Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CLASS-VIII MPC BRIDGE COURSE CLASS - VIII MPC BRIDGE COURSE CONTENTS MATHEMATICS : 2 TO 43 PHYSICS : 44 TO 69 CHEMISTRY : 70 TO 101 KEY : 102 TO 108 NARAYANA GROUP OF SCHOOLS 1 CLASS-VIII MPC BRIDGE COURSE MATHEMATICS DAY-1 : SYNOPSIS Variable: A letter symbol which can take various numerical values is called a variable or literal. Examples: x, y, z etc. Constant : Quantities which have only one fixed value are called constants. Term: Numericals or literals or their combinations by operation of multiplication are called terms. Constant Term : A term of an expression having no literal is called a constant term. TYPES OF ALGEBRAIC EXPRESSIONS: * An expression containing only one term is called a monomial. * An expression containing two terms is called a binomial. * An expression containing three terms is called a trinomial. * An expression containing two or more terms is called a multinomial. * An expression containing one or more terms with positive integral indices (powers) is called a polynomial. Every non-zero number is Note: considered a monomial with degree zero. Degree of polynomial : The highest power of terms in a polynomial is called the degree of a polynomial. Zero polynomial : If all the coefficients in a polynomial are zeroes, then it is called a zero polynomial. Zero of the polynomial : The number for which the value of a polynomial is zero, is called zero of the polynomial. Degree of zero polynomial is not Note: defined. Substitutions : The method of replacing numerical values in the place of literal numbers is called substitution. Like and Unlike terms : * Terms which contain the same literal factors are called like terms or similar terms. In like terms, the numerical coefficient may be different. NARAYANA GROUP OF SCHOOLS 2 2 Ex: 8ab , 5ab , 2 2 8 2 ab ; 2m2n, 4nm2 , m n. 3 3 * Terms which do not have the same literal factors are called Unlike terms. Example : 5a,5a2,5ab; 2xy2 ,2x2 y Addition of algebraic expressions : Addition of algebraic expressions means adding the like terms of the expressions. * Combining the coefficients of like terms of an expression through addition or subtraction is called simplification of an algebraic expression. There are two methods of adding algebraic expressions. They are i) Horizontal method ii) Vertical method. Horizontal method : In this method, like terms should be added and unlike terms should be written separately Subtraction of Algebraic expressions: Additive Inverse of expression : * The additive inverse or the negative of an expression is obtained by replacing each term of the expression by its additive inverse. Example: Additive inverse of -9x is 9x * To subtract 1 st expression from the 2 nd expression, additive inverse of the 1 st expression should be added to the 2 nd expression. If P and Q are two algebraic expressions then P – Q = P + (–Q). Example: Subtract 11a – 6b from 7a + 4b Solution: (7a + 4b) – (11a – 6b) = 7a + 4b – 11a + 6b = –4a +10b Subtraction can also be done in two ways. * Horizontal method * Vertical method Multiplication of Polynomials: Multiply each term of the first Polynomial with each term of the second and add the like terms in the product. Suppose (a+b) and (c + d) are two Polynomials. By using the distributive law of multiplication over addition, we can find their product as given below. 2 CLASS-VIII MPC BRIDGE COURSE 8. Divide 4x – 10x2 + 5x by 2x, then the resultent value is 3 a b c d a c d b c d a c a d b c b d ac ad bc bd Column Method of Multiplication: In this method we write multiplicand and the multiplier in descending powers of, arrange one under another, and multiply the multiplicand by every term of the multiplier and add. DAY-1 : WORKSHEET Conceptual Understanding Questions : 1. The degree of the polynomial 2 4 1 7 x x + x 7 + 3 is 3 5 8 1) 1 2) 4 3) 7 4) 12 2. The zeroes of the polynomial 2x – 3 is 1) 3. 4. 5. 6. 2 2) 3 2 3) 0 4) 3 3 The simplified form of 1.5 x 3 – 1.7 x 3 + 0.2x 3 + 2x 3 is 1) 3.7 x3 2) 3x3 3) 2x3 4) x3 2 The addition of 7x – 4x + 5 and –3x2 + 2x – 1 is 1) 4x2 – 2x + 4 2) 4x2 + 2x + 4 3) 4x2 – 2x – 4 4) 4x2 + 2x – 4 3 2 Subtract 2x – 4x + 3x + 5 from 4x3 + x2 + x + 6, then the resultent value is 1) 6x3 + 5x2 – 2x + 1 2) 2x3 + 5x2 – 2x+1 3) 2x3 – 5x2 – 2x + 1 4) 2x3 – 5x2 + 2x – 1 Additive inverse of ax2 + bx + c is 1) – ax2 – bx – c 2) –ax2 + bx – c 3) ax2 + bx + c 4) – ax2 + bx + c 7.The product of 4 2 3 2 a bc 1) 3 3) 5 2 3 2 a bc 3 6 5 12 ab, bc and abc is 5 6 9 4 2 3 2 2) a b c 3 4) 2 2 3 2 a bc 3 NARAYANA GROUP OF SCHOOLS 2 1) 2x 5x 5 2 2 2) 4x 5x 5 2 2 3) 4x 5x 5 2 2 4) 4x 5x 5 2 Single Correct Choice Type : 9. Two adjacent sides of a rectangle are 3a – b and 6b – a then its perimeter is 1) 2a + 5b 2) 4a + 10b 3) 2a – 5b 4) 4a – 10b 10.The perimeter of a triangle whose sides are 2y + 3z, z – y, 4y – 2z is 1) 5y + 2z 2) 10y + 4z 3) 5y – 2z 4) –5y + 2z 11.The perimeter of a rectangle, 16x3 – 6x2 + 12x + 4. If one of its sides is 8x2 + 3x, then the other side is 1) 16x3 – 14x2 + ax + 4 2) 8x3 – 11x2 + 3x + 2 3) 16x3 + 14x2 + ax – 4 4) 8x3 + 11x2 + 3x – 2 12. Subtract x3 – xy2 + 5x2y – y3 from –y3 – 6x2y – xy2 + x3 1) 2y3 – 8x2y + 3xy2 – 2x3 2) 2x3 – 8xy2 + x2y – 2y3 3) –11x2y 4) –12x2y 13. What must be added to 2x2 – 3xy + 5y2 to get x2 – xy – y2 is 1) x2 + 2xy – 6y2 2) –x2 + 2xy – 6y2 2 2 3) x – 2xy + 6y 4) x2 – 2xy – 6y2 14. The value of a 3 2a 2 4a 5 a 3 2a 2 8a 5 = 1) 2a3 4a2 12a 10 2) 2a3 4a2 12a 10 3) 2a 3 4a2 12a 10 4) 2a3 4a2 12a 10 15. What must be added to x3 + 3x – 8 to get 3x3 + x2 + 6? 1) 2x3 + x2 – 3x + 14 2) 2x2 + x2 + 14 3) 2x3 + x2 – 6x – 14 4) 2x3 + x2 – 14 3 CLASS-VIII MPC BRIDGE COURSE 16.If A 16x 8 4 and B 15x 10 5 , then A – B = 1) 8x 2) 9x 3) 7x 4) 6x 17.The product of 4x 9y 3x 11y is 1)12x2 17xy 99y 2 2) 12x2 17xy 99y 2 3) 12x2 17xy 99y2 4) 12x2 17xy 99y2 18.The product of 1.5x(10x2y – 100xy2) is 1) 15x3y – 150x2y2 2) 15x3y + 150x2y2 3) 150x2y2 – 15x3y 4) 15x2y2 + 15x3y 19. If A B 32x 8 y 4 16x 4 y 3 4x 2 y 2 , 4xy AB 16x 5 y 4 is 2 2 , then 4x 3 y 3 4x y 1) 8x7y2 + 4x3y + x 3) 8x7 + 4x3y – xy2 2) 8x7y2 – 4x3y – xy2 4) 8x7 – 4x3 + xy2 DAY-2 : SYNOPSIS MULTIPLICATION BY USING FORMULAE: 2 * x a x b x a b x ab 2 * x a x b x a b x ab 2 * x a x b x a b x ab 2 * x a x b x a b x ab 2 2 2 * a b a 2ab b 2 2 * a b a 2ab b Conceptual Understanding Questions : 1. The product of (x + 5) and (x + 4) is 1) x2 + 9x + 20 2) x2 – 9x + 20 2 3) x – 9x – 20 4) x2 + 9x – 20 2. (x2 – ay)2 = 1) x4 – a2y2 – 2xay 2) x4 + a2y2 – 2axy 3) x4 – a2y2 + 2ax2y 4) x4 + a2y2 – 2ax2y 3. (2x + 3y) (2x – 3y) = 1) 4x2 + 9y2 2) 2x2 – 3y2 3) 2x2 + 3y2 4) 4x2 – 9y2 3 3 4. 2a 2a = b b 2 1) 4a 9 b2 2 2) 4a 9 b2 3 3 2 4) 2a 2 2 b b 3 5. (2x + 3y) = 1) 8x3 + 27y3 + 18xy (2x +3y) 2) 8x3 + 27y3 + 36x2y + 54xy2 3) 8x2 + 27y3 + 18xy (x + y) 4) Both 1 & 2 6. (x + 2y) (x2 – 2xy + 4y2) = 1) x2 + 2y2 2) (x + 2y)2 3) x3 + 8y3 4) (x + 2y)3 7. If x = 2, y = 3 and z = – 5, then x3 + y3 + z3 = 1) 90 2) –90 3) 0 4) –90xyz 2 3) 2a Single Correct Choice Type : 2 8. Using the identity the value of 497 2 2 * a b a b a b . * * * * * DAY-2 : WORKSHEET 2 is a3+b3=(a+b)(a2-ab+b2) or (a+b)3-3ab(a+b) 1) 247006 2) 247009 3) 257006 4) 2578009 9. The value of 0.768 × 0.768 – 2 × 0.768 × 0.568 + 0.568 × 0.568 is 1) 0.04 2) 0.4 3) 0.004 4) 0.0004 a3-b3=(a-b)(a2+ab+b2) or (a-b)3+3ab(a-b) 10.If P 1162 162 , Q 1242 242 and R = 120, (a+b)3=a3+3a2b+3ab2+b3 or a3+b3+3ab(a+b) (a-b)3=a3-3a2b+3ab2-b3 or a3-b3-3ab(a-b) a3 + b3 + c3 – 3abc 2 2 2 = (a + b + c)(a + b + c – ab – bc – ca) * (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca NARAYANA GROUP OF SCHOOLS then the value of 1) 20 2) 30 P Q R 100 100 3) 40 4) 10 4 CLASS-VIII MPC BRIDGE COURSE 11. If p a 2 2ab b2 and q 4ab , then p + q is 1) 2ab 2) a b 2 2 12.The expansion 4a 2 2 3) a b 4) a b 2 20. 1) 3xyz 2) xyz 21. 9 12a as a perfect 25 5 3 1) 2a 5 3 3) 2a 5 3 2) 2a 5 2 2 3 4) 2a 5 1 12 2) 1 12 3) 1 18 159991 16 1 1 3 2) 1 1 3) abc 3 4) 27abc DAY-3 : SYNOPSIS 4) 1 18 Literal factor (Divisor): If two or more algebraic expressions are multiplied, their products are obtained. The algebraic expressions which multiplied to form the product, are called the factors of the product. Example: 12xy = 3x × 4y 3x and 4y are factors of 12xy 2) Greatest/Highest common factor (G.C.F./ H.C.F): 160001 16 3 15909 4) 4 16 2 2 15. 52 × 48 = (a) – (b) , then the values of a and b are 1) 50, 2 2) 52, 48 3) 4, 50 4) 4, 52 3) 1991 2 2 4 4 16.The value of a b a b a b a b when a = 0, b = 1 is___ 1) 0 2) 1 3) –1 4) 2 1 1 5 , then x 3 3 = x x 1) 100 2) 125 3) 140 4 4) 145 4 18. If x + y = 8; xy = 12 then x y ––––– –––– 1) 1012 2) 1112 3) 1212 4) 1312 1 1 4 then z 2 2 –––––––––– z z 1) 18 2) 16 3) 14 4) 8 If z NARAYANA GROUP OF SCHOOLS G.C.F./H.C.F of two or more monomials is the highest monomial which divides each of the given monomials completely. L.C.M of Monomials: The L.C.M of two or more monomials is a monomial having the least powers of constants and variables such that each of the given monomials is a factor of it. The sign of the coefficient of the L.C.M of the monomials is the same as the sign of the coefficient of the product of the monomials. Factorization: The process of resolving the given expression into factors is called factorization. 17.If x 19. 1 4) 3xyz 2 1 3 14. If 99 100 = 4 4 1) 3 If a 3 b 3 c 3 0 then a b c 1) 0 1 1 7 7 13. The product of 3 6 and 3 6 is 1) 3 4) 1 y z z x 3) 1 22. 2 3 x y 3) 0 1) 3(x–y) (y–z) (z–x) 2) 0 square is 2 If x + y + z = 0 then x 3 y 3 z 3 Different types of Factorization: 1. Taking out common factor: Steps : * Find the H.C.F. of all the terms of the given expression. * Divide each term by this H.C.F. and enclose the quotient within the brackets, keeping the common factor outside the brackets. 5 CLASS-VIII Finding factors of multinomials : To factorize a multinomial, in general we have to express the multinomial as a product of two or more expressions. These two or more expressions whose product is equal to the given multinomial, are called the factors of the multinomial. This is the reverse process of multiplication. Prime multinomial: A multinomial is said to be prime if it is divisible by one and itself only. Common factor: A number or a number letter combination which divides all the terms of a multinomial is called a common factor of the terms of the multinomial. Common Binomial factor: The greatest common factor of the terms of a multinomial need not be a monomial always. It can also be a binomial. Factorisation and rearrangement of terms: If we observe the terms of an algebraic expression in the order they are given, they may not have a common factor. But by rearranging the terms in such a way that each group of terms has a common factor, some algebraic expressions can be factorized. While rearranging the sign and value of each term should not be altered. Example: ax by ay bx = x(a b) y(a b) = (a b)(x y) The factors are (a + b) and (x – y) To find the factors of difference of two squares: The difference of two squares is always equal to the product of the sum and difference of the square roots of the square terms in the expression. Example: a 2 b2 (a b)(a b) Factorization of (a + b)3, (a – b)3 forms : 3 3 2 2 3 * a b a 3a b 3ab b or a b 3 a 3 b3 3ab a b 3 3 3 3 2 2 3 * a b a 3a b 3ab b or a b 3 a b 3ab a b NARAYANA GROUP OF SCHOOLS MPC BRIDGE COURSE Factorization of a3 + b3, a3 – b3 forms 3 3 2 2 * a b a b a ab b (or) 3 a 3 b3 a b 3ab a b 3 3 2 2 * a b a b a ab b (or) 3 a 3 b3 a b 3ab a b DAY-3 : WORKSHEET Conceptual Understanding Questions : 1. The HCF of 2x2 and 12x2 is 1) 2x2 2) 12x2 3) 2x 4) 12x 3 2 2. The HCF of x and –yx is 1) x3 2) –yx2 3) x2 4) –yx5 3. The HCF of numerical coefficient of the given monomials 6x3a2b2c, 8x2ab3c3 and 12a3b2c2 is 1) 12 2) 8 3) 6 4) 2 2 2 4. The factorization of 4x – 9y is 1) (2x – 3y) (2x + 3y)2) (2x – 3y) (3x – 2y) 3) (2x – 3y) (x – y) 4) (x – y) (3x – 2y) 5. The factorization of x4 – y4 is 1) (x + y)2 (x – y)2 2) (x + y) (x – y) (x2 + y2) 3) (x + y) (x – y) (x2 – y2) 4) (x2 + y2) (x + y) (y – x) 6. The factorization of x3 + 8y3 is 1) (x + 2y) (x2 + 2xy + 4y2) 2) (x + 2y) (x2 – 2xy + 4y2) 3) (x – 2y) (x2 + 2xy + 4y2) 4) (x – 2y) (x2 – 2xy + 4y2) 7. The factorization of 8a3 – 27 is 1) (2a + 3) (4a2 + 6a + 9) 2) (2a + 3) (4a2 – 6a + 9) 3) (2a – 3) (4a2 + 6a + 9) 4) (2a – 3) (4a2 – 6a + 9) Single Correct Choice Type : 8. If A = 384 x4y5z3 and B = 256 x2y3z3, then their G.C.D is 1) 27 x 2 y 3z 5 2) 27 x 2 y 5 z 3 3) 27 x 2 y 3z 3 4) 27 x 4 y 5 z 5 9. The factors of 19x – 57y is 1) – 19 (x + 3y) 2) 19 (x + 3y) 3) – 19 (x – 3y) 4) 19 (x – 3y) 6 CLASS-VIII MPC BRIDGE COURSE 10.The H.C.F of 15pq, 20qr, 25rp is 1) 5pqr 2) 25pqr 3) 5 4) 25 11.The factors of – 10ab3 + 30ba3 – 50a2b3 is 1) – 10ab (b2 – 3a2 + 5ab2) 2) 10ab (b2 – 3a2 + 5ab2) 3) – 10ab (b2 + 3a2 + 5ab2) 4) 10ab (b2 + 3a2 + 5ab2) 12.The factors of (x2 + 3x)2 – 5(x2 +3x) – y(x2 + 3x) + 5(x2 + 3x) are 1) (x2 + 3x)(x2 + 3x – y) 2) (x2 + 3x – 5)(x2 + 3x + y) 3) (x2 + 3x + 5)(x2 + 3x + y) 4) (x2 + 3x – 5)(x2 + 3x – y) 2 13.The factors of 2a 6b 3 a 3b are 1) a 3b 2 3a 9b 2) a 3b 2 3a 9b 18. 