6631_Determinant_of_a_3x3_matrix_Example_4

1.1   6572_Relationship_between_roots_and_coefficients_of_a_quadratic_equation_(1)_ExamSolutions_Maths

1.2   6573_Relationship_between_roots_and_coefficients_of_a_quadratic_equation_(2)_ExamSolutions

1.3   6574_Roots_of_a_Cubic_Equation_ExamSolutions

1.4   6575_Finding_a_Cubic_Equation_with_Reciprocal_Roots_of_another_ExamSolutions_maths_problems_answered

1.5   6576_Roots_of_a_Quartic_Equation_ExamSolutions

1.6   6577_Roots_of_a_Quartic_Equation_(example)_ExamSolutions

1.7   6578_Identity_Sum_of_Cubes_of_Roots_of_a_Cubic_Equation_ExamSolutions

1.8   6579_Identity_Sum_of_Product_Pairs_of_Cubes_of_Roots_ExamSolutions

1.9   6580_Roots_of_a_Cubic_Equation_(example)_ExamSolutions

2.1   6858_Mean_average_value_of_a_function

2.2   6859_Mean_value_of_a_function_Transformations

2.3   6860_Mean_average_value_of_a_function_Example

3.1   6855_Trigonometry_The_t_formulae_explained

3.2   6856_Using_the_t_formulae_to_solve_trig._equations

3.3   6857_t_formulae_Taking_t=tan_may_be_easier_than_t=tan_2_SEE_WHY

4.1   6581_Summing_Series_r_and_ar_b

4.2   6582_Summing_Series_Harder_types_for_r_and_ar_b

4.3   6583_Series_Sum_of_Squared_Types_r^2

4.4   6584_Series_Sum_of_Cube_Types_r^3

4.5   6585_Series_Sum_of_Composite_Functions

4.6   6586_Summing_Series_The_difference_method

4.7   6587_Difference_Method_factorial_example

5.1   6588_Mathematical_Induction_Proof_of_r=n_1_2

5.2   6589_Proof_by_Induction_Sum_of_series_r^2

5.3   6590_Mathematical_Induction_Sum_of_series_r^3

5.4   6591_Proof_by_Induction_Sum_of_series_Example

5.5   6592_Proof_by_Induction_Sum_of_series_Example

5.6   6593_Proof_by_Induction_FP1_Edexcel_January_2013_Q8(a)

5.7   6594_Proof_by_Induction_FP1_Edexcel_January_2013_Q8(b)

5.8   6595_Proof_by_Induction_(Series)_FP1_Edexcel_January_2012_Q6(a)

5.9   6596_Summing_Series_FP1_Edexcel_January_2012_Q6(b)(c)

5.10   6597_Mathematical_Induction_Divisibility_Tests_(1)

5.11   6598_Mathematical_Induction_Divisibility_Tests_(2)

5.12   6599_Edexcel_Further_Maths_FP1_June_2014_Q9

5.13   6600_Edexcel_Further_Maths_June_2010_Q7a

5.14   6601_Edexcel_Further_Maths_June_2010_Q7b

5.15   6602_Mathematical_Induction_Matrix_Multiplication_1

5.16   6603_Mathematical_Induction_Matrix_Multiplication_2

5.17   6604_Proof_by_Induction_Recurrence_Relation_Further_Maths_FP1_Edexcel_June_2013_Q9_a

5.18   6605_Proof_by_Induction_Matrices_Further_Maths_FP1_Edexcel_June_2013_Q9_b

6.1   6861_Vectors_Scalar_or_Dot_Product

6.2   6862_Scalar_Product_Example

6.3   6863_Test_for_Perpendicular_Vectors_3D_Scalar_Product

6.4   6864_Scalar_Product_Perpendicular_Vectors_Example

6.5   6865_Vectors_Equation_of_a_Line_Introduction

6.6   6866_Sketching_3D_vector_problems_lines_and_points

6.7   6867_Vector_Equation_of_a_Line_Example_1

6.8   6868_Vectors_equation_of_a_line_through_two_points

6.9   6869_Vectors_Angle_between_two_lines_given_their_equations

6.10   6870_Vectors_Conditions_for_lines_to_be_parallel

6.11   6871_Vectors_Intersecting_and_Skew_Lines

6.12   6872_Vectors_Closest_point_Shortest_distance_to_a_line

6.13   6873_Shortest_distance_of_a_point_to_a_line

6.14   6886_Vectors_Cartesian_Equation_of_a_Line_in_3D

7.1   6799_Complex_Numbers_What_are_they_Imagine_no_more.

