Proof of this factor theorem
Let p(x) be a polynomial of degree greater than or equal to one and a be areal number such that p(a) = 0.
Then, we have to show that (x – a) is a factor of p(x).Let q(x) be the quotient when p(x) is divided by (x – a).
By remainder theorem
Dividend = Divisor x Quotient + Remainder
p(x) = (x – a) x q(x) + p(a) [Remainder theorem]
⇒ p(x) = (x – a) x q(x) [p(a) = 0]⇒ (x – a) is a factor of p(x)
Conversely, let (x – a) be a factor of p(x).
Then we have to prove that p(a) = 0
Now, (x – a) is a factor of p(x)⇒ p(x),
when divided by (x – a) gives remainder zero.
But, by the remainder theorem, p(x) when divided by (x – a) gives the remainder equal to p(a).
∴ p(a) = 0
Proof of remainder theorem.
Let q(x) be the quotient and r(x) be the remainder obtained when the polynomial p(x) is divided by (x–a).
Then, p(x) = (x–a) q(x) + r(x), where r(x) = 0 or some constant.
Let r(x) = c, where c is some constant.
Then p (x) = (x–a) q(x) + c
Putting x = a in p(x) = (x–a) q(x) + c,
we getp(a) = (a–a) q(a) + c ⇒ p(a) = 0 x q(a) + c
⇒ p(a) = c
This shows that the remainder is p(a) when p(x) is divided by (x–a).1. [a is –8.]
1. Chemical reaction: A chemical reaction involves a chemical change in which substances react to form new substances with entirely new properties. Substances that react or take part in the reaction are known as reactants and the substances formed are known as products.
2. Physical change: If a change involves change in colour or state but no new substance is formed, then it is a physical change.
3. Chemical change: If a change involves formation of new substances, it is a chemical change.
4. Chemical equation: The symbolic representation of a chemical reaction is called a chemical equation.
5. Exothermic and endothermic reactions: If heat is evolved during a reaction, then such a reaction is known as exothermic reaction. If heat is absorbed from the surroundings, then such a reaction is known as endothermic reaction
6. Combination reaction: Combination reaction is a reaction in which two or more substances combine to give a single product.
7. Decomposition reaction: In a decomposition reaction, a single reactant decomposes to give 2 or more products. Decomposition reactions require energy in the form of heat, light or electricity
8. Displacement reaction: A reaction in which a more active element displaces less active element from its salt solution.
9. Reactivity series: The Reactivity series is a list of metals arranged in the order of decreasing reactivity. The most reactive metal is placed at the top and the least reactive metal is placed at the bottom.
10.Double displacement reaction: A chemical reaction in which there is an exchange of ions between the reactants to give new substances is called a double displacement reaction.
11.Precipitation reaction: An insoluble solid known as precipitate is formed during a double is placement reaction. Such reactions are also known as precipitation reactions.
12.Redox reaction: A reaction, in which oxidation and reduction takes place simultaneously is known as redox reaction.
13.Oxidation: Oxidation is a chemical process in which a substance gains oxygen or loses hydrogen.
14.Reduction: Reduction is a chemical process in which a substance gains hydrogen or loses oxygen.
15.During a chemical reaction, there is a breaking of bonds between atoms of the reacting molecules to give products.
16. A chemical reaction can be observed with the help of any of the following observations:
a) Evolution of a gas b) Change in temperature
c) Formation of a precipitate d) Change in colour e) Change of state
17. Skeletal chemical equation: A chemical equation which simply represents the symbols and formulae of reactants and products taking part in the reaction is known as skeletal chemical equation
for a reaction. For example: For the burning of Magnesium in the air, Mg + O2 → MgO is the skeletal equation.
18. Balanced chemical equation: A balanced equation is a chemical equation in which number of atoms of each element is equal on both sides of the equation i.e. number of atoms of an element on
reactant side = number of atoms of that element on the product side.
19. As per the law of conservation of mass, the total mass of the elements present in the products of a chemical reaction is equal to the total mass of the elements present in the reactants.
