Mathematics Reference Manual
Quadratic Equations
Forming Quadratic Polynomials
A quadratic polynomial is of the form \(ax^2 + bx + c\).
Example:
Form a quadratic polynomial given the roots \(\alpha = 3\) and \(\beta = -2\).
Sum of roots: \(\alpha + \beta = 1\)
Product of roots: \(\alpha \beta = -6\)
Quadratic polynomial: \(x^2 – (\alpha + \beta)x + \alpha \beta = x^2 – x – 6\)
Sum and Product of Roots
For a quadratic equation \(ax^2 + bx + c = 0\), the sum and product of the roots are given by:
- \(\alpha + \beta = -\frac{b}{a}\)
- \(\alpha \beta = \frac{c}{a}\)
Example:
Given the quadratic equation \(x^2 – 5x + 6 = 0\), find the sum and product of the roots.
Sum of roots: \(\alpha + \beta = 5\)
Product of roots: \(\alpha \beta = 6\)
Discriminant
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 – 4ac\). The nature of roots depends on the discriminant:
- If \(\Delta > 0\), the equation has two distinct real roots.
- If \(\Delta = 0\), the equation has two equal real roots.
- If \(\Delta < 0\), the equation has two complex roots.
Example:
Given the quadratic equation \(x^2 – 4x + 4 = 0\), determine the nature of its roots.
\(\Delta = 4^2 – 4 \cdot 1 \cdot 4 = 16 – 16 = 0\)
Nature of roots: Two equal real roots
Completing the Square
Completing the square is a method used to solve quadratic equations by rewriting the quadratic expression in the form \((x – p)^2 = q\).
Steps to Complete the Square
- Start with a quadratic equation in the form \(ax^2 + bx + c = 0\).
- Divide all terms by \(a\) (if \(a \neq 1\)).
- Move the constant term \(c/a\) to the right side of the equation.
- Add and subtract \((b/2a)^2\) to/from the left side to form a perfect square trinomial.
- Factor the perfect square trinomial.
- Solve for \(x\).
Example:
Solve the quadratic equation \(2x^2 + 8x – 10 = 0\) by completing the square.
Divide all terms by 2: \(x^2 + 4x – 5 = 0\)
Move the constant term: \(x^2 + 4x = 5\)
Add and subtract 4: \(x^2 + 4x + 4 – 4 = 5 \quad \Rightarrow \quad x^2 + 4x + 4 = 9\)
Factor the left side: \((x + 2)^2 = 9\)
Take the square root: \(x + 2 = \pm 3\)
Solve for \(x\): \(x = 1 \quad \text{or} \quad x = -5\)
Solution: The roots of the equation are \(x = 1\) and \(x = -5\).
Polynomials and Roots
Finding the Roots of a Polynomial
The roots of a polynomial are the values of \(x\) that satisfy the polynomial equation.
Example:
Find the roots of the polynomial \(x^2 – 5x + 6\).
Factorize: \((x – 2)(x – 3) = 0\)
Roots: \(x = 2 \quad \text{and} \quad x = 3\)
Factorization
Expressing a polynomial as a product of its factors helps in solving polynomial equations.
Example:
Factorize the polynomial \(x^2 – 4\).
Factorize: \((x – 2)(x + 2) = 0\)
Factors: \(x – 2 \quad \text{and} \quad x + 2\)
Rational Expressions
Simplifying Rational Expressions
Combining and simplifying expressions involving fractions.
Example:
Simplify the rational expression \(\frac{x}{x^2 – 1} + \frac{1}{x – 1}\).
\(\frac{x}{(x – 1)(x + 1)} + \frac{1}{x – 1} = \frac{x + 1}{(x – 1)(x + 1)} = \frac{1}{x – 1}\)
Exponential Equations
Solving Exponential Equations
Equations involving terms with variables in the exponent.
Example:
Solve the equation \(2^{x+1} = 8\).
Rewrite: \(2^{x+1} = 2^3\)
Equate exponents: \(x + 1 = 3\)
Solution: \(x = 2\)
Radical Equations
Solving Radical Equations
Equations involving square roots or higher-order roots.
Example:
Solve the equation \(\sqrt{2x + 3} = x + 1\).
Square both sides: \(2x + 3 = (x + 1)^2\)
Expand and solve: \(2x + 3 = x^2 + 2x + 1 \Rightarrow x^2 – 2 = 0\)
Roots: \(x = \pm \sqrt{2}\)
Systems of Equations
Solving Systems of Equations
Solving for multiple variables by setting up and solving multiple equations.