125 p 3 8q 3 = 3 1) 5 p 2q 30 pq 5 p 2q 3 2) 5 p 2q 30 pq 2q 5 p 2 2 3) 5 p 2 q 25 p 10 pq 4 q 4) both (1) and (3) 19. 1000a 3 8 = 2) 10 a 2 100 a 14.If you factorise 4p2 – (q + r)2 the factors are 1) 2p q r 2) 2p q r 3) 2p q r 4) Both 2 and 3 15.The factors of x6 – y6 is/are 3 3 3 3 1) x y , x y 2 2 4 2 2 4 2) x y , x x y y 2 2 4 2 2 4 3) x y , x x y y 4) Both 1 and 2 16.The factor of 5x2 – 80y2 are 1) 5(x + 4y)(x – 4y) 2) 5(x – 4y)(x – 4y) 3) 5(x + 4y)(x + 4y) 4) 5(4y – 5)(x + y) 17. 20 a 4 2 1) 10 a 2 100 a 20 a 4 4) 10 a 2 100 a 2 20 a 4 2 3) 10a 2 100a 20 a 4 2 DAY-4 : SYNOPSIS 3) a 3b 2 3a 9b 4) a 3b 2 3a 9b Intriduction: A second-degree equation is a polynomial equation in which the highest degree of the variable is 2. In particular, a second-degree equation in one unknown is called a quadratic equation. We define the standard form of a quadratic equation as Ax2 + Bx + C = 0 A 0 The zero-product rule : If a . b = 0, then a = 0 or b = 0 Solving a Quadratic Equation by using the Quadratic Formula: Consider the quadratic equation ax 2 bx c 0 a 0 . By solving this equation with completion of square method we get b b2 4ac 2a The factors of 2x 3 + 5x 2 y -12xy 2 is x 1) x x 4y 2x 3y Let the roots are denoted by say 2) x x 4y 2x 3y b b2 4ac b b2 4ac , 2a 2a We use theseformulas to find the roots of any other quadratic equation. 3) x x 4y 2x 3y 4) x x 4y 2x 3y NARAYANA GROUP OF SCHOOLS 7 CLASS-VIII MPC BRIDGE COURSE NATURE OF ROOTS: Discriminant :b2 – 4ac denoted by or D is called the discriminant of a quadratic equation ax 2 bx c 0 where a, b, c R and a 0 . Thus b2 4ac Nature of the Roots of a Quadratic Equation 2 ax bx c 0 where a, b, c R and a 0 : DAY-4 : WORKSHEET Conceptual Understanding Questions : 1. The solution set of is b d a c 2) a,c b d , a c 4) b, d 1) , 3) 1) If 0, and a,b,c R , then the roots are complex and conjugates. In this case the graph of the curve y = f(x) does not intersect x - axis. 2) If 0, and a,b,c R, then the roots are real and each of the root is called a double or repeated root and is equal to b . In this case the curve y = f(x) touch 2a b ,0 . Also the x - axis in one point 2a quadratic expression will be a perfect square expression when 0 3) If 0, and a,b,c R, then the roots are real and distinct. In this case the curve y = f(x) intersect x - axis in two distinct points 4) If 0 and a,b,c Q and is a perfect square, then the roots are rational and distinct. 5) If 0 and a,b,c Q and is not a perfect square, then the roots are irrational and conjugates. Nature of the Roots of a Quadratic 2 Equation ax bx c 0 where a, b, c C and real part of a not equal to 0 : In this case roots are complex and may or may not be conjugates. 2. The roots of x 2 x 6 0 are 1) –2,3 2) 1,–2 3) –3,–2 4) –3,2 3. The roots of x 2 4x 12 0 are 1) 2,–6 2) –2,6 3) 2,6 4) –2,–6 4. The nature of the roots of the equation x 2 4x 4 0 are 1) Real and rational 2) Real and irrational 3) Real and equal 4) all of these Single Correct Choice Type : 5. x a x b 0 then the value of x 1) a,b 2) 1 1 , a b 2 3) a, b 4) a b , b a 2 6. x a x a 0 , then x is 1) 0 2) 1 4a 3) 1 4a 4) 1 2a 7. If a b the roots of the equation x a x b b2 are 1) real and distinct 2) real and equal 3) real 4) imaginary 8. The number of real roots of the equation x 1 2 2 2 x 2 x 3 0 is 1) 2 2) 1 3) 0 4) 3 9. The value of m for which the equation 1 m x 2 2 1 3m x 1 8m 0 equal roots, is 1) 0 2) 1 NARAYANA GROUP OF SCHOOLS ax b cx d 0 3) 2 h a s 4) 5 8 CLASS-VIII 10. 11. 12. The roots of 5 x 2 3x 2 0 are 1) Rational and equal 2) Rational and not equal 3) Irrational 4) Imaginary If the roots of the equation x2 - 15 - m(2x-8)=0 are equal then m = 1) 3, -5 2) 3, 5 3) -3, 5 4) -3, -5 Only one of the roots of ax 2 bx c 0, a 0 , is zero if 1) c 0 2) c 0, b 0 3) b 0, c 0 4) b 0, c 0 MPC BRIDGE COURSE For example : ( i ) 16 1 16 i 16 i 4 (i 4) 2 16 ( ii ) 36 1 36 i 66 i 6 (i 6) 2 36 Imaginary number: Square root of a negative number is called an imaginary number. For examle : 1, 5, 6, 25 etc. are all imaginary numbers. Integral powers of iota ( i ) : ( i ) Positve Integral Power of i : we know that, i = 1 we can write higher powers of i as follows : 13. 14. 6 6 6 ...... 2 2 2 ...... is 1) 1 2) 2 3) 3 The roots of the equation a c b x 2 2cx b c a 0 1) 1, 3) 1, 15. 16. a c b bca 2) 1, bca a c b 4) 1, 4) 4 i 0 1, i1 i, i 2 1, i 3 i 2 i 1 i i, i 4 (i 2 ) 2 (1)2 1, are bca 2c i 5 i 4 i 1 i i, i 6 (i )4 i 2 1 (1) 1, i 7 i 4 i 3 1 (i ) i, 2c a c b i 8 (i 4 )2 (1)2 1, i 9 i 8 i 1 i i and so on. If a b c x b c a x c a b is a perfect square, then a, b, c are in 1) A.P. 2) G.P. 3) H.P. 4) A.G.P. If the roots of 2 p 2 q 2 x 2 2q p r x q 2 r 2 0 be real and equal, then p, q, r will be in 1) A.P 2) G.P 3) H.P 4) None DAY-5 : SYNOPSIS Complex Numbers: Euler was he first Mathematician. Who intoduced the symbol i ( read as iota ) for 1 with property i 2 1 0i.e. i 2 1 . He also called this symbol as imaginary unit. 1 1 a a 1. a i a a R NARAYANA GROUP OF SCHOOLS I order to compute i p for p> 4, we divide p by 4 and obtain the remainder r. Let q be the quotient, when p is divided by 4. Then, p = 4q + r. were 0 r 4 i p i 4 q r (i 4 )q i r 1. r r i r Thus, the value of i p for p 4 is i r , where r is the remainder when p s divided by 4. ( ii ) Negative integral Powers of i : Complex Numbers: A number of the form a+ib is called a comples number, wher a and b are real numbers and i 1 . Complex number is generally denoted by z i.e. z a ib For example : 3+2i, 5+i,2-3i etc. are complex numbers. 9 CLASS-VIII Real and imaginary parts of complex number : Let a+ib be a complex number then, a is called real and b the imaginary part of z and may be denoted by Re(z) and Im(z) repectively. For example : If z=3+2i, then Re(z) =3, im(z) =2. Purely real and purney imaginary complex numbers: A complex z a ib is called purely real, if b = 0 i.e Im(z)=0 and is called purely imaginary, if a =0. i.e., Re(z) =0. For example : z=3 is purely real and z=2i is purely imaginary Set of complex numbers: The Product set R R consisting of the ordered pairs of real numbers is called the set of complex numbers. The set of complex number is denoted by C i.e. C a ib : a, b R Note: We observe that nthe system of complex numbers includes the system of real numbers i.e. R C. Equality of complex numbers: MPC BRIDGE COURSE Note: Order relations “ greater than” and “less than” are not defined for complex numbers. The inequalities like i>0, 3+i < 2 etc. are meaningless. Addition of two complex numbers: If z1 ( x , y ) C and z2 (a, b) C then z1 z2 ( x, y ) (a, b) ( xa yb, xb ya) . If z1 , z 2 C , then z1 z2 C and z1 z2 is called the sum of two complex numbers z1 and z2 . Multiplication of two complex numbers: The postulate (iii) defines the binary operation of multiplication of two complex numbers. If z1 ( x, y ) C and z2 (a, b) C , then z1 z2 ( x, y ) (a, b) ( xa yb, xb ya ) e.g. (2,5) (3, 9) (2.3 5.9, 2.9 5.3) ( 39, 33) If z1 , z2 C , then z1 z2 C and z1 z 2 is called the product of two complex numbers z1 and z2 . Division of complex numbers: Two complex numbers z1 a ib and The division of a complex number z1 by a z2 c id are said to be equal, if a = c non zero complex number z2 is defined and b=d. Proof: a ib c id a c i (d b) as the multiplicative inverse of z1 by the multiplicative inverse of ( a c ) 2 ( d b ) 2 ( a c ) 2 ( d b ) 2 0 Here, sum of two positive numbers is zero. This is only possible, if each number is zero. (a c) 2 0 a c and (d b)2 0 b d . For example : If a + ib =3+2i, then a=3,b=2 Zero complex number: A complex number z is said to be zero, if its both real and imaginary parts are zero. In other words, z = a+ib = 0, if and only if a = 0 and b = 0. denoted by Therefore, z2 and is z1 z2 . 1 z1 1 = z1 , z2 z1 . . z2 z2 Let z1 a ib1 and z2 a2 ib2 Then = z1 a1 ib1 (a1 ib1 ) (a2 ib2 ) z2 a2 ib2 (a2 ib2 ) (a2 ib2 ) a1a2 b1 b2 i ( a1b2 b1a2 a22 b22 a1a2 b1 b2 a2b1 a1b2 = a 2 b 2 a 2 b2 2 2 2 2 NARAYANA GROUP OF SCHOOLS 10 CLASS-VIII MPC BRIDGE COURSE For example : If z1 2 3i and z 2 5 4i, then z1 2 3i (2 3i ) (5 4i ) 10 8i 15i 12 z 2 5 4i (54i ) (5 4i ) 25 16 1 22 7 i. 22 7i 41 41 41 12.When simplified the value of i 57 (1/ i 25 ) is 1) 0 2) 2i 3) -2i 4) 2 13.If i 2 1 , then i 2 i 2 i 4 i 6 ........ to ( 2n+1 ) terms is equal to 1) 0 2) -i 3) 1 4) -1 14.The value of (1 i ) 2 (1 i) 4 is equal to DAY-5 : WORKSHEET 1) 8 Conceptual Understanding Questions : 1. i 123 __________ . 1) i 2) -i 3) 1 4) -1 2. i120 i 2 ____________. 1) 0 2) -1 3) 1 4) 2 3. If z =1-2i then Real part of z is ___________. 1) i 2) -2 3) 1 4) 0 4. If z = 3+2i then imaginary part of z is _________. 1) 2 2) 1 3) i 4) 3 5. If p+iq=0 then p+q is ___________. 1) 1 2) -1 3) -2 4) 0 6. If z1 a 2i, z2 a 2i then z1 z2 is _________. 1) 2 2) 0 3) 2a 4) 2i 7. If z1 2 i, z2 2 i then z1 z2 is __________. 1) 2 i 2) 0 3) 4 4) -4 3) -2 9. If z1 1 i, z 2 1 i, then ______________. 1) -i 2)1 1) i 1) (3i ) 6 i18 3) 4) ab 2) 2) 1 3) ab 4) 3) -i 4) -1 NARAYANA GROUP OF SCHOOLS 2) 7 i7 i3 2 2) -i 3) 4 4) i 2 Express (4 3i ) 3 in the form of (a+ib) is 1) (-44-11 77i) 3) -44+11 7i) ab i 11. i103 1) i 4) -1 21 48 5 27 i 6 3 4 17 1 34 19. i i 23. ab 3) -i 16 25 i9 19 1 25 18. i i 22. 4) i a . b 1) 2) 1 16.The value of 2 i 2 i 4 i 6 ___________ 1) 0 2) -1 3) 2 4) 1 17.Which of the following is true? Single Correct Choice Type : 10. 4) -4 1) -i 2) -2 3) 2i 4) 2 20. (1-i) (1+2i) (1-3i) = 1) 1-6i 2) 8-6i 3) 6-8i 4) 6+8i 21. Express ( 2+3i) (-4i) in the form of a+ib 1) 16+i 2) 17+i 3) 18+i 4) None 4) 1 z1 z2 is 3) 0 3) -8 i 4 n 1 i 4 n 1 15.If n N , the value of is 2 1) -4 8. If z1 1 i, z 2 1 i, then z1 z2 is ____________. 1) 0 2) 2 2) 5 2) 44-117i) 4) Non e If (3 y 2)i16 (5 2 x )i 0, then 1) x = 5/2 3) Both (1) & (2) 2) y = 2/3 4) None 11 CLASS-VIII 24. If MPC BRIDGE COURSE 3 5 i 2 5 y 2 , then y = 5 1) 0 2) 1 3) 2 4) 3 25. Matrix match type: Column-I Column-II a) If (3y-2) +i (7-2x) =0, then x = p) 33/4 b) If (3y-2) +i (7-2x) =0, then y = q) 3/4 c) If 4x+i (3x-y) =3-6i, then x = r) 2/3 d) If 4x+i (3x-y) =3-6i, then y = s) 7/2 26. If 2x+i4y=2i, then x is ___________ 27. If 2x+4iy = i 3 x-y+3, then 3y = _______ 28. Which of the following is true ? 