Detailed Solutions of IISER Aptitude Test 2024 (Mathematics)
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00:00 Start
00:18 About CDP
03:39 Q1 Consider the following lines in the XY -plane:
L1 : 5x − 2y = 1,
L2 : the line passing through (0, 1) and (100, 101),
L3 : the line passing through (1, 11) and parallel to the vector −bi + 2bj.
Let A = (L1 ∩ L2) ∪ (L2 ∩ L3) ∪ (L3 ∩ L1). What is the total number of elements of A ?
14:58 Q2 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 y2 ?
21:04 Q3 Consider the lines L1 and L2 given below:
L1 : x = 2 + λ, y = 3 + 2λ, z = 4 + 3λ; L2 : x = 4 + λ, y = 4, z = 4 + λ.
If (2, 3, 4) is the point of L1 that is closest to L2, then which point of L2 is closest to L1?
31:54 Q4 Let a1, a2, a3, . . . be a sequence of real numbers. Let sn = a1+a2+· · ·+an. If 2sn = n(c + an) for some real number c and for all n = 1, 2, 3, . . ., then which one of the following statements is Correct?
35:51 Q5 Let f : R → R be a strictly decreasing function with |f(t)| less than π/2 for all t ∈ R. Let g : [0, π] → R be a function defined by g(t) = sin(f(t)). Which one of the following statements is Correct?
39:05 Q6 Let f, g : R → R be functions. If g is continuous, then which one of the following cases implies that f is continuous?
44:14 Q7 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 − x2 and above the X-axis?
48:23 Q8 Let M be the set of all 3×3 matrices with real entries. Consider the relation R on M given by R = {(A,B) ∈ M×M : det(A−B) is an integer}. Which one of the following statements is Correct?
55:50 Q9 What is the value of 23C0 + 23C2 + 23C4 + · · · + 23C22?
58:26 Q10 Let f : Q → Q be a function such that f(x + y) = f(x) + f(y) for all x, y ∈ Q, and f(1) = 10. Which one of the following statements is Correct?
01:05:01 Q11 Let I = int_{e^-pi/2}^{e^pi/2} sin^2 (log x) + sin (log(x^2) dx. What is the value of I?
01:09:04 Q12 Consider the following subset of the XY -plane: S = (|z − iz|, |z|2) : z is a complex number. Which one of the following statements is Correct?
01:11:26 Q13 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?
01:15:36 Q14 Consider the differential equation cos(y) dy/dx+ 1/x sin(y) = x, (x 0);given that y = π/2
at x = √3. Which one of the following is the value of y at x = sqrt3 /2?
01:25:58 Q15 In the given figure, the angles ∠BAQ = ∠CPQ = ∠CBQ = π/2 ; and the lengths QA = 3 unit, AB = 4 unit, and BC = 1 unit. What is the length of PQ?
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