## Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. |

## 1. | IB Standard |

The acceleration, \(a\) ms^{-2} , of an object at time \(t\) seconds is given by

The object is at rest when \(t=1\).

Find the velocity of the object when \(t=7\).

## 2. | IB Analysis and Approaches |

(a) Find \(\int (4x+5) dx\).

(b) Given \(f'(x) = 4x+5\) find \(f(x)\) if \(f(3.4) = 10.12\).

## 3. | A-Level |

(a) Find \( \frac{dy}{dx} \) when:

$$ y= (7-5x^2)^{ \frac12 } $$(b) Find the following integral:

$$ \int (1 - cos3x) dx$$## 4. | IB Analysis and Approaches |

The function \(f\) is defined by \(f(x) = 8 - 5 \sin{x} \), for \( x \ge 0 \).

The diagram shows part of the graph of \(y = f (x) \).

The shaded region is enclosed by the graph of \(y=f(x)\) and the x-axis for the first three periods of the function.

(a) Find the exact value of the x-coordinate of the right side of the shaded region.

(b) Show that the area of the shaded region is \( 48 \pi \).

A hemisphere has a total surface area in square centimetres equal to the shaded area in the previous diagram.

(c) Find the radius of the hemisphere.

## 5. | IB Applications and Interpretation |

The following diagram shows part of the graph of:

$$f (x) = \frac{(9 + 4x) (6 - x)}{10}, x \in \mathbf R$$The shaded region A is bounded by the x-axis, y-axis and the graph of \(f\).

(a) Write down an integral for the area of region A.

(b) Find the area of region A.

The three points A (0 , 0) , B (6 , 9) and C (p , 0) define the vertices of a triangle.

(c) Find the value of p, the x-coordinate of C , such that the area of the triangle is equal to the area of region A.

## 6. | IB Standard |

Consider the graph of the function \(f(x)=x^2+2\).

(a) Find the area between the graph of \(f\) and the x-axis for \(2\le x \le 3\).

(b) If the area described above is rotated 360^{o} around the x-axis find the volume of the solid formed.

## 7. | IB Standard |

This graph represents the function \(f:x\to a \cos x, a\in \mathbf N\)

(a) Find the value of \(a\).

(b) Find the area of the shaded region.

## 8. | A-Level |

The diagram shows a sketch of the curve C with equation:

$$ y=2x^{\frac32}-4x+3 $$The point P with coordinates (4, 3) lies on C.

The line L is the tangent to C at the point P.

The region A, shown shaded, is bounded by the curve C, the line L and the y-axis.

Find the area of A making your method clear.

Note that solutions based entirely on graphical or numerical methods are not acceptable.)

## 9. | IB Analysis and Approaches |

Let \(f(x) = \frac{ln3x}{kx} \) where \( x \gt 0\) and \( k \in \mathbf Q^+ \).

(a) Find an expression for the first derivative \(f'(x) \).

The graph of \(f\) has exactly one maximum point at P.

(b) Find the x-coordinate of P.

The graph of \(f\) has exactly one point of inflection at Q.

(c) Find the x-coordinate of Q.

(d) The region enclosed by the graph of \(f\), the x-axis, and the vertical lines through P and Q has an area of one square unit, find the value of \(k\).

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