1) (3 5) (3 5) 14 ( 44 117i ) 2) ( 2 3) ( 3 2 3) 7 3i 3) (2 3i ) 2 ( 5 12i) 4) ( 5 7i ) 2 ( 44 14 14 5) i DAY-6 : SYNOPSIS Ordered Pair: A pair of numbers a and b listed in a specific order with a at the first place and b at the second place is called an ordered pair (a, b). Y 5 4 3 2 1 O X' -5 -4 -3 -2 -1 -1 1 2 3 NARAYANA GROUP OF SCHOOLS X -3 -4 -5 Y' The point 0 is called the origin. The configuration so formed is called the coordinate system or coordinate plane. Coordinates of a point in a plane: Let P be a point in a plane. Let the distance of P from the y-axis = a units. And, the distance of P from the x-axis = b units. Then, we say that the coordinates of P are (a, b). a is called the x-coordinates or abscissa of P. b is called the y-coordinates or ordinate of P. Y Note that a, b b, a . Thus, (2, 5) is one ordered pair and (5, 2) is another ordered pair. CO-ORDINATE SYSTEM We represent each point in a plane by means of an ordered pair of real numbers, called the coordinates of that point. The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines, called the coordinate axes. On a graph paper, let us draw two mutually perpendicular straight lines X 'OX and YOY ' , intersecting each other at the point O. These lines are known as the coordinate axes or axes of reference. The horizontal line X 'OX is called the x-axis. The vertical line YOY ' is called the y-axis. 4 5 -2 P(a, b) b X' O a X Y' Consider the point P shown on the adjoining graph paper. Draw PM OX . Quadrants: Let X 'OX and YOY ' be the coordinates axes. These axes divide the plane of the graph paper into four regions, called quadrants. The region XOY is called the First Quadrant. The region YOX ' is called the Second Quadrant. The region X 'OY ' is called the Third Quadrant. The region Y 'OX is called the Fourth Quadrant. 12 CLASS-VIII MPC BRIDGE COURSE Using the convention of signs, we have the signs of the coordinates in various quadrants as given below: Region Quadrant Nature of x and y Signs of coordinates XOY I x > 0, y > 0 (+, +) YOX' II x < 0, y > 0 (–, +) X'OY ' III x < 0, y < 0 (–, –) Y 'OX IV x > 0, y < 0 (+, –) Any point on x-axis: If we consider any point on x-axis, then its distance from x-axis is 0. So, its ordinate is zero. Thus, the coordinates of any point on x-axis is (x, 0). Any point on y-axis: If we consider any point on y-axis, then its distance from y-axis is 0. So, its abscissa is zero. Thus, the coordinates of any point on y-axis is (0, y). Slope of x-axis is 0; slope of y-axis not defined. Distance between two points Let A x1 , y1 , B x2 , y2 be any two points on a line not parallel to the axes. From the adjacent figure we have the right angle triangle ABC. Y B x1, y 1 A X O C y2 x 2 x1 y1 x1 AB2 AC2 BC2 But AC x2 x1,BC y2 y1 2 AB 2 2 x2 x1 y2 y1 2 NOTE : The distance to the point A x1 , y1 from origin is x12 y12 NARAYANA GROUP OF SCHOOLS Conceptual Understanding Questions : 1. If the x co-ordinate of a point is 2 and its y co-ordinate is 3, then it is represented as 1) 2, 3 2) 3, 2 3) (2, 3) 4) (3, 2) 2. If the abscissa & ordinate of a point are 3 and 2 respectively then the point is represented as 1) 2, 3 2) 3, 2 3) (2, 3) 4) (3, 2) 3. If a point is at a distance of 2 units from Y – axis and 3 units from X – axis then the point is represented as 1) 2, 3 2) 3, 2 3) (2, 3) 4) (3, 2) 4. A Point (4, 0) lies on 1) X – axis 2) Y – axis 3) Origin 4) X and Y axes 5. A Point (0, 5) lies on 1) X – axis 2) Y – axis 3) Origin 4) X and Y axes 6. The distance between two points (0,0) and ( 2,5) is 1) 10 units 2) 29 units 3) 4) 27 units 7 units 7. The distance between two points (2,2) and ( 5,4) is 1) 13 units 2) 5 units 3) 4) 27 units 7 units 8. The distance between two points (–2,3) and ( 4,0) is x2 AB2 x2 x1 y2 y1 DAY-6 : WORKSHEET 1) 15 units 2) 2 5 units 3) 3 5 units 4) 35 units Single Correct Choice Type : 9. In which of the following quadrant does the given point (3, –8) lie? 1) I quadrant 2) II quadrant 3) III quadrant 4) IV quadrant 10.In which of the following quadrant does the given point (–5, 1) lie? 1) I quadrant 2) II quadrant 3) III quadrant 4) IV quadrant 13 CLASS-VIII MPC BRIDGE COURSE 11.In which of the following quadrant does the given point (–6, –8) lie? 1) I quadrant 2) II quadrant 3) III quadrant 4) IV quadrant 12.The Horizantal axis is called. 1) X axis 2) Y axix 3) Origin 4) I Quardrant 13.The nearest point from the origin is 1) (2, –3) 2) (5, 0) 3) (0, –5) 4) (1, 3) 14.If Q(x,y) lies in the Fourth Quadrant then x is 1) Positive 2) Negetive 3) Both 1 & 2 4) None 15.The triangle formed by (0, 1), (1, 0) and (1, 1) is(through graph) 1) Right angle isoceles triangle 2) Scalene triangle 3) Equilateral triangle 4) Cannot form a triangle 16.The X co ordinate on Y axis is 1) 0 2) 1 3) Undifine 4) None 17.The distance between (4,–3) (–4,3) is 1) 10units 2) 12units 3)14units 4)11units 18. If the distance between the points (5,–2) (1,a) is 5, then the value of a is 1) 5units 2) 2units 3)4units 4) 1unit 19. The point on x-axis which is equidistant from the points (5, 4), (–2, 3) is 1) (2,0) 2) (4,0) 3)(12,0) 4) (5,0) 20. The distance between A(7,3) and B on the x-axis whose abscissa is 11 is 1) 12units 2) 10 units 3) 5units 4) 15units MATRIX MATCH TYPE 21.Column-I Column-II a) Distance between (5, 3), (8, 7) 1) -3 b) Distance between (0, 0), (-4, 3) 2) 5 1 3 3 1 c) Distance between , , , 3) 3 2 2 2 2 4) If (1, x) is at 10 units from(0, 0) then x = NARAYANA GROUP OF SCHOOLS 4) 5 5) 4 DAY-7 : SYNOPSIS Dividing a line segment in a given ratio (section formulae) : P A (P divides AB in the ratio m:n internally.) n m P A B (P divides AB in the ratio m:n externally.) Section formulae : The point ‘P’ which divides the line segment joining the points A x1 , y1 , B x2 , y2 in the ratio m:n mx2 nx1 my2 ny1 , i) internally is ; m n 0 mn mn mx2 nx1 my2 ny1 , ii) externally is ; m n mn mn Mid point of a line segment : The mid point of line segment joining of x1 , y1 x1 x2 y1 y2 , 2 2 and x2 , y2 is NOTE : 1. The point P(x,y) divides the line segment joining A x1 , y1 and B x2 , y2 in x1 x : x x 2 (or) y1 y : y y 2 i.e AP = PB = x1 x x x 2 2. x-axis divides the line segment joining x1 , y1 and x2 , y2 in the ratio y1 : y2 3. y-axis divides the line segment joining x1 , y1 and x2 , y2 in the ratio x1 : x2 Second - order determinant : The expression a b is called the secondc d order determinant. 14 CLASS-VIII MPC BRIDGE COURSE a b It is defined as = ad-bc c d 4 3 4 1 3 2 4 6 2. 2 1 Example : Area of a triangle : 1. The area of the triangle formed by the points A x1 , y1 , B x2 , y2 and C x3 , y3 = 1 2 (or) x y 1 2 y3 (or) 1 x1 x2 2 y1 y2 x1 x3 y1 y3 1 x1 x2 x3 x1 2 y1 y2 y3 y1 sq.units 2. The area of the triangle formed by the points O 0, 0 , A x1 , y1 , B x2 , y2 1 x1 y2 x2 y1 sq.units. = 2 4. x - axis divides the line segment joining (2,–3), (5,7) in the ratio is 1) 1 : 2 2) 3 : 7 3) 4 : 5 4) 3 : 4 5. The area of the triangle formed by the points (0,0), (2,0), (0,2) is 1)4 sq. units 2) 2 sq.units 3) 3 sq.units 4) 0 Single Correct Choice Type : 6. If the points 3, 8 , 4, 11 and 5, k are collinear then, the value of k is 1) 14 2) -8 3) 4 4) 5 7. The triangle formed by (0,1), (1,0) and (1,1) is 1) Right angle isosceles triangle 2) Scalene triangle 3) Equilateral triangle 4) Cannot form a triangle 8. The mid point of the line joining the points 1, 4 1) 5 and x, y is 2) 5 2 2,3 then x y is 3) 7 4) -5 NOTE : 1. Three points A,B,C are collinear if the area of ABC is zero. 9. If the point p 2,3 divides the line joining 2. If D,E,F are the mid points of the sides of the ABC then the area of is 1) 1: 2 internally 2) 1: 2 externally 3) 2 :1 internally 4) 2 :1 externally 10. The coordinates of the point which divides the line segment joining points A (0,0) and B(9, 12) in the ratio 1 : 2 are 1) (–3, 4) 2) (3, 4) 3) (3, –4) 4) None of these 11. The point which divides the line joining ABC = 4 (area of DEF ). 3. If G is the centroid of the ABC then area of ABC = 3(area of GAB ) DAY-7 : WORKSHEET Conceptual Understanding Questions : 1. The vertices of a triangle are A(0,–4) , B(4,0) and C(0,0), so ABC is 1) Right angled triangle 2) Isosceles triangle 3) Right angled, Isoceles triangle 4) Equilateral triangle 2. The mid point of (1,2) and (3,4) is 1) (2,3) 2) (3,2) 3) (2,4) 4) (1,3) 3. The ratio in which (2,3) divides the line segment joining (4,8), (–2,–7) is 1) 2 : 1 externally 2) 2 : 3 internally 3) 4 : 3 externally 4) 1 : 2 internally. NARAYANA GROUP OF SCHOOLS the points 5,6 and 8,9 ,then the ratio the points a b,a b and a b,a b in the ratio a : b is a 2 b2 a b 2 a 2 b2 b2 ab 2) , , 1) ab ab ab ab a 2 b2 a 2 b2 2ab , 3) 4) None of these a b ab 12.The ratio in which the line segment joining the points (3, –4) and (–5, 6) is divided by the x – axis is 1) 2 : 3 2) 3 : 2 3) –2 : 3 4) None 15 CLASS-VIII MPC BRIDGE COURSE 13.Let P and Q be the points on the line segment joining A(–2, 5) and B(3, 1) such that AP = PQ = QB. Then the midpoint of PQ is 1 1 1) ,3 2) ,4 3) 2,3 2 2 4) 1,4 14. The coordinates of points A, B, C are x1 , y1 , x 2 , y 2 and x 3 , y 3 and Exponential form: The product of a number ‘x’ with itself n times (n is natural number) is given by x x x x......... x (n times) and is written as x n which is called the exponential form. Here x is called base and n is called the exponent (or) index of ‘x’. xn can be read as nth power of x (or) x raised to the power ‘n’. point D divides AB in the ratio l : k. If P divides line DC in the ratio Exponential form is also called as power notation. m : k , then the coordinates of P are Example: 2 2 2 2 2 25 kx1 lx 2 mx 3 ky1 ly 2 my 3 , 1) k l m k l m Here 2 2 2 2 2 is called the product form (or) expanded form and 25 is called the exponential form. lx1 mx 2 kx 3 ly1 my 2 ky 3 , 2) l mk l mk mx1 kx 2 lx 3 my1 ky 2 ly 3 , 3) m k l m k l 4) None of these 15. P = (– 5,4) and Q = (–2,–3). If PQ is produced to R such that P divides QR externally in the ratio 1 : 2, then R is 1) 1,10 2) 1, 10 3) 10,1 4) 2, 10 DAY-8 : SYNOPSIS Variable: A symbol which can take various numerical values is called a variable. Example : x, y, z, a, b, c etc. Note: Variables are also known as Note: Note: * The first power of a number is the number itself. i.e., a1 = a. * The second power is called square. Example: Square of ‘3’ is 32 * The third power is called cube. Example: Cube of x is x3 * 1 raised to any integral power gives ‘1’ Example: 1100 = 1 * (– 1)odd natural number = –1 Example: (– 1)375 = –1 Laws of Exponents (or) Indices: 1. The product of the two powers of the same base is a power of the same base with the index equal to the sum of the indices. i.e., If a 0 be any rational number and m, n be positive integers, then a m a n a m n m n literals. Constant: A symbol having a fixed value is called a constant. 2. x 3 Example: 3, 2010, –5, etc. 7 Term: Numericals or literals or their combination formed by operation of multiplication are called terms. 3. Power of Product ab a n bn where Example: 5n, x , 10x 2 ,7l 5 m , y, 5, x 2 , 6 x mn ,Where m,n are positive integers, x 0 n a 0, b 0 , and n is a positive integer 4. Quotient of powers of the same base. a m n am 1 a n nm a if mn if nm etc. NARAYANA GROUP OF SCHOOLS 16 CLASS-VIII MPC BRIDGE COURSE 5. Power of a quotient m m a a i.e., m where a 0, b 0 , and m b b is a positive integer. 6. Powers with exponent zero: If we apply the above laws of indices of evaluate am where m = n, and a 0 ; then an 10. 4 If p = 32 10 A) 32 5 C) 11. 