MATHEMATICS Questions: 40 Marks: 160

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1 MATHEMATICS Questions: 40 Marks: 60. Algebra i) Irrational Numbers () ii) Linear Equations in Two Variables () iii) Polynomials And Rational Expressions () iv) Quadratic Equations (). Commercial Mathematics () Arithmetic Progessions (4) 4 Co-Ordinate Geometry (4). Trigonometry (4) 6. Mensureation of Plane Figures (4) 7. Statistics () 8. Geometry i) Similar Triangles () ii) Circle & Tangents () iii) gm () 9. Ratio And Proportion () 0. percentage and its applications ()

2 Mathematics ALGEBRA IRRATIONAL NUMBERS. Natural Number Numbers starting from and which goes to infinity are said to be natural numbers e.g.,,,.... Whole Number If we include zero to Natural Numbers then numbers are said to be whole numbers. These number are 0,,,, 4,.... Integers The numbers which include zero, natural numbers and negative natural numbers are called Integers e.g. 4,,,, 0,,,, 4, Rational Numbers : Numbers which can be expressed in the form p where p I ; q I and q 0 also q p & q are co-prime i.e. H.C.F of p & q is. e.g.,, etc. The set of rational numbers are denoted by Q. R S T Q p ; q p I, q I, q 0 Every rational number, like all the fraction, can also be expressed in decimal form and can be represented either in the form of terminating decimal or non-terminating decimal.. Conversion of decimal number in rational number of the form p q. Case (I) When the decimal is of terminating nature. U V W Steps () Convert the given decimal in to rational number. () Find position of decimal in the given number. () To remove decimal in numerator, write in the denominator followed by number of zero as many digits after decimal in numerator. (4) Now on dividing numirator and denominator by common divisor we get rational expression in the form p q. Case (II) When decimal representation is nonterminating repeating nature.. Pure recurring decimal A decimal in which all the digits after decimal point are repeated. Steps : () Let the given fraction equal to some constant number. () Remove bar over the given fraction and write repeated number twice. () Find number of repeated digits and multiply both side by 0 n. where n is number of repeated digit. (4) Subtract step from step. () Now, dividing number obtained by coefficient and express fraction into lowest term.. Mixed recurring decimals Steps : () Change the mixed recurring decimal into Pure recurring decimal by multiply 0 n, where n is the number of non-repeating digits after decimal. () Solve the decimal by using pure recurring steps. 6. Irrational Numbers A number is an irrational number, if it has non-terminating and non-repeating decimal

3 Mathematics representation. OR The numbers which are not rational are called irrational number. e.g.,,,... etc. 7. Real Numbers The set of all rational and irrational numbers is called real numbers. 8. Surds (Õðä ÆÁ») Let a be a real number and n be a positives integer such that the n th root of a i.e., a /n or n a is an irrational number then a /n is called a surds or radical of order n and a is called radicand. (i) Every surd is an irrational number but every irrational number is nor surd. (ii) A surd which has unity only as rational factor is called a pure surd. (iii) A surd which has a rational factor other than unity is mixed surd. SET-I. Numbers which can be expressed in the form of p where p and q are integers and q 0 q are called. ( Natural Numbers ( Integers ( Rational Numbers ( Irrational Numbers.. The rational number 7 in the decimal form 6 is (. 6 (. 6 ( 6 ( None of these.. The rational number in the decimal form is (. 697 ( (. 697 ( None of these can be written as 00 ( ( ( both ( and ( ( None of these.. 6. p q ( form of is 7 00 ( 0 p q form of 4. 7 is ( 47 0 ( 4 99 ( 4 0 ( None of these. ( 4 9 ( None of these. 7. Which of the following is a rational number? d i ( d id i d i ( None of these. ( ( 8. Product of a rational and an irrational number is ( rational number ( natural number ( irrational number ( None of these. 9. Every real number is either... or... number ( rational or irrational ( irrational or rational ( both ( and ( ( None of these. 0. m n a can be written as ( mn a ( n m a ( both ( None of these.

4 4 Mathematics ( n m can be represent as n m a m n a ( m a ) n None of these 7. The value of ( is ( 7. The value of 49 7 ( ( ( 7 ( None of these. ( 9 ( 9. The value of. 4. is ( 4 ( 6 ( 4 ( None of these. 4. The value of ( 4 4 is ( 0 ( ( 4. The value of ( 8 is (6 8 ) ( 6 8 ( 6 8 ( None of these If a b 7, then the value of 7 a and b are ( 8, (, 8 ( both ( None of these. 9. Find the value of 4 4 ( 9 6 ( 6 0. Find the value of ( 8 ( ( 9 ( None of these. ( 6 9 ( None of these is 6. The value of ( ( 7 7 ( 7 ( 6.. Find the value of 49 ( ( 7 ( 7 (

5 Mathematics SET-II. The numbers which include zero and natural numbers and negative nautral numbers are called. ( Real numbers ( Rational numbers ( Integers ( None of these.. The set of rational and irrational numbers are called ( Real numbers ( Natural numbers ( Integers ( None of these.. 9 The value of in the decimal form is 000 ( 0 9 ( ( 0.09 (.9 4. The value of 6 is the decimal form is 4 ( 0. ( 0 6 ( 0. ( None of these.. in the decimal form is : 7 (. 487 (. 4 (. 487 ( None of these p q form of 0. 7 is ( 7 ( ( ( None of these. p q form of 0. is : ( ( ( 9 ( None of these. 8.. can be written as : 76 ( ( 6 ( 66 ( can be written as : ( ( ( ( Two rational numbers between and 6 are ( 4 0, 4 ( 6, 6 0 (, 0 ( 7, 7 0. is a : ( rational ( irrational ( both ( & ( ( None of these.. The sum of rational numbers and Irrational number is a ( rational ( irrational ( both ( & ( ( None of these.. Two irrational numbers between & 7. (, 6 (, (, 6 ( All of above is a/an... number. ( rational ( irrational ( natural ( None of these.. The value of 7 49 is ( 7 ( 7 ( 49 ( None of these. 6. The value of is ( 68 ( 68 ( 68 ( None of these.

6 6 Mathematics 7. The value of ( ( 8. The value of ( ( ( ( ( ( The value of is 8 is ( 6 ( 6 ( 6 is ( None of these. 0. The value of 6 7 is ( ( ( ( None of these.. The value of. 4. ( ( ( ( None of these.. Which of the following is a surd. ( 7 ( ( ( The value of ( ( 9 9 is ( (. 4. is a/an... number ( rational ( irrational ( integer ( None of these.. 90 is a/an... number ( rational ( irrational ( natural ( None of these. 6. If 7 4 a b, then the value of a and b is : (, (, (, 0 ( 0,. 7. a b, then the value of a and b is : ( 0, (, 0 (, 0 ( 4, How many rational numbers lie between two given rational numbers : ( infinite ( four ( six ( None of these. 9. Between two given rational numbers &, 4 find two more rational numbers : ( ( 7, 4 48, 6 8 (, ( None of these.

