Roots and Surds

Surds are numerical expressions containing an irrational number. Surds may be quadratic, bi-quadratic, cubic etc. Pure, Mixed, Simple, Compound, Like and Unlike surds are type of surds.

Summary

Surds are numerical expressions containing an irrational number. Surds may be quadratic, bi-quadratic, cubic etc. Pure, Mixed, Simple, Compound, Like and Unlike surds are type of surds.

Things to Remember

Surds are numerical expressions containing an irrational number. Surds may be quadratic, bi-quadratic, cubic etc.

For example: \( \sqrt2\), \(\sqrt[3]{2}\), \(\sqrt [4]{4}\), \(\sqrt [5] {5}\)

The surds cannot be written in the form of \( \frac {p}{q} \) q≠ 0, so they are irrational numbers.

MCQs

No MCQs found.

Subjective Questions

Q1:

Name any five types of simple machine and give two examples of each.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>Any five types of simple machine with two example of each are given below,</p> <ol><li>Lever<br> Examples: see- saw and nut- cracker</li> <li>Wheel and axle<br> Examples: screw- driver</li> <li>Inclined plane<br> Examples: staircase and ladder</li> <li>Screw<br> Examples: nail and driller</li> <li>Wedge<br> Examples: sickle and knife</li> </ol>

Q2:

What is lever? Name its types with two examples of each.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>A lever is a rigid bar which moves freely about a fixed point called the fulcrum. There are three types of lever which are as follows,</p> <ol><li>First class lever<br>Examples: Beam balance and see- saw</li> <li>Second class lever<br>Examples: Wheel- barrow and nut- cracker</li> <li>Third class lever<br>Examples: Shovel and spade</li> </ol>

Q3:

What do you mean by input and output work? Write down the principle of the lever.


Type: Short Difficulty: Easy

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Answer: <p>The work done on a machine is called the input work and the useful work done by the machine is called the output work. The principle of lever is given below,<br>In ideal lever when it is in balanced condition then, <br>Input work = output work<br>Or, Effort \(\times\) effort arm = Load \(\times) load arm.</p>

Q4:

Define wedge. Write down any three examples of wedge.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>A wedge is a simple machine having two or more sloping surfaces that taper either to form a sharp edge or pointed edge. Any three examples of wedge are axe, knife, and needle.</p>

Q5:

Write down any three advantages of using simple machine in our daily life.


Type: Short Difficulty: Easy

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Answer: <p>Any three advantages of using simple machines in our daily life are as follows,</p> <ul><li>It makes or work easier, faster and more convenient.</li> <li>It increases the amount of a force so that we can lift heavier objects or exert greater force with the help of simple machine.</li> <li>It change direction of the force so that we can move objects in another direction.</li> </ul>

Q6:

What is pulley? Write down types of pulley.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>A pulley is a simple machine having a grooved circular disc over which a rope passes. There are three types of pulley which are as follows,</p> <ol><li>Fixed pulley: In fixed pulley, the pulley does not move as the load is raised. It is used to lift water from the well.</li> <li>Movable pulley: If the pulley moves along with the load, then it is called movable pulley. Construction crane and modern elevator are the examples of movable pulley.</li> <li>Combined pulley: Combined pulley is a type of pulley that consists of the combination of fixed pulley and movable pulley. Crane used to lift heavy objects is an example of combined pulley.</li> </ol>

Q7:

A bottle opener is a second class lever. Why?


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>A second class lever is a type of lever in which load is situated between the fulcrum and the effort. Since in bottle opener, the load ( bottle cork) is situated between the fulcrum and effort so the bottle opener is a second class lever.</p>

Q8:

Classify the given simple machines.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>Simple machines: Beam- balance, nut- cracker, flag pole, screw- driver, ladder, nail and axe.<br>The simple machines with their types are given below,</p> <ul><li>Beam- balance: first class lever</li> <li>Nut- cracker: second class lever</li> <li>Flag pole: fixed pulley</li> <li>Screw- driver: Wheel and axle</li> <li>Ladder: inclined plane</li> <li>Nail: screw</li> <li>Axe: wedge</li> </ul>

Q9:

Write down any five simple machine used in your homes and write down their uses.


