2nd Year Mathematics — MCQs & Practice Questions

Correct answer: C — 4

Factorise the numerator as (x−2)(x+2) and cancel to get x + 2, whose value at x = 2 is 4. The function itself is undefined at x = 2, but a limit never asks about the value at the point — only about the approach to it.

Correct answer: B — The left-hand and right-hand limits both exist and are equal

Only the two-sided agreement is required. f(a) need not exist at all — the previous question is exactly that case. Continuity and differentiability are stronger conditions built on top of the limit, not requirements for it.

Correct answer: B — 1

This is the standard limit, and it holds only when θ is in radians — the reason radians are used throughout calculus. Direct substitution gives 0/0, an indeterminate form, which is the signal that a standard result or a factorisation is needed.

Correct answer: D — Does not exist

From the right the function is constantly +1; from the left it is constantly −1. The one-sided limits disagree, so the two-sided limit does not exist. Both 1 and −1 are correct one-sided answers, which is what makes them tempting.

Correct answer: D — f must be differentiable at a

Differentiability is stronger than continuity, not required for it. f(x) = |x| is continuous at 0 but has no derivative there because of the corner. Every differentiable function is continuous; the converse fails.

Correct answer: D — The form is indeterminate and more work is needed

An indeterminate form is not an answer — it is an instruction to factorise, rationalise or cancel and try again. Depending on the function, 0/0 can resolve to any value at all, or to no limit.

Correct answer: B — At x = 3 the curve is rising with slope 5

A derivative is a slope, not a height. f′(3) = 5 says that at the instant x = 3, the curve climbs 5 units of y for every 1 unit of x. The actual height at x = 3 is f(3), which is a completely different number.

Correct answer: A — 12x² − 7

Power rule on each term: 4x³ → 3·4x² = 12x². Then −7x → −7 (the x disappears). The constant +2 → 0, because a constant never changes, so its rate of change is zero.

Correct answer: B — Because the formula would give 0/0, which is undefined

Substituting h = 0 directly gives [f(x) − f(x)]/0 = 0/0 — meaningless. The limit lets us ask what the expression heads toward as h shrinks, without ever dividing by zero.

Correct answer: B — 5 cos(5x)

Chain rule. Outer function sin → cos(5x). Inner function 5x has derivative 5. Multiply them: 5 cos(5x). Forgetting the inner 5 is the single most common slip in calculus.

Correct answer: B — The estimate converges toward the true area

Each rectangle misses a small triangular sliver above it. Narrower rectangles mean smaller slivers, so the total gap shrinks toward zero. That convergence is exactly what the integral is defined as.

Correct answer: B — 8

The antiderivative of 2x is x². Apply the Fundamental Theorem: F(3) − F(1) = 9 − 1 = 8. Sanity check: the region is a trapezium with parallel sides 2 and 6 and width 2, giving ½(2+6)(2) = 8.

Correct answer: C — x⁴/4 + C

Raise the power by one and divide by the new power: x⁴/4 + C. Answer A is the derivative rather than the integral — the classic direction error. Check any integration by differentiating your answer: d/dx (x⁴/4) = x³. ✓

Correct answer: B — It would require dividing by zero

The rule divides by n + 1, which is zero when n = −1. The integral certainly exists — it is ln|x| + C. The modulus signs matter, since 1/x is defined for negative x but the logarithm is not.

Correct answer: A — 8

The antiderivative is x³, so the value is 2³ − 0³ = 8. A common slip is forgetting to subtract the lower limit, or integrating to x³/3 by dividing by the new power while ignoring the 3 already in front.

Correct answer: B — Differentiation destroys constants, so many functions share one derivative

x², x² + 5 and x² − 100 all differentiate to 2x, so reversing the process cannot recover which one you started from. C stands for that whole family. It disappears in definite integrals because it cancels in F(b) − F(a).

Correct answer: B — Integration by substitution

Substitution undoes the chain rule; integration by parts undoes the product rule. Recognising which rule created the expression is usually the fastest route to choosing the right technique.

Correct answer: B — Displacement

Integrating velocity accumulates distance travelled, giving displacement — the same thing as the area under the graph. Differentiating instead would give acceleration, which is the mirror-image operation.

Correct answer: B — −2

m = −a/b = −6/3 = −2. Rearranging gives y = −2x + 3, which agrees.

Correct answer: B — 2

Substitute the point into the general form and divide by √(a² + b²): |3(0) + 4(0) − 10| / √(9 + 16) = 10/5 = 2. The 3–4–5 pattern in the coefficients is why the square root comes out exactly, and examiners choose such coefficients deliberately.

Correct answer: B — perpendicular

4 × (−1/4) = −1, which is the perpendicularity condition. It also makes the denominator of the angle formula zero, so tan θ is undefined and θ = 90°.

Correct answer: A — x/2 + y/5 = 1

The intercept form puts each intercept under its own variable. Checking: at y = 0 we get x = 2 ✓, and at x = 0 we get y = 5 ✓.

Correct answer: B — (2, 3)

Average each coordinate: x = (0+6+0)/3 = 2 and y = (0+0+9)/3 = 3. Option A is the midpoint of the hypotenuse, a different point.

Correct answer: B — (6, 0)

(3×10 + 2×0)/(3+2) = 30/5 = 6. Three parts of the five lie behind it, so it sits closer to the far end — as the answer shows.

Correct answer: B — never — the modulus makes it positive

The numerator is taken in modulus, so the result is always positive. A distance cannot be negative, whichever side of the line the point lies on.