32 10 3 5 5 If 16 8 B) 2 1. If x is the fourth power of 2 and y is the third power of 4, then (x + y)2 + (y – x) + (y + x) is A) 6528 B) 6526 C) 6524 D) 6520 2. If 212a = 224, 33x = 312, then ax × xa is A) 256 B) 64 C) 512 D) 1024 3. If a = 4, b = 16 and c = 32, then a3b4c5 is A) 247 B) 248 C) 245 D) 249 4. If x4 = 16 and y5 = 243, then x × y is A) 6 B) 8 C) 4 D) 9 12 4 3 2 5. If P = 4 × 8 × 16 × 32 × 643, where p = 2x, then (p)10 is A) 2764 B) 2768 C) 2760 D) 2756 6. If xx + 3 = 32, yy + 3 = 729, then x x+y × y x+y is 32 10 5 10 D) 32 5 5 4 5 x 5 5 ÷ = and 16 16 6 6 × 19 19 DAY-8: WORKSHEET 4 m where p5 = 32 , then p × m is 17 am a m n a 0 1 an 5 4 y 6 = , then (x + y)3 19 A) 243 B) 126 C) 123 D) 244 12. If xa – b × xb – c × xc – a = p, then the value of p is A) 1 B) 0 C) 2 D) – 1 12 9 8 6 13. If a = 5 ÷ 5 , b = 4 ÷ 4 and c = 34 ÷ 32, then (a2b3c4)2 is A) (5 × 4 × 3)12 × 33 B) (5 × 4 × 3)12 × 32 C) (5 × 4 × 3)12 × 34 D) (5 × 4 × 3)12 × 35 14. If a 8 × 4 b = (2)8 × (4) 3, p q = 5 2, then ap × qb is aq × qa A) 16 B) 8 C) 32 D) 64 DAY-9 : SYNOPSIS A) (6)5 B) (12)5 C) (3)10 3 D) 2 5 Natural Numbers: Counting numbers 1, 2, 3, 4,..... are called Natural numbers, denoted by N. N = { 1, 2, 3, 4, ........}. 2 7. If (4 –1 2 1 × 3 ) = , then x is x –1 2 A) 24 B) 6 C) 18 D) 12 -3 23 2 8. If (3 – 2 ) × is multiplied by , 5 × 33 3 then the resultant is 2 2 2 A) 1 B) 0 C) – 1 D) 3 3 2 2 2 3 3 9. If (2x y ) × (4a b ) = p where x = 2y, y = 3a, a = 4b and b = 2, then the value of p is A) 256 (48)6(24)4(8)6(2)9 B) 256(48)5(24)4(2)9 C) 256(48)6(24)4(8)5(2)9 D) 256(48)6(74)3(8)5(2)8 NARAYANA GROUP OF SCHOOLS Note: * The smallest Natural numbers is 1 and the largest number can’t be determined. * The number of natural numbers between a and b where a < b is b – a –1. * The number of natural numbers from a to b where a < b is b –a + 1. Whole Numbers : The natural numbers along with zero are called whole numbers, denoted by W. W = { 0, 1, 2, 3, 4, .......}. Note: * The smallest whole number is ‘0’ and the largest whole number cannot be determined * All natural numbers are whole numbers. 17 CLASS-VIII * The difference of any two consecutive whole numbers is ‘1’. Fractions: * Tej has taken a card board which is in the shape of a square. * He has cut the card board as shown. He has observed that he got two pieces of card board and each is ‘a part of the whole’. Hence, we can say that each part is a fraction. * Therefore from the above illustrations we say that “A part of the whole is called Fraction.” Compa rison of fractio ns: By c ross multiplication: If two fractions a and b c are to be compared ,we cross multiply d a c (i) If a d b c, then b d a c (ii) If a d b c, then b d a c (iii) If a d b c, then b d 2 5 Example: Compare the and 3 6 Solution: On cross multiplication we get 2 6 and 3 5 12 and 15 12 < 15 2 MPC BRIDGE COURSE Solution: The L.C.M of 5,4,2,10 = 20 2 2 4 8 1 1 5 5 , Now , 5 5 4 20 4 4 5 20 3 3 10 30 9 9 2 18 , 2 2 10 20 10 10 2 20 Now compare the numerators of like 8 5 30 18 , , , fractions 20 20 20 20 Arranging them in ascending order, we 5 8 18 30 1 2 9 3 get So, 20 20 20 20 4 5 10 2 Fundamental operations on fractions: Addition: While adding like terms, add the numerators and retain the common denominator. In, general a c a c b b b Multiplication of fractions: In, general If a c and are two fractions, then the b d product of these fractions = product of their numerators a c = b d product of their denomi n ators Example: Multiply Solution: By taking the L.C.M: Taking the L.C.M of the denominator of the given fraction. Convert each of the fraction into an equivalent fraction with denominator equal to the L.C.M. Compare their numerators . The higher the value of the numerator, the greater is the fraction Example: Arrange 2 1 3 5 , , , in ascending 5 4 2 10 order. NARAYANA GROUP OF SCHOOLS Example: Add 3 4 34 7 5 5 5 5 Subtraction: While subtracting like terms, subtracting the numerators and retain the common denominator. 5 36 a c a c b b b 3 2 and . 7 5 3 2 3 2 6 7 5 7 5 35 Division of fractions: If fractions, then a c a d = . b d b c Example: Divide Solution: a c and are two b d 5 25 9 3 5 25 5 3 1 9 3 9 25 15 18 CLASS-VIII MPC BRIDGE COURSE DAY-9: WORKSHEET Conceptual Understanding Questions : 2 3 1. Compare and is 3 4 2 3 2 3 2 3 1) 2) 3) 4) Both 1 & 3 3 4 3 4 3 4 2 2. A Fraction greater than is 5 1 1 1 1 1) 2) 3) 4) 3 2 5 4 3. 10 paise as fraction of Rs. 1 is 1 2 3 1 1) 2) 3) 4) 5 5 10 10 2 1 1 4. Ascending order of , ,and is 3 2 6 1 1 2 2 1 1 1) , , 2) , , 6 2 3 3 6 2 1 2 1 2 1 1 3) , , 4) , , 6 3 2 3 2 6 2 1 3 5. Descending order of , and is 5 3 7 2 1 3 1 2 3 1) , , 2) , , 5 3 7 3 5 7 3 2 1 3 1 2 3) , , 4) , , 7 5 3 7 3 5 2 3 6. = 5 5 1) 1 2) 7. 2 3 = 3 2 13 1) 6 7 5 = 8. 12 7 2 1) 7 5 5 3) 3 2 5 5 15 16 and is 8 27 5 10 2 1) 2) 3) 9 9 9 1 5 2 11. The value of 2 of 2 2 15 15 2 1) 2) 3) 1 2 15 10. Product of 4) All the above 4) 5 16 4) 5 6 4) 18 28 4) 7 13 Single Correct Choice Type : 1. Equivalent fraction of 1) 2. 15 16 2) 12 20 Simplest form of 1) 2 3 2) 513 503 3) 3 is 4 21 28 2012 is 2000 3) 503 500 5 m- 2 = ,then the value of m is 10 30 1) 15 2) 17 3) 20 4) 7 4. The sum of three sides of a triangle is 3 cm. If two of its sides measure 16 5 3 7 cm and 6 cm respectively, then 5 4 10 the length of the third side is 3 5 1) 4 cm 2) 2 cm 20 17 7 3 3) 3 cm 4) 4 cm 20 5 3. If DAY-10 : SYNOPSIS 5 2) 5 2) 1 1 3 = 3 2 7 7 1) 2) 6 6 6 3) 6 11 84 3) 6 4) 13 2 19 4) 1 48 9. 2 3) 5 6 NARAYANA GROUP OF SCHOOLS 4) 5 7 Definition: If N and a 1 are any two positive real numbers and for some real x, such that a x N , then x is said to be logarithm of N to the base ‘a’. It is written x as log a N x . Thus, a N log a N x Note: * It should be noted that “log” is abbrevation of the word “logarithm”. * Logarithms are defined only for positive real numbers. 19 CLASS-VIII * There exists a unique ‘x’ which satisfies the equation a x N . 1 If x1, x 2 ,......x n are positive rational numbers, then So log a N is also unique. Logarithmic Function: Functions defined by such equations are called logarithmic functions. We can express exponential forms as logarithmic forms. Exponential form Logarithmic form (i) 24 = 16 4 log 2 16 (ii) 53 125 3 log 5 125 (iii) 32 2 log 3 1 1 9 9 Note: * For any positive real number ‘a’ we have a 1 a . Therefore log a a 1 . x * If a N x log a N ‘x’ by log a x1 , x 2 ,..x n log a x1 log a x 2 log a x 3 .... log a x n 2. If m and n are two positive rational m numbers, then log a log a m log a n n 3. log a m n 4. log a n m 5. log b n a m n. log a m 1 log a m n m log b a n 6. If m 1, b 1 are positive real numbers i.e., The logarithm of any non - zero positive number to the same base is unity. Replacing MPC BRIDGE COURSE GENERALISATION: log a N in a x N log m a then log b a log b m 7. If a 1, b 1 are two positive real number 1 then log b a log b . a a log a N N * The logarithm of unity to any non - zero base is zero. DAY-10: WORKSHEET o Recall that, if a 0, a 1 log a 1 0 LAWS OF LOGARITHMS: 1. If m, n are positive rational numbers, then log a mn log a m log a n . i.e., The ‘log’ of the product of two numbers is equal to the sum of their ‘logs’. Proof: Let log a m x and x log a n y from (1) & (2) mn a x .a y mn a x y Apply log to the base ‘a’ on both sides log a mn x y log a mn log a m log a n NARAYANA GROUP OF SCHOOLS 1. If 24 16 , then which of the following is true ? 1) log 2 4 16 2) log 2 16 2 3) log 4 16 4 4) log 2 16 4 2. log 0.1 0.1 = ______ y m a ....... 1 and n a ....... 2 x log a m and y log a n Conceptual Understanding Questions : 1) 0.1 2) 0.2 3) 0 4) 1 3) 27 4) 275 3. 5 log 5 27 = ______ 1) 5 2) 55 4. log 213 1 ____ 1) 1 2) 213 3) 1 213 4) 0 20 CLASS-VIII MPC BRIDGE COURSE 9. log 3.5 = _______ 5. log 24 = ____ 1) log 4 log 6 2) log 2 log12 3) log 3 log 8 4) All of these 6. log 3 4 log 2 3 = 1) log 32 4 3 3) log 4 3 log 2 3 4) log 2 4 4 2) log 3 2 4 4) 44 log 3 2 3) 4 log 3 2 1) log10 5 x 2 2) 1 3) only (1) 4) Both 1 & 2 11. log10 20 log10 2 ________ 7. log 34 2 ______ 1) (lo g 3 2 ) log 20 1) log10 2) log10 3) log10 (20 2) 20 4) log 2 10 10 Single Correct Choice Type : 1. log 4 x 2 , then x = ___ 1) 14 2) 16 3) 12 2 2. If 3 4) 11 1 , then which of the following is 9 true ? 1) log 2 3 3) log 3 1 9 2) log 3 2 1 1 9 2 4) log 3 1 9 1 2 9 1) 3 2) 2 3) 9 4. The value of log100.0001 is 1) –4 2) 4 3) –10 1) 2 4) 81 4) 10 4 is 2 2) 4 6. The value of x if log 3) 12. The value of 1) 2 2 4) 3 216 x is 6 1) 5 2) 6 3) 4 4) 3 7. Logarithm of any non-zero number to the same base is _______ 1) Same non-zero number 2) 0 3) 1 4) Negative of the non-zero number 8. Logarithm of unity to any non-zero base is _______ 1) Non-zero number 2) Unity 3) 0 4) –1 NARAYANA GROUP OF SCHOOLS 3) 5 13.The value of log 3 27 3 1) 1 2 2) 20 2 log 8 is log 2 2) 4 3 2 3) 4) 6 is 5 2 4) 7 2 14. log b x log a b = 1) logxa 3. The value of log 9 81 = 5. The value of log log 35 3) log10 10. log10 5 log10 2 _____ log 4 log2 2) log 3 log3 4 4 35 2) log 35 log10 10 4) Both 1&2 1) log 2) logxb 3) logax 4) logbx 15. logamn = 1) logam × logan 2) logam + logan m 3) log a n 4) –logam – logan 16. If log10 2 0.3010 , then logarithm of 5 32 with base 10 is 1) 9.725 2) 7.525 3) 7.725 4) 9.525 17. log10 25 log10 4 = 1) 4 2) 3 3) 2 4) 1 18. Express 2 log 5 5 3 log 2 log 50 1 as single logarithm is 1) log 5 2) log 3 3) 1 4) log 2 21 CLASS-VIII DAY-11 : SYNOPSIS Sequence: It is an arrangement of numbers in a definite order according to same rule Example: 5, 10, 15, 20, ...................... etc. and 4, 44, 444, .............. i) Tn = 5n + 2 represents a sequence 7, 12, 17 ............... ii) Tn = 3n - 4 represents a sequence -1, 2, 5 ....... iii) Tn = 2n 3 represents a sequence 2 5 7 9 , , , ................ 2 2 2 Series: An expression consisting of the terms of a sequene, alternating with the symbol “ + ” is called a “ Series ” For example, associated with the sequence 3 5 7 2n 1 , , ....., ........ 5 7 9 2n 3 MPC BRIDGE COURSE Arithmetic mean (A.M.): The arithmetic mean between two numbers is the number which when placed between them forms with them an arithmetic progression. Thus the arithmetic mean between a and b is A.M = a b where a and b are any 2 two positive numbers. If there are ‘n’ A.M’s between a, b then common difference, d ba n 1 Sum of first ‘n’ natural numbers, S1 n n n 1 2 Sum of the squares of first ‘n’ natural numbers, S 2 n 2 n n 1 2n 1 6 Sum of the cubes of first ‘n’ natural 2 3 5 7 2n 1 ........ we have series , , ..... 5 7 9 2n 3 nth term: The number occuring at the nth place of a sequence is called its nth term. Denoted by Tn. Progression: The sequence which obey the definite rule and its general term is always expressible in terms of ‘n’ is called progression. Example: 1) 2, 4, 6, 8, 10, ...... 2) 1, 3, 9, 27 ........... 1 1 1 3) , , ,.......... 5 7 9 Arithmetic progression (A.P):A Sequence is called an arithmetic progression if its terms continually increase or decrease or decrease by the same number. Example:a, a + d, a + 2d, a + 3d, ... is in A.P nth term of A.P is tn = a + (n-1)d Sum of ‘n’ terms of A.P Sn n n 2a n 1 d and S n a 1 2 2 n 2 n 1 2 numbers, S3 n n 4 3 S n t1 t2 ...........tn 1 t n = tn ; tn sn sn 1 DAY-11: WORKSHEET Conceptual Understanding Questions : 1. Choose a sequence from the given 1) 1,3,5,7,9............2) 1,2,4,5,7,9,10,11..... 3) -1,1,7,10,20......4) 1,7,10,20,25,.... 2. Choose a series from the given 1) 1 + 2 + 4 + 5 + 7 + 9............. 2) 1 + 3 + 5 + 7 + 9......................... 3) -1 + 1 + 7 + 10 + 20.............. 4) 1 + 7 + 10 + 20 + 25................... 3. 0,4,8,12,16 forms a / an 1) Arithmetic sequence 2) Geometric sequence 3) Arithmetic progression 4) Geometric progression where l is the last term. NARAYANA GROUP OF SCHOOLS 22 CLASS-VIII 4. The first term of an Ap is ‘a’ and its common difference is 2, then first five terms of given AP are 1) a, a + 2, a + 4, a + 6, a + 8 2) a + 2, a + 4, a + 6, a + 8, a + 10 3) a, a - 2, a - 4, a - 6, a - 8 4) a, 2 - a, 4 - a, 6 - a, 8 - a 4. If the first term of an AP is 3 and its common difference is 5, then its n th term is 1) 3 + 5n 2) 5n - 2 3) 3n - 5 4) 3 - 5n 5. If 11, 6, 1, -4, -9 forms a AP, then its common difference is 1) -5 2) 4 3) 5 4) -4 Single Correct Choice Type : 6. The next term in the series 7, 12, 19 .... 1) 29 2) 28 3) 26 4) 24 7. 