INSTALMENTS-TIPS

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IMPORTANT TIPS

• INSTALMENT PURCHASE SCHEME is a contract between the buyer and the seller by which buyer makes part payment of the price of the commodity (called DOWN PAYMENT) at the time of purchase to the seller and agrees to pay remaining amount in fixed instalments and, in return, the seller gives that commodity to the buyer.



• CASH PRICE : The amount in full (Selling Price) to be paid at the time of purchase is called CASH PRICE of the commodity.



• DOWN PAYMENT : If the buyer does not pay the full price (selling price) of the commodity and prefers to buy it under instalment scheme, then the part payment made by him at the time of purchase is called DOWN PAYMENT.
  
  For example: Cash price of a TV is Rs. 15000/- and the buyer pays Rs. 5000/- at the time of purchase, then Rs. 5000/- is called DOWN PAYMENT.



• RATE OF INTEREST : If a commodity is purchased under instalment scheme, the buyer has to pay some extra amount as interest on the unpaid amount which is calculated at some rate called RATE OF INTEREST.



• INSTALMENT : An amount paid regularly after definite time period (till the total dues,i.e.principal + interest, are paid in full) by the buyer to the seller is called INSTALMENT.



• For calculating simple interest we use the following formula:
  
  I = PRN/100
  
  where I = simple interest, P = principal amount, R = rate of interest( % ) and N = number of years (time period in years).



• For calculating the rate of interest we use:
  
  R = (I x 100)/PN.

ARITHMETIC PROGRESSION-TIPS

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IMPORTANT TIPS

• Observe the following sets of numbers carefully:
  
  
  
  
  1. 1, 2, 3, 4, 5, ...
  2. 2, 4, 6, 8, 10, ...
  3. 1, 4, 9, 16, 25, ...
  4. 1, 11, 111, 1111, 11111, ...
  5. 1, 1/2, 1/3, 1/4, 1/5, ...
  
  
  
  
  Here, in each set, we find numbers arranged in some order and there is an obvious (definite) rule by which we can obtain the next number and as many subsequent numbers as we wish to find.
  
  Such sets are called SEQUENCES ( PROGRESSION ) and each number of the set is called a TERM of the sequence.



• A sequence refers to an ordered set of numbers in which each number (term) can be obtained by a definite rule.



• The rule by which a sequence is formed may be written as a formula for n^th term of the series but all sequences need not have a formula.



• The n^th term of 1, 2, 3, 4, 5, ... is n.



• The n^th term of 1, 4, 9, 16, 25, ... is n².



• The n^th term of 3, 4, 7, 12, 19, ... is (n - 1)² + 3.



• 2, 3, 5, 7, 11, 13, ... is the progression of prime numbers.



• The n^th term of a progression is denoted by either a_n or T_n



• If a progression starts with a definite real number and if any successive term is obtained by adding a constant nonzero real number to the previous term, then the progression is called an ARITHMETIC PROGRESSION (written in short as A.P.)



• Sequences (a) and (b) in the examples given in the beginning are examples of Arithmetic progression whereas sequences (c), (d) and (e) are progressions but they are not Arithmetic progressions.



• The difference between any two consecutive terms of an A.P. is a nonzero constant. This difference is called the COMMON DIFFERENCE.



• The general form of an A.P. whose first term is a (a ∈ R) and the common difference is d (d≠ 0) is:
  
  a, a + d, a + 2d, a + 3d, ...



• The n^th term (T_n) of an A.P. is
  
  T_n = a + (n - 1)d



• The sum of the first n terms of an A.P. is given by
  
  S_n = (n/2)(a + l)
  
  where a = first term of the A.P. and l = last term of the A.P.
  
  Since l = T_n = a + (n - 1)d,
  
  S_n = n/2 [ 2a + (n - 1)d]



• T_n - T_n-1 = d



• S_n - S_n-1 = T_n



• T_m - T_n = (m - n)d [where m>n,&m,n ∈ N].

QUADRATIC EQUATION-TIPS

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IMPORTANT TIPS

• ax² + bx + c is a quadratic polynomial or second degree polynomial where a, b, c are real constants and a ≠ 0.



