TOPIC-1 REAL NUMBERS. Rational Numbers WORKSHEET-1 WORKSHEET-2 P-1 SECTION. \ x = [CBSE Marking Scheme, 2012] S O L U T I O N S CHAPTER
|
|
- Kristopher Hood
- 6 years ago
- Views:
Transcription
1 SECTION CHAPTER REAL NUMERS TOPIC- Rational Numbers WORKSHEET So, it is a rational number Yes, zero is a rational number. Zero can be epressed as 0 0, 0 6, 00 etc, which are in the form of, p q where p and q are integers and q ) \ 0.8 [CSE Marking Scheme, 0] [CSE Marking Scheme, 06] 6. Since LCM of and is, \ and Hence, three rational numbers between and 9 are : 6, 8,. Any eample & verification of eample : Let m /, n 9/ Difference 9 0 (Rational Number) Sum (Rational Number) Product (Rational Number) Division 9 8 (Rational Number) WORKSHEET (Decimal point is shifted three 000 places to the left) [CSE Marking Scheme, 0] S O L U T I O N S P-
2 .. 6 Terminating [CSE Marking Scheme, 0] Alternative Method :. 6 6) (Terminating). Two rational numbers between and are 0. and 0. i.e. and and 0. [CSE Marking Scheme, 06] 6. LCM of and is \ and so, < The required three rational numbers are, and. [CSE Marking Scheme, 0]. Let a & b Here, we find si rational numbers, i.e., n 6 So d b a n + 6+ st rational number a + d + nd rational number a + d + rd rational number a + d + th rational number a + d + th rational number a + d + 6 th rational number a + 6d So, si rational numbers are,,,, &. 8. Since LCM of and 6 is 0 \ and Hence, four rational numbers between 6 and are : 6 0, 8 9 0, 0, [CSE Marking Scheme, 0] WORKSHEET-. and 8 So three rational numbers and are 6,, [CSE Marking Scheme, 06]. and i.e., 0 and The numbers are and [CSE Marking Scheme, 06] P- M A T H E M A T I C S IX
3 . Let, ( ) ( ) 9 9. Let, ( ) ( ) 9 9. Let, (...) (0...) Alternative Method : Let [CSE Marking Scheme, 0] (...) (0...) 999 \ Let Let, (i) multiplying 0 on both the sides, we get, (ii) From (ii) (i), we get [CSE Marking Scheme, 0] TOPIC- Irrational Numbers WORKSHEET-. 0. is a terminating number. So, it is not an irrational number , is repeating continuously, so it is not an irrational number , is repeating continuously, so it is not an irrational number , non-terminating and non recurring decimal. Hence, it is an irrational number. So, is an irrational number.. No, it may be rational or irrational.. Required two irrational number are : (i) (ii) and [CSE Marking Scheme, 0] S O L U T I O N S P-
4 Hence three inrrational numbers between and 9 can be : [CSE Marking Scheme, 06] Let A C unit length Using Pythagoras theorem, we see that OC + Construct CD unit length perpendicular to OC, then using Pythagoras theorem, we see that OD ( ) + Using a compass with centre O and radius OD, draw an arc which intersects the number line at the point Q, then Q corresponds to. Let two irrational numbers are : 6 and, (i) 6 Difference is an irrational number. (ii) 6 + sum is an irrational number. (iii) 6 8 product is an irrational number. (iv) 6 / division is an irrational number. WORKSHEET-. Sum of and +.. Since,.6 Hence, the irrational number between and. is. ( ) ( ) + ( ) Given, (Irrational Number) Hence, required number can be [CSE Marking Scheme, 0] 6. Marks the distance 9. units from a fied point A on a given line to obtain a point such that A 9. units from, marks a distance of unit and mark the new point as C. Find the mid point of AC and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through and intersecting the semi-circle at D. then D 9. To represent 9. on the number line, let us treat the line C as the number line, with as zero (c) as and so on. Draw an arc with centre and radius D which intersect the number line at E \ E represents 9. [CSE Marking Scheme, 06]. Mark the distance. units from a fied point A on a given line to obtain a point such that A. units. From, mark a distance of unit and mark the new point as C. Find the mid-point of AC P- M A T H E M A T I C S IX