7 Mathematics 7 / 0. The value of () 6 is : ( 0 / ( 0 / ( 0 /4 ( None of these.. as a surd o th order can be expressed as ( ( ( ( none of these. The value of is ( 6 ( 4 ( 4 (. Which of the following number is the biggest ( (++) ( [(+) ] ( ( x x ) ( (4) 4 (9) -(8) is completely divisible by ( 7 ( ( 0 ( 0. The rationalising factor of 49 is ( 49 ( 7 ( 0 ( none 6. If 4 x =, then the value of x is ( x= ( x= 0 ( x=6 None 7. Write down in the form of ( ( p q 7 00 ( 7 ( 9 Answers (Set I). (. (. ( 4. (. ( 6. ( 7. ( 8. ( 9. ( 0. (. (. (. ( 4. (. ( 6. ( 7. ( 8. ( 9. ( 0. (. ( Answers (Set II). (. (. ( 4. (. ( 6. ( 7. ( 8. ( 9. ( 0. (. (. (. ( 4. (. ( 6. ( 7. ( 8. ( 9. ( 0. (. (. (. ( 4. (. ( 6. ( 7. ( 8. ( 9. ( 0. (. ( ( ( 4 ( ( 6 ( 7 (

8 8 Mathematics ALGEBRA LINEAR EQUATIONS IN TWO VARIABLES Point to remember. Linear Equation : Equation of the form ax + by + c=0, a 0, b 0 where a, b, c are real numbers is called a linear equation in two variables x and y.. Solution of a linear equation : x = l and y = m is a solution of ax + by + c = 0 if al + bm + c = 0. Graph of a linear equation ax + by + c = 0. The graph of a linear equation can be drawn by the following method : i) Draw two mutually perpendicular lines X OX and YOY as coordinate axes. ii) Obtain three convenient pairs of values of x and y satisfying the given equation. iii) Plot these ordered pairs on the graph paper and join the points. The straight line is the required graph 4. Checking whether the given points lie on the graph of the given equation. Substitute the values of x and y in the given equation. If it satisfies the equation the point lies on the straight line and if it does not satisfy the equation, the point does not like on the straight line.. Solving a System of equations graphically: Let the given system of linear equations be ax+ by+ c=0 and px+qy+m=0 where a,b,p and q 0. Draw the graphs of the two equations and let these be represented by lines l and l. There are three following cases. i. When L and L are parallel In this case, the system of equations has no solution. Such system of equations is called inconsistent. ii) When L and L are coincident. In this case, the given system of equations has an infinite number of solutions. iii) When L and L intersect In this case, the system of equations has a unique solution given by the point of intersection of the two lines L and L 6. Solving system of simultaneous Linear equations Algebraically. Method of substitution i) Find y in terms of x from one of the two equations. ii) Substitute this value of y in the other equation. Solve this equation for x. iii) Substitute this value of x in any of the given equations and solve it for y. Alternatively we may find x in terms of y. then we shall get value of y in step (ii) above and value of x in step (iii) above. Comparison Method Find the value of x or y from both the equations and eqvate to get value of one variable. Substitution of this value in any one equation will give value of the second variable. Method of equalizing the co-efficient. i) Multiply the given equations by some suitable constants so an to make the co efficients of one of the variables numerically equal. ii) Solve the equation obtained in step in one variable. iii) Solve the equation obtained in step (ii) in one variable.

9 Mathematics 9 iv) Substitute the value of this variable in any of the two equations and solve for the second variable. A special case when coefficients of x and y are interchanged in two equations let the two equations be px + qy = m and qx + py = x i) Add the two equations to get (p + q) x + (q + p) y = m +n or (p + q) (x + y) = m + n or x + y m n p q...(a) ii) Subtract the equations to get (p q) x + ( p + q) y = m n or (p q) (x y) = m n or x y m n...(b) p q iii) Add A and B to get x and subtract B from A to get y. e) Cross Multiplication method :- The cross multiplication rule to solve the system of equations. a x + b y + c = 0 a x + b y + c = 0 is given by b c or x x b c a b a b c a y c a bc bc ab ab and y c a c a a b a b 7. Let a x + b y + c = 0 and a x + b y + c = 0 Then, the system of equations has no solution if a a b b c c The system of equations has a unique solution if a a b b c c The system of equations has an infinite number of solutions, if a a LINEAR EQUATIONS:- b b c c. The graph of equation ax + by = c is Circle Straight line Square Parabola. If the point P(m, n) lies on the graph of ax + by = c, then solution of ax + by = c is x = m, y = n x = 0, y = 0 y = 0 x = 0. The graph of the equation y =a is line Parallel to x- axis Parallel to y axis Passing through origin None of these. 4. The graph of the equation x = a, a>0 is on the Right side of y- axis left side of y- axis Right side of x- axis Left side of x- axis. If (a, 4) lies on the graph of x+y=0, then the value of a is : The value for b so that equation x+y =b may have x =, y = as a solution. b = b = b = 0 b = 7. If the equations a x + b y = c and a x + b y = c represents two parallel line then they have

10 0 Mathematics No point of intersection One point of intersection Many points of intersection None of these. 8. Solution for equations 6x + 9y = and 6x + 0y = 4 is x = - and y = -7 x= and y = 7 x=- and y=7 x= 7 and y = 9. The value of k for which the system of equations x+y= and 4x+ky=0 has infinetly many solutions, then / 6 0. Values of k, for which the system of equation kx + y = and x + y = has unique solution. k 6 k - 6 k k 4. Determine the values of a and b for which the following system of linear equations hs infinite many solutions k = 6 k = -6 k = k = -4. A linear equation in two variables has one solution two solution three solutions infinete solution. Value of a for which system of equations ax + y = and x y = has infinitely many solution. a = -6 a = 6 a = a = - 4. If three times of the age of son (in years) is added to the age of father, the sum is 8 and if the twice the age of father is added to the age of son the sum is 9. then the age of father and son is. (40, ) (0, 0) (0, ) (0, ). The sum of two numbers is and their difference is then numbers are (,4) (4, ) (, ) (4, ) 6. pens and 6 pencils costs Rs. 9 and pens and pencils costs Rs.. Find cost of one pen and one pencil separately is Rs., Rs 4 Rs. 4, Rs. Rs., Rs. Rs. Rs chairs and tables cost Rs. 00/- and chairs and tables costs 70/-. Find the cost of a chair and a table Rs. 00, Rs. 0 Rs. 0, Rs. 00 Rs. 0, Rs. 0 Rs. 00, Rs tables and chairs together cost Rs. 70 and 8 tables and chairs together cost rs. 700 then cost of chair and table separately is Rs. 6000, Rs Rs. 600, Rs Rs. 6400, Rs Rs. 6600, Rs Five years ago, A was trice as old as B and 0 years hence A shall twice as old as B then present ages of A and B are 0 years, 0 years 0 years, 0 years