20. Decomposition reaction: In a decomposition reaction, a single reactant decomposes to give 2 or more products. Decomposition reactions require energy in the form of heat, light or electricity
Types of decomposition reactions:
a. Decomposition reactions which require heat are known as thermolytic decomposition reactions
b. Decomposition reactions which require light are known as photolytic decomposition reactions
c. Decomposition reactions which require electricity are known as electrolytic decomposition reactions
Chemical reaction and Equation: Solved Test paper -1
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10th Chemical and chemical Equations (13 pages visit blog )
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X_similar_triangle_guess_questions test Paper-5
X EBOOk Chapter Similar Triangle
X Maths: Similar Triangle:Optional Exercise Solved
I. Fill in the blanks:
1. Green plants are called ___________ as they can prepare their own ________.
2. Most ________ and ________ get their food from dead and decaying plants and animals. They are called_____________.
3. In some cases two organisms that live together help each other. They are called ____________.
4. The relationship between these two organisms is called __________.
II. State whether True or False:
1. All living things get energy from food.
2. Stomata are small pores present mainly in roots of plants.
3. Leaves are green because of chlorophyll.
4. Some plants are heterotrophic and depend on other organisms for
5. Saprophytes are green in colour.
6. Lichen has an alga and fungi living together and helping each other.
7. Plants need nitrogen in addition to glucose to synthesis proteins.
III. Short answer questions:
1. Which processes help the body get energy from food?
2. What factors are essential for photosynthesis to take place?
3. What willl happen when you add a drop of iodine to starch?
4. Why are green plants called autotrophs?
5. Name any two non-green plants.
6. What do you mean by symbiosis?
7. What is a saprophyte?
8. What are decomposers?
9. What are insectivorous plants?
10. Give one example of parasite, symbiont, saprophyte, insectivorous plant
IV . Long answer questions:
1. How will you test the presence of starch in leaves?
2. Explain briefly what happens during photosynthesis.
3. Some plants are heterotrophs. Explain with example.
4. What are autotrophs?How is an autotroph different from heterotroph.?
5. How are nutrients put back in soil?
Chapter: Current Electricity Worksheet by Jsunil Tutorial
1. Define electric current, state its unit, list the equation defining electric current.
2. List three types of energy that may be used to produce current, and list three devices that provide this type of energy.
3. List three types of loads and list the energy conversions taking place inside each one.
4. What is the definition of voltage? What is its unit?
5. Prove that 1 J is equivalent to 1 Vx A x s.
6. How long does it take a current of 5.0 mA to deliver 15 C of charge?
7. What is the potential difference between two points if 1.0 kJ of work is required to move 0.5 C of charge between the two points?
8. What is the voltage of a source which provides 12.0 J to each Coulomb of charge present?
9. What is the energy of an electron accelerated through a potential difference of 100.0 kV? (charge of an electron 1.6 x 10 -19Coulomb)
10. What is the potential difference between two points when a charge of 80.0 C has 4.0 x 10^3J of energy supplied to it as it moves between the two points?
11. There is a current of 0.50 A through an incandescent lamp for 5.0 min, with a voltage of 115 V. How much energy does the current transfer to the lamp? What is the power rating of the lamp?
12. If there is a current of 2.0 A through a hair dryer transferring 15 kJ of energy in 55 s, what is the potential difference across the dryer?
13. An electric drill operates at a potential difference of 120V and draws a current of 6.0 mA. If it takes 45 s for the drill to make a hole in a piece of wood, how much energy is used by the drill?
14. An electric toaster operating at a potential difference of 115 V uses 34 200 J of energy during the 20 sec it is on. What is the current through the toaster?
15. A motor draws a current of 2.0 A for 20.0 sec in order to lift a small mass. If the motor does a total of 9.6 J of work calculate the voltage drop across the motor.
16. In a lightning discharge, 30.0 C of charge moves through a potential difference 10^8 V in 20 min. Calculate the current of the lightning bolt.
17. How much energy is gained by an electron accelerated through a potential difference of 3.0 x 10^2 V?
18. A 12V car battery can provide 60.0 A for 1.0 h. how much energy is stored in the battery?
19. How much energy is required to dry your hair if the hair dryer draws 12.0 A from a 110 V outlet for 12.0 min?
1. a) Describe the difference between current in a series circuit and current in aparallel circuit.
b) Describe the difference between voltage in a series circuit and voltage in aparallel circuit.