Correct answer: A — 2x − y − 1 = 0

Parallel means the same gradient of 2, so only the constant changes. Substituting (1,1) into 2x − y + k = 0 gives k = −1.

Correct answer: B — a solid line

The ≤ sign includes the boundary itself, so the line is solid. A dashed line would mean the points on it are excluded.

Correct answer: C — (0, 0)

The origin makes the arithmetic trivial and the line does not pass through it. Options A and B lie on the line itself and so decide nothing.

Correct answer: B — satisfy every constraint

Feasibility requires all constraints to hold at once; it is the intersection of the half-planes, not their union.

Correct answer: B — at a corner point

The objective line sweeps across with a constant gradient, so it leaves a convex polygon at a vertex.

Correct answer: B — an unbounded region and a meaningless answer

Without the non-negativity constraints the region can extend into quadrants where the variables are negative, which cannot represent a number of items produced.

Correct answer: B — solving pairs of boundary equations

A hand-drawn graph is not accurate enough for exact coordinates, and the marks are for exact values. Solving simultaneously gives them.

Correct answer: C — 18 at two points

The values are 0, 10, 18 and 18. Two corners tie, which means the objective line is parallel to the edge joining them and every point on that edge is optimal.

Correct answer: B — redundant

A redundant constraint is automatically satisfied whenever the others are, so removing it changes nothing. Noticing one saves you drawing a line.

Correct answer: C — exactly 1

A parabola is defined by being equidistant from the focus and the directrix, so the ratio of those distances is exactly 1.

Correct answer: B — (2, −5)

2g = −4 gives g = −2, and 2f = 10 gives f = 5. The centre is (−g, −f) = (2, −5).

Correct answer: B — (0, 2)

Comparing with x² = 4ay gives 4a = 8, so a = 2. Since x is the squared variable, the parabola opens upward and the focus is on the y-axis at (0, 2).

Correct answer: B — along the y-axis

The larger denominator, 25, sits under y², so the ellipse is taller than it is wide and the major axis is vertical. The foci are then on the y-axis.

Correct answer: B — a² + b²

The hyperbola adds; the ellipse subtracts. That is why a hyperbola always has e > 1 — c is bigger than a.

Correct answer: C — no real circle

r² = g² + f² − c = 1 + 1 − 10 = −8, which is negative. No real points satisfy the equation, and saying so is the correct answer.

Correct answer: B — y = ±(3/2)x

y = ±(b/a)x with a = 2 and b = 3 gives y = ±(3/2)x. Note a comes from the term with the plus sign.

Correct answer: B — a circle

When c = 0 the eccentricity is 0, both foci sit at the centre, and a = b — the equation becomes x² + y² = a², a circle.

Correct answer: C — Velocity

Velocity is speed together with a direction. Speed on its own would be a scalar, which is precisely the distinction being tested.

Correct answer: A — ⟨7, 2⟩

Add componentwise: 2 + 5 = 7 and 3 + (−1) = 2.

Correct answer: B — 10

√(36 + 64) = √100 = 10. This is the 3–4–5 triple scaled by 2, which examiners use frequently.

Correct answer: B — ⟨3, −4⟩

AB = b − a = ⟨5 − 2, 3 − 7⟩ = ⟨3, −4⟩. Option C is BA, the vector in the opposite direction.

Correct answer: B — ⟨0, 1⟩

The magnitude is 4, so dividing gives ⟨0, 1⟩, which points the same way and has length 1.

Correct answer: A — ⟨2, −3⟩

⟨6, −9⟩ = 3⟨2, −3⟩, a scalar multiple, so they are parallel. Option B is perpendicular to it, not parallel.

Correct answer: B — a parallelogram

Equal vectors mean the sides are the same length and parallel, which defines a parallelogram. It might additionally be a rectangle or rhombus, but nothing here establishes that.

Correct answer: A — ⟨4, −11⟩

3⟨2, −1⟩ = ⟨6, −3⟩ and 2⟨1, 4⟩ = ⟨2, 8⟩. Subtracting gives ⟨6 − 2, −3 − 8⟩ = ⟨4, −11⟩.

Correct answer: B — 7

Square each component, add, then take the root: √(2² + (−3)² + 6²) = √(4 + 9 + 36) = √49 = 7. The minus sign on the j component disappears when it is squared, so a negative component never reduces the magnitude.

Correct answer: B — 0

a · b = |a||b| cos 90° = 0. This is the standard test for perpendicularity.

Correct answer: B — a vector perpendicular to both

The cross product returns a vector at right angles to the plane containing a and b, with direction given by the right-hand rule.

Correct answer: B — parallel

|a × b| = |a||b| sin θ, which vanishes only when sin θ = 0, so θ is 0° or 180° — the vectors are parallel. Perpendicular is the condition for the dot product.

Correct answer: A — 0

i and j are perpendicular unit vectors, so their dot product is 1 × 1 × cos 90° = 0. Note that i × j = k, which is a different product entirely.

Correct answer: B — ½|a × b|

The cross product magnitude gives the parallelogram area, and a triangle is half of that parallelogram.

Correct answer: B — coplanar

The triple product is the volume of the solid they span. Zero volume means they all lie in one plane.

Correct answer: B — a · b = b · a but a × b = −(b × a)

The dot product is commutative because multiplication of components is. The cross product reverses direction when the order is swapped, which is why the order must be preserved exactly as the question gives it.