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Sample CAPESA ISE cycle long / AS Practice Questions
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1What is the limit of ln(1 + 3x)/x as x approaches 0?
A.3
B.1
C.0
D.+∞
Explanation: Write the quotient as 3 ln(1 + 3x)/(3x). The standard logarithmic limit is 1, so the required limit is 3.
2What is the definite integral of x/(1 + x²) from 0 to 1?
A.ln 2
B.(ln 2)/2
C.1/2
D.2 ln 2
Explanation: An antiderivative is ln(1 + x²)/2, since the derivative of 1 + x² is 2x. Evaluating at 1 and 0 gives (ln 2)/2.
3What is the general real solution of y'' − 5y' + 6y = 0?
A.C1 cos(2x) + C2 sin(3x)
B.(C1 + C2x)e^(2x)
C.C1e^(2x) + C2e^(3x)
D.C1e^(−2x) + C2e^(−3x)
Explanation: The characteristic polynomial factors as (r − 2)(r − 3). Its distinct real roots give independent solutions e^(2x) and e^(3x), with arbitrary real coefficients.
4Using integration by parts, evaluate the integral of xe^(2x) from 0 to 1.
A.(e² − 1)/4
B.e²/2
C.(e² − 1)/2
D.(e² + 1)/4
Explanation: Integration by parts gives the primitive xe^(2x)/2 − e^(2x)/4. Its values at 1 and 0 are e²/4 and −1/4, so their difference is (e² + 1)/4.
5What is the limit of x(√(x² + 4) − x) as x tends to +∞?
A.2
B.4
C.0
D.+∞
Explanation: Rationalizing gives 4x/(√(x² + 4) + x). Dividing by positive x gives 4/(√(1 + 4/x²) + 1), whose limit is 2.
6For x > −1/2, what is the derivative of ln(1 + 2x)?
A.1/(1 + 2x)
B.2/(1 + 2x)
C.2 ln(1 + 2x)
D.1/(2x)
Explanation: For positive differentiable u, the derivative of ln u is u'/u. Taking u = 1 + 2x gives 2/(1 + 2x).
7The sequence u0 = 1, u(n+1) = √(2 + un) is increasing and bounded above by 2. What is its limit?
A.√2
B.3/2
C.2
D.+∞
Explanation: Monotonicity and boundedness ensure a real limit L. Continuity gives L² − L − 2 = 0; since every term is at least 1, the admissible root is 2.
8In space, D1 consists of (t, t − 1, t) and D2 of (s, 1, s), for real t and s. Which statement is correct?
A.They are parallel and distinct.
B.They are skew.
C.They coincide.
D.They intersect at (2, 1, 2) and determine a plane.
Explanation: Equality of the first coordinates gives t = s, and the second gives t − 1 = 1, hence t = s = 2. Their direction vectors (1, 1, 1) and (1, 0, 1) are not proportional, so these distinct intersecting lines determine a plane.
9What is the modulus of 3 − 4i?
A.5
B.7
C.25
D.−1
Explanation: The modulus of a + bi is √(a² + b²). Thus |3 − 4i| = √(9 + 16) = 5.
10Evaluate the integral of sin²x from 0 to π/2.
A.1
B.π/4
C.π/2
D.1/2
Explanation: Use sin²x = (1 − cos 2x)/2. The primitive x/2 − sin(2x)/4 gives π/4 on this interval.
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