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>Any five simple machines used in our homes are as follows,</p> <ol><li>Knife: It is used to cut vegetables.</li> <li>Beam- balance: It is used to measure the weight or mass.</li> <li>Bottle- opener: It is used to open the cork of the bottle.</li> <li>Paper- cutter: It is used to cut the papers.</li> <li>Lemon- squeezer: it is used to squeeze the lemon.</li> </ol>

Q10:

Why is shovel not called second class lever?


Type: Short Difficulty: Easy

Show/Hide Answer
Answer: <p>In second class lever, load is situated between the fulcrum and the effort and in third class lever the effort is situated between the fulcrum and the load. Since in shovel, the effort is situated between the fulcrum and load so it is called third class lever but not a second class lever.</p>

Q11:

What are the differences between simple machines and complex machines?


Type: Long Difficulty: Easy

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Answer: <p>The differences between simple machines and complex machines are as follows,</p>
<table width="629">
<tbody>
<tr>
<td width="315">
<p><strong>Simple machines</strong></p>
</td>
<td width="315">
<p><strong>Complex machines</strong></p>
</td>
</tr>
<tr>
<td width="315">
<p>They are simple in structure.</p>
</td>
<td width="315">
<p>They are complex in structure.</p>
</td>
</tr>
<tr>
<td width="315">
<p>Their mechanism is very simple and is very easy to understand and operate.</p>
</td>
<td width="315">
<p>Their mechanism is very complex which is difficult to understand and operate.</p>
</td>
</tr>
<tr>
<td width="315">
<p>Examples: scissors and beam balance</p>
</td>
<td width="315">
<p>Examples: car and truck</p>
</td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>

Q12:

What are the differences between first class lever and third class lever?


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>The differences between first class lever and third class lever are as follows,</p>
<table width="638">
<tbody>
<tr>
<td width="319">
<p><strong>First class lever</strong></p>
</td>
<td width="319">
<p><strong>Third class lever</strong></p>
</td>
</tr>
<tr>
<td width="319">
<p>In the first class lever, the fulcrum is situated between the effort and the load.</p>
</td>
<td width="319">
<p>In the third class lever, the effort is situated between the fulcrum and the load.</p>
</td>
</tr>
<tr>
<td width="319">
<p>Effort arm of a first class lever is longer than the load arm.</p>
</td>
<td width="319">
<p>Effort arm of a third class lever is shorter than the load arm.</p>
</td>
</tr>
<tr>
<td width="319">
<p>Example: beam balance and see- saw.</p>
</td>
<td width="319">
<p>Example: fishing rod and shovel.</p>
</td>
</tr>
</tbody>
</table>

Q13:

What are the differences between input work and output work?


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>The differences between input work and output work are as follows,</p>
<table width="641">
<tbody>
<tr>
<td width="321">
<p><strong>Input work</strong></p>
</td>
<td width="321">
<p><strong>Output work</strong></p>
</td>
</tr>
<tr>
<td width="321">
<p>The work done on a machine is called input work.</p>
</td>
<td width="321">
<p>The useful work done by a machine is called output work.</p>
</td>
</tr>
<tr>
<td width="321">
<p>It is done by the persons.</p>
</td>
<td width="321">
<p>It is done by the machine.</p>
</td>
</tr>
<tr>
<td width="321">
<p>Example: flour, salt, and eggs are put in a machine by men to makes noodles.</p>
</td>
<td width="321">
<p>Example: Noodles are made by machine using flour, salt, and eggs</p>
</td>
</tr>
</tbody>
</table>
<p>&nbsp;</p>

Q14:

What are the differences between fixed pulley and movable pulley?


