyzx3+y3+z3−3xyz

Approach:

1. Use the identity (x+y+z)3=x3+y3+z3+3(x+y)(y+z)(z+x)(x+y+z)3=x3+y3+z3+3(x+y)(y+z)(z+x).
2. Given x+y+z=15x+y+z=15, find x3+y3+z3x3+y3+z3.
3. Also, find (x+y)(y+z)(z+x)(x+y)(y+z)(z+x).
4. Substitute the values in the expression x3+y3+z3−3xyzx3+y3+z3−3xyz.

Step-by-Step Solution:

1. Find x3+y3+z3x3+y3+z3:

• Using the identity (x+y+z)3=x3+y3+z3+3(x+y)(y+z)(z+x)(x+y+z)3=x3+y3+z3+3(x+y)(y+z)(z+x), where x+y+z=15x+y+z=15.
• (15)3=x3+y3+z3+3(x+y)(y+z)(z+x)(15)3=x3+y3+z3+3(x+y)(y+z)(z+x)
• 3375=x3+y3+z3+3(xy+yz+zx+3xyz)3375=x3+y3+z3+3(xy+yz+zx+3xyz) (Expanding (x+y+z)3(x+y+z)3)
• x3+y3+z3=3375−3(xy+yz+zx)x3+y3+z3=3375−3(xy+yz+zx) (Subtracting 3xyz3xyz from both sides)
2. Find (x+y)(y+z)(z+x)(x+y)(y+z)(z+x):

• Given x+y+z=15x+y+z=15, let's find xy+yz+zxxy+yz+zx.
• Squaring x+y+z=15x+y+z=15:
• (x+y+z)2=(15)2(x+y+z)2=(15)2
• x2+y2+z2+2(xy+yz+zx)=225x2+y2+z2+2(xy+yz+zx)=225 (Expanding (x+y+z)2(x+y+z)2)
• 83+2(xy+yz+zx)=22583+2(xy+yz+zx)=225 (Given x2+y2+z2=83x2+y2+z2=83)
• xy+yz+zx=225−832=71xy+yz+zx=2225−83=71
• Using (x+y)(y+z)(z+x)=(xy+yz+zx)+xyz(x+y)(y+z)(z+x)=(xy+yz+zx)+xyz:
• (x+y)(y+z)(z+x)=71+xyz(x+y)(y+z)(z+x)=71+xyz
3. Substitute values into the expression:

• x3+y3+z3−3xyz=3375−3(71)−3xyzx3+y3+z3−3xyz=3375−3(71)−3xyz
• x3+y3+z3−3xyz=3375−213−3xyzx3+y3+z3−3xyz=3375−213−3xyz
• x3+y3+z3−3xyz=3162−3xyzx3+y3+z3−3xyz=3162−3xyz

• x3+y3+z3−3xyz=3162−3xyzx3+y3+z3−3xyz=3162−3xyz

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