Tag: mathematical modelling

Questions Related to mathematical modelling

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.