2. Draw a schematic diagram of the following circuit: One power source and a resistor are connected in series with a combination of 3 light bulbs connected in parallel with each other. Include a fuse, 4 switches, a voltmeter, and an ammeter. The fuse should protect the whole circuit, one switch should shut off the whole circuit and the other switches should control the individual bulbs. The ammeter should read I and the voltmeter the voltage of the resistor.
3. a) What is a short circuit?
b) Why is it dangerous?
c) Give two ways to protect against short circuits.
4. Describe the effect on the rest of the bulbs in problem two when one burns out. Will the remainder glow brighter? dimmer? What will be the effect on the source?
5. Describe resistance, list 4 factors affecting resistance.
6. A conductor has a length of 2.0 m and a radius of 3.0 mm. If the resistance is R = 100 S, calculate the new resistance if the same material has:
a) length = 6.0 m and r = 6.0 mm
b) length = 1.0 m and r = 1.0 mm
7. List two ways to increase the current drawn by a circuit.
8. Draw a graph of V-I for 2 resistors and indicate which has the greatest resistance and why.
9. A voltmeter measures a voltage drop of 60.0 V across a heating elementwhile an ammeter reads the current through it as 2.0 A. What is the resistance of the heating coil?
10. How much current flows through a 7.5 S lightbulb with a potential difference of 1.5 V?
11. What is the voltage drop across an element which draws a current of 5.0 A and has a resistance of 40 ohms?
12. A set of 6 motors is connected in series to a 120 V source drawing 1.0 A of current. Find:
a) R total
b) R of each motor
c) Voltage drop across each load.
13. Do resistors in parallel increase or decrease total resistance?
14. What resistance must be added in series to a circuit containing a 33 ohm resistor in in order to draw 2.0 A of current from a 120V source?
15. How much energy is dissipated in 10 minutes when a current of 4.0 A is flowing through a potential drop of 60.0 V?
Secondary School Curriculum
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SCALARS & VECTORS
Physical quantities which can completely be specified by a number (magnitude) having an appropriate unit are known as "SCALAR QUANTITIES".
Scalar quantities do not need direction for their description.
Example: Work, energy, electric flux, volume, refractive index, time, speed, electric potential, potential difference, viscosity, density, power, mass, distance, temperature, electric charge, electric flux etc
Physical quantities having both magnitude and direction with appropriate unit are known as "VECTOR QUANTITIES".
We can't specify a vector quantity without mention of direction.
Velocity, electric field intensity, acceleration, force, momentum, torque, displacement, electric current, weight, angular momentum etc.
"Kinematics is the branch of Physics in which we discuss bodies at rest or motion without the reference of external agent that causes motion or rest."
"The branch of physics which deals with the description of motion of objects without reference to the force or agent causing motion in it, is called Kinematics."
"If a body does not change its position with respect to its surroundings then
the body is said to be in a state of rest."
"If a body continuously changes its position with respect to its surrounding
than it is said to be in a state of motion."
Motion of objects can be divided into three categories.
(i) TRANSLATIONAL MOTION (ii) ROTATIONAL MOTION (iii) VIBRATIONAL MOTION
"Motion of a body in which every particle of the body is being displaced by the same amount is called Translational Motion".
EXAMPLE: (i) Motion of a person on a road. (ii) Motion of a car or truck on a road.
"Type of motion in which a body rotates around a fixed point or axis is called Rotational Motion."
EXAMPLE: (i) Motion of wheel (ii) Motion of the blades of a fan
"Type of motion in which a body or particle moves to and fro about a fixed point or mean position is called Vibratory Motion."
EXAMPLE: (i) Motion of simple pendulum (ii) Motion of the wires of guitar (iii) Motion of swing
Q.1. Based on Euclid’s algorithm: a = bq + r ; Using Euclid’s algorithm: Find the HCF of 825 and 175.
Ans:Since 825>175, we division lemma to 825 and 175 to get825 = 175 x 4 + 125.
Since r ≠ 0, we apply division lemma to 175 and 125 to get
175 = 125 x 1 + 50
Again applying division lemma to 125 and 50 we get,
125 = 50 x 2 + 25.
Once again applying division lemma to 50 and 25 we get.