11 Mathematics 40 years, 0 years 60 years, 0 years 0. In a two digit number, unit s digit is twice the ten s digit. If the digits are reversed new number is 7 more that the original number then the number is Ten years ago, father was twelve times as old as his son and ten years hence he will be twice as old as his son will be. Present ages are 4 years, years 4 years, years 0 years, years 6 years, years. Solution of following pair of equation is 4 y = 4 and 4 y = x x x = /, y = x =, y = / x =, y = x =, y = /. Solution of following pair of equations is x y = and x x = /, y = / x =, y = x = /, y = / x =, y = y 6 4. A two digit number is four times the sum of its digits and twice the product of the digits, then the number is Solution of following pair of equation is x y = and x y 6 x =, y = x=, y = x =, y = x =, y = 6. Solution of the following equation is x y = 0 and x 8 y = 0 x =, y = x = 0, y = 0 x = 0, y = x =, y = - 7. If x y = 7 and (a + x (a + b ) y = 4a + b represent coincident lines, then a and b satisfy the equation : a + b = 0 a + b = 0 a b = 0 a b = 0 8. Lines representing the following pairs of linear equations are x+ y + 0 = 0 and 0x + y + = 0 Intersecting Lines Parallel Lines Coincident Lines None of these 9. Lines representing the following pairs of linear equations are 6x y + 0 = 0 and x y + 9 = 0 Intersecting Lines Parallel Lines Coincident Lines None of these 0. For which values of k will the following pair of linear equations have no solution x + y = and (k )x + (k-)y = k + k = and k -, - k = - and k k = and k, - k = - and k -, -. If the system of equations x + y = and (x ) x + (k ) y = k + is inconsistent, then k is 0

12 Mathematics. Solve the following pair of equations = and = x y x y and x, y x =, y = x =, y = x =, y = x =, y =. A fraction becomes when subtracted from the numerator and it becomes /4 when 8 is added to its denominator, then the fraction is 7 4. For which value of a the system of equations 4x + 9y + = 0 and x + y + 7 = 0 has exactly one solution. a 6 a -6 a a -. If (4,9) is a solution of the equation y = ax+ then a = If (x-) is a factor of x + ax + bx + 8 and a+ b = - 7 then values of a and b are a =, b = 4 a =, b = 4 a = 4, b = a =, b = 4 a b c 7. On comparing the rations a b c which pair of linear equations are inconsistent x + y = and x y = 7 x y = 7 and 9x 0y = 4 4 x + y = 8 and x + y = x y = 8 and 4x 6y = 9 8. If x-y = and x+y =, then x =, y = - x = -, y = x = -, y = - none of these 9. The value of k for which the system of equations x + y = 0 and kx + 0y = 0 has a non-zero solution is : Algebraic interpretation for the following pair of equations x y = 0 and 6x 0y = 40 is Have unique solution Have infinitely many solutions Have no solution None of these 4. If the graph of two linear equations meet at a point then equations are Inconsistent Consistent Dependent None of these 4. If the graph of two linear equations are parallel then equations are Inconsistent Consistent Dependent None of these 4. If two linear equations have unique solution, then their graphs. Meet at a point Meet nowhere Are Parallel None of these

13 Mathematics 44. If two linear equations have no solution, then their graphs. Meet at single point Are Parallel Meet no where Are coincident lines 4. The graph of these equations x + y = 6, 7x 4y = will Meet at one point Meet no where Parallel lines Be coincident lines. 46) If x, then find the value of x Following the equations x + y =, x - y = and x = 0, the vertex of triangle on x-axis will be :- 0,, 0 0, - 0, Sum of two numbers is and their difference is. Find the numbers. 4, 6, 4, 9 0, 49. Determine the values of a and b for which the following system of linear equations has infinite solutions x (a 4) y = b + and 4x (a ) y = b 4, 6, 4 7,, 7 0. Find the value of k for which the system of equations has a unique solution x + y =, x + ky + 7 = 0 k 0 k k k 0 LINEAR EQUATION b a b d 4 b a d a b 4 a a b c d 4 a 4 a 4 a 4 a 4 a 44 a c b a b 4 d 6 b 6 a 6 c 6 c 46 b 7 a 7 b 7 c 7 d 47 b 8 c 8 b 8 c 8 a 48 a 9 d 9 b 9 b 9 a 49 c 0 b 0 d 0 a 40 b 0 a

14 4 Mathematics POLYNOMIALS AND RATIONAL EXPRESSIONS Points to remember i) A polynomial of degree, and is called a linear polynomial, quadratic and cubic polynomial respectively. ii) iii) iv) A real number is a zero of the polynomial p(x) if p() = 0. The zeroes of polynomial p(x) are precisely the x- coordinates of the points where the graph of y = p(x) intersects the x axis. If and are the zeroes of a quadratic polynomial ax + bx + c = 0, a 0 then Sum of zeroes = + = b a and Product of zeroes c a v) A polynomial of degree n can have at most n zeroes. So, a quadratic polynomial has at most zeroes and a cubic polynomial can have at most zeroes. vi) If,, are the zeroes of the cubic polynomial ax + bx + cx + d= 0 then + + b a, + + c a and d a vii) Divided = Quotient x Divisor + Remainder viii) p(x) = q(x). g(x) + (x) Where, either (x) = 0 or degree of r (n) < degree of g(x). This result is known as division algorithm for polynomials. Rational Expression An expression of the form p(x) f ( x ) g( x ) where f(x) and g(x) are polynomials and g(x) 0, is called a rational expression. Operations on Relational Expressions i) Addition ii) iii) iv) p( x ) q( x ) r ( x ) s( x ) Subtraction p( x ) r ( x ) q( x ) s( x ) Multiplication p( x ) q( x ) Division r ( x ) s( x ) p( x ) q( x ) r ( x ) s( x ) p( x ) q( x ) s( x ) r( x ) p( x )s( x ) q( x )r ( x ) q( x )s( x ) p( x )s( x ) q( x )r ( x ) q( x )s( x ) p( x )r ( x ) q( x )s( x ) p( x )s ( x ) q( x )r( x ) v) Opposite or additive inverse of p( x ) q( x ) inverse of p( x ) q( x ) and reciprocal or multiplicative p( x ) q( x ) is q( x ) p( x ) vi) The sum, difference, product and division of two rational expressions is always a rational expression except where divisor in case of division is a zero polynomial here division is not defined. vii) A rational expression is not defined for those points which makes the denominator zero. REMAINDER THEOREM The remainder obtained an dividing a given polynomial f(x) by x-a is given by f (. is