• x² - √ 5 x + 7, 3x² - 2x + 1, x² - 3, 4x², etc. are examples of second degree polynomials.



• The values of the variable(x) for which the value of the polynomial is zero, are called ZEROS OF A POLYNOMIAL.



• For a second degree polynomial we can have at the most two values of the variable for which the value of the polynomial is zero, i.e. there are at the most two zeros of a second degree polynomial.



• For p(x) = x² - 6x + 8, we have x = 4 and x = 2 for which the value of p(x) is 0. Thus, 4 and 2 are the zeros of p(x).



• For p(y) = y² + 4, we cannot find a value of y for which p(y) is 0. Thus, a polynomial may not have zeros.



• If p(x) is a second degree polynomial, then p(x) = 0 is called a QUADRATIC EQUATION.



• ax² + bx + c = 0 is the general form of a quadratic equation, where a, b, c ∈ R and a ≠ 0.



• The values of x which satisfy the equation ax² + bx + c = 0 (a ≠ 0) are called the roots of that quadratic equation.



• Zeros of polynomial ax² + bx + c are the roots of the quadratic equation ax² + bx + c = 0.



• If ax² + bx + c = (x - α) (x - β) then α and β are the roots of the equation ax² + bx + c = 0.



• All polynomials of the form ax² + bx + c cannot be factorized by normal methods of factorization into factors like (x - α) and (x - β). In such cases, equation ax² + bx + c = 0 is solved by the METHOD OF PERFECT SQUARE ( also known as the Method of Discriminant or the Method of formula).



• Method of Perfect Square can be used to solve any quadratic equation which has a real solution, i.e. solution in R.



• By the method of perfect square, we obtain the roots α and β of the quadratic equation ax² + bx + c = 0 (a ≠ 0) as
  
  α = ( - b - √ D)/2a
  
  β = ( - b + √ D)/2a
  
  where D = b² - 4ac and it is called the DISCRIMINANT of the given quadratic equation.



• If D ≥ 0, the roots of the equation are real.



• If D > 0, the roots are real and distinct.



• If D > 0, D is not a perfect square ( for example : D = 22), then the roots are irrational and distinct.



• If D > 0, D is a perfect square ( for example : D = 25), then the roots are rational and distinct.



• If D = 0, the roots 0f the equation are equal ( also called 'repeated' or 'identical' roots), i.e. the equation has only one root. The value of this root is ( - b )/2a .



• If D < 0, ( i.e. D is negative ), then the given equation has no real roots. ( We say real roots do not exist ).

RATIONAL EXPRESSION-TIPS

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IMPORTANT TIPS

• ax + b (a ≠ 0) is a linear polynomial whereas ax² + bx + c (a ≠ 0)is a polynomial of second degree or a quadratic polynomial.



• If p(x) and q(x) are polynomials over R and q(x) ≠ 0, then p(x)/q(x) is called a rational expression.



• If the HCF of p(x) and q(x) is 1, i.e. there is nothing common between p(x) and q(x) except 1 , then p(x)/q(x) is called a rational expression in its REDUCED FORM.



• If p(x) = h(x) · p'(x) and q(x) = h(x) · q'(x)where h(x) is the HCF of p(x) and q(x), then the rational expression p(x)/q(x) can be reduced to p'(x)/q'(x), i.e. p'(x)/q'(x) is the reduced form of p(x)/q(x).



• To simplify p(x)/q(x), factorize both p(x) and q(x) into their prime factors; then cancel out like terms in both the numerator and the denominator.



• x + x = 2x (Some students write x + x = x². In fact, x · x = x²).



• 1 + 2 + x = 3 + x (Some students write 1 + 2 + x = 3x which is not true).



• Two polynomials can be added, subtracted, multiplied, and divided in the same manner as numbers.



• ADDITION:Two terms of same sign are added and the sign is retained. Thus, (-2) + (-3) = (-5) and (+4) + (+3) = (+7) or 4 + 3 = 7.
  
  The smaller term is subtracted from the bigger term if two terms have opposite signs and then the sign of the bigger term is retained in the result. Thus, (-7) + (+3) = (-4) and 7 + (-3) = 4.