50 = 25 x 2 + 0.
Since remainder has now become 0, this implies that HCF of 825 and 125 is 25.
Q.2. Based on Showing that every positive integer is either of the given forms:
Prove that every odd positive integer is either of the form 4q + 1 or 4q + 3 for some integer q.
Ans: Let a be any odd positive integer (first line of problem) and let b = 4. Using division Lemma we can write a = bq + r, for some integer q, where 0≤r<4. So a can be 4q, 4q + 1, 4q + 2 or 4q + 3. But since a is odd, a cannot be 4q or 4q + 2. Therefore any odd integer is of the form 4q + 1 or 4q + 3.
Q.Find H C F (26,91) if LCM(26,91) is 182
Sol: We know that LCM x HCF = Product of numbers.
or 182 x HCF = 26 x 91
or HCF = 26 x 91 = 13
Hence HCF (26, 91) = 13.
Q. prove that √5 is irrational.
Solution: let us assume on the contrary that √5 is rational. That is we can find co-primes a and b b (≠0) such that √5 = a/b.
Or √5b = a.
Squaring both sides we get
5b2 = a2.
This means 5 divides a2. Hence it follows that 5 divides a.
So we can write a = 5c for some integer c.
Putting this value of a we get
5b2 = (5c)2
Or 5b2 = 25c2
Or b2 = 5b2.
It follows that 5 divides b2. Hence 5 divides b.
Now a and b have at least 5 as a common factor.
But this contradicts the fact that a and b are co-primes.
This contradiction has arisen because of our incorrect assumption that √5 is rational. Hence it follows that √5 is irrational.
Q. prove that product of three consecutive positive integers is divisible by 6?
Ans: Let three consecutive positive integers be, n, n + 1 and n + 2.
Whenever a number is divided by 3, the remainder obtained is either 0 or 1 or 2.
∴ n = 3p or 3p + 1 or 3p + 2, where p is some integer.
If n = 3p, then n is divisible by 3.
If n = 3p + 1, then n + 2 = 3p + 1 + 2 = 3p + 3 = 3(p + 1) is divisible by 3.
If n = 3p + 2, then n + 1 = 3p + 2 + 1 = 3p + 3 = 3(p + 1) is divisible by 3.
So, we can say that one of the numbers among n, n + 1 and n + 2 is always divisible by 3.
⇒ n (n + 1) (n + 2) is divisible by 3.
Similarly, whenever a number is divided 2, the remainder obtained is 0 or 1.
∴ n = 2q or 2q + 1, where q is some integer.
If n = 2q, then n and n + 2 = 2q + 2 = 2(q + 1) are divisible by 2.
If n = 2q + 1, then n + 1 = 2q + 1 + 1 = 2q + 2 = 2 (q + 1) is divisible by 2.
So, we can say that one of the numbers among n, n + 1 and n + 2 is always divisible by 2.
⇒ n (n + 1) (n + 2) is divisible by 2.
Since, n (n + 1) (n + 2) is divisible by 2 and 3. ∴ n (n + 1) (n + 2) is divisible by 6.
Q. Express HCF of F 65 and 117 in the form of 65m +117n
Ans: Among 65 and 117; 117 > 65
Since 117 > 65, we apply the division lemma to 117 and 65 to obtain
117 = 65 x 1 + 52 … Step 1
Since remainder 52 ≠ 0, we apply the division lemma to 65 and 52 to obtain
65 = 52 x 1 + 13 … Step 2
Since remainder 13 ≠ 0, we apply the division lemma to 52 and 13 to obtain
52 = 4 x 13 + 0 … Step 3
In this step the remainder is zero. Thus, the divisor i.e. 13 in this step is the H.C.F. of the given numbers
The H.C.F. of 65 and 117 is 13
From Step 2:
13 = 65 – 52 x 1 … Step 4
From Step 1:
52 = 117 – 65 x 1
Thus, from Step 4, it is obtained
13 = 65 – (117 – 65 x 1) x 1
⇒13 = 65 x 2 – 117
⇒13 = 65 x 2 + 117 x (–1)
In the above relationship the H.C.F. of 65 and 117 is of the form 65m + 117 n, where m = 2 and n = –1
1oth maths term-1