5 . and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through and intersecting the semicircle at D. Then, D. To represent. on the number line, let us treat the line C as the number line, with as zero, C as, and so on. Draw an arc with centre and radius D, which intersects the number line at E. \E represents. TOPIC- n th Root of Real Number ` Mark the distance 9. units from a fied point A on a given line to obtain a point such that A 9. units from, mark a distance of unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through and intersecting the semi-circle at D. Then, D 9. To represent 9. on the number line,let us treat the line C as the number line, with as zero, C as, and so on. Draw an arc with centre and radius D, which intersects the number line at E. \ E represents 9. [CSE Marking Scheme, 0] WORKSHEET-6 ( )( ). + S O L U T I O N S { ( ) } { } 0 [CSE Marking Scheme, 0] ( 8 + ) [CSE Marking Scheme, 0] [CSE Marking Scheme, 0] Alternative Method : 0 0 [CSE Marking Scheme, 0] P-
6 8. LCM of and is [CSE Marking Scheme, 0] Alternative Method LCM of and is 6 \ and ( ) ( ) Now, ( ) ( ) 6 [CSE Marking Scheme, 0] WORKSHEET-. b a a b. ( )( ) ( ). a+ b a b ( a) b a b [CSE Marking Scheme, 06] \ ( + ) ( ) ( ) ( ) [CSE Marking Scheme, 0, 0] 6. ( ) ( ) + ( ) [Using (a b) a + b ab] ( 8 ). [CSE Marking Scheme, 0] ( )( + ) [CSE Marking Scheme, 0] Alternative Method : ( )( + ) ( ) ( ) ( ) ( ) ( + ) ( ) P-6 M A T H E M A T I C S IX
7 Alternative Method : [CSE Marking Scheme, 0] ( ) + ( ) + + TOPIC- Laws of Eponents with Integral Powers + ( ) + ( ) ( ) + ( ) WORKSHEET (6) 0.6 (6) 0.09 (6) 0. (6) / ( ) /. ( + + ) / ( ) / (6) / [(6) ] / [CSE Marking Scheme, 0]. Given, a and b. a b + b a [CSE Marking Scheme, 0] ( ) + ( + ) ( ). ( ) ( )() + () + ( ) + ( ) ( ) + ( ) ( ) () +( ) () Alternative Method : ( y ) y ( ) y ( y) y y y 9 y 9 [CSE Marking Scheme, 0] y. y ( ) ( ) y ( y) y ( y) + + y 9 y 9 y [CSE Marking Scheme, 0] S O L U T I O N S P-
8 . ( a b ) a+b. ( b c ) b+c. ( c a ) c+a a b b c c a.. a b b c c a (Any number to the power 0 is ) [CSE Marking Scheme, 0] ( ) ( ) + ( ) + ( ) [CSE Marking Scheme, 06] WORKSHEET-9. [(6) ] ( ) [ ] / /. 6 ( 6 ) ( 6 ) / ( 6 ) / 6 [CSE Marking Scheme, 06]. Given, a and b. (i) (a b + b a ) ( + ) (8 + 9). (ii) (a a + b b ) ( + ) ( + ) [CSE Marking Scheme, 0] 8 [CSE Marking Scheme, 0]. ( 6) ( 6) 6 ( ) ( ) [CSE Marking Scheme, 0, 0] Alternative Method : 6 ( ) ( ) 6 ( ) ( ) (i) ( + ) (ii) ( + ) 9 [CSE Marking Scheme, 0] Alternative Method : (i) ( y +y ) ( + ) ( + ) (). (ii) ( +y y ) ( + ) ( + ) (9) 9 P-8 M A T H E M A T I C S IX
9 0 ( ) [CSE Marking Scheme, 0] WORKSHEET ( ) ( ) ( ) ( 8 ) 6 / + ( ) ( ) ( ) / + ( ) [CSE Marking Scheme, 0] / 6 +. ( 6) ( 6) 6. / 6. [(8 / + 6 / ) / ] [( + ) / ] [ / ] [ / ] [ / ]. 8 + ( + ) ( ) [CSE Marking Scheme, 0] Alternative Method : 8 + ( ) + ( ). ( + ) ( ) ( ). ab c ( b ) c ( ) ba ( c) a ab ac ( ) ( ab ac ba+bc ) ( bc ac ) ac ba bc bc Alternative Method : ab ( c) b a ba ( c) c ab ac ac ba bc bc ( ab ac ba+bc ) ( bc ac ) ( bc ac ) ( bc ac ). ( ) ( ) ( ) \ [CSE Marking Scheme, 0] Alternative Method : Given, ( 8) ( ) ( ) ( ) ( ). + + comparing the power of both sides, we get 0 [CSE Marking Scheme, 0] S O L U T I O N S P-9
10 \ [CSE Marking Scheme, 0] Alternative Method : 8 6 Comparing the eponents, we get.. abc ( a ) bc (y) bc (y b ) c (z) c abc 8. ab ab a b+ a + a b a Alternative Method : WORKSHEET- abc [CSE Marking Scheme, 0] b ab+ b + ab b b a b a [CSE Marking Scheme, 0] 9. a a + b + a a b a + a + a b a b a a a+ b + b a ab ab b b a+ b + b a ( b a ) + bb ( + a ) ( b+ a)( b a) b ab+ b + ab b a b b ( a b ) a b [CSE Marking Scheme, 06]. ( 8) ( 8) ( ) ( ). 8. P-0 M A T H E M A T I C S IX
11 . a b [CSE Marking Scheme, 06] a b 8 b a [ 8] a b [ 8] 9 [CSE Marking Scheme, 0].,, 6,, ( ),( ),( 6 ) 9, 6, 6 In ascending order ( 6),( 6),( 9) i.e., 6,,.. [CSE Marking Scheme, 0] Alternative Method : Since LCM of,, is.,, 6,, 6 6, 6, 6, 6,, 6 6, 9, 6. In ascending order ( 6),( 6),( 9) a b ( b c) ( c a) ( + ) a b c ( ) 6,, + a+ b b+ c c+ a.. a b c.. a+ b+ c a+ b+ c [CSE Marking Scheme, 0] 6. a b 0, + 0 ( ) ( + ) [CSE Marking Scheme, 06]. ( ) ( ) \ [CSE Marking Scheme, 0] TOPIC- Rationalization of Real Numbers WORKSHEET So, rationalizing factor is ( 6 ) 6 8 S O L U T I O N S P-
