LFM Pure and Mechanics

isi-entrance 2009 Q5 Geometric or applied optimisation problem View

Find the maximum volume of a rectangular box (with a lid) that can be inscribed in a cylinder of radius $30$ cm and height $60$ cm.

isi-entrance 2010 Q13 Find absolute extrema on a closed interval or domain View

The minimum value of $x_1^{2} + x_2^{2} + x_3^{2} + x_4^{2}$ subject to $x_1 + x_2 + x_3 + x_4 = a$ and $x_1 - x_2 + x_3 - x_4 = b$ is
(a) $(a^{2} + b^{2})/4$
(b) $(a^{2} + b^{2})/2$
(c) $(a+b)^{2}/4$
(d) $(a+b)^{2}/2$

isi-entrance 2011 Q9 Geometric or applied optimisation problem View

Consider the diagram below where $ABZP$ is a rectangle and $ABCD$ and $CXYZ$ are squares whose areas add up to 1. The maximum possible area of the rectangle $ABZP$ is
(a) $1 + 1 / \sqrt{2}$
(b) $2 - \sqrt{2}$
(c) $1 + \sqrt{2}$
(d) $( 1 + \sqrt{2} ) / 2$

isi-entrance 2011 Q13 Find critical points and classify extrema of a given function View

Consider the function $f ( x ) = x ( x - 1 ) e ^ { 2 x }$ if $x \leq 0$ $f ( x ) = x ( 1 - x ) e ^ { - 2 x }$ if $x > 0$ Then $f ( x )$ attains its maximum value at
(a) $1 - 1 / \sqrt{2}$
(b) $1 + 1 / \sqrt{2}$
(c) $- 1 / \sqrt{2}$
(d) $1 / \sqrt{2}$

isi-entrance 2012 Q9 Find absolute extrema on a closed interval or domain View

Let $f(x) = \dfrac{2x^2 + 3x + 1}{2x - 1}$. Find the maximum and minimum values of $f$ on $[2, 3]$.

isi-entrance 2012 Q18 Find absolute extrema on a closed interval or domain View

If $xy = 1$, find the minimum value of $\dfrac{4}{4-x^2} + \dfrac{9}{9-y^2}$.

isi-entrance 2013 Q18 4 marks Geometric or applied optimisation problem View

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

isi-entrance 2013 Q29 4 marks Geometric or applied optimisation problem View

The maximum of the areas of the isosceles triangles with base on the positive $x$-axis and which lie below the curve $y = e^{-x}$ is:
(A) $1/e$
(B) 1
(C) $1/2$
(D) $e$

isi-entrance 2013 Q38 4 marks Determine intervals of increase/decrease or monotonicity conditions View

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

isi-entrance 2013 Q64 4 marks Determine intervals of increase/decrease or monotonicity conditions View

If $f(x) = \cos(x) - 1 + \frac{x^2}{2}$, then
(A) $f(x)$ is an increasing function on the real line
(B) $f(x)$ is a decreasing function on the real line
(C) $f(x)$ is increasing on $-\infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f(x)$ is decreasing on $-\infty < x \leq 0$ and increasing on $0 \leq x < \infty$

isi-entrance 2014 Q13 Determine intervals of increase/decrease or monotonicity conditions View

Let $f(x) = ax^3 + bx^2 + cx + d$ be a strictly increasing function with $a > 0$. Define $g(x) = f'(x) - 6ax - 2b + 6a$. Then $g(x)$ is
(A) always negative (B) always positive (C) sometimes positive sometimes negative (D) always zero

isi-entrance 2015 QB6 Find critical points and classify extrema of a given function View

Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.

isi-entrance 2015 QB8 Find absolute extrema on a closed interval or domain View

Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.

isi-entrance 2015 QB10 Find critical points and classify extrema of a given function View

Find the maximum among $1,2 ^ { 1/2 } , 3 ^ { 1/3 } , 4 ^ { 1/4 } , \ldots$.

isi-entrance 2015 QB6 Find critical points and classify extrema of a given function View

Show that the function $f ( x )$ defined below attains a unique minimum for $x > 0$. What is the minimum value of the function? What is the value of $x$ at which the minimum is attained? $$f ( x ) = x ^ { 2 } + x + \frac { 1 } { x } + \frac { 1 } { x ^ { 2 } } \quad \text { for } \quad x \neq 0 .$$ Sketch on plain paper the graph of this function.

isi-entrance 2015 QB8 Find absolute extrema on a closed interval or domain View

Find the maximum value of $x ^ { 2 } + y ^ { 2 }$ in the bounded region, including the boundary, enclosed by $y = \frac { x } { 2 } , y = - \frac { x } { 2 }$ and $x = y ^ { 2 } + 1$.

isi-entrance 2015 QB10 Find critical points and classify extrema of a given function View

Find the maximum among $1,2 ^ { 1/2 } , 3 ^ { 1/3 } , 4 ^ { 1/4 } , \ldots$.

isi-entrance 2015 Q13 4 marks Determine intervals of increase/decrease or monotonicity conditions View

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(a) $f ( x )$ is an increasing function on the real line
(b) $f ( x )$ is a decreasing function on the real line
(c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$
(d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.

isi-entrance 2015 Q13 4 marks Determine intervals of increase/decrease or monotonicity conditions View

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(a) $f ( x )$ is an increasing function on the real line
(b) $f ( x )$ is a decreasing function on the real line
(c) $f ( x )$ is increasing on the interval $- \infty < x \leq 0$ and decreasing on the interval $0 \leq x < \infty$
(d) $f ( x )$ is decreasing on the interval $- \infty < x \leq 0$ and increasing on the interval $0 \leq x < \infty$.

isi-entrance 2015 Q23 4 marks Determine intervals of increase/decrease or monotonicity conditions View

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(a) $\alpha \geq 2$
(b) $\alpha < 2$
(c) $\alpha < - 1$
(d) $\alpha > 2$.

isi-entrance 2015 Q23 4 marks Determine intervals of increase/decrease or monotonicity conditions View

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(a) $\alpha \geq 2$
(b) $\alpha < 2$
(c) $\alpha < - 1$
(d) $\alpha > 2$.

isi-entrance 2016 Q9 4 marks Geometric or applied optimisation problem View

The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is
(A) $a b$
(B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$
(C) $\frac { ( a + b ) ^ { 2 } } { 2 }$
(D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$

isi-entrance 2016 Q9 4 marks Geometric or applied optimisation problem View

The minimum area of the triangle formed by any tangent to the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the coordinate axes is
(A) $a b$
(B) $\frac { a ^ { 2 } + b ^ { 2 } } { 2 }$
(C) $\frac { ( a + b ) ^ { 2 } } { 2 }$
(D) $\frac { a ^ { 2 } + a b + b ^ { 2 } } { 3 }$

isi-entrance 2016 Q18 4 marks Geometric or applied optimisation problem View

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

isi-entrance 2016 Q18 4 marks Geometric or applied optimisation problem View

A truck is to be driven 300 kilometres (kms.) on a highway at a constant speed of $x$ kms. per hour. Speed rules of the highway require that $30 \leq x \leq 60$. The fuel costs ten rupees per litre and is consumed at the rate $2 + \left( x ^ { 2 } / 600 \right)$ litres per hour. The wages of the driver are 200 rupees per hour. The most economical speed (in kms. per hour) to drive the truck is
(A) 30
(B) 60
(C) $30 \sqrt { 3.3 }$
(D) $20 \sqrt { 33 }$

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40