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Proof by contradiction - class-X

Description: proof by contradiction
Number of Questions: 23
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Tags: mathematical modelling similar triangles
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To understand objects that are too small or too large to see,

mathematical modelling proof by contradiction similar triangles

  1. physical model is used

  2. mathematical model is used

  3. conceptual model is used

  4. mechanical model is used

Show answer
Correct Option: A
Explanation:

In order to see too small objects , we use microscope 

We might use Physical model to see the microscopic cells as well as the large cells
Therefore option $A$ is correct

State the following statement is True or False
The terms  $(x+y)^2$ and $x^2$ + $ y^2$  are equivalent.

mathematical modelling proof by contradiction similar triangles

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

The statement is False as ${ \left( x+y \right)  }^{ 2 }={ x }^{ 2 }+{ y }^{ 2 }+2xy$ which is not equivalent to ${ x }^{ 2 }+{ y }^{ 2 }$.

Scientists who study universe are known as

mathematical modelling proof by contradiction similar triangles

  1. Astrologers

  2. Astronomers

  3. Geologists

  4. Meteorologists

Show answer
Correct Option: B
Explanation:

A: Astrologers $\rightarrow$ People who study of stars and other celestial objects.

B: Astronomers $\rightarrow$ People who study the universe.
C: Geologists $\rightarrow$ People who studies the solid and liquid matter that constitute the earth as well as the processes that shape it.
D: Meteorologists $\rightarrow$  People who study atmosphere. 



Scientists use models to help guide their search for

mathematical modelling proof by contradiction similar triangles

  1. life on earth

  2. existence of living organisms

  3. other worlds

  4. new information

Show answer
Correct Option: D
Explanation:

Scientists use models to help them get new information and use these information in their research.

The contradiction of the statement to prove by contradiction method of the following will be
" If function is continuous then it is differentiable."

mathematical modelling proof by contradiction similar triangles

  1. Assume function is differentiable.

  2. Assume function is not continuous.

  3. Assume function is continuous.

  4. Assume function is not differentiable.

Show answer
Correct Option: D
Explanation:

To prove an implication by contradiction, we need to assume the inverse of its conclusion and them show that our assumption is wrong.

Hence $D$ is correct

Which of the following is the correct steps to take when proving a statement using proof by contradiction?

mathematical modelling proof by contradiction similar triangles

  1. 1) Assume that your statement is true.

    2) Show this is the case using definitions and theorems.

    3) State that the statement is true.

  2. 1) Assume your statement is true for a certain instance.

    2) Show that it is true in more than one instance.

    3) State that your statement must be true.

  3. 1) Assume your statement to be false.

    2) Proceed as you would in a direct proof.

    3) Come across a contradiction.

    4) Use the contradiction to state that your assumption of the statement being false can't be the case, so your statement must be true.

  4. None of the answers are correct.

Show answer
Correct Option: C
Explanation:

Proof by contradiction can be used to prove any kind of statements.

Steps to be followed:
     $\rightarrow$ Assume the statement we want to prove to be false.
     $\rightarrow$ Then start proving from that statement. 
     $\rightarrow$ We end up seeing our assumption to be wrong.
     $\rightarrow$ Now we can conclude our statement to be true. 

Scientists who study earth's atmosphere have developed

mathematical modelling proof by contradiction similar triangles

  1. physical climate models

  2. conceptual climate models

  3. mathematical climate models

  4. aerial climate models

Show answer
Correct Option: C
Explanation:

Scientists who study about atmosphere develop mathematical climate models which depict longitudinal and latitudinal measures and area covered by a region.

A pattern, plan, representation or description designed to show structure is known as a

mathematical modelling proof by contradiction similar triangles

  1. sample

  2. model

  3. design

  4. structure

Show answer
Correct Option: B
Explanation:

Sample: It is a small amount of an element or a compound.

Model: A 3D plan or representation, designed to show structure.
Design: Pictorial representation of a structure.
Structure: An arrangement of the relations between the parts or elements of something complex.

So, the correct answer is "Model".

