11 Asa Math Annualplan Hema

# July - August

• Carry out the process of completing the square for a quadratic polynomial $ax^2+ bx + c$, and use this form, e.g. to locate the vertex of the graph of $y=ax^2 + bx + c$ or to sketch the graph;
• Find the discriminant of a quadratic polynomial $ax^2+ bx + c$ and use the discriminant, e.g. to determine the number of real roots of the equation $ax^2+ bx + c=0$
• Solve quadratic equations, and linear and quadratic inequalities, in one unknown;
• Solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic;
• Recognize and solve equations in x which are quadratic in some function of $x$, e.g. $x^4 – 5x^2 + 4 = 0$.

## Forces and Newton’s laws of motion

• Apply Newton’s laws of motion to the linear motion of a particle of constant mass moving under the action of constant forces, which may include friction;
• use the relationship between mass and weight;
• solve simple problems which may be modeled as the motion of a particle moving vertically or on an inclined plane with constant acceleration;
• solve simple problems which may be modeled as the motion of two particles, connected by a light inextensible string which may pass over a fixed smooth peg or light pulley.

## Work and Energy

• understand the concept of the work done by a force, and calculate the work done by a constant force when its point of application undergoes a displacement not necessarily parallel to the force (use of the scalar product is not required);
• understand the concepts of gravitational potential energy and kinetic energy, and use appropriate formulas;
• understand and use the relationship between the change in energy of a system and the work done by the external forces, and use in appropriate cases the principle of conservation of energy;
• use the definition of power as the rate at which a force does work, and use the relationship between power, force and velocity for a force acting in the direction of motion;
• solve problems

# September - October

## Functions

• understand the terms function, domain, range, one-one function, inverse function and composition of functions;
• identify the range of a given function in simple cases, and find the composition of two given functions;
• determine whether or not a given function is one-one, and find the inverse of a one-one function in simple cases;
• illustrate in graphical terms the relation between a one-one function and its inverse

## Equilibrium

• identify the forces acting in a given situation;
• understand the vector nature of force, and find and use components and resultants;
• use the principle that, when a particle is in equilibrium, the vector sum of the forces acting is zero, or equivalently, that the sum of the components in any direction is zero;
• understand that a contact force between two surfaces can be represented by two components, the normal component and the frictional component;
• use the model of a ‘smooth’ contact, and understand the limitations of this model;
• understand the concepts of limiting friction and limiting equilibrium; recall the definition of coefficient of friction, and use the relationship $F=\mu R$ or $F\le \mu R$, as appropriate;
• use Newton’s third law.

## Kinematics of motion in a straight line.

• understand the concepts of distance and speed as scalar quantities, and of displacement, velocity and acceleration as vector quantities (in one dimension only);

# October - November

## Coordinate Geometry

• find the length, gradient and mid-point of a line segment, given the coordinates of the end-points;
• find the equation of a straight line given sufficient information (e.g. thecoordinates of two points on it, or one point on it and its gradient);
• understand and use the relationships between the gradients of parallel and perpendicular lines;
• interpret and use linear equations, particularly the forms $y = mx + c$ and $y – y_1 = m(x – x_1)$
• understand the relationship between a graph and its associated algebraic equation, and use the relationship between points of intersection of graphs and solutions of equations (including, in simple cases, the correspondence between a line being tangent to a curve and a repeated root of an equation).

# December

## Circular measure

• understand the definition of a radian, and use the relationship between radians and degrees;
• use the formulae $s=r\theta$ and $A=\frac{r^2\theta}{2}$ in solving problems concerning the arc length and sector area of a circle

## Trigonometry

• sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, and using either degrees or radians);
• use the exact values of the sine, cosine and tangent of 30°, 45°, 60°, and related angles,
• use the notations $\sin^{-1}(x),\cos^{-1}(x),\tan^{-1}(x)$ to denote the principal values of the inverse trigonometric relations;
• Use the identities

# January - February- March

## Series

• use the expansion of $(a+b)^n$ , where n is a positive integer
• recognize arithmetic and geometric progressions;
• use the formulas for the nth term and for the sum of the first n terms to solve problems involving arithmetic or geometric progressions;

*use the condition for the convergence of a geometric progression, and the formula for the sum to infinity of a convergent geometric progression.

## Differentiation

• understand the idea of the gradient of a curve, and use the notations like $f’(x), f’’(x)$
• apply differentiation to gradients, tangents and normals, increasing and decreasing functions and rates of change (including connected rates of change);
• locate stationary points, and use information about stationary points in sketching graphs (the ability to distinguish between maximum points and minimum points is required, but identification of points of inflexion is not included).

## Integration

• understand integration as the reverse process of differentiation,

and integrate $(ax + b)^n$ (for any rational n except –1), together with constant multiples, sums and differences;

• solve problems involving the evaluation of a constant of integration,
• evaluate definite integrals
• use definite integration
• the area of a region bounded by a curve and lines parallel to the axes, or between two curves.