10 Question MCQ (including Assertion)

Question 1 of 10

The locus of a point which moves such that its distance from the \(y\)-axis is twice its distance from the \(z\)-axis is given by:

A \(x^2 + z^2 = 4(x^2 + y^2)\) B \(x^2 + z^2 = 2(x^2 + y^2)\) C \(x^2 + y^2 = 4(x^2 + z^2)\) D \(y^2 + z^2 = 4(x^2 + y^2)\)

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Question 2 of 10

Three vertices of a parallelogram \(ABCD\) are \(A(3, -1, 2)\), \(B(1, 2, -4)\), and \(C(-1, 1, 2)\). The coordinates of vertex \(D\) are:

A \((1, -2, 8)\) B \((1, -2, -4)\) C \((-3, 4, -4)\) D \((3, 2, 8)\)

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Question 3 of 10

The perpendicular distance of any general point \(P(x, y, z)\) from the \(x\)-axis is:

A \(|x|\) B \(\sqrt{y^2 + z^2}\) C \(\sqrt{x^2 + z^2}\) D \(\sqrt{x^2 + y^2}\)

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Question 4 of 10

The maximum length of a rod that can be placed inside a cuboidal container with side lengths \(a, b,\) and \(c\) aligned along coordinate axes is:

A \(a + b + c\) B \(\sqrt{a^2 + b^2 + c^2}\) C \(\sqrt{ab + bc + ca}\) D \(2\sqrt{a^2 + b^2 + c^2}\)

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Question 5 of 10

If a sphere has diameter endpoints at \((2, 3, -1)\) and \((4, -1, 5)\), its center coordinates are:

A \((3, 1, 2)\) B \((6, 2, 4)\) C \((1, -2, 3)\) D \((3, 2, 2)\)

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Question 6 of 10

Assertion (A): The reflection of the point \(P(2, -3, 5)\) across the \(XY\)-plane is \((-2, 3, 5)\).
Reason (R): When a point is reflected in the \(XY\)-plane, its \(z\)-coordinate changes sign while its \(x\) and \(y\) coordinates remain unchanged.

A Both (A) and (R) are true and (R) is the correct explanation of (A). B Both (A) and (R) are true but (R) is NOT the correct explanation of (A). C (A) is true but (R) is false. D (A) is false but (R) is true.

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Question 7 of 10

A point \(P\) lies on the \(x\)-axis and is equidistant from \((1, 2, 3)\) and \((2, 3, 4)\). The \(x\)-coordinate of \(P\) is:

A \(\frac{15}{2}\) B \(\frac{13}{2}\) C \(7\) D \(6\)

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Question 8 of 10

Assertion (A): The distance of the point \(P(3, -4, 12)\) from the origin \(O(0,0,0)\) is \(13\text{ units}\).
Reason (R): The distance of any point \(P(x, y, z)\) from the origin is given by \(\sqrt{x^2 + y^2 + z^2}\).

A Both (A) and (R) are true and (R) is the correct explanation of (A). B Both (A) and (R) are true but (R) is NOT the correct explanation of (A). C (A) is true but (R) is false. D (A) is false but (R) is true.

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Question 9 of 10

Assertion (A): If a point \(P(a, b, c)\) lies in Octant V, then \(a > 0\), \(b > 0\), and \(c < 0\).
Reason (R): Octant V consists of all points in space where the \(x\) and \(y\) coordinates are positive while the \(z\) coordinate is negative.

A Both (A) and (R) are true and (R) is the correct explanation of (A). B Both (A) and (R) are true but (R) is NOT the correct explanation of (A). C (A) is true but (R) is false. D (A) is false but (R) is true.

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Question 10 of 10

If \(A(1, 3, 0)\), \(B(2, 4, k)\), and \(C(3, 5, -2)\) are collinear points in 3D space, then the value of \(k\) is:

A \(1\) B \(-1\) C \(2\) D \(-2\)

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10 Question MCQ (including Assertion)