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>The differences between fixed pulley and movable pulley are as follows,</p>
<table width="635">
<tbody>
<tr>
<td width="318">
<p><strong>Fixed pulley</strong></p>
</td>
<td width="318">
<p><strong>Movable pulley</strong></p>
</td>
</tr>
<tr>
<td width="318">
<p>Fixed pulley is a type of pulley in which pulley does not move with the load.</p>
</td>
<td width="318">
<p>Movable pulley is a type of pulley in which pulley moves along with the load.</p>
</td>
</tr>
<tr>
<td width="318">
<p>It does not multiply the force.</p>
</td>
<td width="318">
<p>It multiplies the force.</p>
</td>
</tr>
<tr>
<td width="318">
<p>Examples: pulley used to lift the water from the well.</p>
</td>
<td width="318">
<p>Examples: pulley used in construction crane and modern elevators.</p>
</td>
</tr>
</tbody>
</table>

Q15:

A load of 100N can be lifted by applying an effort of 25N. If the load arm is 25cm, calculate the effort arm.


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>Solution:<br> We have,<br> Load = 100N<br> Load arm = ?<br> Effort = 25N<br> Effort arm = 25cm<br> Now, by using principle of lever<br> Input work = Output work<br> Or, Effort \(\times\) effort arm = Load \(\times\) load arm<br> Or, 25 \(\times\) effort arm = 100 \(\times\) 25<br> Or, Effort arm = \(\frac{100 \ times 25}{25}\)<br> Or, Effort arm = 100cm<br> \(\therefore\), The load arm is 100cm.</p>

Q16:

An effort of 25N is applied to lift the load. If the load arm and effort arm is 20cm and 25cm respectively, then calculate the load.


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>Solution:<br> We have,<br> Effort = 25N<br> Effort arm = 20cm<br> Load arm = 25cm<br> Load = ?<br> Now, by using principle of lever<br> Input work = Output work<br> Or, Effort \(\times\) effort arm = Load \(\times\) load arm<br> Or, 25 \(\times\) 20 = load \(\times\) 25<br> Or, Load = \(\frac{25 \ times 20}{25}\)<br> Or, Load = 25N<br> \(\therefore\), The load is 25N.</p>

Q17:

An effort of 50N is applied to lift the load of 75N. If the load arm is 100cm then find the effort arm.


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>Solution:<br> We have,<br> Load = 75N<br> Load arm = 100cm<br> Effort arm = ?<br> Effort = 50N<br> Now, by using principle of lever<br> Input work = Output work<br> Or, Effort \(\times\) effort arm = Load \(\times\) load arm<br> Or, 50 \(\times\) effort arm = 75 \(\times\) 100<br> Or, effort arm = \(\frac{ 75 \times 100}{50}\)<br> Or, effort arm = 150cm<br> \(\therefore\), The effort arm is 150cm.</p>

Q18:

To lift the load of 100N, an effort is applied. If the load arm is 50cm and the effort arm is 20cm then what is the effort applied?


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>Solution:<br> We have,<br> Load = 100N<br> Load arm = 50cm<br> Effort arm = 20cm<br> Effort = ?<br> Now, by using principle of lever<br> Input work = Output work<br> Or, Effort \(\times\) effort arm = Load \(\times\) load arm<br> Or, Effort \(\times\) 20 = 100 \(\times\) 50<br> Or, Effort = \(\frac{ 100 \times 50}{20}\)<br> Or, Effort = 250N<br> \(\therefore\), The effort applied is 250N</p>

Q19:

Calculate the input work and output work if effort applied is 10N and effort arm is 20cm, and load is 25N and load arm is 5cm.


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>Solutions:<br> We have,<br> Load = 25N<br> Load arm =10cm<br> Effort arm = 20cm<br> Effort = 10N<br> Now, by using principle of lever<br> Input work = Effort \(\times\) effort arm<br> = 10 \(\times\) 20<br> = 200.<br> Output work = Load \(\times\) load arm<br> = 25 \(\times\) 5<br> = 50.<br> \(\therefore\), The input work is 200 and output work is 50.<br><br></p>

Q20:

An effort of 5N is applied to lift the load of 50N. If the effort arm is 10m, calculate load arm, input work and output work.


