JEE MATH APEX

IAT 2024

Mathematics Questions

JEE MATH APEX
Consider the following lines in the XY-plane:

L_1: 5x - 2y = 1,

L_2: the line passing through (0,1) and (100,101),

L_3: the line passing through (1,11) and parallel to the vector -\hat{\imath} + 2\hat{\jmath}.

Let A = (L_1 \cap L_2) \cup (L_2 \cap L_3) \cup (L_3 \cap L_1). What is the total number of elements of A?
(A) 3
(B) 0
(C) 1
(D) 2
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JEE MATH APEX
Let A be the set of points in the XY-plane which are equidistant from P(-1,0) and Q(1,0). Let B be the set of points in the XY-plane which are equidistant from A and Q. If (5,y) is a point in B, then what is the value of y^2?
(A) 9
(B) 1
(C) 4
(D) 16
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JEE MATH APEX
Consider the lines L_1 and L_2 given below:
L_1: x = 2 + \lambda, y = 3 + 2\lambda, z = 4 + 3\lambda;
L_2: x = 4 + \lambda, y = 4, z = 4 + \lambda.
If (2,3,4) is the point of L_1 that is closest to L_2, then which point of L_2 is closest to L_1?
(A) (3, 4, 3)
(B) (3, 4, 4)
(C) (5, 4, 5)
(D) (4, 4, 4)
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JEE MATH APEX
Let a_1, a_2, a_3, \ldots be a sequence of real numbers. Let s_n = a_1 + a_2 + \cdots + a_n. If 2s_n = n(c + a_n) for some real number c and for all n = 1, 2, 3, \ldots, then which one of the following statements is correct?
(A) a_1, a_2, a_3, \ldots is an Arithmetic Progression.
(B) a_1, 2a_2, 3a_3, \ldots is an Arithmetic Progression.
(C) a_1, a_2, a_3, \ldots is a Geometric Progression.
(D) a_1, 2a_2, 3a_3, \ldots is a Geometric Progression.
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JEE MATH APEX
Let f: \mathbb{R} \rightarrow \mathbb{R} be a strictly decreasing function with |f(t)| < \pi/2 for all t \in \mathbb{R}. Let g: [0, \pi] \rightarrow \mathbb{R} be a function defined by g(t) = \sin(f(t)). Which one of the following statements is Correct?
(A) g is decreasing on [0, \pi].
(B) g is increasing on [0, \pi].
(C) g is increasing on (0, \pi/2) and decreasing on (\pi/2, \pi).
(D) g is decreasing on (0, \pi/2) and increasing on (\pi/2, \pi).
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JEE MATH APEX
Let f, g: \mathbb{R} \rightarrow \mathbb{R} be functions. If g is continuous, then which one of the following cases implies that f is continuous?
(A) g(x) = (f(x))^3
(B) g(x) = |f(x)|
(C) g(x) = (f(x))^2
(D) g(x) = \sin(f(x))
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JEE MATH APEX
What is the largest area of a rectangle, whose sides are parallel to the coordinate axes, that can be inscribed under the graph of the curve y = 1 - x^2 and above the X-axis?
(A) \frac{4}{3\sqrt{3}}
(B) \frac{2}{3\sqrt{3}}
(C) \frac{4}{3}
(D) \frac{1}{3}
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JEE MATH APEX
Let M be the set of all 3 \times 3 matrices with real entries. Consider the relation R on M given by R = \{(A, B) \in M \times M : \det(A - B) \text{ is an integer}\}. Which one of the following statements is Correct?
(A) R is reflexive and symmetric, but not transitive.
(B) R is reflexive, but neither symmetric nor transitive.
(C) R is an equivalence relation.
(D) R is symmetric and transitive, but not reflexive.
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JEE MATH APEX
What is the value of {}^{23}C_0 + {}^{23}C_2 + {}^{23}C_4 + \cdots + {}^{23}C_{22}?
(A) 2^{22}
(B) 2^{22} - 1
(C) 2^{23} + 1
(D) 2^{23}
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JEE MATH APEX
Let f: \mathbb{Q} \rightarrow \mathbb{Q} be a function such that f(x + y) = f(x) + f(y) for all x, y \in \mathbb{Q}, and f(1) = 10. Which one of the following statements is Correct?
(A) f is bijective.
(B) f is injective but not surjective.
(C) f is surjective but not injective.
(D) f is neither injective nor surjective.
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JEE MATH APEX
Let
I = \int_{e^{-\pi/2}}^{e^{\pi/2}} (\sin^2(\log(x)) + \sin(\log(x^2))) \, dx.
What is the value of I?
(A) e^{\pi/2} - e^{-\pi/2}
(B) 0
(C) \frac{\pi e^{\pi/2}}{2}
(D) e^{\pi} - 1
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JEE MATH APEX
Consider the following subset of the XY-plane:
S = \{ (|z - iz|, |z|^2) : z \text{ is a complex number} \}.
Which one of the following statements is Correct?
(A) S is a parabola.
(B) S is a circle.
(C) S is an ellipse but not a circle.
(D) S is a hyperbola.
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JEE MATH APEX
A ship sets off on a voyage with three engines, labelled A, B, and C, which work independently. The ship can complete the voyage only if at least two of these engines keep working. The probability that engine A breaks down is 1/4, that engine B breaks down is 1/4, and that engine C breaks down is 1/2. What is the probability that the ship can complete the voyage?
(A) \frac{3}{4}
(B) \frac{1}{2}
(C) \frac{1}{32}
(D) \frac{1}{4}
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JEE MATH APEX
Consider the differential equation
\cos(y) \frac{dy}{dx} + \frac{1}{x} \sin(y) = x, \quad (x > 0);
given that y = \frac{\pi}{2} at x = \sqrt{3}. Which one of the following is the value of y at x = \frac{\sqrt{3}}{2}?
(A) \frac{\pi}{6}
(B) \frac{\pi}{3}
(C) \frac{\pi}{2}
(D) \frac{\pi}{4}
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JEE MATH APEX
In the given figure, the angles \angle BAQ = \angle CPQ = \angle CBQ = \frac{\pi}{2} and the lengths QA = 3 unit, AB = 4 unit, and BC = 1 unit. What is the length of PQ?
(A) 2.2 unit
(B) 2 unit
(C) \sqrt{2} unit
(D) 3 - \sqrt{2} unit
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