12 ( ) ( ) ( + ) ( ) ( ) ( ) 9 8. ( ) ( ) ( ) ( ) + [CSE Marking Scheme, 0] Alternative Method : 0 ( ) ( ) ( ) ( ) ( + ) ( + ) (rationalizing) ( ) 0 + ( ) ( ) ( ) + + ( ) ( ) + ( ( + ) ( ) ( + ) ( ) ( ) ( ) ( + + ) + + ( + )+ ( + ) ( + ) ( + + 6) ( ) ( 6) WORKSHEET-. ( ) ( + ) ( ) [CSE Marking Scheme, 0] Alternative Method : + ( ) ( + ) ( ) + 6 ( ) ( ) () + 6 P- M A T H E M A T I C S IX
13 \ Given epression [CSE Marking Scheme, 0] Alternative Method : ( ( ) + ( ) ( ) ( 6 ) ( ) ( ( + ) ( 6 ) ( 6) S O L U T I O N S ( ) and ( )( ) ( ) ( + ) + + ( + ) + ( ) + 0 8( + ) [CSE Marking Scheme, 0] Alternative Method : ( + ) ( + ) ( + ) ( ) ( ) ( + ) LHS 8 88 ( + ) ( ) ( 8) ( 8) ( ) + 6 ( + 6) + ( ) ( 6) ( 6 + ) ( 6) ( ) ( 6 + ) + ( ) ( ) + 8 ( 8 + ) ( + 6) ( 6 + ) ( + ) [ a b (a b) (a + b)] RHS 6. Denominator \ ( 0 ) ( 0 + ) ( 0 ) [CSE Marking Scheme, 0] Alternative Method : ( 0 + ) 0 ( 0 + ) ( 0 + ) 0 (. +. ). P-
14 WORKSHEET-.... (rationalizing) p [CSE Marking Scheme, 0] ( ) ( + ) ( 0 ) ( ) ( 0 + ) 8 + ( 0 + ) 0 ( 0 + ) ( 0 + ) ( ) + ( ) [CSE Marking Scheme, 06] 9. ( + ) ( )+ ( + ) 9 ( ) ( 6+ )+ ( + ) ( ) (Rationalizing) 9( ) ( 6+ )+. +. [CSE Marking Scheme, 0] + ( + ) ( ) ( ) + ( ) + ( ) ( ) ( + ( + ). (appro). 6. a a a b [CSE Marking Scheme, 0] ( ) ( ( + ) ) ( ) ( ) ( ) b b 9 9 On comparing both sides, we get a 9, b 9. [CSE Marking Scheme, 0] P- M A T H E M A T I C S IX
15 WORKSHEET Alternative Method : (. ) + (. ) ( + ) + ( ) ( )( + ) +.06 [CSE Marking Scheme, 0] Squaring, both sides, we get b and y y+ y (+ ) ( ) a ( + ) 9+ ( ) b + 9 \ a + b [CSE Marking Scheme, 06] \ + ( + + ) (8) 6 S O L U T I O N S P-
16 SECTION CHAPTER POLYNOMIALS TOPIC- Polynomials WORKSHEET-6... Not defined. Linear polynomial.. Not a polynomial.. Constant polynomial is. 6. Linear polynomial +, degree Quadratic polynomial +, degree Cubic polynomial,, degree. f() + then, f() + 6 then, f() + 0 f() f() p() + 6 (i) When (ii) When then, p() f() + Then, f() + f( ) ( ) ( ) + and f f() f( ) + f then, p( ) ( ) + 6 WORKSHEET-. inomial. Degree of a polynomial is 0. Degree of + Degree of Degree of ( ) ( ) Every real number is a zero of the zero polynomial.. No. of zeroes of cubic polynomial. 6. p (y) y y + p (0) or y + y. 8. Co-efficient of in epression + π is π. 9. p() + 6 Then, p( ) ( ) ( ) ( ) f(y) y y + ay + b f() () () + a() + b 0 Þ a + b 0 Þ a + b...(i) f(0) b 0 from (i) a + 0 Þ a \ a, b 0 [CSE Marking Scheme, 06]. f() + + f(0) f( ) ( ) ( ) + ( ) ( ) f() () () + () () \ f(0) f( ) ¹ f() [CSE Marking Scheme, 06] P-6 M A T H E M A T I C S IX
17 . f() + f() () () + () f() f( ) ( ) ( ) + ( ) 8 6 f( ) 0 f(0) \ f() + f( ) + f(0) 0 6 [CSE Marking Scheme, 06]. p() + + p() () + () () p( ) ( ) + ( ) ( ) p(0) p() + p( ) p(0) [CSE Marking Scheme, 0] TOPIC- Remainder Theorem WORKSHEET-8. Let, p(y) y y y +, then remainder p(0) p() Put, 0 Remainder p S O L U T I O N S in p() [CSE Marking Scheme, 0]. Factors of (±, ±, ±, ±, ± 6, ± ) p() 6 + p() 6() () is a zero of p() or ( ) 6 + is a factor of p() 6 ( ) ( ) + 6( ) ( ) (6 + 6) ( ) ( ) ( ) [( ) ( )] ( ) ( ) ( ).. Let p() a + and g() + a R p() R g() + p() a() + () R 8a + 8a R () () + a a R 6 + a R R 8a 6 + a a a [CSE Marking Scheme, 0]. p() a p() leave the same remainder when divided by ( + ) and ( +). p( ) ( ) + 8( ) + ( ) + a ( ) 8 + a 0 a p( ) ( ) + 8( ) + ( ) + a( ) + 8 a 0 a Remainders are equal So, 0 a 0 a a 0 a 0 6. f() + a + b Put, 0 or in f(), we get f() () () + () a + b + a + b a + b a b a b...() P-
18 Again put, + 0 or in f(), we get f( ) ( ) ( ) + ( ) a ( ) + b a + b 9 6 a + b a + b...() Adding equations () and (), a 0 a y equation (), + b b 8 f() Again put, 0 or in f() f() () () + () WORKSHEET-9. Let, f() + 0 Put, + 0 Then remainder is : f( ) ( ) Here, p() a + 6 a, and the zero of a is a. So, p(a) a a.a + 6a a a. So, by the remainder theorem a is the remainder when a + 6 a, is divided by a.. Let, p() + k, and q() + + k Put, + 0 or in p () and q () P( ) ( ) ( ) + k( ) 0 k k 6 q( ) ( ) ( ) + ( ) +k 8 +k 0 +k p() and q() leave the same remainder when divided by +. k 6 k 0 k 6 k. Let p() a + + and q() + a Put, 0 or in p() and q() p() a() + () + a a + q() () () + a + a + a According to the question, p() q() a + + a a a 6a 6 a. [CSE Marking Scheme, 0]. p() + Put, 0 or in p() p() () + + Hence, must be subtracted from + so that it is eactly divisible by ( ). Resultant polynomial to be divisible by ( ) quotient + ) Thus, quotient + + and remainder. Let, p() Zero of + is / y remainder theorem, Remainder p P-8 M A T H E M A T I C S IX