Example:
Solve the system of equations:
\[ \begin{cases} 2x + 3y = 5 \\ x – y = 1 \end{cases} \]
Multiply the second equation by 3 and add to the first: \(2x + 3y + 3(x – y) = 5 + 3 \Rightarrow 5x = 8 \Rightarrow x = \frac{8}{5}\)
Substitute \(x\) into the second equation: \(\frac{8}{5} – y = 1 \Rightarrow y = \frac{3}{5}\)
Arithmetic and Geometric Progressions
Sum of Arithmetic Progressions
Using the formula for the sum of the first \(n\) terms of an arithmetic sequence.
Example:
Find the sum of the first 5 terms of the arithmetic progression with first term \(a = 2\) and common difference \(d = 3\).
\(S_n = \frac{n}{2}(2a + (n – 1)d)\)
\(S_5 = \frac{5}{2}(2 \cdot 2 + 4 \cdot 3) = \frac{5}{2}(4 + 12) = 40\)
Logarithmic Equations
Solving Logarithmic Equations
Using properties of logarithms to solve equations involving logarithmic expressions.
Example:
Solve the equation \(\log_2(x + 3) = 4\).
Rewrite: \(x + 3 = 2^4 = 16\)
Solution: \(x = 13\)
Symmetry and Geometry
Properties of Triangles and Circles
Using geometric properties and theorems to solve problems involving triangles and circles.
Example:
Find the hypotenuse of a right triangle with legs 3 cm and 4 cm.
Hypotenuse: \(c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5 \, \text{cm}\)
Algebraic Manipulations
Simplifying Complex Expressions
Using algebraic identities and techniques to simplify and solve complex expressions.
Example:
Simplify the expression \(\frac{x^2 – 1}{x – 1}\).
\(\frac{(x – 1)(x + 1)}{x – 1} = x + 1\)
Specific Concepts and Formulas Required
Roots of Unity
Using the properties of roots of unity in polynomial equations.
Example:
Find the cube roots of unity: \(1, \omega, \omega^2\), where \(\omega = e^{2\pi i / 3}\).
Sum of Inverses of Roots
For roots \(\alpha\) and \(\beta\), the sum of their inverses is given by \(\alpha^{-1} + \beta^{-1} = \frac{\alpha + \beta}{\alpha \beta}\).
Example:
Given the quadratic equation \(x^2 – 5x + 6 = 0\), find the sum of the inverses of its roots.
Roots: \(\alpha = 2, \beta = 3\)
Sum of inverses: \(\alpha^{-1} + \beta^{-1} = \frac{1}{2} + \frac{1}{3} = \frac{3 + 2}{6} = \frac{5}{6}\)
Example Problems
Questions
- Form a quadratic polynomial, one of whose zeroes is $2 + \sqrt{5}$ and the sum of zeroes is 4.
- Form a quadratic polynomial, one of whose zeroes is $\sqrt{5}$ and the product of the zeroes is $-2 \sqrt{5}$.
- Determine if 3 is a zero of the polynomial \(P(x) = \sqrt{x^2 – 4x + 3} + \sqrt{x^2 – 9} – \sqrt{4x^2 – 14x + 6}\).
- \(\alpha, \beta\) are zeroes of the quadratic polynomial \(x^2 – (k + 6)x + 2(2k – 1)\). Find the value of \(k\) if \(\alpha + \beta = \frac{1}{2} \alpha \beta\).
- \(m, n\) are zeroes of \(ax^2 – 5x + c\). Find the value of \(a\) and \(c\) if \(m + n = 5\) and \(mn = 10\).
- Form a quadratic polynomial whose zeroes are \(\frac{3 – \sqrt{3}}{5}\) and \(\frac{3 + \sqrt{3}}{5}\).
- For what values of \(k\), \((4 – k)x^2 + (2k + 4)x + 8k + 1 = 0\) is a perfect square?
- A polygon of \(n\) sides has \(\frac{n(n – 3)}{2}\) diagonals. How many sides does a polygon with 54 diagonals have?
- Mr. Prakash was born in 1809 A.D. In year \(x^2\) A.D., he was \(x – 3\) years old. Find the value of \(x\).
- A rectangle of perimeter 34 units is inscribed in a circle of diameter 13 units. Find its sides.
