Maths Y1 Pure Bicen

• What are the algebraic laws for multiplying and dividing with the same base?

  • Multiplying: \( a^n \cdot a^m = a^{n+m} \)
  • Dividing: \( \frac{a^n}{a^m} = a^{n-m} \)
• What does a negative power indicate in algebraic laws?

  • It means to take the reciprocal: \( a^{-n} = \frac{1}{a^n} \)
• What is the value of any number raised to the power of zero?

  • It is equal to one: \( a^0 = 1 \)
• How is the logarithmic relationship expressed between \( \log_a n = x \) and its exponential form?


  • It is equivalent to \( a^x = n \).
• What is the logarithmic property for the product of two numbers?


  • \( \log_a (xy) = \log_a x + \log_a y \)
• What is the logarithmic property for the quotient of two numbers?


  • \( \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \)
• How can you express the logarithm of a power?

  • \( \log_a (x^k) = k \cdot \log_a x \)
• What is the value of \( \ln e \) and \( \ln 1 \)?

  • \( \ln e = 1 \) and \( \ln 1 = 0 \)
• What does the factor theorem state?

  • If \( f(a) = 0 \), then \( x - a \) is a factor of \( f(x) \).
• What happens to the inequality symbol when negating an inequality?

  • The symbol flips to the opposite direction.
• What is the cosine rule formula?

  • \( a^2 = b^2 + c^2 - 2bc \cos A \)
• What is the sine rule formula?


  • \( \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \)
• What is the formula for the area of a triangle?


  • Area = \( \frac{1}{2}ab \sin C \)
• What are the formulas for arc length and sector area in a circle?

  • Arc length = \( r\theta \)
  • Sector area = \( \frac{1}{2}r^2\theta \)
• What is the formula for the segment area of a circle?

  • Segment area = \( \frac{1}{2}r^2\theta - \frac{1}{2}r^2\sin\theta \)
• What is the definition of a function?

  • A relation where each input has exactly one output.
• What is the difference between the domain and range of a function?

  • Domain: input values (x-values); Range: output values (y-values).
• How do you find the inverse of a function?


  • Switch x and y, then rearrange to solve for y.
• What is the formula for the midpoint of a line segment?


  • Midpoint = \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \)
• What is the formula for the gradient of a line?


  • Gradient \( m = \frac{\Delta y}{\Delta x} \)
• What is the equation of a line in point-slope form?


  • \( y - y_1 = m(x - x_1) \)
• What is the discriminant formula and its significance?

  • Discriminant = \( b^2 - 4ac \); Determines the number of roots.
• What does it mean if the discriminant is greater than zero?

  • There are two distinct real roots.
• What does it mean if the discriminant is equal to zero?

  • There is one repeated real root.
• What does it mean if the discriminant is less than zero?

  • There are no real roots.
• What is the equation of a circle?


  • \( (x - a)^2 + (y - b)^2 = r^2 \)
• What are the key characteristics of sine, cosine, and tangent graphs?

  • Sine: starts at origin, ranges from -1 to 1.
  • Cosine: starts at 1, ranges from -1 to 1.
  • Tangent: can take any value.
• What is the transformation for \( f(x + a) \)?

  • It shifts the graph upwards by \( a \).
• What is the transformation for \( f(x - a) \)?

  • It shifts the graph to the left by \( a \).
• What is the transformation for \( af(x) \)?

  • It stretches the graph in the y-direction by a scale factor of \( a \).
• What is the transformation for \( f\left(\frac{x}{a}\right) \)?

  • It stretches the graph in the x-direction by a scale factor of \( a \).
• What is the transformation for \( -f(x) \)?


  • It reflects the graph in the x-axis.
• What is the transformation for \( f(-x) \)?


  • It reflects the graph in the y-axis.
• What is the formula for position vectors?


  • \( \vec{m} = \vec{b} - \vec{a} \)
• What is the condition for two vectors to be parallel?

  • \( \vec{a} = \lambda \vec{b} \) where \( \lambda \) is a constant.
• How do you calculate the magnitude of a vector \( \vec{a} = xi + yj + zk \)?

  • Magnitude = \( \sqrt{x^2 + y^2 + z^2} \)
• What is the formula for a binomial coefficient \( \binom{n}{r} \)?


  • \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
• What is the binomial expansion for \( (1 + x)^n \)?


  • \( 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \ldots \)
• What is the formula for the nth term of an arithmetic sequence?

  • \( a_n = a + (n-1)d \)
• What is the formula for the sum of the first n terms of an arithmetic series?


  • \( S_n = \frac{n}{2}(2a + (n-1)d) \)