19 WORKSHEET-0. 0 Þ y remainder theorem, if f() is divided by, the remainder is f \ f Hence required remainder is.. p() So, [CSE Marking Scheme, 06] + ) ( ) (+) 9 ( ) (+) + + (+) ( ) (Remainder) Quotient + Remainder. p() + p + p Put, + 0 or in p(), we get p( ) ( ) ( ) + ( ) p( ) + p p + p 9 p 9 p 0 p The polynomial p() Put, + 0 or in p() p( ) ( ) ( ) + ( ) ( ) ) ( Hence, Quotient + + and Remainder (Remainder) Verification : Let, f() + 0 y remainder theorem, remainder f( ) ( ) + ( ) ( ) [CSE Marking Scheme 0] S O L U T I O N S P-9
20 TOPIC- Factor Theorem WORKSHEET-. ( ). a b 6ab 6ab(a b). Since, f 0 is a zero of polynomial f() So, + or + is a factor of f().. f() + k ( + ) is a factor of f() + k Þ f( ) 0 Þ ( ) + ( ) k( ) 0 Þ + + k 0 Þ k 0 Þ k. ( ) is a factor of f() + k + k f() 0 Þ () + k() + k 0 Þ + k + 0 Þ + k 0 Þ k [CSE Marking Scheme, 06] 6. Let, p() 9, p ( ) Since, ( + ) is a factor of 9 ( 9 ) ( + ) ( ) + ( ) + ( + ) ( + ) ( ) Factors are : ( + ) ( ) [CSE Marking Scheme, 0]. Let p m(n p ) + n(p m + p(m n ) p(m n) n(n p ) + n(p n ) + p(n n ) n(n p ) n(n p ) m n is a factor of p Similarly p(n p) 0 & p(p m) 0 n p is a factor of p. and p m is a factor of p. [CSE Marking Scheme, 0] 8. Let p() + a + 6 p() is divisble by Þ p 0 p p a Þ a 0 Þ a Þ a 9 0 Þ a 9 \ p() ) \ ( + )( + ) Hence p() ( )( + )( + ) WORKSHEET-. + is a factor of + + m p() p( ) m 0 m. is a factor of p(), then p() 0 p() () + k k 0 k. P-0 M A T H E M A T I C S IX
21 . is a factor of p() 6 + k p k. 0 k [CSE Marking Scheme, 0]. ( ) is a factor of f() a a + a f() 0 Þ a () a() + a 0 Þ a a + a 0 Þ a 0 Þ a ± [CSE Marking Scheme, 06]. Given, p() k ( + ) is a factor of p(), then p( ) 0 k ( ) ( ) 0 k + 0 k [CSE Marking Scheme, 0] 6. p() m n If ( a) is a factor of p(), then p(a) 0 (a) m a n a 0 a[a m n] 0, a 0 a m n 0 a m+ n. [CSE Marking Scheme, 0]. p() 9 Factor of ± p() 9 0 Þ is a factor of p() \ 9 9 ( ) + 6( ) + ( ) ( )( ) ( )( + ) \ p() ( )( + )( + ) [CSE Marking Scheme, 06] 8. Let f() The factors of the constant term are ±, ±, ± and ± 6, The factor of coefficient of is Hence possible rational factors are ±, ±, ± ±, ± We have f( ) ( ) + ( ) ( ) 9( ) andf( ) ( ) + ( ) ( ) 9( ) So, + and + are factors of f() Þ ( + )( + ) is a factor of f() Þ + + is a factor of f() Let us now divide f() by + + to get the other factos of f(). + + ) \ ( + +)( ) ( + )( + )( 6 + ) ( + )( + )[( ) + ( )] ( + )( + )( )( + ) WORKSHEET-. Since, a + ab 6 a (a 6 + b 6 ). Factors are a and (a 6 + b 6 ) ( ) ( ) ( ) ( ) ( ) + ( ) ( )( + ) [CSE Marking Scheme, 06] S O L U T I O N S. f() + 6 Put, 0 or in f() Thus, f() Hence, ( ) is a factor of f(). [CSE Marking Scheme, 0] P-
22 . ( + ) is a factor of p() k p ( ) 0 ( ) k ( ) + ( ) k k [CSE Marking Scheme, 0] 6. Let, p() 6 a + a + a + ( a) is a factor of the polynomial p(), then p(a) 0 a 6 a a + a a a + a a + 0 a 6 a 6 + a a + a a + 0 a a.. Factors of (±, ±) p() () () 9() 0 is a zero of p() or ( ) is a factor of p() Then, 9 ( ) + ( ) + ( ) ( ) ( + + ) ( ) ( ) ( ) (( + ) + ( + )) ( ) ( + ) ( + ) 8. Factor of 6 (±, ±, ±, ± 6) p() + 6 p( ) ( ) + ( ) ( ) is zero of p() of ( + ) is a factor of p() \ + 6 ( + ) + ( + ) 6( + ) ( + )[ + 6] ( + )[ + 6] ( + )[( + ) ( + )] ( + )( + )( ) TOPIC- Algebraic Identities WORKSHEET-. y ( ) 9 y y y + [CSE Marking Scheme, 0] (9) () (9 + ) (9 ) [CSE Marking Scheme, 0]. 9 (0 ) (0 + ) (0) () [CSE Marking Scheme, 0]. ( y + z) + y + z y + yz + z [CSE Marking Scheme, 0] Alternative Method : y using the identity, (a + b + c) a + b + c + ab + bc + ca ( + ( y) + z) () + ( y) + z + ()( y) + ( y) (z) + (z)() + y + z y yz + z 6. (a b) [ (a b) ] [() {(a b)} ] [ + (a b)] [ (a b)]. [CSE Marking Scheme, 0]. a 6 b 6 (a ) (b ) (a b ) (a + b ) (a b) (a + b + ab) (a + b) (a + b ab) (a b) (a + b) (a + b + ab) (a + b ab). [CSE Marking Scheme, 0] 8. a + ab 6 a(a 6 + b 6 ) a[(a ) + (b ) ] a(a + b ) [(a ) + (b ) a b ] a(a + b ) (a + b a b ). [CSE Marking Scheme, 0] P- M A T H E M A T I C S IX