The contrapositve of the statement If Mohan works hard, then he gets a first class is

mathematical modelling proof by contradiction similar triangles

  1. If Mohan gets first class, then he does not works hard

  2. If Mohan does not get a first class, then he works hard

  3. If Mohan does not get a first class, then he does not work hard

  4. If Mohan does not work hard, then he does not get a first class

Show answer
Correct Option: A

To prove : "The integers can be of the form $4n,4n+1,4n+2  \ or \ 4n+3$" by direct method, we shall start the proof by the assumption 

mathematical modelling proof by contradiction similar triangles

  1. Let integers not of form $4n,4n+1,4n+2 \ or \ 4n+3$.

  2. Let integers be not of form $4n$.

  3. Let $z$ be any integer.

  4. Let $z<0$

Show answer
Correct Option: C
Explanation:

As we are solving by direct method we need to assume what is given in the hypothesis, which is in this case z be any integer.

$C$

To prove " If  $x,x\in N$  is even then $x^2$ is even". By direct method, we must start with the assumption:

mathematical modelling proof by contradiction similar triangles

  1. Let $x ^2 $ not even

  2. Let $x$ not even

  3. Let $x$ be even natural number.

  4. Let $x\notin N$

Show answer
Correct Option: C
Explanation:

To prove an implication by straight forward approach or direct method, we have to take the hypothesis of the statement as an assumption and then reach at the conclusion. 

In the statement, "If $p$ then $q$",  $p$ is hypothesis and $q$  is conclusion.
So, here $C$ is correct

To prove: "The perpendicular from centre of a circle to the chord, bisects the chord." The proof started from assumption "Let OM be the perpendicular to chord AB". 

This method of proof is  

mathematical modelling proof by contradiction similar triangles

  1. The proof by contradiction

  2. The Direct method.

  3. Induction method.

  4. The proof by contrapositive method.

Show answer
Correct Option: B
Explanation:

In direct method we have to assume the hypothesis as we have assumed the hypothesis "Let OM be the perpendicular to chord AB" in this case hence the option is $B$

To prove: "If $f,g $ are continuous functions then $f+g$ is continuous." The proof started from assumption " Let $f,g$ be continuous functions."  

This method of proof is 

mathematical modelling proof by contradiction similar triangles

  1. The Direct method.

  2. The proof by contradiction.

  3. The proof by contrapositive method.

  4. Induction method.

Show answer
Correct Option: A
Explanation:

In direct method we have to assume the hypothesis as we have assumed the hypothesis f,gf,g be continuous functions in this case,hence it is proved by direct method

$A$

________ modelling where equations are developed and tested withinstated assumptions

mathematical modelling proof by contradiction similar triangles

  1. Optimistic

  2. Mathematical

  3. Scale

  4. Simulation

Show answer
Correct Option: B
Explanation:

A mathematical model is a description of a system using mathematical concepts and language . The process of developing a mathematical model is termed mathematical modeling.
It helps in testing assumption.

The given equation $4xy-x-y=z^2$ has:

mathematical modelling proof by contradiction similar triangles

  1. three positive integer solutions

  2. one positive integer solutions

  3.  two positive integer solutions

  4.  no positive integer solutions

Show answer
Correct Option: D
Explanation:
Suppose all the solution of the given equation are positive integers
We write the equation in the equivalent form

$(4x-1)(4y-1)=4z^2+1$.

Let $p$ be a prime divisor of $4x-1$. Then

$4z^2+1\equiv 0$(mod $p$)

or

$(2z)^2\equiv -1$ (mod $p$).

On the other hand, Fermat's theorem yields

$(2z)^{p-1}\equiv 1$ (mod $p$)

hence

$(2z)^{p-1}\equiv (2z^2)^{\frac{p-1}{2}}\equiv (-1)^{\frac{p-1}{2}}\equiv 1$(mod $p$)

This implies that $p \equiv 1$ (mod $4$). It follows that all prime divisors of $4x 1$ are congruent to $1$ modulo $4,$ hence $4x 1 1$ (mod $4$), a contradiction.

Hence they are no positive integer solutions.