7.2   6800_Complex_Numbers_Addition_Subtraction_and_Multiplication

7.3   6801_Complex_Numbers_Complex_Conjugates

7.4   6802_Complex_Numbers_Division_Example

7.5   6803_Complex_Numbers_The_Argand_Diagram

7.6   6804_Complex_Numbers_Examples

7.7   6805_Complex_Numbers_Modulus_and_Argument

7.8   6806_Complex_Numbers_Equations_Equating_real_imag._parts

7.9   6807_Complex_numbers_Finding_Square_roots_of

7.10   6808_Complex_Numbers_Roots_of_a_quadratic_equation_conjugate_pairs

7.11   6809_Complex_Numbers_Roots_of_a_cubic_equation

7.12   6810_Complex_Numbers_Roots_of_a_quartic_equation

7.13   6811_Edexcel_FP1_Further_Maths_June_2014_Q1_a

7.14   6812_Edexcel_FP1_Further_Maths_June_2014_Q1_b

7.15   6813_Complex_Number_Operations_Further_Maths_FP1_Edexcel_June_2013_Q7_a_b

7.16   6814_Complex_Number_Operations_Further_Maths_FP1_Edexcel_June_2013_Q7_c

7.17   6815_Argument_of_a_Complex_Number_Further_Maths_FP1_Edexcel_June_2013_Q7_d

7.18   6816_Complex_Numbers_FP1_Edexcel_January_2013_Q2

7.19   6817_Complex_Numbers_FP1_Edexcel_January_2012_Q1

7.20   6818_Complex_Numbers_FP1_Edexcel_January_2011_Q7

7.21   6819_Complex_Numbers_Reflection_in_the_Real_Axis

7.22   6820_Example_to_try_on_Reflecting_a_Complex_Number_in_the_Real_Axis

7.23   6821_Complex_Numbers_roots_FP1_Edexcel_January_2012_Q5

7.24   6822_Complex_Numbers_Roots_FP1_Edexcel_January_2013_Q5

7.25   6823_Roots_of_a_Cubic_Equation_Further_Maths_FP1_Edexcel_June_2013_Q3

7.26   6824_Edexcel_FP1_Further_maths_June_2014_Q3

7.27   6825_Complex_Numbers_Mod_Arg_form_r_cos_x_isin_x

7.28   6826_Complex_Numbers_Exponential_Form_or_Euler_s_Form

7.29   6827_Multiplication_Rule_for_the_Mod_Arg_of_two_Complex_Numbers

7.30   6828_Division_Rule_for_the_Mod_Arg_of_two_Complex_Numbers

7.31   6829_Multiplication_Division_of_Complex_Numbers_in_Mod_Arg_form_Examples

7.32   6830_Mod_Arg_of_Complex_Numbers_Further_Maths_FP2_Edexcel_June_2013_Q2

7.33   6831_Complex_Numbers_de_Moivre_s_Theorem

7.34   6832_Using_de_Moivre_s_Theorem_Example_1

7.35   6833_Using_de_Moivre_s_Theorem_Example_2

7.36   6834_using_de_Moivre_s_theorem_to_express_sin_n_and_cos_n_in_terms_of_sin_and_cos

7.37   6835_Example_using_de_Moivre_s_theorem_Trig._identities

7.38   6836_Exam_Question_FP2_Further_Maths_June_2014_Q4_a

7.39   6837_Edexcel_FP2_Further_Maths_June_2014_Q4_b

7.40   6838_Exam_Question_FP2_Edexcel_June_2011_Q7_a

7.41   6839_Trig_Equation_FP2_Edexcel_June_2011_Q7_b

7.42   6840_Another_identity_using_de_Moivre_s_theorem

7.43   6841_Powers_of_cos_expressed_in_multiple_angles

7.44   6842_Odd_Powers_of_sin_expressed_in_multiple_angles

7.45   6843_Even_Powers_of_sin_expressed_in_multiple_angles

7.46   6844_Complex_Numbers_Cube_Roots_of_1_unity

7.47   6845_Complex_Numbers_How_to_find_the_nth_root

7.48   6846_Mod_Arg_form_of_a_Complex_Number_FP2_Edexcel_June_2012_Q3_a

7.49   6847_4th_Roots_of_a_complex_number_FP2_Edexcel_June_2012_Q3_b

7.50   6848_Loci_in_the_Complex_Plane_Circle

7.51   6849_Complex_Numbers_Loci_Perpendicular_bisector

7.52   6850_Complex_Numbers_Loci_Half_lines

7.53   6851_Complex_Numbers_Loci_Regions

7.54   6852_Complex_Numbers_Loci_FP2_Edexcel_June_2012_Q8_a

7.55   6853_FP2_Edexcel_June_2012_Q8_b_c

7.56   6854_Complex_numbers_Further_Maths_IB_HL_Specimen_questions

8.1   6887_Polar_Coordinates_Defining_the_position_of_a_point

8.2   6888_Cartesian_to_Polar_Coordinates_Exercise

8.3   6889_Polar_to_Cartesian_Coordinates_(Exercise)