- Solve the equation: \(2(x – 3)^2 + 3(x – 2)(2x – 3) = 8(x + 4)(x – 4) – 1\).
- Abhishek takes 6 days less than the time taken by Anubhav to finish a piece of work. If both of them together finish the work in 4 days, find the time taken by Anubhav alone to finish the work.
- Solve:
- (i) \(\frac{x – 1}{x – 2} + \frac{x – 3}{x – 4} = \frac{10}{3}\)
- (ii) \(\frac{1}{x + 1} + \frac{2}{x + 2} = \frac{4}{x + 4}\), \(x \neq -1, 2\)
- A two-digit number is four times the sum and three times the product of its digits. Find the number.
- Find the value of \(k\) for real and equal roots.
- (i) \((k + 1)x^2 – 2(k – 1)x + 1 = 0\)
- (ii) \(k^2 x^2 – 2(2k – 1)x + 4 = 0\)
- If \(\alpha, \beta\) are the roots of the quadratic equation \(x^2 + ax + b = 0\), find the value of \((\alpha^2 – 3)(\beta^2 – 3)\).
- If \(\alpha, \beta\) are the roots of \(2x^2 + x + 7 = 0\), find the value of:
- (i) \(\alpha^2 \beta + \alpha \beta^2\)
- (ii) \(\alpha^3 + \beta^3 + 9\alpha \beta (\alpha^2 + \beta^2) + 3(\alpha^2 \beta + \alpha \beta^2)\)
- (iii) \(\alpha^4 + \alpha^2 \beta^2 + \beta^4\)
- If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 – 6x + k = 0\), find the value of \(k\) such that \(\alpha^2 + \beta^2 = 40\).
- In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/hr and the time of flight increases by 30 minutes. Find the duration of the flight.
- Find two consecutive natural numbers whose product is 20.
- Find the whole number which, when decreased by 20, is equal to 69 times the reciprocal of the number.
- Find the value of \(a\) such that the quadratic equation \((a – 12)x^2 + 2(a – 12)x + 2 = 0\) has equal roots.
- Two circles touch externally. The sum of their areas is \(130\pi \, \text{sq cm}\) and the distance between their centers is 14 cm. Find the radii of the two circles.
- The speed of the boat in still water is 15 km/hr. It can go 30 km upstream and return downstream to the original point in 4 hrs. and 30 minutes. Find the speed of the stream.
- \(ABCD\) is a square. \(F\) is the midpoint of \(AB\). \(BE\) is one-third of \(BC\). If the area of \(\triangle AFB E\) is 108 sq cm, find the length of \(AC\).
- Find the value of \(\sqrt{8 + 2\sqrt{8 + 2\sqrt{8 + \ldots}}}\).
- In a group of children, each child gives a gift to every other. If the number of gifts is 132, find the number of children.
- A sum of Rs.1200 becomes Rs.1333 in 2 years at compound interest compounded annually. Find the rate of interest.
- The roots \(\alpha\) and \(\beta\) of the quadratic equation \(x^2 – 5x + 3(k – 1) = 0\) are such that \(\alpha – \beta = 1\). Find \(k\).
- The difference of the squares of two numbers is 45. The square of the larger number is 4 times the square of the smaller number. Determine the numbers.
- If the equation \((1 + m^2)x^2 + 2mcx + (c^2 – a^2) = 0\) has equal roots, prove that \(c^2 = a^2 (1 + m^2)\).
- Solve for \(x\): \[ x = \frac{1}{1 – \frac{1}{1 – \frac{1}{2 – \frac{1}{2 – x}}}} \]
- Solve:
- (i) \( x^2 + \left(\frac{a + b}{a} + \frac{a}{a + b}\right)x + 1 = 0\)
- (ii) \( \frac{2x}{x – 3} + \frac{x}{2x + 3} + \frac{3x + 9}{(x – 3)(2x + 3)} = 0\)
- The area of an isosceles triangle is 60 cm\(^2\) and the length of each one of its equal sides is 13 cm. Find its base.
- Solve: \(3^{x+1} – 2 \times 3^{x+2} = 81\).
- Solve: \[ \frac{1}{x + 1} + \frac{1}{x + 5} = \frac{1}{x + 2} + \frac{1}{x + 4} \]
- Solve the equations:
- (i) \(\sqrt{2x + \sqrt{2x + 4}} = 4\)
- (ii) \(4x^2 – 4x^2 – 7x^2 – 4x + 4 = 0, \, x \neq 0\)
- Solve for \(x\): \(4x^2 – 2(a^2 + b^2)x + a^2 b^2 = 0\).