15 Mathematics Factor Theorem x-a is a factor of the polynomial f(x) if and only if f( = 0. SOME FORMULA i) (a+ = a + ab + b ii) (a- = a - ab + b iii) a b = (a+ (a- iv) (a+ = a + a b + ab + b v) (a- = a - a b + ab - b vi) a + b = (a+ (a -ab +b ) vii) a b = (a- (a +ab +b ) viii) x + y + z xyz = (x + y + z) (x + y + z xy yz zx) = ½ (x + y + z) [(x y) + (y z) + (z x) ] ix) a 4 +a +=(a -a+)(a +a+). If there is no common factor then HCF of polynomials is 0 ( (. H.C.F of polynomials 4(x-) (x-) (x+) and 6 (x-) (x+) (x+7) is (x ) (x + ) (x ) (x + ) (x ) (x + ) (x ) (x + ). H.C.F of p(x) and q(x) where p(x) = x x+6 and q(x) = x + x-4 (x-) (x-) (x-) (x+) 4. Non-Zero constant is a polynomial of degree 0 Infinite. Degree of polynomial x 4 + x 7 is Degree of polynomial a 0 + a x +a x +a x + +a n x n where n>0 is an n n 7. Degree of polynomial (x-) (x-) (x-) is 0 8. H.C.F of polynomials 0 (x ) and (x+) (x-) (x + ) (x ) (x + ) (x ) (x + ) (x ) (x + ) (x ) 9. L.C.M of p(x) = 4(x-) (x +6x+8) and q(x) = 0(x-) (x+) (x +7x +0) is 0 (x ) (x + ) (x + 4) (x + ) 0 (x ) (x + ) (x + 4) (x + ) 0 (x ) (x + ) (x + 4) (x + ) 0 (x ) (x + ) (x + 4) 0. L.C.M of p(x) = (x 4 x ) and q(x) = 8 (x 4 -x +x ) is 6x (x ) (x ) 6x (x ) (x ) x (x ) (x ) 6x (x ) (x ). L.C.M of p(x) = (x +x+) (x+) and q(x) = (x-) is (x + ) (x + x + ) (x + ) (x + x + ) (x ) (x + ) (x ) (x + x + ). If p(x) and q(x) are two polynomials and q(x) 0, then p( x ) is known as q( x ) Rational Number Rational expression Polynomial None of these

16 6 Mathematics. Every polynomial is a rational expression, but each rational expression.. Is also a polynomial Need not be a polynomial Must be a polynomial None of these 4. x x is not a rational expression x because Its numerator is not a polynomial Its denominator is not a polynomial Neither numerator nor denominator is polynomial None of these. Rational expression x x terms x x x 9 x x 9 x 7 9 x x in its lowest x 9 x x 9 x 6. Additive inverse of algebraic expression x is x 4 x x 4 x x 4 x 4 x x x 4 7. Additive Inverse of algebraic expression x 4 4 x 6 x 4 x 4 4 x 6 4 x 6 x 4 x 4 4 x 6 4 x 6 8. Which of the following is a polynomial? x -xy+7y x+y 7x -x +x-4 x -x+/x 9. The value of the polynomial x-4x + at x = - is Which of the following is a rational polynomial. x x x x x x x. Factors of a --x-x are (a-x+)(a-x-) (a+x-)(a-x+) (a+x+)(a-x-) None of these. The rational expression x x form is x x x 4 x x 4 x x x. Write the rational expression simplest form : x x x x x x x x 4 x x x 8 4 in its lowest x 4 x x 4 x x ( x ) x x x None of these in its 4. The rational expression x 4 x ( x ) x is x 6 x 4 x 6 x 4 x 6 x 4 x 6 x 4

17 Mathematics 7. The rational expression ( x x ) _ x x x ( x x ) _ x x x 6. The rational expression x x x x x x x x x x x is x ( x ) x ( x x ) _ x x ( x x ) x x x x x x x x x 7. Which rational expression must be added in x x x x x x x x 7 so that their sum is x x 6 x x x x 8. The factors of x 4 +x + are (x +x+)(x -x+) (x +x+)(x +x-) (x +x+)(x -x+) None of these 9. Write multiplicative inverse of x x x x x x x 0. If one of the factors of x 4 + x 0 is (x + ), the other factor is x x + 4 x 4 x + is. If (x-) is a factor of x 6x + x + k, then the value of k 0 -. Find the value of a if x=½ is a root of the polynomial 8x + ax -4x + is a=4 a=-4 a= a=-. Divide the polynomial p(x) = x x + 4x by factor (x- ) then remainder is Using remainder theorem find remainder if (x 6x + 9x + ) is divided by (x-) The factors of x 7x + 6 are x(x-6)(x-) (x -6)(x-) (x+)(x+)(x-) (x-)(x+)(x-) 6. Zeros of a polynomial x are 0 None of these 7. H.C.F of 8(x x + x) and 8 (x +) is x + x + x x + 4(x x +) 8. L.C.M of ( x ), (+x) and (-x + x ) is ( x ) (+x ) (- x 6 ) x + 9. H.C.F of (x-) (x -) and (x -) is (x ) (x - ) (x - ) x +

18 8 Mathematics 40. The remainder when x ax + 6x a is divided by x a a 4a 0 6a 4. The factors of x 4 +4 are (x +x+)(x -x+) (x +x+)(x +x-) (x -x-)(x -x+) (x +)(x -) 4. Write factors of (x 4 + x + ) (x x + ) (x + x + ) (x + ) (x + ) (x ) (x + x + ) (x + ) (x x + ) 4. If one root of the polynomial f(x) = x + x + k is reciprocal of the other, then the value of k is 0 / Find the reminder when polynomial f(x) = x 4-6x +x -x+ is divided by If (x -) is factor of (x + ax- then a = When x + is devided by (x + ), then remainder is 0 None of these 47. If p(t)=t -t+, q(t)=t +t-,then degree of polynomial p(t)-q(t) is 0 undefined 48. If p(x)=x +x -x+, q(x)=x -x +x- then the degree of p(x)q(x) is: If P(x) = 7x +4x -x+ then the degree of P(x)is 4 0. The area of a square is (49x + 8x + 4) it's side is : (7x + ) (7x - ) (7x + ) (x + 7) POLYNOMIAL AND RATIONAL EXPRESSION B B c d 4 a C B a b 4 a C B c d 4 b 4 a 4 A 4 d 4 a 44 a A d a d 4 d 6 C 6 d 6 b 6 a 46 c 7 C 7 a 7 c 7 d 47 c 8 A 8 a 8 c 8 c 48 d 9 A 9 b 9 c 9 b 49 b 0 A 0 c 0 c 40 a 0 c