3, 7, 15, 31, 63, (..............) 1) 92 2) 115 3) 127 4) 131 8. If 5x + 2, 4x - 1, x + 2 are in A.P, then the first term is 1) 18 2) 17 3) 19 4)16 9. If the nth term of AP is 4n - 1 then the 18th term is 1) 72 2) 73 3) 74 4) 71 10. If 8k + 4, 6k - 2, 2k + 12 are in AP then the value of k is 1) -10 2) -11 3) 9 4) 10 th 11. The 25 term of the A.P. 10, 6, 2, -2, -6, -10, ........ is 1) - 86 2) 106 3) 96 4) 46 12. If first term is 3 and 38th term is 114 ehn the 68th term is 1) 204 2) 202 3) 200 4) 208 rd th 13. If the 3 and 7 terms of an A.P. are 17 and 27 respectively, then the first term of the A.P is 1) 9 2) 12 3) 14 4) 16 14. If a, x1, x2, x3.......xn b are in A.P then xn 1) an b an b a nb a b 2) 3) 4) n 1 n 1 n 1 n 13.If the nth term of the A.P 23, 25, 27, 29 ... and -17, -10, -3, 4 ..... are equal, then the value of n = _________ 1) n = 10 2) n = 8 3) n = 9 4) n = 12 NARAYANA GROUP OF SCHOOLS MPC BRIDGE COURSE DAY-12 : SYNOPSIS Geometrical Progression: A sequence (finite or infinite) of non-zero number in which every term, except the first one, bears a constant rati with its preceding term, is called a geometric progression, abbreviated as G.P. Illustration: The sequences given below: i) 2, 4, 8, 16, 32,........... ii) 3, -6, 12, -24, 48,............. iii) 1 1 1 1 1 , , , , ,........ 4 12 36 108 324 Note: In a G.P. any term may be obtained by multiplying the preceding term by the common ratio of the G.P. Therefore, if any one term and the common ratio of a G.P. be known, any term can be written out, i.e., the G.P. is then completely known. In particular, if the first term and the common ratio are known, the G.P. is completely known. The first term and the common ratio of a G.P. are generally denoted by a and r respectively. General term of a G.P.: Let a be the first term and r 0 be the common ratio of a G.P. Let t1, t2, t3, ........, tn denote 1st, 2nd, 3rd, ...., nth terms, respectively. Then, we have t 2 = t1r, t 3 = t 2 r, t 4 = t 3 r, ..., tn = t n-1r. On multiplying these, we get t 2 t 3 t 4 ... tn = t1t 2 t 3 ... t n-1r n-1 t n = t1r n-1, but t 1 = a. General term = tn = arn-1. Thus, if a is the first term and r the common ratio of a G.P. then the G.P. is a, ar, ar2, ... ar2-1 or a, ar, ar2........ according as it is finite or infinite. If the last term of a G.P. consisting of n terms is denoted by l, then l = arn-1. Note: * If a is the first term and r the common ratio of a finite G.P. consisting of m terms, then the nth term from the end is given by arm -n. 23 CLASS-VIII th * The n term from the end of a G.P. with the last term l and common ratio r i s l/r n-1. * Three numbers in G.P. can be taken as a/r, a, ar, four numbers in G.P. cn b e taken as a/r3, a/r, ar, ar3, five numbers in G.P. can be taken as a/r2, a/r, a, ar, ar2, etc..... * Three numbers a, b, c are in G.P. if and only if b/a = c/b, i.e., if and only if b 2 = ac. DAY-12: WORKSHEET Conceptual Understanding Questions : 1. 1,2,22,23,24,......... forms a / an 1) Arithmetic sequence 2) Geometric sequence 3) Arithmetic progression 4) Geometric progression 2. If the first term of a GP is 3 and its common ratio is 3, then first five terms of the GP are 1) 3,6,9,12,15 2) 3,5,7,9,11 3) 3,9,27,81,243 4) 3,7,11,15,19 3. if t7 of a GP is 128 and its common ratio is 2, then its first term is 1) 1 2) 2 3) 3 4) 4 4. The first term of a GP is 1, and common ratio is 2. Then the sum of its 5 terms is 1) 20 2) 25 3) 27 4) 31 5. Geometric Mean of 27 and 3 is 1) 4 2) 6 3) 7 4) 9 Single Correct Choice Type: 6. If x, 2x + 2, 3x + 3, ..... are in geometric progression, the fourth term is 1) -27 2) 13 1 2 3) 12 4) 13 7. If the 10 th term of a G.P is 9 and 14 th term is 4, then the 7th term is 1) 2 2) 6 3) 4 4) 8 th th th 8. If 5 , 8 , and 11 term of G.P. are p, q and s respectively, then 1) p2 = qs 2) q2 = ps 3) s2 = pq 4) s = pq 9. If x + 9, x - 6, 4 are in G.P then the value of x is 1) 8 2) 12 3) 16 4) 20 NARAYANA GROUP OF SCHOOLS MPC BRIDGE COURSE 10. The first term of the progression G.P. is 50 and the 4th term is 1350, then the 5th term is 1) 8050 2) 5050 3) 4050 4) 6050 11.Sum of three consecutive terms in a G.P. is 42 and their product is 512. Find the largest of these numbers 1) 28 2) 16 3) 32 4) 30 12.The sum of the 8 terms of 3, 6, 12, 24,....... is 1) 765 2) 465 3) 565 4) 645 13. Number of terms of a G.P. 3,3 2,33,..... are needed to give the sum 120 is 1) 4 2) 2 3) 8 4) 10 14. Sum to ‘n’ terms of the series 0.5 + 0.55 + 0.555 + ....... equals to 5 10n 1 1) n 9 9.10n 1 10n 1 2) n 9 9.10n 5 10n 1 n 3) 9 9.10n 1 10n 1 n 4) 9 9.10n 15. The sum to n terms of the progression 1, -1, 1, -1, 1,...........(-1)n+1 = 1) 0 if n is even 2) -1 if n is even 3) 0 if n is odd 4) -1 if n is odd DAY-13 : SYNOPSIS 1. Introduction : In everyday life, we have to deal with the collections of objects of one kind or the other. For Example : i) The collection of even natural numbers less than 12 i.e., the numbers 2, 4, 6, 8 and 10. ii) The collection of vowels in the English alphabet i.e., the letters a, e, i, o, u. In each of the above collections, it is definitely known whether a given object is to be included in the collection or not to be included. Each one of the above collections is an example of a set. However, the collection of all intelligent students in a class of a given school is not a set. Here it is difficult to decide who is intelligent and who is not. The same student may be intelligent in the eyes of one teacher and may not be intelligent in the eyes of another. We say that such a collection is not well defined. With this basic notion, we have the following. 24 CLASS-VIII Definition : Any well defined collection of objects is called a set. By ‘well-defined collection’ we mean that given a set and an object, it must be possible to decide whether or not the object belongs to the set. The objects are called the members or the elements of the set. Sets are usually denoted by capital letters and their members are denoted by small letters. We write the elements of set with in the bracess { }. If x is a member of the set S, we write x S (read as x belongs to S) and if x is not a member of the set S, we write x S (read as x does not belong to S). If x and y both belongs to set S, we write x, y S. Representation of Sets : There are two ways to represent a given set. 1) Roster or Tabular Form or list form. In this form, list all the members of the set, separate these by commas and enclose these within braces (curly brackets) For example : i) The set S of even natural numbers less than 12 in the tabular form is written as S = { 2, 4, 6, 8, 10 }. Note that 8 S while 7 S. ii) The set S of prime natural numbers less than 20 in the tabular form is written as S ={2,3,5, 7, 11, 13, 17, 19 } iii)The set N of natural numbers in the tabular form is written as N = { 1, 2, 3, ... }, the dots indicating infinitely many missing positive integers. 2. Set Builder or rule form : In this form, write one or more (if necessary) variables (say x, y etc.) representing an arbitrary member of the set, this is followed by a statement or a property which must be satisfied by each member of the set. For example : i) The set S of even natural numbers less than 12 in the set builder form is written as S = { x/x is an even natural number less than 12}. ii) The set of prime natural number less than 20 in the set builder form is written as {x/x is a prime natural number less than 20}. NARAYANA GROUP OF SCHOOLS MPC BRIDGE COURSE The symbol ‘|’ stands for the words ‘such that’ or ‘where’. Sometimes we use the symbol ‘;’ or ‘:’ in place of the symbol ‘|’. iii)The set N of natural numbers in the set builder form is written as N = { x :x is a natural number}. Some Standard Sets : We enlist below some sets of numbers which are most commonly used in the study of sets : i) The set of natural numbers (or positive integers). It is usually denoted by N. i.e. N = { 1, 2, 3, 4, .... } ii) The set of whole numbers. It is usually denoted by W. i.e. W = { 0, 1, 2, 3, ... }. iii) The set of integers. It is usually denoted by Z . i.e. Z = { ...., -3, -2, -1, 0, 1, 2, 3,...} iv) The set of rational numbers. It is usually denoted by Q i.e. Q = { x : x is a rational number} or m Q x : x , where m and n are int egers and n 0 n v) The set of real numbers. It is usually denoted by R. i.e. R = { x : x is a real number } or R = { x : x is either a rational number or an irrational number } Note :- i 1 Types of sets : The Empty set : A set containing no element is called the empty set. It is also called the null set or void set . There is only one such set.It is denoted by or by { }. For example : i) The collection of all integers whose square is less than 0 is the empty set. ( Square of an integer cannot be negative) ii) The collection of all girl students in a boys college is the empty set. iii) The collection of all the real roots of the equation x2 + 5 = 0 is the empty set. ( The equation x2 + 5 = 0 is not satisfied by any real number, for if ‘a’ is a real root of x2 + 5 = 0, then a2 + 5 = 0 a2 = – 5, which is not possible as square of a real number cannot be negative ). The order of the empty set is zero. 25 CLASS-VIII Singleton set : A set is said to be a singleton set if it contains only one element. The set { 7 }, { – 15 } are singleton sets. { x : x + 4 = 0, x Z } is a singleton set, because the set contains only one integer namely, –4. The set { x : x + 4 = 0, x N } is a null set, because there is no natural number which may satisfy the equation x + 4 = 0. A set whose order is 1 is called a singleton set. Thus, a singleton set is a set which contains only one distinct element. For example : i) If A = { x : x is a positive divisor of 20 }, then n(A) = 6 as A = { 1, 2, 4, 5, 10, 20 } ii) If B = { x : x is a positive even prime }, then n(B) = 1 as B = { 2 }. Note that B is a singleton set. iii) If C = { x : x is an integer neither positive nor negative }, then n(C) = 1 as C = {0}. C is a singleton set. Finite and Infinite sets : A set is called finite if the process of counting of its different elements comes to an end; otherwise, it is called infinite. The empty set is taken as finite. For example : i) The set S = { 2, 4, 6, 8 } is a finite set. ii) The set of all students studying in a given school is a finite set. iii) The set N of all natural numbers is an infinite set. iv) The set of divisors of a given natural number is a finite set. v) The set of all prime numbers is an infinite set. Order of a finite set : The number of different elements in a finite set S is called order of S, it is denoted by O(S) or n(S). Note : The order of an infinite set is not defined. Equivalent sets : Two finite sets A and B are said to be equivalent written A ~ B (or A B), iff they contain the same number of distinct elements i.e., iff n(A) = n(B). NARAYANA GROUP OF SCHOOLS MPC BRIDGE COURSE For example : i) The sets { 1 } and {2, 2, 2} are equivalent. ii) The sets { 3, 4 } and { x : x2 = 4 } are equivalent sets. iii) The sets { a, b, c, d, e }, { 1, 2, 3, 4, 5 } and { a, e, i, o, u } are equivalent sets as each of these sets contains 5 distinct elements. Equal sets :Two sets A and B are said to be equal, written as A = B, iff every member of A is a member of B and every member B is a member of A.Remember that equal sets are always equivalent but equivalent sets may not be equal. For example :i) The sets { – 1, +1 } and { x : x2 = 1 } are equal. ii) The sets { 0, 0 } and { 3 } are not equal, but they are equivalent. DAY-13: WORKSHEET Conceptual Understanding Questions : 1. Which of the following not a well defined collection of objects 1) The collection of days in week 2) The Colloection of all even integers 3) The Vowels in the English Alphabet 4) The collection of ten most talented in your Class 2. Write the set {x : x is a positive integer 2 and x 40 } in the roster form 1) {1,2,3,4,5,6} 2) {1,2,3,4,5,6,7} 3) {2,3,4,5,6,7} 4) {0,1,2,3,5,6} 3. Roster form of set of ‘S’ odd natural numbers less than 15 is 1) {1,3,5,7,9,11} 2){1,3,5,7,9,11,13} 3){x/x is odd natural numbers } 4){1,2,4--------} 4. Set builder or roster form of set S of odd natural numbers lessthan 15 is 1) {1,3,5,7,9,11} 2){1,3,5,7,9,11,13} 3){x/x is odd natural numbers } 4){1,2,4--------} Single Correct Choice Type: 1.Which of the following collections are sets. i) The collection of all prime numbers between 7 and 19. ii) The collection of all rich persons in India. 