• (a - b)² = (b - a)² BUT (a - b)³ ≠ (b - a)³. In general, (a - b)ⁿ = (b - a)ⁿ if n is even and (a - b)ⁿ = - (b - a)ⁿ if n is odd.



• x/x = 1. Similarly, (x + 2y)/(x + 2y) = 1. (Remember: x/x ≠ 0).



• (2x - 4)² ≠ 2(x - 2)² BUT (2x - 4)² = 4(x - 2)² and (2x - 4)³ = 8(x - 2)³.



• 16(x + 2)² ≠ (16x + 32)² BUT 16(x + 2)² = (4x + 8)² and 27(x - 3)³= (3x - 9)³.



• (2x + 10)/2 ≠ x + 10 (2 cancelled with 2 of 2x).
  
  (2x + 10)/2 ≠ 2x + 5 (2 cancelled with 2 of 10).
  
  Actually, (2x + 10)/2 = 2(x + 5)/2 = (x + 5).



• (4x + 5)/(2x + 5) ≠ 2.[You cannot cancel x + 5].



• While simplifying rational expression, strictly follow the rule of the order of operations: BODMAS → Brackets of Division, Multiplication, Addition and Subtraction.

HCF AND LCM OF POLYNOMIALS-TIPS

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IMPORTANT TIPS

• Make sure that you know how to find HCF(Highest Common Factor) and LCM(Lowest Common Multiple) of given set of numbers.



• p(x)=a_nxⁿ + a_n-1x^n-1 + a_n-2x^n-2 + ... + a₁x+a₀ is called the general form of a polynomial in x, where a_i∈ R(i=0,1,2,3,...) and a_n≠ 0.



• n is called the degree of polynomial.



• p(x) is the symbol for a polynomial over x and q(y) is the symbol for a polynomial over y.Similarly, we use r(x), m(x), n(y), p(t), s(t), etc. to denote polynomials symbolically.



• If p(x)= 0, then it is called the ZERO POLYNOMIAL.



• If p(x)= k, (k∈ R, a constant), then it is called a CONSTANT POLYNOMIAL.



• If n = 1, the polynomial is called a LINEAR POLYNOMIAL. Thus, p(x)= 3x + 2 is a linear polynomial.



• If n = 2, the polynomial is called a SECOND-DEGREE POLYNOMIAL or a QUADRATIC POLYNOMIAL. Thus, p(x)=2x² - 3x - 3 is a second-degree polynomial.



• The highest exponent(index) of the variable denotes the degree of a polynomial.Thus, m(y)= 6y⁵ - 4y³ + x² - 3x + 1 is a polynomial of degree 5.



• The standard method of writing a polynomial is to write it either in the ascending order or the descending order of the exponent of its variable.Thus, p(x)= x⁴ - 3x³ + 2x² + 5x + 6 (descending order) or p(x)= 6 + 5x + 2x² - 3x³ + x⁴ (ascending order).



• In the general form of a polynomial, a_i is the coefficient of xⁱ (i=1,2,3,...). Thus, for i = 0, we have a₀x⁰ = a₀ ( because x⁰ = 1). ∴ a₀ is called the CONSTANT TERM.



• If a polynomial p(x) is the product of polynomials g(x) and h(x), then g(x) and h(x) are aclled the factors of p(x).



• HCF (GCF) OF POLYNOMIALS:If h(x) is a common factor of the given set of polynomials and every common factor of the given polynomials is a factor of h(x), then h(x) is said to be the Highest(Greatest) Common Factor of the given set of polynomials.



• LCM OF POLYNOMIALS:If m(x) is a common multiple of the given set of polynomials and every common multiple of given polynomials is also a multiple of m(x), then m(x) is said to be the Least Common Multiple of the given set of polynomials.



• If h(x) and m(x) are the HCF and LCM, respectively, of two polynomials p(x) and q(x), then
  
  

  p(x) · q(x) = ± h(x) · m(x)

  
  


• ALERT:
  
  

  p(x) · q(x) · r(x) ≠ ± h(x) · m(x)
  

  
  
  
  The relation holds only for two polynomials.