Tag: similar triangles

Quizzes (14) Questions (23)

Questions Related to similar triangles

Multiple choice mathematical modelling proof by contradiction similar triangles

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

  1. Let $x ^2 $ not even
  2. Let $x$ not even
  3. Let $x$ be even natural number.
  4. Let $x\notin N$
Reveal answer Fill a bubble to check yourself
C Correct answer
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

Multiple choice mathematical modelling proof by contradiction similar triangles

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  

  1. The proof by contradiction

  2. The Direct method.

  3. Induction method.

  4. The proof by contrapositive method.
Reveal answer Fill a bubble to check yourself
B Correct answer
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 $B$

Multiple choice mathematical modelling proof by contradiction similar triangles

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 

  1. The Direct method.

  2. The proof by contradiction.

  3. The proof by contrapositive method.

  4. Induction method.
Reveal answer Fill a bubble to check yourself
A Correct answer
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

$A$

Multiple choice mathematical modelling proof by contradiction similar triangles

________ modelling where equations are developed and tested withinstated assumptions

  1. Optimistic

  2. Mathematical

  3. Scale

  4. Simulation
Reveal answer Fill a bubble to check yourself
B Correct answer
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.

Multiple choice mathematical modelling proof by contradiction similar triangles

The given equation $4xy-x-y=z^2$ has:

  1. three positive integer solutions

  2. one positive integer solutions

  3. two positive integer solutions

  4. no positive integer solutions
Reveal answer Fill a bubble to check yourself
D Correct answer
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.
Multiple choice mathematical modelling proof by contradiction similar triangles

A simple market model is an example of

  1. Static physical model

  2. Dynamic physical model

  3. Static mathematical model

  4. Dynamic mathematical model
Reveal answer Fill a bubble to check yourself
C Correct answer
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.

Multiple choice mathematical modelling proof by contradiction similar triangles

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

  1. a rational number

  2. an integer

  3. a natural number

  4. a real number
Reveal answer Fill a bubble to check yourself
A,B,C,D Correct answer
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.

Multiple choice mathematical modelling proof by contradiction similar triangles

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.

  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
Reveal answer Fill a bubble to check yourself
C Correct answer
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.




Multiple choice mathematical modelling proof by contradiction similar triangles

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

  1. The Direct method.

  2. The proof by contradiction.

  3. Induction method.

  4. The proof by giving counter example.
Reveal answer Fill a bubble to check yourself
D Correct answer
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

$D$

Multiple choice mathematical modelling proof by contradiction similar triangles

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

  1. One

  2. Two

  3. Three

  4. more than three
Reveal answer Fill a bubble to check yourself
A Correct answer
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.