19 Mathematics 9 QUADRATIC EQUATIONS POINTS TO REMEMBER i) The general term of a quadratic equation is ax + bx + c = 0 where a,b,c, are real numbers and a 0 ii) If we can factorise ax + bx + c = 0, a 0 into a product of linear factors then the roots of the above quadratic equation can be found by equating each factor to zero. iii) iv) By the method of completing the square:we can find whether the given quadratic equations has real roots and find them if they exist. Quadratic formula : The roots of a quadratic equation ax + bx +c = 0, a 0 are given by x b b 4 ac, provided b - 4ac 0 a v) Nature of Roots:-A quadratic equation ax + bx +c =0, a 0 has Two distinct real roots, if b 4ac >0 Two real and equal roots, if b 4ac = 0 No real roots, if if b 4ac < 0 The quantity b 4ac is called discriminant. vi. vii. If, are the roots of quadratic equation ax +bx +c =0 then sum of roots = + = b -coeff.of x a = coeff. Of x and product of the roots = c constant terms = a coeff. of x Formation of an equation when roots are known. If known roots are and then the quadratic equation would be x - (sum of roots)x + product of roots = 0 i.e. x ( + ) +. = 0 i.e. x Sx + P = 0 viii) If a quadratic equation has equal roots, then quadratic equation is a perfect square : ix) + = ( + ) - + = ( + ) - ( + ) = ( ) + ( ) + = ( + ) ( + ) - ( - ) = 4. The degree of quadratic equation is 0. Which of the following is quadratic equation x - 4x + = 0 x = (x 0) x x = 0 (x 0) x x (x + ) +. Which of the following is quadratic equation. x -6x+4 = 0 x -7x = 0 x+/x = x x +/x = 4. How many real roots of a quadratic equation are possible : 4. How many zeros of a cubic polynomial can have : 4 6. Solutions of the quadratic equation 4x = 0 x = / x = / x = / x = /

20 0 Mathematics 7. Solutions of quadratic equation x +6x+= 0 are -,-,, -,- 8. If equation x +4x+k=0 has real and distinct roots then K > 4 K 4 9. Solution of quadratic equations x = 4 are x = 6 x = x= 6, 6 x = 0. Solution of quadratic equation z + k = 0 are z = k, k z = k z = k No real roots of z. Find the discriminant of the quadratic equation x x + 6 = 0 0. If discriminant of a quadratic equation is less then zero then which of the following statement is true. Roots are real Roots are real but unequal Roots are imaginary Roots are real and equal. The roots of equation x 7x - 8 = 0,-4 -, 8, -8, 7 4. The roots of quadratic equation ax + bx + c = 0, a 0 are real if b 4ac 0 b 4ac < 0 b 4ac 0 b b 4ac 0. If p and q the root of the equation x -px+q = 0 then p =, q = - p = 0, q = p = -, q = 0 p = -, q = 6. If x + k(4x+k-)+ = 0 has equal roots then k=,,,, 7. The values of k for which x = is a solution of the equation kx + x = 0 k = /4 k = /7 k = /4 k = /7 8. The sum of the squares of two natural numbers is 4. if the first number is one less than twice the second number then the numbers are. (, ) (, 4) (6, -) (, ) 9. The sum of two natural numbers is 8. if the sum of their reciprocal is 8/ then numbers are (,6) (7, ) (, ) (4, 4) 0. The sum of the squares of three consecutive the integers is 0. the integers are, 4,,, 4 4,, 6 7, 8, 9. Roots of the quadratic equation x 7 x 0 F HG F HG,, I KJ I KJ F HG F HG,, I KJ I K J

21 Mathematics. Two consecutive positive integers whose sum of squares is 6 are (, ) (, 4) (, ) ( 9,0). If a rectangular mango grove has its length twice of its breadth and area 800 m then its length and breadth are (0m, 0m) (00m, 400m) (00m, 40m) (0m, 60m) 4. If two numbers are differ by and their product is 4. then the numbers are (6, 9) (, 6) (7, 0) (, 8). Find the value of K for which the given equation (k + ) x (k-) x + = 0 has real and equal roots. k = k = k = k = 4 6. Find the values of k for which the following equation has equal roots (k ) x + (k ) x + = 0 k =, 4 k = 6, k = 6, 4 k = 0, 8 7. The equation whose roots are and - is : x -4 x-6=0 x -4 x- = 0 x - = 0 x - 6 = 0 8. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is cm, then the other two sides. cm, 0 cm 4 cm, cm cm, 9 cm cm, cm 9. If the sum of the squares of consecutive natural numbers is 0. then the smallest number is The roots of the equation 4 x x 9 = are 0, 0, - 0, - 0,. If, are roots of the quadratic equation x + x + = 0 then value of 0. If, are the roots of the quadratic equation x x+ = 0, then value of + /4 /4 /4 /4. If, are the roots of equation x 4x = 0 then value of + is If, are the roots of equation x 4x = 0 then value of - is If root of the equation x +ax+ = 0 is then its other root is If one root of x + x + k = 0 is reciprocal of the other then k = 0 k = k = /6 k = 6

22 Mathematics 7. If are the roots of the equations x + x + = 0, then the quadratic equation whose roots are and is x 0x + 0 = 0 x 0x 0 = 0 x + 0x + 0 = 0 x + 0x + 0 = 0 8. The value of is 4-9. If the sum of roots of a quadratic equation is 0 and the difference is. then the quadratic equation is x + 0x + 4 = 0 x +0x-4=0 x 0x 4 = 0 x 0x + 4 = The condition that the one root of ax + bx + c = 0 is thrice the other root is b = 6a b = 6ac b = 6ac b = 6c 4. The set of values of k for the given quadratic equation has no real roots x +kx+6 = 0 k k k k 8 4. The sum of two numbers is. if the sum of their reciprocal is /0 then two numbers are, 0 6, 9 8, 7, 4. If p and q are the roots of the equation x px + q = 0, then p =, q = p = 0, q = p =, q = 0 p =, q = 44. The quadratic equation with sum of roots ¼ and product of its roots - x x = 0 4x x + 4 = 0 4x x 4 = 0 4x + x+ 4= 0 4. If, are the roots of quadratic equation x +6x +4 =0 then find thesum of roots + is If ax + bx + c = 0 has equal roots then the value of c is b a b 4a b a b 4a 47. The solutions of quadratic equation x = are : None of these Quadratic Equation d 4 c 7 c 40 b a a 8 d 4 b a 6 b 9 d 4 a 4 b 7 c 0 c 4 a c 8 d a 44 c 6 a 9 c b 4 d 7 a 0 a a 46 d 8 a a 4 b 47 d 9 d b a 0 d c 6 b d 4 a 7 d a b 8 b b 6 a 9 d