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>Solution:<br> We have,<br> Load = 50N<br> Load arm = ?<br> Effort arm = 10cm<br> Effort = 5N<br> Now, by using principle of lever<br> Input work = Output work<br> Or, Effort \(\times\) effort arm = Load \(\times\) load arm<br> Or, 5 \(\times\) 10 = 50 \(\times\) load arm<br> Or, load arm = \(\frac{ 5 \times 10}{50}\)<br> Or, load arm = 1m<br> \(\therefore\), The load arm is 1m.</p> <p>Now,<br> Input work = effort arm \(\times\) effort<br> = 10 \(\times\) 5<br> = 50.</p> <p>Output work = load arm \(\times\) load<br> = 2 \(\times\) 50<br> = 100.<br> \(\therefore\), The input work is 100 and output work is also 100.</p> <p></p> <p></p>

Q21:

An effort of 5N is applied to lift the load of 50N. If the effort arm is 10m, calculate load arm, input work and output work.


Type: Long Difficulty: Easy

Show/Hide Answer
Answer: <p>Solution:<br> We have,<br> Load = 50N<br> Load arm = ?<br> Effort arm = 10cm<br> Effort = 5N<br> Now, by using principle of lever<br> Input work = Output work<br> Or, Effort \(\times\) effort arm = Load \(\times\) load arm<br> Or, 5 \(\times\) 10 = 50 \(\times\) load arm<br> Or, load arm = \(\frac{ 5 \times 10}{50}\)<br> Or, load arm = 1m<br> \(\therefore\), The load arm is 1m.</p> <p>Now,<br> Input work = effort arm \(\times\) effort<br> = 10 \(\times\) 5<br> = 50.</p> <p>Output work = load arm \(\times\) load<br> = 2 \(\times\) 50<br> = 100.<br> \(\therefore\), The input work is 100 and output work is also 100.</p> <p></p> <p></p>

Q22: Define Simple Machine.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: The machine which is simple in structure and makes wok easier, faster and convenient is called a simple machine.

Q23: Give any two examples of simple machine ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: The two examples of simple machines are nut- cracker and jack screw.

Q24: How many types of simple machines are there ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: There are six types of simple machine.

Q25:

Define lever.


Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: <p>A lever is&nbsp;defined as a rigid bar that moves freely about a fixed point called the fulcrum.</p>

Q26: Give two examples of first class lever.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: The two examples of first class lever are beam balance and see- saw.

Q27: Give any two examples of second class lever.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: The two examples of second class lever are nut- cracker and paper- cutter.

Q28: Give any two examples of third class lever.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Any two examples of third class lever are shovel and spade.

Q29: What is pulley ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: A pulley is a simple machine with grooved circular disc over which a rope passes.

Q30: What is wheel and axle ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: A wheel and axle is a simple machine having two cylinders of different radius.

Q31: Give any two examples of wheel and axle.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Any two examples of wheel and axle are screw- driver and paddle of bicycle.

Q32: Give any two examples of screw.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Any two examples of screw are nail and driller.

Q33: What is wedge ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: A wedge is a simple machine having two or more sloping surfaces that taper either to form a sharp edge or pointed edge.

Q34:

Give any two examples of wedge.


Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: <p>Any two examples of wedge are axe and knife.</p>

Q35: Define inclined plane.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Inclined plane is a sloping surfaces or a wooden plank which is used to push things upward.

Q36: Which type of simple machine is broom and fishing rod ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Broom and fishing rod is a third class lever.

Q37: Which type of machine is wheel- barrow and beam balance ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Wheel barrow is a second class lever and beam balance is a first class lever.

Q38: Which type of machine is screw- driver and paddle of bicycle ?
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Screw- driver and paddle of bicycle is a wheel and axle.

Q39: Give any two examples of inclined plane.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Any two examples of inclined plane are staircase and ladder.

Q40: Define movable pulley.
Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: Movable pulley is a type of pulley in which pulley moves with the load.

Q41:

What is third class lever?