A simple market model is an example of

mathematical modelling proof by contradiction similar triangles

  1. Static physical model

  2. Dynamic physical model

  3. Static mathematical model

  4. Dynamic mathematical model

Show answer
Correct Option: C
Explanation:

In $simple-market-model$  generally there is a balance between supply and demand. Both factors depend on price. Demand for the commodity will be low when the price is high and it will increase as the price drops. If we take the simplistic linear case the relationship between demand (𝑄) and price (𝑃) might be represented by the straight line.


Therefore,  $simple-market-model$  is a $static-mathematical-model$ as it doesn't vary with time.

$\forall n\in N$, value of $\displaystyle \frac{n^{4}}{24}+\frac{n^{3}}{4}+\frac{11n^{2}}{24}+\frac{n}{4}$ is

mathematical modelling proof by contradiction similar triangles

  1. a rational number

  2. an integer

  3. a natural number

  4. a real number

Show answer
Correct Option: A,B,C,D
Explanation:

 Given$ \dfrac{[n^{4}+6n^{3}+11n^{2}+6n]}{24}$


            $\dfrac{[n(n+1)(n+2)(n+3)]}{24}$

             $=^{(n+3)}{C _{4}}$
Thus the above number is always divisible by $24$ Thus all four options are correct.

If a triangle is equiangular, then it is an obtuse angled triangle. Which of the following statements doesn't convey the same meaning as of this mentioned sentence.

mathematical modelling proof by contradiction similar triangles

  1. A triangle is equiangular only if it is an obtuse angled triangle

  2. If a triangle is not obtuse angled triangle then it is not an equiangular triangle.

  3. Equiangularity is a sufficient condition for triangle to be obtuse angled.

  4. A triangle is only obtuse is obtuse angled if it is equiangular

Show answer
Correct Option: C
Explanation:

Consider the given statements to be in the form of $p\rightarrow{q}$

Options A, B and C represents $q\rightarrow{p}, \sim{p}\rightarrow\sim{q}$ and $\sim{q}\rightarrow\sim{p}$ respectively.

Hence, option C, which is in the form of $p\rightarrow{q}$ is the correct answer.




To prove "" All prime numbers are not odd."  we showed that "$2$ is even and prime"
This method is  

mathematical modelling proof by contradiction similar triangles

  1. The Direct method.

  2. The proof by contradiction.

  3. Induction method.

  4. The proof by giving counter example.

Show answer
Correct Option: D
Explanation:

counterexample is a special kind of example that disproves a statement or proposition.hence it disproves that all prime numbers are odd hence it is proof by counter example

$D$

To prove any preposition by "giving counter example" we must give at-least ______ example(s).

mathematical modelling proof by contradiction similar triangles

  1. One

  2. Two

  3. Three

  4. more than three

Show answer
Correct Option: A
Explanation:
In order to prove any given statement wrong, we need to specify atleast one example against that example.

Hence, to prove any preposition by "giving counter example" we must give at least ONE example.

The proposition $(p \, \Rightarrow \, ~p)\,\wedge \, (~p \, \Rightarrow  \, p)$ is a

mathematical modelling proof by contradiction similar triangles

  1. Tautology

  2. neither tautology nor contradiction

  3. contradiction

  4. None of these

Show answer
Correct Option: C
Explanation:
p ~p $P\Rightarrow   ~P$ $~P\Rightarrow P$ $(P\Rightarrow ~P)\wedge (~P\Rightarrow P)$
T F  F   T                    F
F T  T F F

Which of the following is true for counter example.

mathematical modelling proof by contradiction similar triangles

  1. A counter example is an exception to a proposed general rule or law

  2. A counter example is a specific instance of the falsity of a universal quantification (a "for all" statement).

  3. Any hard-working student is a counter example to "all students are lazy"

  4. None of these

Show answer
Correct Option: A,B,C
Explanation:

All three a,b,c are true

A counter example is an exception to a proposed general rule or law is a defination.Counterexamples are often used in math to prove the boundaries of possible theorems.option C is an example of counterexample and option b and option a are more or less same

Counter example to the statement "All prime numbers are odd." is

mathematical modelling proof by contradiction similar triangles

  1. The prime number $2$

  2. The prime number $3$

  3. Number $1$

  4. None of these

Show answer
Correct Option: A
Explanation:

$2$ is only even number which is prime number.

So, option A is correct.

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