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Free Practice Questions for CAPESA ISE — Option Mathématiques
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Sample CAPESA ISE — Option Mathématiques Practice Questions
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1Which condition characterizes compact subsets of finite-dimensional Euclidean space ℝⁿ?
A.Closed and bounded
B.Closed but not necessarily bounded
C.Bounded but not necessarily closed
D.Open and bounded
Explanation: In ℝⁿ, a set is compact exactly when it is closed and bounded. Both conditions matter: an open ball is bounded but not compact, while ℝⁿ is closed but unbounded for n ≥ 1.
2What is the radius of convergence in x of Σ(n≥1) 3^n x^(2n)/n²?
A.3
B.1/√3
C.1/3
D.+∞
Explanation: The ratio of consecutive absolute terms tends to 3x². Absolute convergence holds for 3x² < 1, giving radius 1/√3; endpoint behavior does not change the radius.
3Let fn(x) = nx/(1 + n²x²) on [0,1]. Which statement correctly describes convergence to zero?
A.Uniform on [0,1] because each fixed x has limit zero
B.Not pointwise at x = 0
C.Pointwise on [0,1], uniform on every [a,1] with 0 < a ≤ 1, but not uniform on [0,1]
D.Uniform on [0,1] because every fn is continuous
Explanation: Each fixed x has limit zero, but fn(1/n) = 1/2, so the supremum on [0,1] does not tend to zero. On [a,1], |fn(x)| ≤ 1/(na), which tends uniformly to zero.
4For f(x,y) = x³y + 2xy² − 3x + 4, what is ∇f(1,2)?
A.(14,9)
B.(11,5)
C.(11,1)
D.(11,9)
Explanation: The partial derivatives are 3x²y + 2y² − 3 and x³ + 4xy. At (1,2), these are 11 and 9 respectively.
5What is the minimum of x² + 4y² subject to x + y = 1, for real x,y?
A.4/5, at (4/5,1/5)
B.1, at (1,0)
C.5/4, at (1/2,1/2)
D.0, at (0,0)
Explanation: Substitute y = 1 − x to obtain 5x² − 8x + 4 = 5(x − 4/5)² + 4/5. The square is minimized at x = 4/5, giving y = 1/5 and minimum 4/5.
6For which real α does the improper integral ∫₀^∞ x^(α−1)/(1 + x) dx converge?
A.α > 1
B.0 < α < 1
C.α < 0
D.−1 < α < 1
Explanation: Near zero the integrand behaves as x^(α−1), requiring α > 0. At infinity it behaves as x^(α−2), requiring α < 1; both conditions must hold.
7Let F(t) = ∫₀^∞ e^(−x)sin(tx)/x dx, with the integrand extended continuously at zero. Differentiation under the integral is justified by the integrable bound e^(−x). What are F'(t) and F(t)?
A.t/(1+t²) and ln(1+t²)/2
B.1/(1−t²) and arctanh t
C.1/(1+t²) and arctan t
D.e^(−t) and 1−e^(−t)
Explanation: Differentiation gives ∫₀^∞ e^(−x)cos(tx) dx = 1/(1+t²), obtained by integrating the real part of e^((−1+it)x). Since F(0)=0, integration in t gives F(t)=arctan t.
8Solve y' + 2y = 4x on ℝ.
A.Ce^(2x) + 2x − 1
B.Ce^(−2x) + 4x − 2
C.Ce^(−x) + 2x + 1
D.Ce^(−2x) + 2x − 1
Explanation: The homogeneous solution is Ce^(−2x). Trying a particular solution ax+b gives 2a=4 and a+2b=0, so a=2 and b=−1.
9What is the limit of (x³ + y³)/(x² + y²) as (x,y) tends to (0,0) away from the origin?
A.0
B.1
C.No limit because the axes differ
D.+∞
Explanation: Set x=r cos θ and y=r sin θ. The absolute quotient is at most 2r, which tends to zero independently of θ, proving the limit is zero.
10What is the degree-three Taylor polynomial of ln(1+2x) at zero?
A.2x−4x²+8x³
B.2x−2x²+(8/3)x³
C.2x−x²+(2/3)x³
D.2x+2x²+(8/3)x³
Explanation: Substitute u=2x into ln(1+u)=u−u²/2+u³/3+o(u³). The resulting polynomial is 2x−2x²+(8/3)x³.
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