Case study (4 Marks)

Question 14 Marks

Aarushi and Amin are playing with match-sticks by making different geometrical and other figures. Avni kept one match-stick horizontally and then two match-sticks vertically as shown in Figure and then asks Aarushi to join the open ends of horizontally and vertically placed strings by a thread. Avni's eleder sister Mira comes and ask them to find the length of the thread if each matchstick is of unit length.
Aarushi replies that the length of the thread can be found by using Pythagoras Theorem and it is equal to $\sqrt{1^2+2^2}=\sqrt{4+1}=\sqrt{5}$ units using your knowledge about numbers, answer the following questions.

(i) $\sqrt{5}$ is
(a) a rational number
(b) an irrational number
(c) non-terminating non-recurring
(d) not possible
(ii) The decimal representation of an irrational number is
(a) terminating $\quad$(b) non-terminating recurring $\quad$(c) an integer $\quad$(d) a whole number $\quad$
(iii) The decimal representation of a rational number cannot be
(a) terminating
(b) non-terminating
(c) non-terminating repeating
(d) non-terminating non-repeating
(iv) the sum of any two irrational numbers is
(a) always an irrational number
(b) always a rational number
(c) always an integer
(d) sometimes rational, sometimes irrational

Answer

(i) (b): $\sqrt{5}$ is an irrational number, because the square root of a prime number is an irrational number.
(ii) (c)
(iii) (d)
(iv) (d).

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Question 24 Marks

Ravish and Aarushi dedcided to visit world book fair which is organised every year. During their visit Aarushi was fascinated by the cover page of a book with $\pi / e$ written on it. $\pi$ and e are mathematical constants. In Euclidean geometry $\pi$ is defined as the ratio of a circle's circumference to its diameter. It is also referred to as Archimede's constant. The constant e is known as Euler's number and it is the limiting value of $\left(1+\frac{1}{n}\right)^n$ as $n$ approches infinity. Using the knowledge of rational and irrational numbers answer the following questions.
(i) $\pi$ represents
(a) an integer
(b) a rational number
(c) an irrational number
(d) a natural number
(ii) e represents
(a) a natural number
(b) an integer
(c) a rational number
(d) an irrational number
(iii) The product of any two irrational numbers is
(a) always an irrational number $\quad$(b) not necessarily an irrational number $\quad$
(c) never an irrational number $\quad$ (d) always an integer $\quad$
(iv) A rational number between $\sqrt{2}$ and $\sqrt{3}$ is
(a) $\frac{\sqrt{3}-\sqrt{2}}{2}$$\quad$(b) $\frac{\sqrt{3}+\sqrt{2}}{2}$$\quad$(c) $1 . \overline{6}$ $\quad$(d) $0 . \overline{2}+0 . \overline{3}$$\quad$

Answer

(i) (c): $\pi$ is an irrational number its value is $3.1415926538 \ldots$
(ii) (d): $e$ is an irrational number and its value is $2.71828182845 \ldots$
(iii) (b): The product of irrational numbers $2+\sqrt{3}$ and $2-\sqrt{3}$ is 1 , which is rational. However, the product of rational numbers $(2+\sqrt{3})$ and $\sqrt{3}$ is $2 \sqrt{3}+3$ which is an irrational number. Hence, the product of two irrational numbers is sometimes rational and sometimes irrational.
(iv) (c): $\frac{\sqrt{3}-\sqrt{2}}{2}$ and $\frac{\sqrt{3}+\sqrt{2}}{2}$ are irrational numbers. So, options (a) and (b) are not correct.
We find that $0 . \overline{2}+0 . \overline{3}=0 . \overline{5}$, which does not lie between $\sqrt{2}$ and $\sqrt{3}$.
We have $\sqrt{2}=1.412 \ldots$ and $\sqrt{3}=1.736 \ldots$
Clearly, $\sqrt{2}<1 . \overline{6}<\sqrt{3}$. Hence, option (c) is correct.

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Maths Case study (4 Marks)

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