8.4   6890_Converting_Polar_Curves_to_Cartesian_Form

8.5   6891_Converting_Cartesian_Curves_to_Polar_Form

8.6   6892_Polar_Graphs_the_Spiral

8.7   6893_Polar_Curves_Half_Lines

8.8   6894_Polar_Curves_Circles_and_Arcs

8.9   6895_Polar_Curves_Cardioid

8.10   6896_Polar_Curves_r=asin2(theta)

8.11   6897_Area_bounded_by_a_Polar_Curve

8.12   6898_Polar_Curves_Area_bounded_by_a_Cardioid

8.13   6899_Area_of_a_Loop_in_a_Polar_Curve

8.14   6900_Tangents_to_a_Polar_Curve_Parallel_to_the_Initial_Line

8.15   6901_Tangents_to_a_Polar_Curve_Perpendicular_to_the_Initial_Line

9.1   6619_Matrices_Introduction

9.2   6620_Matrices_Addition_Subtraction_and_Scalar_Multiplication

9.3   6621_Matrices_Matrix_Multiplication

9.4   6622_Matrices_Identity_and_Inverse_of_a_2x2_Matrix

9.5   6623_Singular_Matrix_Example

9.6   6624_Solving_a_Matrix_Equation

9.7   6625_Matrix_Identity_Example

9.8   6626_Matrices_Transposing_Matrices_and_Symmetric_Matrices

9.9   6627_Determinant_of_a_3x3_Matrix

9.10   6628_Determinant_of_a_3x3_matrix_Example_1

9.11   6629_Determinant_of_a_3x3_matrix_Example_2

9.12   6630_Determinant_of_a_3x3_matrix_Example_3

9.14   6632_Determinant_of_a_3x3_matrix_Example_5

9.15   6633_Determinant_of_a_3x3_matrix_Example_6

9.16   6634_Matrices_Minors_of_elements_in_a_3x3_matrix

9.17   6635_Matrices_Finding_the_cofactor_matrix

9.18   6636_Matrices_Inverse_of_a_3x3_matrix

9.19   6637_Matrices_Working_with_Inverse_Matrices_Example

9.20   6638_Inverse_Matrices_on_a_Casio_ClassWiz_Calculator

9.21   6639_Simultaneous_Equations_Matrix_Method

9.22   6640_Solution_of_3_linear_simultaneous_equations_by_matrices

10.1   6641_Transformation_Matrices_Rotation_90_degrees

10.2   6642_Transformation_Matrices_Rotation_180_degrees

10.3   6643_Transformation_Matrices_Rotation_270_degrees

10.4   6644_Transformation_Matrices_Rotation_theta_degrees

10.5   6645_Transformation_Matrices_Reflection_x_axis

10.6   6646_Transformation_Matrices_Reflection_y_axis

10.7   6647_Transformation_Matrices_Reflection_the_line_y=x

10.8   6648_Transformation_Matrices_Reflection_the_line_y=_x

10.9   6649_Transformation_Matrices_Enlargement_scale_factor_k_positive

10.10   6650_Transformation_Matrices_Enlargement_scale_factor_k_negative

10.11   6651_Matrix_Transformations_Test

10.12   6652_Combining_Matrix_Transformations

10.13   6653_Matrices_Reversing_a_transformation

10.14   6654_Determinant_of_a_matrix_as_the_area_scale_factor

10.15   6655_Proof_Determinant_of_a_matrix_as_the_area_scale_factor

11.1   6938_Taylor's_Series_Expansions_Derivation_ExamSolutions_Maths_Revision

11.2   6939_Taylor_Series_example_in_ascending_powers_of_x

11.3   6940_Taylor_Series_example_in_ascending_powers_of_(x_a)

11.4   6941_Solving_differential_equations_by_Taylor's_series_ExamSolutions_Maths_Revision