23 9. (i) 0 0 (00 + ) (00 + ) 00 + ( + ) (ii) (0) (00 + ) (00) (00 + ) WORKSHEET-. ( ) () () ( ) Co-efficient of in the epansion of ( ) 6.. a + b + c abc (a + b + c) (a + b + c ab bc ca) a + b + c abc 0, as a + b + c 0 a + b + c abc.. ( y + z) [ + ( y) + z] () + ( y) + z + ( y) + ( y) z + z + 9y + z y 6yz + z.. a b + a b + + a b + + () + a b + b () + a () a b ab a + + b Area of rectangle a a + a 0a a + a(a ) (a ) (a ) (a ) length breadth Length and breadth are (a ) and (a ) respectively y [ 6y ] [() (6y) ] ( 6y) [() + (6y) + 6y] a b (a b) (a + b + ab) ( 6y) ( + 6y + 0y).. pq + ( ) 9 8. (i) + pq pq ( pq) a + b (a + b) (a + b ab) 9 pq + pq , + pq 9 9 pq 9 () (ii) y ( ) (y ) ( y ) ( + y ) ( y) ( + y) ( + y ). ++ WORKSHEET-6.. Given, + + y + y + y y ( ) () 6. + y y + y + y 0 y ( y) ( + y + y) ( y) ( ) ( 8+ ) ( ) a + b (a+ b) (a + b ab) y y y y S O L U T I O N S P-
24 8 y y y 8y y y (00 + ) (00 + ) [CSE Marking Scheme, 0] 6. ( + y) + y + y ( + y) ± ±.. Given Ep ( y) ( + y) ( y) + ( + y) ( y) ( + y) ( y) ( + y) ( y) + ( + y) ( y) [ y y] ( + y) [ y y] ( y) [ y ] ( + y) [ y] ( y ) [ y ( + y)] (y + ) ( y) (y + ) ( + y) ( + y) (y + ) 8. + Cubing both sides ( ) ( ) + 0 [CSE Marking Scheme, 06] WORKSHEET as(a + b) (a b) a b [CSE Marking Scheme, 0, 0]. Given,. p 6 abc abc a+ b+ c 0 a + b + c abc. 9 p + p (p) 6.(p) + (p) 6 6 p 6 p p p [CSE Marking Scheme, 00, 0, 0]. On squaring both sides, we get + + a bc ( ) +. [CSE Marking Scheme, 0] b c + + a + b + c ac ab abc. y + z + yz () + ( y) + (z) () ( y) (z) ( y + z) [() + ( y) + (z) ()( y) ( y) (z) () (z)] a + b + c abc (a + b + c)(a + b + c ab bc ca ) ( y + z) [ + 9y + z + y + yz z] [CSE Marking Scheme, 0] P- M A T H E M A T I C S IX
25 + (6) WORKSHEET-8. Let, + p and q, then given epression p 8pq q p 8pq + 0pq q 6p(p q) + q(p q) (p q) (6p + q) [( + ) ( )] [6( + ) + ( )] ( ) ( ) ( 6 + ) ( ).. a, b, c a + b + c + 0 a + b + c abc () + () + ( ) ( ). + y 8. ( + y) y + y ( + y) 8 + y y y.. a + b a b (a b) () a b ab(a b) a b ab a b 9ab.. (00 + ) (00) + (00) () + (00) () + () 66 [CSE Marking Scheme, 0] 6. ( + ) + ( ) ( ) + ( ) + + ( ) + ( ) ( ). Add, and [CSE Marking Scheme, 0] S O L U T I O N S P-
26 SECTION CHAPTER LINEAR EQUATIONS IN TWO VARIALES TOPIC- Introduction of Linear Equation WORKSHEET-9. a + by + c 0, where a, b, c are real numbers and both a, b 0.. No. ( a, b 0 for a linear equation). True.. Let the runs scored by Raina & Dhoni are & y respectively then, + y 98. So, when y 0 0 y 0 6. The equation of two lines on the same plane which are intersecting at point (, ) are : + y y. Let father s present age years Son s present age y years After years father s age will be ( + ) years After years son s age will be (y + ) years According to the question, + (y + ) + y + 0 y 0 WORKSHEET-0. Let the number of goats & hens in herd are & y respectively. then, + y 0. 6y. Given equation : y The point (0, 0) satisfies the given equation, hence answer is yes.. No. ( 0 0). According to question, 00 + y y Put and y, then y () () So, (, ) is the solution of the equation. Again put and y, then, y. 8. The line passing through (, ) is y or, y Infinitely many lines are there. The equation in the form a + by + c 0 is y y y + 0 Putting 0, y + 0. \ (0, ) is not the solution of given equation. Putting, y , which is correct \ (, 9) is a solution of the given equation. 0. Let larger number be, then times of larger number and smaller number be y Quotient and remainder 9 So, according to the question, y + 9 y 9 0. y ( ) ( ) So, (, ) is not a solution of the given equation.. (, ) lies on the graph y + k 0 \ ( ) + k 0 k + k 8. Given, A(, ) and C(, ) Then, (, ) and D(, ) Also, equations of sides of square are, A : Y C : X CD : Y DA : X P-6 M A T H E M A T I C S IX
