R_1=(x,y)∣x,y∈R, x^2+y^2≤ 25 and R_2=(x,y)∣x,y∈R, 9y≥ 4x^2. Then R_1∩R_2 is

D. None of these. Step 1: R_1 is every point on or inside the circle x^2+y^2=25 (centre origin, radius 5). Step 2: R_2 is every point on or above the parabola y=4x^29. Step 3: So R_1∩R_2 is the shaded region inside the circle and above the parabola (see figure). Step 4: 'One–one' and 'onto' are words used only for functions. A relation is a function only if each x has exactly one y. Step 5: Take x=0. The points (0,0), (0,1), (0,2), …, (0,5) all lie in the region, so one x is paired with many y values. Step 6: Hence R_1∩R_2 is not a function at all, so it can be neither one–one nor onto.

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$R_{1} = (x, y) ∣ x, y \in R, x^{2} + y^{2} \leq 25$ and $R_{2} = (x, y) ∣ x, y \in R, 9 y \geq 4 x^{2}$ . Then $R_{1} \cap R_{2}$ is

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Answer: Option D — None of these

Detailed Solution

$R_{1}$ is every point on or inside the circle $x^{2} + y^{2} = 25$ (centre origin, radius 5).

$R_{2}$ is every point on or above the parabola $y = \frac{4 x ^{2}}{9}$ .

So $R_{1} \cap R_{2}$ is the shaded region inside the circle and above the parabola (see figure).

'One–one' and 'onto' are words used only for functions. A relation is a function only if each $x$ has exactly one $y$ .

Take $x = 0$ . The points $(0, 0)$ , $(0, 1)$ , $(0, 2)$ , …, $(0, 5)$ all lie in the region, so one $x$ is paired with many $y$ values.

Hence $R_{1} \cap R_{2}$ is not a function at all, so it can be neither one–one nor onto.

Quick tip

A relation is a function only if every $x$ has exactly one $y$ (vertical line test)

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This question tests your understanding of Relations & Mappings, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

A relation from set A to set B is a subset of the Cartesian product A x B. A mapping, or function, is a special relation in which every element of A is related to exactly one element of B. This topic covers types of relations (reflexive, symmetric, transitive and equivalence relations), domain, codomain and range, and types of functions such as one-one (injective), onto (surjective) and bijective functions, along with composition and inverse functions. Exam questions test identifying the type of a given relation or function, counting the number of relations or functions between two sets, and finding the domain or range of a function. Useful counting facts: if A has m elements and B has n elements, there are 2^(mn) relations and n^m functions from A to B. Practise the relations and mappings questions below, each with a step-by-step solution.

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R1​=(x,y)∣x,y∈R,x2+y2≤25 and R2​=(x,y)∣x,y∈R,9y≥4x2. Then R1​∩R2​ is

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$R_{1} = (x, y) ∣ x, y \in R, x^{2} + y^{2} \leq 25$ and $R_{2} = (x, y) ∣ x, y \in R, 9 y \geq 4 x^{2}$ . Then $R_{1} \cap R_{2}$ is