2D Cartesian system Practice Questions - NDA Mathematics Question Bank

Question 1 of 10 Authentic UPSC NDA Question

What is the maximum numberof points ofintersection of 5 circles?

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Solution

Option (c) is correct.

Explanation:

The maximum number of intersection points between two circles is 2. Thus, for \( n \) circles, the maximum number of intersection points \( P \) can be calculated using the formula:

\[ P = n(n-1) \]

where \( n \) is the total number of circles. The factor of 2 is included due to the two intersection points per circle pair. Therefore, we rewrite:

\[ P = 2 \cdot \frac{n(n-1)}{2} \]

For \( n = 5 \):

\[ P = 2 \cdot \frac{5(5-1)}{2} = 2 \cdot \frac{5 \cdot 4}{2} = 5 \cdot 4 = 20 \]

Thus, the maximum number of intersection points of 5 circles is 20.

📌 Hints / Properties Used:

Maximum intersections between two circles = 2
General formula for circles' intersections = \( P = 2 \cdot \frac{n(n-1)}{2} \)

📊 Visual Diagram Suggestion:

A diagram illustrating five overlapping circles with highlighted intersection points demonstrating a total of 20 intersections would clarify the solution visually.

📖 Factual Verification & Reference:

Verified against standard combinatorial geometry principles.

Question 2 of 10 Authentic UPSC NDA Question

For the following two (02) items:

Suppose S is the sphere with the smallest radius that passes through the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1). On

which one of the following planes does the centre of S lie?

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Question 3 of 10 Authentic UPSC NDA Question

For the following two (02) items: Suppose S is the sphere with the smallest radius that passes through the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1).

What is the radius of S?

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Question 4 of 10 Authentic UPSC NDA Question

For the following two (02) items:

A plane P is parallel to the line having direction ratios ⟨1, 3, 2⟩ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0.

What is the equation of the plane P?

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Question 5 of 10 Authentic UPSC NDA Question

For the following two (02) items:

A plane P is parallel to the line having direction ra- tios ⟨1, 3, 2⟩ and contains the line of intersection of the planes 6x + 4y - 5z = 2 and x - 2y + 3z = 0.

Which of the following are the direction ratios of the line of intersection of the given planes?

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Question 6 of 10 Authentic UPSC NDA Question

If any point on an ellipse is (3sinα, 5cosα), then what is the eccentricity of the ellipse?

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Question 7 of 10 Authentic UPSC NDA Question

What is the distance between the two foci of the hyperbola 25x² – 75y² = 225?

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Question 8 of 10 Authentic UPSC NDA Question

A tangent to the parabola y² = 4x is inclined at an angle of 45° with the positive direction of x-axis. What is point of contact of the tangent and the parabola?

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Question 9 of 10 Authentic UPSC NDA Question

A square is inscribed in a circle x² + y² + 2x + 2y + 1 =0 and its sides are parallel to coordinate axes. Which one of the following is a vertex of the square?

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Question 10 of 10 Authentic UPSC NDA Question

For the following two (02) items:

Let A(1, –1, 0), B(-2, 1, 8) and C(-1, 2, 7) are three consecutive vertices of a parallelogram ABCD.

What is the fourth vertex D?

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2D Cartesian system Questions

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