23 Mathematics COMMERCIAL MATHEMATICS i.) Points to remember If selling price (S.P) > cost price (C.P) then profit = S.P C.P If S.P < C.P, then Loss = C. P S.P Profit or loss is expressed as percent of the C.P Profit % = Profit x 00 C.P Loss % = Loss x 00 C.P ii.) Discount : Discount = List price + rate of discount S.P = list price discount iii) iv) Simple Interest : When the principal remains the same for the entire loan period or time, then interest is called simple interest. S.I P R T 00 P = Principal Amount R% = Rate of interest per annum T = Times in years Compound Interest: When a certain interval of time due to addition of (compounding) of interest to the principal, new principal is formed which does not remain the same for the entire loan period, then the interest so obtain on new principal is called the compound interest. R A = P 00 Where A = Amount P = Principal N = Time R% = Rate of Interest. N Compound Interest ( C.I) = A P C. I. = P R 00 N P v) Cash Price : If we purchase an object by paying whole of its price in cash, the price is called is cash price. vi) Cash Down Payment: The amount of cash to be paid under the installment plan at the time of purchase is called cash down payment. vii) Installments: The payments which the buyer has to make at regular intervals are called installments. viii) Classification of taxes Mainly the taxes may be classified in two ways Direct Taxes Indirect Taxes Direct Taxes : A tax that is imposed on an individual or a group of individual which they have to pay to Government directly. e.g. Income Tax, Wealth tax, Property tax etc. Indirect tax : Those taxes which are paid indirectly by individuals. e.g. Sales tax, Octroi tax, Education Cess, Central Excise Tax etc. ix) Income Tax: When income from all sources exceeds a certain specified limit then he/ she/they has to pay a part of his/their income as Income Tax to the government. Rate of Income Tax : Rate for single man (Age Less than 6 years)

24 4 Mathematics Total Income Tax Upto,0,000 Nil,0,000 to 0% of amount (Rs. 4000),0,000 exceeding,0,000,0,000 to Rs % of amount,0,000 exceeding Rs.,0,000 (Rs 0000) Exceeding Rs Rs Rs.,0,000 0% amount exceeding Rs.,0,000 Rate for Single Woman (Age Less than 6 years) Total Income Upto,4,000 Tax NIL,4,000 to 0% amount exceeding Rs.,0,000,4,000,0,000 to Rs % amount,0,000 exceeding Rs.,0,000 Exceeding Rs Rs % Rs.,0,000 of amount exceeding Rs.,0,000 Rate of Senior Citizens (age greater or equal to 6 years) Total Income Upto,8,000 Tax NIL,8,000 to 0% amount exceeding Rs.,0,000,8,000 Above Rs. Rs % amount,0,000 exceeding Rs. 0,000 Note () if a person has total income more than 0,00,000 he has to pay 0% surcharge.. Each individual has to pay % education cess on his total income tax. Rebates in Income tax under section of income tax. An individual is also allowed a rebate in income tax on account of savings in following instruments. i) General Provident Fund (GPF) ii) Public Provident Fund (PPF) iii) Contributory Provident Fund (CPF) iv) Group Insurance plan (GIS) v) Life Insurance premium (LIC) including (ULIP) vi) Tuition fee of two children vii) Postal life Insurance (PLF) viii) Interest paid towards housing loan ix) National saving certificate (NSC) Etc.. PAN : Permanent Account Number. A car was purchased for Rs.,80,000 and sold for Rs.,00,000 then loss % 7.7% 9% 8.7% 8%. If the C.P of 6 articles is equal to the S.P of 4 articles then the gain percent. 0% 00% 00% %. An toys dealer buys a toy for Rs. 4 and spend Rs. on transporting it to his shop if he sells the toy for Rs. 0, then his profit percent is. 0% % 7% 0% 4. If the C.P of watches be equal to the S.P of watches then the loss percent is 0% 0% % 40%. A price of a bike is Rs It is sold at 400 after allowing certain discount. Then rate of discount is

25 Mathematics 6. Find the single discount equivalent to successive discount of Rs. % and 0% 0 % 8% % % 7. The single discount equivalent to a discount series 0% 0%, 40% 0% 0% 40% 0% 8. If the list price of product is Rs. 00 and discount series %, % then S.P is Rs Rs Rs. 40. Rs The list price of a briefcase is Rs. 600 the rate of discount is % then the amount to be paid by the customer for the briefcase is Rs. 0 Rs. 9 Rs. 70 Rs A laptop is available for Rs. after 9% then its list price is Rs. 64. Rs Rs Rs C.I on Rs. 800 for ½ years at 0% per annum, interest compound half yearly Rs Rs. 44. Rs Rs C.I on Rs. 400 compound annually for years 6 months at the rate of % per annum. Rs..0 Rs.. Rs..00 Rs..7. A calculator is available for Rs. 70 cash or Rs. 00 C.D.P along with 6 monthly installments of Rs. 00 each. Then the rate of interest charged under the installment plan is % pa % pa % pa % pa 4. A person bowered Rs. 00 from the bank and promised to return the amount in three equal annual installments. If the rate of interest is 0% p.a correspond annually then amount of the installment is. Rs. 000 Rs. 0 Rs. 40 Rs. 0. A bicycle is sold for Rs. 700 cash or for Rs. 900 C.D.P alongwith two monthly installments of Rs. 9 each. Then the rate of interest charged under the installment scheme is.4 % p.a.4% p.a 4.4 % p.a.4 % p.a 6. PAN stands for Permanent Account Number Permanent Address Number Pre Account Number None of these 7. Which of the following is not including in rebate under section 80C. National Saving Certificate Postal Life Insurance Group Insurance Plan Direct Investment in stock exchange. 8. HDFC stands for Human Development Finance Corporation Housing Develoment Finance Corporation Home development Finance Corporation None of these 9. NSC stands for National saving certificate National school council National student council National sports council

26 6 Mathematics 0. % education cess is paid on Total Income Total Savings Total Tax payable LIC premium. Find Income Tax, Mr. Gupta has to pay during the year Total Income Rs.,40,000 P.F Rs. 000 P.M LIC Rs. 000 Quarterly N.S.C Rs. 0,000 Rs.,,000 Rs.,0,000 Rs.,00,000 Rs.,0,000. Find the single discount equivalent to discounts of % & 0% 4.% 6% % 0%. IFFCO stands for Indian Farmer Furtilizer Co-operative Indian Former Furtilizer Co-operative Indian Farmer Furtilizer Coperation None of these 4. Total Income of Mrs. Asha is Rs During year she donates Rs towards CM s state relief fund. She contributes Rs. 600 per year in her GPF and deposits Rs. 000 in PPF. Then Income tax to be paid by her is Raj Rani get Rs.,80,000 as salary this year. If she deposit Rs. 000 in provident fund then her income tax is Rs. 600 Rs. 400 Rs Rs What is the present value of Rs. 000 due years hence if the rate of interest is 6% per annum compound half yearly (Given (.0) 0 = 0 744) Rs Rs. 600 Rs. 800 Rs Raj Rani is a pensioner getting a pension of Rs per month. Calcualte the income tax paid by her, when she does nto make any saving Rs. 400 Rs Rs Rs A senior citizen s annual pension is Rs Calculate wheather he is required to pay any income tax. If yes howmuch tax will he have to pay? Rs. 400 Rs. 800 Rs. 00 No tax 9. A person deposits in a bank Rs. 0,000 after every six months. If the interest is 0% per annum per annum compound half yearly, What would be the sum standing to his credit at the end of years? (Given.0) 4 =.) Rs. 400 Rs. 400 Rs Rs Amount on Rs. 00 compoundded annually for years at the rate of % per half year is Rs. 04 Rs. 0 Rs. 0 Can t be determined COMMERCIAL MATHEMATICS c a b 4 c d 6 c 7 d 8 a 9 c 0 a d a c 4 d a 6 a 7 d 8 b 9 a 0 c a a a 4 a b 6 a 7 c 8 d 9 a 0 c