12.1   6612_Maclaurins_Series_Expansion

12.2   6613_Maclaurins_Series_Expansion_for_e_x

12.3   6614_Maclaurins_Series_Expansion_for_sin_x_cos_x

12.4   6615_Maclaurins_Series_Expansion_for_ln_1_x

12.5   6616_Further_series_using_Maclaurins_Series

13.1   6874_Differentiating_inverse_sin_x_a

13.2   6875_Differentiating_inverse_cos_x_a

13.3   6876_Differentiating_inverse_tan_x_a

13.4   6877_Differentiating_Inverse_Trig_Functions_Example_1

13.5   6878_Differentiating_Inverse_Trig_Functions_Example_2

13.6   6879_How_to_differentiate_Inverse_Trig_Functions_Example_3

13.7   6880_Standard_Integrals_1_a_2_x_2_and_1_root_a_2_x_2

13.8   6881_Integrals_which_require_completing_the_square

13.9   6882_Integration_using_completing_the_square_arctan_type_2

13.10   6883_Integration_using_completing_the_square_arcsin_type_2

13.11   6884_How_to_integrate_1_root_a_2_x_2_with_limits

13.12   6885_How_to_integrate_1_a_2_x_2_with_limits

14.1   6902_Hyperbolic_Functions_Definitions_ExamSolutions

14.2   6903_Graphs_of_sinh_x_cosh_x_and_tanh_x_ExamSolutions

14.3   6904_Graphs_of_sech_x_cosech_x_and_coth_x_ExamSolutions

14.4   6905_Graphs_of_Inverse_Hyperbolic_Functions_ExamSolutions

14.5   6906_arsinh(x)_as_a_natural_logarithm_ExamSolutions

14.6   6907_arcosh(x)_as_a_natural_logarithm_ExamSolutions

14.7   6908_artanh(x)_as_a_natural_logarithm_ExamSolutions

14.8   6909_Solving_simple_Hyperbolic_Equations_using_a_Casio_Calculator_ExamSolutions

14.9   6910_Solving_hyperbolic_equations_using_identities_ExamSolutions

14.10   6911_Solving_Hyperbolic_Equations_using_Exponentials_ExamSolutions

14.11   6912_Osborns_Rule_for_Hyperbolic_Identities_ExamSolutions

14.12   6913_Proof_of_the_differentials_of_sinh_cosh_and_tanh_x_ExamSolutions

14.13   6914_Differentiating_hyperbolic_functions_containing_sinh_cosh_tanh(x)_ExamSolutions

14.14   6915_Proof_of_the_differentials_of_cosech_sech_and_coth_x_ExamSolutions

14.15   6916_Differentiating_hyperbolic_functions_containing_sech_cosech_coth(x)_ExamSolutions

14.16   6917_Differentiating_hyperbolic_functions_(short_exercise)_ExamSolutions

14.17   6918_Hyperbolic_Functions_How_to_differentiate_arsinh_(x_a)_ExamSolutions

14.18   6919_Hyperbolic_Functions_How_to_differentiate_artanh_(x_a)_ExamSolutions

14.19   6920_Hyperbolic_Functions_How_to_differentiate_arcosh_(x_a)_ExamSolutions

15.1   6617_Convergent_Integrals

15.2   6618_Divergent_Integrals

16.1   6606_Volume_of_Revolution_about_the_x_axis_1

16.2   6607_Volumes_of_Revolution

16.3   6608_Volume_of_Revolution_about_the_x_axis_between_curves_2

16.4   6609_Volume_of_Revolution_about_the_y_axis_1

16.5   6610_Volume_of_Revolution_about_the_y_axis_between_curves_2

16.6   6611_Volumes_of_Revolution_for_Parametric_Equations_x_axis

17.1   6921_Differential_Equations_Exact_Type

17.2   6922_Integrating_factor_Differential_Equations

17.3   6923_Differential_Equations_Integrating_factor_type_(Examples)

17.4   6924_Differential_Equations_Substitution_Type_(1)

17.5   6925_Differential_Equations_Substitution_Type_(2)

17.6   6926_2nd_Order_Linear_Differential_Equations_Auxillary_Equation_Introduction

17.7   6927_2nd_Order_Linear_Differential_Equations_Auxillary_Equation_Real_Different_Roots

17.8   6928_2nd_Order_Linear_Differential_Equations_Auxillary_Equation_Equal_Roots

17.9   6929_2nd_Order_Linear_Differential_Equations_Auxillary_Eqn_Imaginary_Roots

17.10   6930_2nd_Order_Linear_Differential_Equations_P.I._=_Constant

17.11   6931_2nd_Order_Linear_Differential_Equations_P.I._=_Linear_type

17.12   6932_2nd_Order_Linear_Differential_Equations_P.I._=_Quadratic_type

17.13   6933_2nd_Order_Linear_Differential_Equations_P.I._=_Exponential_type

17.14   6934_2nd_Order_Linear_Differential_Equations_P.I._=_trig_type

17.15   6935_Special_Case_Particular_Integral_(k)_2nd_Order_Linear_Differential_Equation

17.16   6936_Special_Case_Particular_Integral_(Exp)_2nd_Order_Linear_Differential_Equation

17.17   6937_2nd_Order_Linear_Differential_Equations_Particular_Solutions