SATHEE: Permutations and Combinations 4 Question 2

Permutations and Combinations 4 Question 2

2. Let $S$ be the set of all triangles in the $x y$ -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is

(2019 Main, 9 Jan II)

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Answer:

Correct Answer: 2. (a)

Solution:

According to given information, we have the following figure.

(Note that as $a$ and $b$ are integers so they can be negative also). Here $O (0, 0), A (a, 0)$ and $B (0, b)$

are the three vertices of the triangle.

Clearly, $O A = | a |$ and $O B = | b |$ .

$∴$ Area of $△ O A B = \frac{1}{2} | a | | b |$ .

But area of such triangles is given as 50 sq units.

$\begin{aligned} ∴ \frac{1}{2} | a | | b | = 50 \\ \Rightarrow | a | | b | = 100 = 2^{2} \cdot 5^{2} \end{aligned}$

Number of ways of distributing two 2’s in $| a |$ and $| b | = 3$

$\| a \|$	$\| b \|$
0	2
1	1
2	0

$\Rightarrow 3$ ways

Similarly, number of ways of distributing two 5’s in $| a |$ and $| b | = 3$ ways.

$∴$ Total number of ways of distributing 2’s and 5’s $= 3 \times 3 = 9$ ways

Note that for one value of $| a |$ , there are 2 possible values of $a$ and for one value of $| b |$ , there are 2 possible values of $b$ .

$∴$ Number of such triangles possible $= 2 \times 2 \times 9 = 36$ .

So, number of elements in $S$ is 36 .

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Permutations and Combinations 4 Question 2

2. Let $S$ be the set of all triangles in the $x y$ -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is

Answer:

$\Rightarrow 3$ ways

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नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Permutations and Combinations 4 Question 2

2. Let S be the set of all triangles in the xy-plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in S has area 50 sq. units, then the number of elements in the set S is

Answer:

⇒3 ways

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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2. Let $S$ be the set of all triangles in the $x y$ -plane, each having one vertex at the origin and the other two vertices lie on coordinate axes with integral coordinates. If each triangle in $S$ has area 50 sq. units, then the number of elements in the set $S$ is

$\Rightarrow 3$ ways