27 Mathematics 7 ARITHMETIC PROGESSION Points to Remember i) Arithmetic Progression (AP ) : A sequence a, a, a,... a n... is called an arithmetic progression, if there exists a constant number d such that a = a + d a = a + d a 4 = a + d a n = a n + d & so on. The constant d is called common difference of the A.P. Thus if the first term is a andthe common difference is d, then a, a + d, a + d, a + d,... is an arithmetic progression. ii) General Term of an A.P a n = a +(n-) d or l = a+(n-) d where l is last term. iii. Sum to n terms of an A.P n S n a l ( ) n or S n a n d ( ) iv) The sum of first n positive integer is given by S n = n ( n + ) v) Arithmetic mean between two nos. a and b is A = a b. Which of the following sequences is A.P sequence.,, 8,,..., 9, 7, 8,...,, 4, 9,..., 4, Series whose nth term is n,,, 7,..., 4, 6, 8,...,,, 7,..., 4, 6,.... Which of the following sequences is A.P sequence., 4, 6, 8,...,, 4,,... 0,, 6, 9,... a, a +, a +, a + 6, Find the series whose nth term is n, 7,, 7,..., 4, 6, 8,...,, 4,..., 4, 6,.... If the 7 th term of an A. P. is 4 and th term is 64, then find the 8 th term of A.P th term of A.P,, 7, 9 is st Term of A.P,8,, is

28 8 Mathematics 8. Is 4 a term of the A.P 4,9,4,9.? 4 th th rd 6 th 9. Find n th term if first term on an AP is -7 and common differernce find 8th term If nth term of an A.P is (n-). Then its 7 th term is n + 9. How many terms are there in the AP? 0,,,,..., Which term of the sequence -, -7, -.. is -40? The third term of the series of an A.P is / and th term is / then ist term is / - -/ 4. If k +, 4k 6 and k are the three consecutive terms of an A.P. then value of k is 4. The common difference of an A.P whose nth term is -6n is Which term of the series +7++ is 47 th No term 0 th 7 th 7. The sum of series terms is The sum of the sequence 8, - 6, is None 9. How many terms of the series 9 6. must be taken so that series sum is The sum of m numbers of an A.P is m + 4m, its pth term is 6P+ P + 4P+ 6P. Find number of terms of the series 0... which amount to be S n of an A.P = n + n, then its rth term is 6r + 6r 6r None of these. Sum of first n natural number is n( n ) n( n n( n ) n( n How many terms of the AP, 4, 7,... are needed to given sum 7?

29 Mathematics x = 48 then x =? The sum of the series 8 terms is -/ -/ -/ 7. Sum of all positive integers between 00 to 000 which are divisible by 7 is Find the common difference of the AP for which 0th term is 0 more than the 8th term Sum of all the numbers between 0 and 00 which are exactly divisible by In boring a well 0 meters deep, the cost is Rs. 6/- for first meter and additional Rs..0 for each subsequence meter. What is the boring th meter and also the cost of boring the entire well Rs.. Rs. 7. Rs.. Rs. 6.. A man saves Rs. 00 in the st year in each succeeding year, he saved Rs. 0 more than the previous year, how much has he saved at the end of 40 years. Rs Rs Rs Rs The three numbers in A.P whose sum and product are and 48 respectively., 4, 6,,,6,0,6,9. Four numbers in A.P whose sum is 0 and sum of whose square is 0,,,7,6,0,4,4,6,8,,7,9 4. The sum of three numbers in AP is - and product is 8. Find numbers,-,-4,,4,4, 4,,. Sum of all two digit positive number between 0 and If 7 times the 7 th term of an A.P is equal to times the th term then 8 th term of the A.P is None of these 7. The sum of odd numbers between 0 and 0 is In an A.P the sum of st n th term is n n, find its th term Number of three digit numbers between 00 and 000 which are divisible by Which term of the sequence 0, 8, 7,... 4 terms? 9, 4 is the first negative

30 0 Mathematics 6th term 0th term 8th term 9th term 4. Which term of the AP,, 7, 9,... will be more than its 4th term? 66th term 6th term 7th term 76th term 4. How many three digit numbers are divisible by 7? If x, x + 0, x + are in A.P., then find the value of x If the sum of n terms of AP is S n = n + n, write its common difference If a,b,c are in AP then arithmetic mean between a and b are a+b (a+(a- a b a b 46. Arithmetic mean of 8,0,,4,6,...is Find the sum of th term from the last of A.P. 0,7,4,..., 6 is If 0 times the 0 th term of an A.P. is equal to times the th term then th term of an A.P. is If the sum of p th terms of an A.P. is q and sum of q th term is p then sum of p+q th term will be 0 p-q p+q -(p+q) AIRTHMATIC PROGRESSION a 6 a b 46 a c 7 b a 47 b c 8 b c 48 a 4 a 9 b 4 a 49 d c 0 a d 0 d 6 c c 6 c 7 b b 7 d 8 b b 8 b 9 c 4 c 9 a 0 c a 40 c a 6 d 4 b b 7 b 4 d d 8 d 4 a 4 c 9 a 44 c c 0 b 4 b

31 Mathematics CO-ORDINATE GEOMETRY Important facts and formulae: XOX is called X axis YOY is called Y axis Mid Point Formula:-Let D be the middle point of P(x, y ) and Q (x, y ) Then coordinates of D are F HG x x y y, I K J If P is any point in the plane, then its X coordinate i.e. distance from X axis is called abscissa and the Y coordinate i.e. distance from Y axis is called ordinate Distance Formula : To find the distance between two points whose coordinates are given, say p(x, y ) and q( x, y ) PQ ( x x ) ( y y ) The distance of a point P (x,y) from the origin O (0,0) ( x 0) ( y 0) Section formula (Internal Division) : To find the coordinates of the points which divide internally, the straight line joining the (x, y ) and (x, y ) in the ratio m:n Coordinates are F HG m x m m x m m y, m m y m Coordinates of centroid of triangle : Let the vertices of a ABC be A (x, y ), B((x, y ) and C (x, y ) respectively coordinates of centroid of the triangle are G F HG x x x y y y, I KJ I K J Area of ABC = [x (y y ) + x (y y ) + x (y - y )] Three points (x,y ), (x, y ) and (x,y ) are collinear when [x (y -y ) + x (y y ) + x (y y )] = 0 i.e. Area of ABC = 0 Three points A (x, y ), B (x, y ) and C (x, y ) are collinear or lie on a line if any one of the following takes place. AB + BC = AC AC + CB = AB CA + AB = CB Area of ABC = 0 Four points will form A parallelogram, if its opposite sides are equal. A rectangle, if opposite sides are equal and two diagonals are also equal. A rhombus, if all the four sides are equal and diagonals are perpendicular to each other.

