Logical Reasoning — Chapter Test | Class 11 IMO

Q1

The locus of a point for which y = 0 and z = 0 is:

When both y and z coordinates are zero, the x-coordinate can be any real number. This is the exact definition of the x-axis.

Q2

What is the focal distance of a point (x₁, y₁) on the parabola x² = -4ay?

For x² = -4ay, focus is (0, -a) and directrix is y = a. By definition, focal distance equals distance to directrix. Distance from (x₁, y₁) to y = a is |y₁ - a|.

Q3

If A and B are equal sets, then:

Equal sets satisfy both conditions.

Q4

Which among the strictly matched pairing of spatial points reliably confirms their existence identically within the same absolute geometric octant?

Points sharing an octant practically display universally identical sign signatures. Evaluating strictly: (2, 3, −4) yields (+, +, −). Similarly, (5, 1, −2) delivers identically (+, +, −). Both firmly inhabit Octant V.

Q5

For any two sets A and B, the Commutative Law for union states that:

The Commutative Law implies that the order of sets does not change the result of the operation. So, A ∪ B = B ∪ A.

Q6

The compound statement (p → q) ∧ (q → p) is logically equivalent to:

The biconditional statement 'p if and only if q' strictly means 'if p then q' AND 'if q then p'.

Q7

If statement p is true, then the statement p ∨ ~p is a:

p ∨ ~p means 'True OR False', which always evaluates to True. A statement that is always true is called a tautology.

Q8

If z = 0 + 9i, then z is:

Real part is zero.

Q9

For a point P on the ellipse x²/49 + y²/25 = 1, find the sum of its focal distances.

Sum of focal distances = 2a. Here a²=49 => a=7. Sum = 2(7) = 14.

Q10

Which of the following sets are disjoint?

No common element exists.

Q11

Which among the strictly following defined spatial points sits nearest to the absolute spatial origin (0, 0, 0)?

Compare the squared absolute distances: A is 4+4+1 = 9. B is 9. C is 1+1+1 = 3. D is 0+4+4 = 8. The smallest squared distance is 3, proving (1, 1, 1) is absolutely closest.

Q12

The connective used in conjunction is:

Conjunction corresponds to the word 'and'.

Q13

Which equation represents an ellipse?

Sum of positive squared terms equals 1.

Q14

If |x + 2| ≤ 9 and x is an integer, how many solutions exist?

−9 ≤ x + 2 ≤ 9 → −11 ≤ x ≤ 7. Number of integers = 7 − (−11) + 1 = 19.

Q15

If sides are proportional to 3 : 4 : 5, then the triangle is:

3² + 4² = 5², so it is right-angled.