27 9. (, ) lies on the graph of ay + \ ( ) a ( ) + a + a 0 a. 0. (a) when y, then + (b) when, then () + y y 8 (c) when, then + y y \ One more solution is (, ). WORKSHEET-. Sol y. + y 0. As the line intersects y-ais, put 0 in the given equation, we get (0) + y \ y 6 The required point is (0, 6).. Given, + \ ( ) \. + y + k 0 If, and y is the solution of the linear equation + y + k 0 then () + () + k 0 k. 6. y 8 y 8 For y 8 8 \ (, ) lies on the line.. Equation y or y \ y On -ais y 0 \ 0 6 At point (6, 0) the given line cuts the -ais. On y-ais 0 \ y 0 y At point (0, ) the given line cuts the y-ais. 8. The linear equation is + ky 8 At, y, () + k() 8 + k 8 \ k If, then () + y y 8 y 0 \ y 0 9. The equation is y 9 A (, ); 9() ; True (, 6); 6 9 ( ) 9 6; True C (0, ); 9 (0) 0 ; True. WORKSHEET-. Given point lies on the line i.e., () a() + a a... (0, ) is the solution of given equation \ it satisfies the equation S O L U T I O N S \ (0) + () k \ k 6. y 6. (, ). 6. If point (, ) lies on y a + \ a + a a. P-
28 . Equation : + y For intersection with -ais y 0 \ or \ Co-ordinates are (, 0) For intersection with y-ais or y 0 \ y \ Co-ordinates are (0, ). 8. Substituting and y in 9k + ky 6, we get 9k ( ) + k ( ) 6 9k k 6 k 6 k 9. Writing in standard form On -ais, y 0 + y 0 0 \ Point on the -ais (, 0) On y-ais, 0 y 0 y 6 \ Point on the y-ais (0, 6). 0. Given equation is (p + ) (p + )y 0...(i) If, y is the solution of the equation (i), then (p + ) (p + ) 0 p + 6p 9 0 p 8 0 p Put the value of p in the equation (i), then + y 0 or y + 0, is the required equation. TOPIC- Graphical Representation of Linear Equation in Two Variable WORKSHEET-. y k.. k.. y-ais. Given y + y At y-ais 0; \ (0) y 0 y \ y Given line intersects the y-ais at y.. Given equation + y Put 0, then 0 + y y Hence point on y-ais is (0, ) Now put y 0, then + 0 Hence point on -ais is (, 0) y O (0, ) (, 0) 6. Let Sita contribute ` and Gita contribute ` y. According to the question, + y y y 00 P-8 M A T H E M A T I C S IX
29 (0, 00) y y The points are (0, ) & (, ) y (0, ) (, ) y (00, 00) y'. (i) One Variable (00, 0) y 0 (ii) Two Variables 0 + y y 0 y ' y' WORKSHEET-. a > 0.. (0, ). Parallel, 6. Let the cost of a toy elephant, ball y \ y y 6 9 y P O 0 (i) Equation in one variable (Number line) : A point P at a distance of units to left of O on the number line. (ii) In two variables (Cartesian plane) : A line A parallel to y-ais at a distance of units to the left of y-ais.. y 0 Þ y 6 9 y 0 S O L U T I O N S P-9
30 WORKSHEET-., y 0 i.e., (, 0). (, ) O X. + 6 y ais From graph it is clear that line meets -ais at (, 0) and y-ais at (0, ).. + y 8 y y From the graph it is clear that Line intersects -ais at (, 0) and y-ais at (0, 8). -ais Plot on the cartesian plane. On cartesian plane, is a line parallel to y-ais.. y + ; three solutions are, y 0;, y and 0, y. WORKSHEET-6. y + 0. The given equation can be written as, + 6y 0 y ( + 0)/6 0 y 0 0 P-0 M A T H E M A T I C S IX
31 . 8 6 (i) Point P (, 0) represents the solution on the number line p 0 (ii) Line A represents the solution in the cartesian plane. y. Equation of the sides are, A : Y 0 C : X CD : Y DA : X Area sq. units ' (, 0) 0 y' S O L U T I O N S P-
32 SECTION CHAPTER COORDINATE GEOMETRY TOPIC- Cartesian System WORKSHEET-. Origin. P (, ) and Q (, ) lie in IV and II quadrants.. (a, b) (0, ) a 0, b., 0,,. [CSE Marking Scheme, 0]. The point on y-ais has -co-ordinate 0. Since it lies at a distance of units in the negative direction of y-ais. The point is (0, ). 6. (A) II quadrant () III quadrant (C) IV quadrant (D) I quadrant.. (A) IV quadrant () II quadrant (C) III quadrant (D) I quadrant 8. C(, ), distance from ais distance from y ais D(, ), distance from ais distance from y ais WORKSHEET-8. Since P(a, b) lies in IV quadrant \ a > 0, and b < 0 \ a > b [CSE Marking Scheme, 06]. In II quadrant, < 0 Points (, 0), (, 0). II Quadrant.. (A) II quadrant () III quadrant (C) I quadrant (D) II quadrant.. (A) P(0, ) () Q(0, ) (C) R(, 0) (D) S(, 0) 9. In a point (, ), < 0 and y > 0 Point (, ) lies in II quadrant. In a point (, ), > 0 and y < 0 Point (, ) lies in IV quadrant In a point (, 0), > 0 and y 0 Point (, 0) lies on -ais In a point (6, 6), > 0 and y > 0 Point (6, 6) lies in I quadrant In a point (, ), < 0 and y < 0 Point (, ) lies in III quadrant 0. (i) E(, ) (ii) D(, ) (iii) Co-ordinates of A (, ) Co-ordinates of (, ) The abscissa of A abscissa of (iv) Co-ordinates of C (, ) Co-ordinates of F (, ) The ordinate of C + ordinate of F + ( ) ' y 6 P(0, ) (, 0) R(, 0) O 6 Q(0, ) y' 6. (i) A point which lies on and y-aes is (0, 0) i.e., origin (ii) A point whose abscissa is and ordinate is 6 i.e., and y 6 is (, 6) (iii) A point whose ordinate is 6 i.e., y 6 and lies on y-ais is (0, 6) (iv) A point whose ordinate is and abscissa is i.e., y and is (, ) P- M A T H E M A T I C S IX