- Solve for \(x\): \(4x^2 – 4a^2 x + (a^4 – b^4) = 0\).
- Solve for \(x\): \(9x^2 – 9(a + b)x + [2a^2 + 5ab + 2b^2] = 0\).
- Using the quadratic formula, solve the following quadratic equation for \(x\): \[ p^2 x^2 + (p^2 – q^2)x – q^2 = 0 \]
- Using the quadratic formula, solve the following quadratic equation for \(x\): \[ x^2 – 2ax + (a^2 – b^2) = 0 \]
- Using the quadratic formula, solve the following quadratic equation for \(x\): \[ x^2 – 4ax + 4a^2 – b^2 = 0 \]
- Solve for \(x\): \(9x^2 – 6a^2 x + (a^4 – b^4) = 0\).
- Solve for \(x\): \(9x^2 – 6ax + (a^2 – b^2) = 0\).
- Solve for \(x\): \(16x^2 – 8a^2 x + (a^4 – b^4) = 0\).
- Solve for \(x\): \(36x^2 – 12ax + (a^2 – b^2) = 0\).
- If the roots of the equation \((a – b)x^2 + (b – c)x + (c – a) = 0\) are equal, then:
- (A) \(2b = a + c\)
- (B) \(2a = b + c\)
- (C) \(2c = a + b\)
- (D) \(\frac{1}{b} = \frac{1}{a} + \frac{1}{c}\)
- If one of the roots of \(x^2 + ax + 4 = 0\) is twice the other root, then the value of \(a\) is:
- (A) \(-3\sqrt{2}\)
- (B) \(8\sqrt{2}\)
- (C) \(\sqrt{2}\)
- (D) \(-2\sqrt{2}\)
- In the equation \(2x^2 – hx + 2k = 0\), the sum of the roots is 4 and the product of the roots is \(-3\). Then \(h\) and \(k\) respectively, have the values:
- (A) 8 and 6
- (B) 4 and \(-3\)
- (C) \(-3\) and 8
- (D) 8 and \(-3\)
- \(\alpha\) and \(\frac{1}{\alpha}\) are zeroes of the polynomial \(4x^2 – 2x + (k – 4)\). Find the value of \(k\).
- If \(\alpha, \beta\) are zeroes of \(x^2 + 5x + 5\), find the value of \(\alpha^{-1} + \beta^{-1}\).
- If \(\alpha, \beta\) are zeroes of \(x^2 + 7x + 7\), find the value of \(\frac{1}{\alpha} + \frac{1}{\beta} – 2 \alpha \beta\).
- Find the zeroes of \(\sqrt{3}x^2 + 10x + 7\sqrt{3}\).
- Write a quadratic polynomial whose one zero is \(3 – \sqrt{5}\) and the product of zeroes is 4.
- Solve \(x^{2/3} – 2x^{1/3} = 15\).
- Solve the following by reducing to quadratic equations:
- (i) \(x^4 – 26x^2 + 25 = 0\)
- (ii) \(2x – \frac{3}{x} = 5\)
- (iii) \(\sqrt{3x^2 – 2} = 2x – 1\)
- (iv) \(\sqrt{3x + 10} + \sqrt{6 – x} = 6\)
- (v) \(4\left(x^2 + \frac{1}{x^2}\right) + 8\left(x – \frac{1}{x}\right) – 3 = 0\)
- (vi) \(\sigma\left(x^2 + \frac{1}{x^2}\right) – 25\left(x – \frac{1}{x}\right) + 12 = 0\)
- (vii) \(x^4 + 2x^3 – 13x^2 + 2x + 1 = 0\)
- (viii) \((x – 7)(x – 3)(x + 1)(x + 5) – 1680 = 0\)
- Solve \(\frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x}\), \((a + b) \neq 0\).
- If \(-4\) is a root of the quadratic equation \(x^2 + kx – 4 = 0\) and the quadratic equation \(x^2 + px + k = 0\) has equal roots, find the values of \(k\) and \(p\).
- If \(x = 2\) and \(x = 3\) are the roots of the equation \(3x^2 – 2kx + 2m = 0\), find the values of \(k\) and \(m\).