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PMS 2023 Mathematics Paper 1
(a) For the function f(x) graphed here, find the following limits or explain why they do not exist.
(i) lim f(x), x→1
(ii) lim f(x), x→2
(iii) lim f(x), x→2.5
(iv) lim f(x) x→2.5
(b) Find the values of a and b that make the following function differentiable for all x-values.
f(x) =
For the function f(x) graphed here, find the following limits or explain why they do not exist.
(i) lim f(x),
x→1
(ii) lim f(x),
x→2
(iii) lim f(x),
x→2.5
(iv) lim f(x)
x→2.5
Find the values of a and b that make the following function differentiable for all x-values.
f(x) =
ax + b, x > -1
bx²-3, x ≤-1
(a) Discuss the validity of Rolle's theorem of f(x) = x(x+3)e-x/2 on [-3,0]. Find 'c'
(if possible).
(b) Use Mean Value Theorem to show that |sin x - sin y| ≤ |x - y| for any real numbers x, y.
Discuss the validity of Rolle's theorem of f(x) = x(x+3)e-x/2 on [-3,0]. Find 'c'
(if possible).
Use Mean Value Theorem to show that |sin x - sin y| ≤ |x - y| for any real numbers x, y.
(a) Find the area of region bounded by the curve y = x² - 4x, the x-axis, and the lines x = 1
and x = 3.
(b) Using rectangular rule for n = 5, approximate the value of the definite integral
Find the area of region bounded by the curve y = x² - 4x, the x-axis, and the lines x = 1
and x = 3.
Using rectangular rule for n = 5, approximate the value of the definite integral
$$ \int_{0}^{1} \frac{dx}{1+x^3} $$
(a) Find the volume of the tetrahedron bounded by the coordinate planes and the plane
$$ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 $$, where a, b, c are positive.
(b) The area in the first quadrant bounded by the parabola y² = 4ax and its latus rectum is
revolved about the x-axis. Find the volume of the solid generated.
Find the volume of the tetrahedron bounded by the coordinate planes and the plane
$$ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 $$, where a, b, c are positive.
The area in the first quadrant bounded by the parabola y² = 4ax and its latus rectum is
revolved about the x-axis. Find the volume of the solid generated.
(a) Solve x $$ \frac{dy}{dx} $$ + y = y² ln x.
(b) An oscillator moves under the forces:
restorative force = -kx
damping force = -2μ
driving force = F0e-at each force being per unit mass.
Set up and solve the equation of motion completely.
Solve x $$ \frac{dy}{dx} $$ + y = y² ln x.
An oscillator moves under the forces:
restorative force = -kx
damping force = -2μ
driving force = F0e-at each force being per unit mass.
Set up and solve the equation of motion completely.
(a) Prove that the function f(z) = $$ \sqrt{|xy|} $$ is not differentiable at origin although Cauchy-
Riemann conditions are satisfied at origin.
(b) Evaluate $$ \oint_{C} \frac{z}{(z-1)(z+2i)}dz $$ where (i) C:|z| = 2, (ii) C:|z| = 3/2.
Prove that the function f(z) = $$ \sqrt{|xy|} $$ is not differentiable at origin although Cauchy-
Riemann conditions are satisfied at origin.
Evaluate $$ \oint_{C} \frac{z}{(z-1)(z+2i)}dz $$ where (i) C:|z| = 2, (ii) C:|z| = 3/2.
(a) Transform x² + y² - z = 9 into spherical coordinates.
(b) The tangent at any point on the curve x³ + y³ = 2a³ makes intercepts p and q on the
coordinate axes respectively. Show that p-3/2 + q-3/2 = 2-1/2 a-3/2
Transform x² + y² - z = 9 into spherical coordinates.
The tangent at any point on the curve x³ + y³ = 2a³ makes intercepts p and q on the
coordinate axes respectively. Show that p-3/2 + q-3/2 = 2-1/2 a-3/2
(a) Find the tangent line and normal plane to the curve $$ \vec{x} = t\hat{e}_1 + t^2 \hat{e}_2 + t^3 \hat{e}_3 $$ at t = 1.
(b) Find the curvature and torsion of $$ \vec{r} = (a \cos \theta, a \sin \theta, a \cot \alpha) $.
Find the tangent line and normal plane to the curve $$ \vec{x} = t\hat{e}_1 + t^2 \hat{e}_2 + t^3 \hat{e}_3 $$ at t = 1.
Find the curvature and torsion of $$ \vec{r} = (a \cos \theta, a \sin \theta, a \cot \alpha) $.
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