33 (v) A point whose abscissa is i.e., and lies on -ais is (, 0) (vi) A point whose abscissa is and ordinate is i.e., and y is (, ). (i) Required points are A(0, ) and L(0, ). (ii) Required points are G(, 0) and I(, 0). (iii) Required points are D(, ) and H(, ). WORKSHEET-9. The distance of a point from the y-ais is called its -co-ordinate, or abscissa.. The P is on -ais y 0 P is at a distance of units from y-ais to its left. In second quadrant, the co-ordinates of the point P (, 0).. (, y) (y, ).. The co-ordinate on -ais (, 0) 6. Co-ordinates of A (, 0) Co-ordinates of (, ) Co-ordinates of C (, ) Co-ordinates of D (0, ).. (i) Since the point lies on -ais at a distance of 9 units from y-ais. Hence its co-ordinates are (9, 0) (ii) According to the question, the required coodinates are (0, 9). 8. The vertices of the rectangle OAC are O(0, 0), A( 6, 0), ( 6, ), C(0, ) 9. (i) (, ) lies in III quadrant, as < 0 and y < 0. (ii) (, ) lies in IV quadrant, as > 0 and y < 0. (iii) (, ) lies in II quadrant, as < 0 and y > Verification : The points (, ), (, ) and (, ) are plotted as shown in figure : Result is verified : (i) (, ) lies in III quadrant. (ii) (, ) lies in IV quadrant. (iii) (, ) lies in II quadrant. Y X' (, ) (II) O Y' (I) (III) (IV) (, ) (, ) X TOPIC- Plotting a Point in a Plane WORKSHEET-0. The co-ordinate of point Q (, ). on ais C, F, G on y ais A, D, E [CSE Marking Scheme, 06]. For plotting a point A(, ), we will take a distance of units in the negative direction of y-ais and a distance of units in the positive direction of -ais, which is shown in the figure given below. Similarly, we plot all the points i.e., ( 6, 0), C(, ) and D(, ). D(, ) ( 6, 0) X' X 6 O 6 A(, ) C(, ) Y' S O L U T I O N S Y. From the graph it is clear that the verte of C(, ) Y (, ) 6 A(, ) X 6 6 X C(, ) 6 D(, ) Y [CSE Marking Scheme, 06] P-
34 . Y D(0, ) 6 C(, ) X A(0, 0) (, 0) X Y [CSE Marking Scheme, 06] 6. (i) Plotting of points M(, ) and N(, ) on the graph paper is shown in the diagram. (ii) Length of MN + 8 units (iii) From figure, A(, ), (, ), C(, ) + + y ' 6 N(, )C 6 8 A M(, ) y' WORKSHEET-. Plotting of P(, 6) y ' 6 M (, 6) N 6 P Co-ordinates of M(, 0) Co-ordinates of N(0, 6). y'. Middle point of line segment A is (0, 0) Y A(, ) X X (, ) O(0, 0). Plotted graph shows that P, Q and R are collinear. Y K(, ) X Q(0, ) P(, 0) X Y [CSE Marking Scheme, 06]. Point of intersection of diagonals is (, ) Y 6 P(, ) S(, ) A(, ) X Q(, ) R(, ) X Y [CSE Marking Scheme, 06] Y [CSE Marking Scheme, 06] P- M A T H E M A T I C S IX
35 . Y (, )C D (, ) A(, ) X' O X 6. Plot the three vertices of the rectangle as A(, ), (, ), C(, ). Y' We have to find the co-ordinates of the fourth verte D so that ACD is a rectangle. Since the opposite sides of a rectangle are equal, so the abscissa of D should be equal to abscissa of A, i.e., and the ordinate of D should be equal to the ordinate of C, i.e.,. Co-ordinates of D are given by (, ). WORKSHEET-. Drawing the points, A(, ), (, ), C(, ) and D(, ). Joining A, C, CA, AD, we get a quadrilateral ACD. y. (,) 6 C ' 6 y' 6 (,) A (,) 6 8 (, ) D After plotting these two points A line segment is formed. + + Mid-point, (, ). The points A(, ), (, 0), C(, ) and D(, 0) are plotted as shown below : (II) C (, ) Y (, 0) D (, 0) X' X 0 (I) X' Y (,) 0 0 A(,) 0 0 O 0 0 X 0 0 Y' (III) (IV) A (, ) Y' (i) A(, ) lies in III quadrant. (ii) (, 0) lies on -ais. (iii) C(, ) lies in II quadrant. (iv) D(, 0) lies on -ais S O L U T I O N S P-