26 CLASS-VIII iii) Collection of all factors of 50 which are greater than 6. 1) (i) and (ii) 2) (i), (ii) and (iii) 3) (ii) and (iii) 4) (i) and (iii) 2. Which of the following collections are sets. i) The collection of all months of a year, beginning with letter J. ii) The collection of most talented writers of India. iii) The collection of all natural numbers less than 100. iv) The collection of most dangerous animals of the world. 1) Only (i) 2) Only (ii) 3) Both (i) and (iii) 4) Both (ii) and (iv) 3. The solution set of the equation x 2 x 2 0 in roster form is 1) 1, 2 4) 2) 1, 2 3) 1, 2 x : x R,1 x 2 4. If A 6,12 then set builder form is 1) A x : x R and 6 x 12 2) A x : x R and 6 x 12 3) A x : x R and 6 x 12 4) A={6,7,8,9,10,11,12} 5. Which of the following sets are singleton sets ? i) A = { x : 3x – 2 = 0, x Q } ii) B = { x : x3 = 0, x R } iii) C = { x : 30x– 59 = 0, x N } iv) D = { x : |x| = 1, x Z } 1) (i) and (iii) 2) Only (iii) 3) (i) and (ii) 4) Only (iv) 6. Which of the following sets are null sets? i) A = { x : x < 1 and x > 3 } ii) B = { x : x2 = 9 and 3x = 7 } iii) C = { x : x2 – 1 = 0, x R } iv) D = { x : x is an even prime number } 1) (i) and (iii) 2) (i) and (ii) 3) (ii) and (iv) 4) (iii) and (iv) NARAYANA GROUP OF SCHOOLS MPC BRIDGE COURSE 7. Which of the following statements are not true ? i) { x : 3x + 1 < 10, x N } = { 1, 2, 3, 4 } ii) { x : 2x < 50, x N } = {1, 2, 3, 4, 5 } 1) Only (ii) 2) Both (i) & (ii) 3) Only (i) 4) none of these 8. Which of the following is an empty set ? 1) { x : x R and x2 – 1 = 0 } 2) { x : x R and x2 + 1 = 0 } 3) { x : x R and x2 – 9 = 0 } 4) { x : x R and x2 = x + 2 } 1 9. If Q = x : x = , where x, y N , then y 2 Q 3 10. Which of the following sets are finite. i) Prime numbers set ii) Human beings set in the world iii) A real number set between 1 and 2. iv) The set of multiples of 3. 1) Only (ii) 2) Only (iii) 3) (i) and (ii) 4) (iii) and (iv) 11.Which of the following is an infinite set. i) Natural number set less than one million. ii) The set which can be written by using the digit 1, repeated any number of times. iii) The set of elephants in the world. iv) Spheres set passing through a given point. 1) Only (ii) 2) (ii) and (iv) 3) Only (iii) 4) (i) and (iv) 1) 0 Q 2) 1 Q 3) 2 Q 4) DAY-14 : SYNOPSIS Cardinal number of set : The number of distrinct elements contained in a finite set is called its cardinal number and is denoted by n(A). Examples : If A = {1, 2, 3, 4, 5} then n(A) = 5, if B = {1, 2, 3} then n(B) = 3. Subsets: Let A, B be two sets such that every member of A is a member of B, then A is called a subset of B, it is written as A B. Thus, A B iff (read as ‘if and only if’) x A x B. 27 CLASS-VIII If (read as ‘there exists’) atleast one element in A which is not a member of B, then A is not a subset of B and we write it as A B. For example: i) Let A = {–1, 2, 5} and B = {3, – 1, 2, 7, 5}, then A B. Note that B A ii) The set of all even natural numbers is a subset of the set of natural numbers. Some properties of subsets : i) The null set is subset of every set. Let A be any set. A , as there is no element in which is not in A. ii) Every set is subset of itself. Let A be any set. xA xA A A. iii) If A B and B C , then A C . Let xA . x B x C A C. iv) A = B iff A B and B A . Let A = B. A B A B . Similarly, x B x A A B B A. Conversely, let A B and B A . x A x B A B and x B x A Ex 2 : A = { 1, 2, 3, 4, 5 }, B = { 2, 3, 4 } Every element of B i.e., 2, 3 and 4 is also an element of A. B A Further we note that there are two more elements that are in A and not in B. They are 1 and 5. Then A B . In such circumstances we say that B is a proper subset of A. Ex 2 : N W Z Q R Remark 1 : If A B then every element of A is in B and there is a chance that A may be equal to B i.e., every element of B is A, but if A B , then every element of A is in B and there is no chance that A may be equal to B i.e., there will exist at least one element in B which is not in A. A B A B, A B i.e., A B, B A . A B B C x A x B MPC BRIDGE COURSE Ex 1 : If A = { 1, 2, 3 }, then proper subsets of A are , { 1 },{ 2 },{ 3 }, { 1, 2 }, { 1, 3 }, { 2, 3 } B A A=B Note:1. Two sets A and B are equal iff A B and B A. 2. Since every element of a set A belongs to A, it follows that every set is a subset of itself. Proper subset: Let A be a subset of B. We say that A is a proper subset of B if A B i.e., if there exists atleast one element in B which does not belong to A. A subset, which is not proper, is called an improper subset. Observe that every set is an improper subset of itself. If a set A is non-empty, then the null set is a proper subset of A. NARAYANA GROUP OF SCHOOLS Remark 2 : If A B , we may have B A , but if A B , we cannot have B A . Power Set: The set formed by all the subsets of a given set A is called the power set of A, it is usually denoted by P(A). For example: i) Let A = {0}, then P(A) = { , {0}}. Note that n(P(A)) = 2 = 21. ii) Let A = {a, b}, then P(A) = { , {a}, {b}, {a, b}}. Note that n(P(A)) = 4 = 22 iii) Let A = {1, 2, 3}, then P(A) = { , {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, A}. Note that O(P (A)) = 8 = 23. In all these examples, we have observed that n(P(A)) = 2n(A). Rule to write down the power set of a finite set A: First of all write . Next, write down singleton subsets each containing only one element of A. In the next step write all the subsets which contain two elements from the set A. Continue this way and in the end write A itself as A is also a subset of A. Enclose all these subsets in braces to get the power set of A. 28 CLASS-VIII Comparable Sets : Two sets A and B are said to be comparable iff either A B or B A. For example : i) The sets A = {1, 2} and B = {1, 2, 4, 5} are comparable as A B. ii) The sets A = {0, 1, 3} and B = {1, 3} are comparable as B A. iii) The sets A = {–1, 1} and B = {x : x2 = 1} are comparable as A B and also B A. Clearly, equal sets are always comparable. However, comparable sets may not be equal Universal Set: In any application of the theory of sets, all sets under investigation are regarded as subsets of fixed set. We call this set the universal set, it is usually denoted by X or U or . OPERATIONS OF SETS UNION OF SETS The union of two sets A and B is the set of all those elements, which are either in A or in B (including those which are in both) In symbolic form, union of two sets A and B is denoted as, A B . It is read as “A union B”. Clearly, x A B And, x A B x A or x B . x A and x B . It is evident from definition that A A B; B A B SOLVED EXAMPLES (i) A a,e,i,o,u , B a,b,c (ii) A 1,3,5 , B 1,2,3 Solution:- (i) We have, A B a,e,i,o,u a,b,c MPC BRIDGE COURSE Here, the common elements 1 and 3 have been taken only once, while writing A B . UNION OF THREE OR MORE SETS The union of n n 3 for sets A 1 , A2........., An is defined as the set of all those elements which are in Ai 1 i n for atleast one value of i. The union of n A1, A 2, A 3 ,........A n is denoted A i i 1 In symbols, we write n A i { x : x A i for at least one value of i 1 i, 1 i n } SOLVED EXAMPLE :- If A = {1, 2, 3, 4} B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}, find (i) A B (ii) A B C (iii) B C D Solution :(i) We have, A B 1,2,3,4 3,4,5,6 A B 1,2,3,4,5,6 (ii) We have, A B C 1,2,3,4 3,4,5,6 5,6,7,8 A B C 1,2,3,4,5,6,7,8 (iii) We have, B C D 3,4,5,6 5,6,7,8 7,8,9,10 B C D 3,4,5,6,7,8,9,10 INTERSECTION OF SETS The intersection of two sets A and B is the set of all those elements which belong to both A and B. Symbolically, we write A B = {x : x A and x B } and read as “A intersection B”. Let x A B x A and x B and A B a,b,c,e,i,o,u x A B x A or x B Here, the common element a has been taken only once, while writing A B . It is evident from the definition that A B A, A B B (ii) We have A B 1,3,5 1,2,3 A B 1,2,3,5 NARAYANA GROUP OF SCHOOLS SOLVED EXAMPLES Example 1 : (i) If A = {3,5,7,9,11}, B = {7,9,11,13}, find A B . 29 CLASS-VIII MPC BRIDGE COURSE (ii) If A = {a, b, c}, B = , find A B . Solution. (i) We have, A B = {3,5,7,9,11} {7,9,11,13} Since, 7,9,11 are the only elements which are common to both the sets A and B. A B 7,9,11 (ii) We have, A B a,b,c . Since, there is no common element. Example 2. Let A = {x : x is a natural number} and B = { x : x is an even natural number}. Find A B Solutions : We have, A = { x : x is a natural number}. A = {1, 2, 3, 4,......} B = {x : x is an even natural number}. B = {2, 4, 6,.......} We observe that 2, 4, 6,.. are the elements which are common to both the sets A and B. Hence, A B ={2, 4, 6...} = B. INTERSECTION OF MORE SETS The intersection of n n 3 se ts A1,A2,A3.........An is defined as the set of all those elements which are in A i 1 i n for each i. The intersection of A 1 ,A 2 .......,A n is n denoted by A i i 1 n In symbols, we write, A i { x : x A i i 1 for all i, 1 i n} DISJOINT SETS : Two sets A and B are said to be disjoint, if A B = , . If A B then A and B are said to be inter- secting sets or overlapping sets. e.g. Let A = {1, 2, 3}, B = {a, b, c} A B= , hence A and B are disjoint. Solved example : 1. Which of the following pairs of sets are disjoint ? i) A = {1, 2, 3, 4}, B = { x : x is a natural number and 4 x 6 }. ii) A = { a, e, i, o, u}, B= { c, d, e, f} iii) A = {x : x is an even integer}, B = {x : x is an odd integer} NARAYANA GROUP OF SCHOOLS Solution :i) We have, A = {1, 2, 3, 4}; B = {x : x is a natural number and 4 x 6 } B = { 4, 5, 6 } A B = { 1, 2, 3, 4 } { 4, 5, 6 } = { 4 } Here, A B , hence A and B are not disjoint sets, but are intersecting sets. ii) We have, A B = { a, e, i, o, u } { c, d, e, f } = { e } Here, A B , hence A and B are not disjoint sets but are intersecting sets. iii) We have, A = { x : x is an even integer} = { ..... – 2, 0, 2, .......} and B = { x : x is odd integer } = { ...–3, –1, 1, 3, ....} Hence, A B , hence A and B are disjoint sets DAY-14: WORKSHEET Conceptual Understanding Questions : 1. _______ set is subset of every set 1) U 2) 3) finite 4) infinite set 2. A B 1) {x : x A or x B} 2) {x : x A and x B} 3) {x : x A or x B} 4){x : x A and x B} 3. If A ={1,2,3,4}, then the no.of elements of power set of A is ____. 1)8 2)1 3)16 4)10 4. If A ={1,4,6} and B = {1,2,3,4,5,6} then A B 1){1,4,5} 2){1,4,6} 3){1,2,3,4,5,6} 4){1,2,3,4,5} 5. If A={1,2,3}, B={2,3,4} and C = {3,4,5,6} then A B C 1){1,2,3,4} 2){2,3,4,5,6} 3){1,2,3,4,6} 4){1,2,3,4,5,6} 6. If A={1,4,6} and B= {1,2,3,4,5,6} then A B ____. 1){1,2,3,4} 2){1,4,6} 3){1,2,3,4,5,6} 4){2,4,6} 7. If A={1,2,3,4}, B={5,6,7}, then A B ____. 1)A 2)B 3) 4){2,3,5} 30 CLASS-VIII MPC BRIDGE COURSE 8. The cardinal number of A = {1, 2, 3, 4, 5, 6, 7, 8} is 1) 7 2) 8 3) 6 4) 5 9. A x / x 5, x N then n(A) 10. If A 2,3, 4,8,10 , B 3, 4,5,10,12 , C 4,5, 6,12,14 then A B A C 2 1) 7 2) 6 3) 5 4) 4 Single Correct Choice Type: 1. If A B, then A B = 1) A 2) B 3) 2. If A = , then A B = 4) A B 1) A 2) B 3) 4) A B 3. If A={1, 2, 3, 4, 5, 6, 7} then number of proper subsets of A is 1) 27 2) 26 3) 27 1 4) 28 1 4. Howmany elements has P(A),if A 1) 1 2) 2 3) 3 4) 0 5. A = {x / x = y + 1, y N, y < 6}, B = {x / x = y –1, y N, 2 < y < 5} then which of the following is correct. 