Type: Very_short Difficulty: Easy

Show/Hide Answer
Answer: <p>The lever in which the effort is situated between the fulcrum and the load is called the third class lever.</p>

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Simple Machine
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Roots and Surds

Roots and Surds

Roots

.

\begin{align*} \text {We know that} \: 3 \times 3 &= 9 \\ or, \: 3^2 &= 9 \\ or, \: 3^{2 \times \frac {1}{2}} &= 9^{\frac{1}{2}} \\ or, \: 3 &=9^{\frac{1}{2}} \\ or, \: 3 &= \sqrt{9} \\ \end{align*}

The square root of 9 is 3.

\begin{align*} similarly, \: 5 \times 5\times 5 &= 125 \\ or, \: 5^3 &= 125 \\ or, 5^{3 \times \frac {1}{3}} &= (125)^{\frac{1}{3}} \\ or, \: 5 &= (125)^{\frac{1}{3}} \\ or, \: 5 &= \sqrt[3]{125} \end{align*}

The cube root of 125 is 5.
In case of equation, root indicates the value of variable.
e.g. x + 2 = 0, x = -2,
So, -2 is the root of x
e.g. x2 - 4 = 0,
(x - 2)(x +2) = 0

Either, (x - 2) = 0 or, x = 2
or, (x + 2) = 0, x = -2

So, -2 and 2 are roots of x2 -4 =0

Surds

.

Surds are numerical expressions containing an irrational number. Surds may be quadratic, bi-quadratic, cubic etc.

For example: \( \sqrt{2}, \sqrt[3]{3}, \sqrt [4]{4}, \sqrt [5] {5}\)

The surds cannot be written in the form of \( \frac {p}{q} \) q≠ 0, so they are irrational numbers.

Types of surds

  1. Pure surds:
    If the natural number is completely inside the root or radical, the surd is called pure surd. \( \sqrt {2}, \sqrt [3]{5} \) are pure surds.
  2. Mixed surd:
    If the integers are inside and outside the radical, the surds is called mixed surd. 2\( \sqrt [2] {5} \) is a mixed surd.
  3. Simple surd:
    The single surd which may be pure or mixed is called simple surd. \( \sqrt [4]{4} \: and \:3 \sqrt [4]{6} \) are simple surds.
  4. Compound surd
    The sum or difference of two pure or mixed surds is called compound surd. \( \sqrt {5} + 2 , \: \sqrt[3]{2} + 6 \) are compoud surds.
  5. Like surds
    If the power (degree) of surds and number inside the root is same the surds are called like surds.\( \sqrt {5} , 2\sqrt {5} \) are like surds.
  6. Unlike surds
    If a power of root is different or numbers inside the root are different the surds are called, unlike surds.\( \sqrt {5} , \sqrt [4]{4}\:and \sqrt {2}\) are unlike surds.

Four fundamental operations on surds

.
  1. Additional and subtraction of surds:
    The addition and subtraction of like surds can exist, unlike surds are neither added nor subtracted. For example: \( 7 \sqrt{2} + 8 \sqrt{2} = (7 + 8) \sqrt{2} = 15 \sqrt{2} \)
  2. Multiplication and division of surds:
    If the order of surds is same, we can put them within common root and perform the multiplication and division just like in arithmetic.
    For example: a. \(\sqrt {2} \times \sqrt{3} \\ = \sqrt {2 \times 3} \\ = \sqrt {6}\)   b. \( \sqrt{10}÷ \sqrt{2} \\ = \sqrt {\frac{10}{2}} \\ = \sqrt{}5 \)
    If the order of surds are not same then we shall reduce these surds in the same order.
    For example: \(\sqrt{3} \times \sqrt[3]{4} \\ = \sqrt [2 \times 3]{3^3} \times \sqrt [3 \times 2]{4^2} \\ = \sqrt[6]{27} \times \sqrt [6] {16}\\ = \sqrt [6]{27 \times 16} \\ = \sqrt[6] {432} \)

\(\boxed{\text {Note: Same process can be applied for division also}}\)

 

Lesson

Algebra

Subject

Compulsory Mathematics

Grade

Grade 10

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