32 Mathematics A square, if all the sides are equal and diagonals are also equal. Three points will form i) An equilateral triangle, if all the three sides are equal. ii) An isosceles triangle. If any two sides are equal. iii) A right angled triangle, if the sum of the squares of any two sides is equal to the square of the third largest side. Co-ordinate Geometry Q. Three points are collinear is Area of = 0 Area of <0 Area of >0 None of these. Four points will form a parallelogram if Any two sides are equal All sides are equal Its opposite sides are equal None of these. Joining of three points will term a isosceles triangle if Any two sides are equal All sides are equal Its opposite sides are equal None of these 4. If A (x, y ), B (x, y ) C (x, y ) are vertices of ABC then co-ordinates of centroid of triangle are. F HG F HG F HG x x y y, x x x y y y, x x y y, I K J I K J I K J F HG x x x y y y,. Three points A (x, y ), B (x, y ) and C (x, y ) are collinear if AB + BC + AC = 0 AB + AC = BC AB = AC AB + BC = AC 6. Distance between the two points (cos, sin) and (sin, -cos) is 7. If end points of a diameter are (7,-) and (-,) then radius of the circle is For what value of a these points (, ), (, and (, -) are collinear 0 9. If the mid-point of the segment joining A (x, y + ) and B (x +, y + ) is c (/, /), then find x, y. (, ) (, ) (, ) (, ) 0. Let A (,) and B ( 4, 9) be two points then mid point of AB is (-, -) (, ) (-, ) (,-). For what value of K these three points (k,) (,) and (,9) are collinear. 4 I K J

33 Mathematics. If the distance between the points (,0) and (0,y) is, what are the value of y? If point (a, 0), (0, and (, ) are collinear, then what will be value of? a b 0 4. Co-ordinates of the points, which divides the join of (,) and (4,7) in the ratio : F8 7I, HG K J F7 I, HG K J F HG F HG 6,, 6 7. If A (-, ), B (P,q) and C (-, 4) are the vertices of an isosceles triangle and AB = BC then p+q = p-q = p + q=0 P+q = - 6. What will be centroid of the triangle, whose vertices are given as (4, -8) (-9, 7) and (8, ) (, 4) (, 4) (4, 4) (4,) 7. The distance between the points A(0cos, 0) and B(0, 0 sin) If two vertices of triangle are (-,) and (0,) and centroid is at origin then third vertex is (, ) (, ) (, ) (4, ) I K J I K J 9. If the point (a, 0) is equidistant from (, 4) and (-, -) then value of a is The point on y-axis which is equidistant from (4, 0) and (4, ) is (0 4) (0, 6) (0, -6) (0, -4). The point on x axis which is equidistant from (,4) and (, -) is (4, 0) (6, 0) (-6, 0) (-4,0). The length of line segment AB is 4. if point A is (, -4) & B is (-6, K) then value of K is The distance between points p(,-)and q(x,) is 0 unit then value of x = 8,4 8,-4,4,4 4. In what ratio the x-axis divide the line segment joining the points (6,4) and (, -7) 4 : : 7: 4 7:. If the centroid of the triangle formed by (7, x), (y, 6) and (9, 0) is at (6, ) ; then (x, y) is : (4, ) (, 4) (, ) (, 4) 6. If A(,),B(-4,-4) and C(,-8) are the vertices of a triangle then the length of the medien through vertex C is 6 7 8

34 4 Mathematics 7. If two vertices of a triangle are (, -) and (-7,4) and co-ordinates of the centroid of a triangle are (,-) then co-ordinates of third vertex. (0,) (-0, ) (0, -) (-0, -) 8. The end points of the diameter of a circle are (, -) and (, 4) then the co-ordinates of its centre F I, HG K J F 7 I, HG K J F HG F HG 7, 7, 9. One end of diameter is (,) if the centre of circle is (,4) then other end of diameter is (-, -6) (-, 6) (, 6) (, -6) 0. For what value of P the points (-, ), (, P) and (, -) are collinear p = p = p = - p = -. X, Y and Z are seated at A (, ), B(6,K) and C (8,6) if they are seated in line then value of K will be 4. The area of the triangle formed by the points (a,b+,(b,c+ and (c,a+ is a+b+c abc (a+b+ 0. If the area of the triangle formed by the points (x, x),(-, 6) and (,) is square units then x is equal to I K J I K J 4.. The ratio in which line segment joining the points (-, 0) & (6, -8) is divided by (-, 6) 7 : : 7 : 7 : - 7. A point P (, -) is equidistant from the points A ( a, 7) and B (-, 9) then value of a is Three points P (,), Q (9,0) and R (,) are the vertices of a Isosceles triangle Equilateral triangle Right angled triangle None of these 7. If A(,), B(4,), C(6,6) are the three vertices of a parallogram ABCD. Find the cordinate of 4th vertex D (,) (,6) (6,) (,) 8. If p(,6) is the mid point of the line segment joining A(6,), B(4,) then y= The value of K, if the point A (0,) is equidistant from (,m) and (m, ) Three points (a, 4, (a, 6 and ( a a, are vertices of An equilateral triangle Isosceles triangle Right angled triangle None of these

35 Mathematics 4. The area of the triangle, whose vertices are (, ), (-, 0), (, -4) sq units sq units sq units sq units 4. If the centroid of the triangle formed by the points (a,, (b, and (c, is at the origin, then a +b +c is equal to abc 0 a+b+c abc 4. The four points (0,0), (0, ), (6, )and (6, 0) represents a rhombus a square rectangle Parallelogram 44. If the points (k,)(6,-)(-,4) are collinear then k = 4. If the distance between the points A (4, p) & B (, 0) is units, then the value of p is Find the co-ordinates of the points which divides the line segment joining A (-,) and B (4,-7) in the ratio :4 points are , and6, , and6, , and 6, 8,6 and, Find the ratio inwhich y-axis divides the line segment joining A (, 4) and B (, ). : : : : 48. If two vertices of a triangle are (-,) and (0,-) and centroid is at the origine, then find rd vertex. (,) (,) (,) (,) 49. The area of triangle formed by A(0,) B(0,), C(,4) is 6 sq.unit 8 sq.unit 4 sq.unit 6 sq.unit 0. Find the relation between x and y such that point (x,y) is equally distance from the point (,6)(-,4) the relation is x+y=4 x+y= 4x+y= x+y= CO-ORDINATE GEOMETRY a c b d 4 c c b a d 4 d c a b a 4 c 4 b 4 d 4 c 4 b 44 c d a d c 4 a 6 b 6 b 6 c 6 c 46 a 7 a 7 a 7 c 7 b 47 c 8 d 8 c 8 d 8 b 48 c 9 d 9 d 9 c 9 b 49 d 0 a 0 b 0 a 40 a 0 b