1) n A n B 2) n A n B 3) n A n B 4) n A n B 6. A = {1, 2, 3}, B = {1, 2, 3, 4, 5} then which of the following is true 1) A is subset of B 2) A is super set of B 3) A is impropersubset of B 4) A is equal set to B 7. Let A={1,2,3},B={3,4},c={4,5,6} then A (B C ) 1){3} 2){1,2,3,4} 3){1,2,5,6} 4){1,2,3,4,5,6} 8. A = { 1, 2} then which of the following is true 1) 1 P A 3) 1,2 , P A 2) 2 P A 4) all of these 9. Given N 1, 2,3,....,100 . Then subset B of N, whose elements are represented by x 2 where x N 1) {3, 5, 6, ..., 100} 2) {3, 4, 5, 6, .., 100} 3) {1, 3, 4, 5, 6,....,100} 4) {2, 4, 6, 8, ..., 100} NARAYANA GROUP OF SCHOOLS 1) 2,3, 4,5,8,10,12 2) 2, 4,8,10,12 3) 3,8,10,12 4) 2,8,10 11.Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in A B ? 1) 3 2) 6 3) 9 4) 18 12. If A = {x : x = 3n, n Z } and B = {x : x = 4n, n Z }, then A B = 1) 12n 2) 15n 3) 10n 4) n2 13.If A {2, 4} and B {3,5} then A B = 1) {2, 3,4,5} 3) {2,3, 4} 14.If 2) {3, 5} 4) {3, 4} n(u) 60, n(A) 21, n(B) 43 th en greatest value of n A B and least value of n A B are 1) 60, 43 2) 50, 36 3) 70, 44 4) 60, 38 15.Let A and B be two sets such that A B A . Then A B is equal to 1) 2) B 3) A 4) A B DAY-15 : SYNOPSIS Difference of Sets : If A and B are two sets, then their difference A – B is the set of all those elements of A which do not belong to B Let x A – B x A and x B. Similarly, the difference B – A is the set of all those elements of B that do not belong to A. i.e., B – A = { x : x B and x A} x B – A x B and x A Solved Examples : Example - 1 If X = {a, b, c, d} and Y = { f, b, d, g }, find (i) X – Y (ii) Y – X (iii) X Y Solution : i) We have, X – Y = {a, b, c, d} – {f, b, d, g} = {a, c} 31 CLASS-VIII Since, the only elements of X which do not belong to Y are a and c ii) We have, Y – X = {f, b, d, g} – {a, b, c, d} = {f, g} Since, the only elements of Y which do not belong to X are f and g iii) We have,X Y = {a, b, c, d} {f, b, d, g} = {b, d} Since, b and d are common elements of X and Y. Example - 2 If A = {1, 2, 3, 4, 5, 6} and B = { 2, 4, 6, 8}. Show that A – B B – A. Solution :- We have, A – B = {1, 2, 3, 4, 5, 6} – {2, 4, 6, 8} = { 1, 3, 5} ........(1) ( 2, 3, 6 B) B – A = {2, 4, 6, 8} – {1, 2, 3, 4,5, 6}= { 8 } ........(2) ( 2, 4, 6 A) From (1) and (2), we get A – B B – A Symmetric Difference of two sets : Let A and B be two sets. The symmetric difference of sets A and B is the set (A – B) (B – A) and is denoted by A B A B = (A – B) (B – A) or A B A B For example : Let A = {1, 3, 5, 7, 9}, B = {2, 3, 5, 7, 11} Then, A – B = {1, 9}; B – A = {2, 11} A B = {A – B} {B – A} = {1, 9} {2, 11} = {1, 2, 9, 11} Complement of a set : Let U be the universal set and A is a subset of U. Then, the complement of A with respect to (w.r.t) U is the set of all elements of U which are not the elements of A. Complement of A with respect to U is denoted by A ' or Ac. In symbolic form A’ = {x : x U and x A} clearly, A’ = U – A MPC BRIDGE COURSE Some results on complementation : 1. U’ = {x 2. : x U} = ’ = {x U : x } = U 3. (A’)’ = {x U : x A’} = {x U : x A } = A 4. A A’ = {x U : x A} {x U : x A } = U VENN DIAGRAM A Venn diagram is merely a closed figure and the points of the interior of the closed figure represents the elements of the set under consideration. Generally given below closed figures are used to represent the sets Note: i) It brings out relationship in sets and their use in simple logical problems. ii) Universal set is represented by a rectangular region. iii) Each subset is represented by a closed bounded figure placed within these rectangular regions. Venn -diagr ams ill ustrati ng var ious relationships in sets 1. To represent A where is the universal set : The universal set is represented by a rectangle and its subset A is represented by a circle placed inside this rectangle as shown. Clearly, the shaded region (which lies within the rectangle but outside the circle) represents A or Ac. For example : If U = {1, 2, 3, 4, 5, 6} and A = {2, 4, 6} then, A’ = U – A = {1, 2, 3, 4, 5, 6} – {2, 4, 6} A’ = {1, 3, 5} NARAYANA GROUP OF SCHOOLS 32 CLASS-VIII MPC BRIDGE COURSE 2. To represent two subsets A and B of a universal set such that A B : The universal set is represented by a rectangle, a circle representing the set B is placed in this rectangle and a smaller circle representing the set A is placed wholly inside the circle representing the set B. In this Venn - diagram, we can represent the various operations –(A B), (A B), (A –B) and (B – A). These operations are shown by the shaded portions in the following figures. 4. To represent two disjoint subsets of a universal set : The universal set is represented by a rectangle and its two disjoint subsets A and B are represented by two disjoint circles labelled as A and B respectively. In this Venn - diagram, we can represent the various operations – (A B), (A B), (A – B) and (B – A). These operations are shown by the shaded portions in the following figures : (A B) B AB A (A B) B (B A) 3. To represent two intersecting subsets of a universal set. The universal set is represented by a rectangle and its two intersecting subsets A and B are represented by two intersecting circles labelled as A and B respectively, 5. To represent three intersecting subsets of a universal set : The universal set is represented by a rectangle and its three intersecting circles labelled as A,B and C respectively. In this Venn diagram, we can represent the various operations (A B), (A B), (A – B), (B – A), A B , A B etc. These operations are shown by the shaded portions in the following figures. NARAYANA GROUP OF SCHOOLS 33 CLASS-VIII MPC BRIDGE COURSE In this Venn - diagram, we can represent the various operations such as (A B C), (A B C) etc., as shown by the shaded portions in the following figures : 2. The shaded part represents 1) A B 2) A B 3) A B 4) B A n 3. If A = { x : x = 2 – 1, n 5, n N } and B = { 2, 4, 8, 16 }, then A B = 1) {1, 2, 3, 4, 7, 8} 2) { 1, 2, 3, 4, 7, 8, 15, 16, 31 } 3) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } 4) { 0 } 4. From the figure A B = DAY-15: WORKSHEET 1. If A ={1,2,3,4,5} and B = {4,5} then A–B = ____. 1){1,2,3} 2){2,3,4} 3){3,4,5} 4){1,4,5} 2. If A ={1,2,3,4,5} and B = {4,5} then A B = ____. 1){1,2} 2){3,4} 3){1,2,5,6} 4){1,2,3,6} 3. If 1) {1, 2, 3, 4, 5} 2) {1, 2, 4, 5} 3) { 3 } 4) {1, 2, 3, 4, 5, 6} 5.If A, B are any two sets, then A – (A – B) = 1) A 2) B 3) A – B 4) A B 6.From the figure A B' = ={1,2,3,4,5,6,7,8,9,10} and A={1,2,3,4,5} then A| =_____. 1){1,2,3,4,5,6} 2){6,7,8,9,10} 3){1,2,3,6,7} 4){1,4,9,10} 4. If A B then A B =_____. 1)A 2)B 3) 4) Single Correct Choice Type: 1. If A a, b,c,d,e ,B a,c,e, g and c a, b, g , then 1) A B C A B A C 2) A B C A B A C 3) A B C A B A C 4) both (2) & (3) NARAYANA GROUP OF SCHOOLS 1) {1, 2, 4, 5} 2){1, 2, 4, 5, 6} 3){1, 2, 4, 5, 6, 7, 8} 4){1, 2, 4, 5, 6, 7} REASONING TYPE: 7. Statement I : A = {1, 2, 3, 4}, B = {5, 6, 7, 8} , then A B { } StatementII : The pictorial representation of sets is known as venn diagram 1) Both statements I and II are true 2 ) Both statements I and II are false 3) Statement – I is true but statement – II is false 4) Statement – I is false but statement – II is true 34 CLASS-VIII MPC BRIDGE COURSE LAWS OF RADICALS: COMPREHENSION TYPE: * If n a and b are two radicals of same n order ‘n’ then Example: * If n 4 n 3 4 2 4 32 4 6 a and n b are two radicals of same n 8. In the above figure, A B is 1) { 11, 9, 7} 2) { 2, 3, 5} 3) {3, 9, 3} 4) {2, 5, 3, 7, 9} 9. In the above figure A – B is 1) { 7, 9, 11} 2) {9, 11} 3) {7, 10, 9, 11} 4) {4, 6, 8, 10} 10. In the above figure A B ' is 1) {4, 6, 8, 10, 7, 9, 11} 2) {4, 6, 8} 3) {4, 6, 8, 10, 9, 11, 12} 4) {7, 9, 11} DAY-16 : SYNOPSIS order ‘n’ then 3 Example: * n 1 a or a n is not number(irrational)then n a rational a is called a surd of order ‘n’ n is called radical sign. * ‘n’ is called the order of the surd. * ‘a’ is called the radicand. * n a can be read as “ nth root of a” 25 3 4 10 , etc., are surds. Example: 3, 2, 11, 3 Note: * n b 3 n a b 10 3 5 2 4 3 6 12 6 * If m,n are two natural numbers and ‘a’ is any positive rational number then p m a n m a pm n a pm m n ap Illustration: 6 729 23 729 2 3 93 9 3 Entire surd or pure surd: A surd, expressed in the form a n b , where a 1 , is called an entire surd or a pure surd. Illustration: * The other name of a ‘surd’ is radical * The symbol 10 3 a we have n m a mn a Example: Definition: Let ‘a’ be a positive rational If n 2 If ‘m’ and ‘n’ are two natural numbers, then for any positive rational number ‘a’ n m number and ‘n’ be a positive integer 1 a n b n ab an a * In general 2 a is written as a . * Every surd is an irrational number, but every irrational number need not be a surd. Example: 0.5454454.............. is not a surd Example: 2 is a surd and also irrational, but is only irrational and not a surd. 6, 20, 3 5, 3 25 , etc. Simples t form of a surd: A surd, expressed in the form a n b , where ‘b’ is the least positive rational number, is called the simplest form of the given surd. Example: (i) The entire form of 2 10 10 4 40 (ii) The simplest form of 32 2 16 4 2 ADDITION AND SUBTRACTION OF SURDS: Two surds can be added or subtracted one from the other by using distributive law, only when they are similar surds. We cannot add or subtract dissimilar surds. Example-1: (i) a c b c a b c ( distributive law) (ii) NARAYANA GROUP OF SCHOOLS 2 6 4 6 2 4 6 6 6 35 CLASS-VIII MPC BRIDGE COURSE (iii) 4 2 3 2 10 2 4 3 10 2 3 2 Example-2: Simplify 8 3 4 75 3 300 Solution: 8 3 4 3 25 3 3 100 2010 3 6. Express 4 64 as a pure surd 1) 3 2010 2) 64 8 3 4 3 52 3 3 102 7. Express 8 3 4 5 3 3 10 3 8 3 20 3 30 3 1) 8 20 30 3 18 3 Conceptual Understanding Questions : 1. An example of a surd is 1) 2) 4 3) 9 4) 3 8 2 2. A cubic surd among the following is 1) 8 2) 3) 3 4 12 4) 3. The order of the surd 7 is 1) 1 2) 2 3) 1 / 2 4. The order of 1) 1 3 2 3 1 4) - 1 6 is 2) 2 3) 3 4) 6 Single Correct Choice Type : 1. 2 1) 2. 2 3 4 10000 2) 100 3 3. 3 3) 4 100 4) 1000 3) 3 2 4) 2 2 512 = 1) 2 2) 6 2010 2) 25 125 3) 2 2 2009 = 2010 2010 4) 2010 16 201 402 2 4) 5 25 5 simplified the A) 14 5 B) 14 5 C) 15 5 D) 16 5 9. If 2 3 189 33 875 7 3 56 is simplified, then the resultant answer is A) 8 3 7 B) 6 3 7 C) 7 3 7 D) 9 3 7 10. If 54 162 74 32 + 4 1250 is simplified ,then the resultant value is A) 63 2 B) 64 2 C) 65 2 D) 0 11. If 53 54 + 23 16 + 4 3 686 is simplified, then the resultant answer is A) 93 2 B) 83 2 10 2 1000 = 3 2010 402 2 as a pure surd 125 125 8. 3 45 20 + 7 5 is resultant answer is DAY-16: WORKSHEET 3 2010 3) C) 83 2 D) 93 2 7 1 1 is simplified , × + 3 27 3 then the resultant answer is 12. If 3 147 + A) 198 3 B) 199 3 C) 197 3 D) 195 3 9 9 9 9 13. If 4 128 + 4 75 5 162 is simplified ,then the resultant answer is 3 2009 2009 1) 2) 2010 2010 3) 3 2009 4) 1 2010 4. Which of the following is of an Irrational number but not surd is 22 1) 121 2) e 3) 7 4) 3.24317094237890 5. If two surds are different multiples of the same simple surd, then the surd is 1) Compound surd 2) Dissimilar surd 3) Similar surd 4) Complex surd NARAYANA GROUP OF SCHOOLS A) 20 3 13 2 B) 20 3 13 2 C) 13 2 20 3 D) 13 2 20 3 1 is simplified then 6 the resultant answer is 14.If 252 + 294 + 48 A) 6 7 15 6 B) 6 7 15 6 C) 15 6 6 7 D) 6 7 15 7 36 CLASS-VIII 15. MPC BRIDGE COURSE Illustration-2: As, If x = 2 363 5 243 + 192 , y = 44 5 176 + 2 99 , then x + y is A) 15 3 12 11 16. If x = 24 , y = 150 , B = 96 , A = x+y is A +B A) 1 B) 0 C) –1 D) 2 17. If irrational numbers have the same irrational factors ,then they are known as A) similar irrational numbers B) dissimilar irrational numbers C) a and b D) neither a nor b 18. Sum of 3 2 and 5 2 is 19. B) 8 2 C) 15 2 D) 12 5 × 5 = ____________ 20. 8 3 ÷ 2 3 is _______________ numbers 21. Simplified form of 4 2 + 3 32 is A) 16 2 B) 7 32 C) 7 2 22. Simplified form n 2 B) 3 C) 5 3 rationalising factor of Illustration-1: Because, 3 6 3 6 n a is given by a1- 3,3 3 are R.F.’s of 6 3 1 n 3 3 6 3 3 6 3 3 6 3 18 R 3 3 18 3 54 R NARAYANA GROUP OF SCHOOLS 3 4 and 4 is a R.F. of 3 3 2. DAY-17: WORKSHEET 1. If rationalizing factor of 3 5 is multiplied by rationalizing factor of resultant is 5 23 , then the A) 15 512 20 B) 15 510 26 C) 15 510 28 D) 15 510 29 1 , B= 5- 3 2. If A = 1 4 B) 1 8 3 2 -2 3 2 +2 22 4 4. If A = B) 1 , then A × B is 5+ 3 C) , B= 22 5 1 2 D) 3 2 +2 3 2 -2 C) 1 2 , then A + B is 21 3 D) 22 7 5+2 2 is rationalized, then the 3-2 2 simplified form is A) 23 17 2 B) 23 18 2 C) 23 19 2 D) 23 16 2 5. If p = n a x then its rationalizing factor is A) 2 3 4 3 2 4 The R.F. of a given surd is not unique. A surd has infinite number of R.F.’s. A) Rationalising Factor (R.F.): If the product of two surds is a rational number, then each of them is called a rationalising factor (R.F.) of the other. The 7 3 Note: 3. If A = DAY-17 : SYNOPSIS and 3 2 is a R.F. of A) 7 3 7- = 2, a rational number. D) 14 2 D) 2 a n b n ab 3 8 48 72 27 + 2 18 is A) 2 7 73 3 4 R 7 3 Illustration-3: As, 54 , then A) 3 2 7 + 3 is a R.F. of 7 3 is a R.F. of B) 15 3 12 11 C) 15 3 12 11 D) 15 3 12 11 7 3 6. If n 5 ax B) C) ax 82 × 5 83 = x , 15 n a n-x D) n a x-n 154 × 15 1511 = y , then x 2 + y 2 is A) composite number B) even prime C) irrational D) prime number 37 CLASS-VIII MPC BRIDGE COURSE 7. If P 3 122 = 12 then the value of P × 3 18 is A) 8 B) 9 C) 6 D) 4 8. If 4 a12 b16 c11 × 4 a 3 b 7 c8 × 4 abc is simplified then the resultant is A) a 4 b3 c 4 B) a 4 b6 c 4 C) a 4 b5 c6 D) a 4 b6 c5 9. Match the following a) 7 2 32 1) Trinomial b) 2 3 10 2) Monomial d) 5 3 2 2 9 5 3) Rational d) 4) Binomial 3 343 4 256 1) a 2 2) b 4 3) c 1 4) d 3 a 1 b2 c 1 d3 a 1 b2 c3 d4 a 1 b4 c3 d2 10. 4 1250 × 4 8 +3 75 × 3 48 is simplified then the resultant is A) 550 B) 560 C) 540 D) 570 Note : 1) If the rotation of the terminating ray is anti clockwise direction, then the angle is regarded as positive. 2) If the rotation of the terminating ray is clockwise direction, then the angle is regarded as negative. DAY-18 : SYNOPSIS INTRODUCTION The word ‘Trigonometry’ is derived from three Greek words ‘tri’ meaning ‘three’, ‘gonia’ meaning ‘an angle’ and ‘metron’ meaning ‘measure’. The basic task of trigonometry is the solution of triangles, finding unknown quantities of a triangle from given values of other quantities. The study of trigonometry is of great importance in several fields, like in Surveying, Astronomy, Navigation and Engineering. In recent times trigonometry is widely applied in many branches of Science and Engineering such as seismology, design of electrical circuits, estimating the heights of tides in the ocean etc. Angle : The amount of rotation of moving ray with reference to fixed ray is called an angle. An angle is usually denoted by , , ... etc. NARAYANA GROUP OF SCHOOLS Units of measurement of angles : For measurement of angles, there are three systems : They are : i) The sexagesimal (English) system. ii) The centesimal (French) system, iii) The radian (or) circular measure. The Sexagesimal system : In this system unit of measurement of an angle is ‘degree’. Degree: In this system one complete rotation is divided into 360 equal parts. Each parts is called ‘a degree’, denoted as 10 . Minute : A degree is further divided into 60 equal parts and each part is called one minute, denoted as 1' . Second: A minute is further divided into 60 equal parts and each part is called one second, denoted as 1'' . 38 CLASS-VIII 1 NOTE: 1) 10 60' or 1 60 MPC BRIDGE COURSE In this way, the circumference of the circle subtends at the centre, an angle whose measure is 360 , the we get 0 ' 1 2) 1' 60'' or 1 60 2 c 360 0 i.e., one complete angle = 2 c and c 1800 i.e., one straight angle = ' '' 3) In this system, one right angle = 900 . The Centesimal system : In this system unit of measurement of angle is ‘grade’. Grade: In this system one complete rotation is divided into 400 equal parts. Each part is called ‘a grade’, denoted as 1g . Minute : A grade is divided into 100 equal parts and each part is called a minute, denoted as 1' . Second: A minute is divided into 100 equal parts and each part is called a second, denoted as 1'' . NOTE : 1) 1g 100 2) 1 100 1 or 1 100 g 1 100 or 1 3) In this system, one right angle = 100 g 4) Though we have used the same name ‘minute’ (or ‘second’) in both ‘sexagesimal system’ and ‘centesimal system’, it can be easily observed that are not the same. Circular system : In this system unit of measurement of an angle is ‘radian’. Radian : ‘The angle subtended by an arc of length equal to the radius of the circle at its centre is called one radian, denoted as 1c . c c 0 90 i.e., one right angle = and 2 2 c Approximate values of 1c and 10 : we know that c 1800 Also 1800 c c 1800 1c ( where 3.1415... ) 180 10 0.01745c 10 \ 1c = 57o17 '45 '' (Approximately) Note: 1) A radian is a measure of an angle. Hence it is different from the radius of the circle. 2) Measure of an angle is a real number. 3) When no unit of measurement is specified for an angle, it is assumed as radian. 4) The formula connecting the three systems can be stated as follows: D G C o g 90 100 2c Where D denotes degrees, G denotes grades and C denotes radians Illustration: 1) Express the sexagesimal measure 300 as radian measure and centesimal measure. Given D 300 r o r G 30 2C 30 and 100 90 90 1 1 100 G G 100 and C 3 3 2 3 From (1) , 1 r Relation between degrees and radians : The angle which is subtended by an arc of length 2r at the centre is 2c . NARAYANA GROUP OF SCHOOLS D G 2C , 90 100 30 G 2C ....(1) 90 100 Solution : We know that, 100 g 30 3 0 C 6 c 30 6 0 39 CLASS-VIII MPC BRIDGE COURSE Introduction : In the figure given below, ABC is a triangle, right angled at B. The side opposite to the right angle is AC and it is called hypotenuse. Consider angle , the side opposite A It is written as ‘sine ’ or ‘sin ’. sin side opposite to ' ' y Hypotenuse r OM side adjacent to ' ' x , This ratio is OP Hypotenuse r called cosine of an angle ’ ’. It is written as ‘cosine ’ or ‘cos ’. cos C B Consider a system of rectangular coordinate axes OX and OY. Draw a circle with centre O and radius r. Choose a point P(x,y) on the circle such that the line OP makes an angle radians with MP side opposite to ' ' y OM side adjacent to ' ' x , This ratio is called tangent of an angle ‘ ’. It is written as ‘tan ’. OX (positive X-axis) measured in anticlock wise direction. perpendicular PM to In OPM Draw a tan OX , OP= Hypotenuse=r PM side opposite to ‘ ’ = y OM = side side adjacent to ' ' x Hypotenuse r side opposite to ' ' y side adjacent to ' ' x OP Hypotenuse r . This ratio is MP side oppositeto ' ' y called cosecant of an angle . It is written as cosec (or) csc . adjacent to ‘ ’ = x csc Hypotenuse r side oppositeto ' ' y OP Hypotenuse r , This is called OM side adjacent to ' ' x x secant of an angle . It is written as ‘sec ’. sec The ratios of different pairs of sides of the right angled triangle are called triginometrical functions or trigonometrical ratios with respect to an angle’ ’, these ratios having following names and designations. Hypotenuse r side adjacent to ' ' x OM side adjacent to ' ' x MP side opposite to ' ' y , This ratio is called cotangent of an angle . It is written as ‘cot ’. cot side adjacent to ' ' x side opposite to ' ' y MP Side oppositeto ' ' y , This ratio is OP Hypotenuse r called sine of an angle ’ ’. NARAYANA GROUP OF SCHOOLS 40 CLASS-VIII MPC BRIDGE COURSE Relations : 1) sin cos ec MP OP 1 OP MP sin cos ec 1 2) cos sec OM OP 1 OP OM cos .sec 1 3) tan cot MP OM 1 OM MP tan .cot 1 4) sin MP OP MP OP MP tan cos OM OP OP OM OM tan 5) sin cos cos OM OP OM OP OM cot sin MP OP OP MP MP cot cos sin NARAYANA GROUP OF SCHOOLS NOTE : 1. Since the six trigonometrical ratios discussed above represent the ratios of sides of a right angles triangle, they are all real numbers. 2. The six trigonometrical ratios are defined with respect to a certain angle . Hence sine, cosine,tangent, etc., by themselves do not have any meaning. They are meaningful only when they are associated with an angle like ‘ ’. 3. Sin is an abbrevation for sine and it is not the product of sin and ‘ ’. Similar is the case of cos and tan . 4. Cosec , Sec and cot are reciprocals of sin , cos and tan respectively. 5. We use the notation sin2 , cos2 , tan2 , etc., in place of (sin )2, (cos )2, (tan )2 respectively. 6. We write cosec =(sin )-1 not as sin-1 which has a different meaning (sine inverse ). 7. All the values of trigonometric ratios depend just on the angles but not on the sides. 41 CLASS-VIII MPC BRIDGE COURSE DAY-18: WORKSHEET 3. The Conceptual Understanding Questions : 1. If the terminal side completes one revolution about its vertex, then theangle made is 2) 3600 1) 900 3)1800 4)1200 2. Express the sexagesimal measure 1350 as radian measure. 1) 4 2) 3 4 3) 3. Express the 3 4) 6 circular measure 2) 900 3) 300 6 4)1200 3 4 2) 4 3) 5 3 4) 5 5 3 5. In the ABC , B 90 0 , AB = 4 cm, BC =3 cm, then sin A= 1) 5 2) 3 3 3) 4 3 4) 5 4 5 6. In the ABC , B 90 0 , AB = 4 cm, BC =3 cm, then tan A= 1) 3 2) 5 5 3) 3 4 4) 3 1) 300° 3 4 1. The sexagesimal measure, = 720 in radian measure is 1) 2 3 2) 2 5 c 3) 3 4 c 4) None in 2) 225° 3) 270° 4) 180° 5 , and ‘ ’ is acute angle. Then 12 the value of sec cosec 1) 22 60 2) 221 60 3) 22 7 4) None sin A cos A sin A cos A is 1) 7 2) 2/11 3) 1/2 4) 1 6. In a right angled triangle ABC, B 900 , tan A = 5/12, then 5 13 2) sin A 3) Sin A 513 5 13 4) cos A 13 5 7. If 3sinθ - 4cosθ = 0 , then the value of 3sinθ + 4cosθ is 1) 4.8 2) 0.75 3) 1.33 4) 12 8. The ‘sine’ value of an angle of a triangle is 1 , then the sum of other two angles 2 is 1) 1500 2) 600 3) 900 4) 1200 9. If sinA : cosA = 3 : 4, then secA + cosecA is 1) 35 12 2) 12 35 3) 1 4) 3 4 10. If sin = – 7/25 and is the third 7 cot 24 tan quadrant, then 7 cot 24 tan 1) 17/31 2) 16/31 3) 15/31 4) none 2. 200 g = 1) 1800 5c 4 4. If tan Single Correct Choice Type : c sexagerimal measure is 1) CotA 4. If 5sin 3 then tan = 1) measure 5. Given, 4 cotA = 3, the value of sexagesimal measure. 1) 600 circular 2) 1000 3) c 4 NARAYANA GROUP OF SCHOOLS 4) c 2 42 CLASS-VIII MPC BRIDGE COURSE DAY-19 : SYNOPSIS To find the values of trigonometric functions of any angle : Quadrant Q1 Tr. Ratios 900 cos sin cot tan cos ec sec Sin Cos Tan Cot Sec Cosec Q3 Q2 900 1800 sin cos cos sin cot tan tan cot cos ec sec sec cos ec 1800 sin cos tan cot sec cos ec Q4 2700 cos sin cot tan cos ec sec 2700 cos sin cot tan cos ec sec 3600 sin cos tan cot sec cos ec DAY-19: WORKSHEET Conceptual Understanding Questions : 0 1. If A 30 then sin 2A is 2 1) 3 2) 3 2 3) 1 4)1 2 5. The value of sin0 0 + cos30 0 – tan45 0 + cosec 600 + cot900 is 1) 7 3 1 6 2) 7 3 1 6 3) 7 3 6 6 4) 7 3 6 6 2. The value of sin2 30 0 cos 2 600 is 1) 1 2 2) 3 2 3)1 4) 1 2 2 6. The value of sin 3. The value of is cos 00 sin 900 2 sin 450 is 1)2 2)1 3)–2 sin A 1. If A 450 , B 300 , then cos A sin A.sin B 2) 2/3 3) 1/4 4) 1/5 2. sin 30 0 cos 0 0 tan 45 0 cot 45 0 sec 60 0 cos ec 30 0 1) 0 2) 1/2 3) 3/2 4) 1/4 3. Sin 60° + cos0° - Tan 45° +cot 45° - sin 90°= 1) 2 2) 3 2 4. Sin 0° + cos 90° + Tan 45° 1) sin 45° 3) tan 45° 1 3) 2 4) 15 13 9 11 2) 3) 4) 2 2 2 2 0 0 7. The value of Sin 45 × Cos30 + Cos 450 × Sin300 is 1) 4)3 Single Correct Choice Type : 1) 1/2 π π π + se c 2 + tan 2 4 3 3 1) 3 +1 3 1 2) 3) 2 2 2 2 3 1 2 8. The value of tan245 + 2tan2 60 is 1) 5 2) 6 3) 7 4) 8 9. cosec 2 30° cot 2 30° = 3 2 cos 45° + cot 45° - 3 1 4) 2 1) 1 2) 2 3) 1 2 4) 1 4 2) cosec 45° 4) cos 45° NARAYANA GROUP OF SCHOOLS 43