36 . y 6. Y D(, ) C(, ) A( p, 0) O (0, 0) X' ( p) ( q) X ' A(, ) (, ) y' From figure length of line segment A is 6 units.. 6 (,) (,) (, ) (0,.) 6 (6, ) 6 (, ) 6 ( p, q) Y' C (0, q) From figure the co-ordinate of vertices of a rectangle are : TOPIC- Graph of a Linear Equation O(0, 0) A( p, 0) ( p, q) C(0, q) WORKSHEET-. -ais.. Given equation + y 0 Y y +. The given equation is : Putting 0, we get y 0 Putting, we get y Putting, we get y Thus, we have the following table : 0 y 0 Now, plot the points R(0, 0), P(, ) and a(, ) on the graph paper. X' a (, ) (0, 0) R P(, ) Y' X P-6 M A T H E M A T I C S IX
37 . OA O 6 units and C AD units. Co-ordinates of A (, 0) Co-ordinates of (, 0) Co-ordinates of C (, ) Co-ordinates of D (, ).. (i) y ' y' No. of Pairs of Shoes Price in Hundred of Rupees (ii) Graph is a straight line. 6. (i) Plot the points A(, ) and (, ). WORKSHEET-. Given equation, + 6y 0... Line intersect the -ais... y (0) 0 S O L U T I O N S So, the point where line intersect the -ais is (, 0).. Given, y + 6 Put 0, we get y Similarly at, y 9 Points (0, 6), (, 9) lies on the line y The given equation is : y + Putting 0, we get y Putting, we get y Putting, we get y Putting, we get y Thus, we have the following table : 0 y Now Let plot the points A(0, ), (, ), C(, ) and D(, ) on the graph paper. X' Joining these points, we get the line segment A. We can take the point C(, ) between the points A and. (ii) We can take the point D(, ) lying on the A produced. Y X' 8 D(, ) 6 (, ) C(, ) O 6 8 A(, ) 6 Y' Y D(, ) 6 C(, ) (, ) (0, )A 6 X Y'. The given equation is: y Putting 0, we get : y ( ) Putting, we get y Putting, we get y Putting, we get y Thus, we have the following table 0 y Now plot the points A(0, ), (, ), C(, ) and D(, ) on the graph paper. Join A,, C, D and etend it in both the directions. Then, line AD is the graph of the equation y. X P-
38 ' y D(, ) 6 C(, ) (, ) 0 6 A(0, ) y'. (i) Draw X OX and Y OY as the co-ordinate aes and mark their point of intersection O as the origin (0, 0). In order to plot the points (, 8), we take units on OX and then 8 units parallel to OY to obtain the point A(, 8). Similarly, we plot the point (, ). In order to plot (0,.), we take. units below the -ais on the y-ais to obtain C(0,.). In order to plot (, ), we take unit on OX and then units parallel to OY to obtain the point D(, ) Y A(, 8) 8 (, ) 6 D(, ) X' X C E(, ) (0,.) Y' In order to plot (, ), we take units on OX and then unit below -ais parallel to OY to obtain the point E(, ) (ii) Co-ordinate geometry. (iii) Co-ordination among people is good for progress. P-8 M A T H E M A T I C S IX
39 SECTION CHAPTER INTRODUCTION TO EUCLID S GEOMETRY TOPIC- Euclid s Geometry WORKSHEET-. Theorem requires a proof.. Euclid s aioms () Things which are equal to the same thing are equal to one another. () If equals are added to equals, the wholes are equal. + S O L U T I O N S [CSE Marking Scheme, 0, 0] If equals are added to equals, the wholes are equal. [CSE Marking Scheme, 0]. Given, and Again, Aiom : Things which are equal to the same or equal things are equal to one another... AC DC (Given) C CE Adding, AC + C DC + CE A DE If equals are added to equals, the wholes are equal. WORKSHEET-6. Let, First thing [CSE Marking Scheme, 0] Second thing y then, y. D A C 6. Here, and and. Euclid s first aiom says, the things which are equal to same things are equal to one another. So,.. Since and, therefore adding both equation + + Þ AD CD Þ A C If equals are added to equal, the wholes are equal Two more aioms : Things which are equal to the same thing are equal to one another e.g., if A PQ and PQ XY, then A XY If equals are subtracted from equals, the remainders are equal. e.g., if m m then m m m m [CSE Marking Scheme, 06] A AD AC AD A AC Things which are equal to the same thing are equal to one another. [CSE Marking Scheme 0]. Let kg be the weight of Ram and Ravi each. On gaining kg, weight of Ram and Ravi will be ( + ) kg each. According to Euclid s second aiom, when equals are added to equals, the wholes are equal. So, weights of Ram and Ravi are again equal. P-9
40 . + 0 Þ If equals are subtracted from equals, the remainder are equal.. Here PQ QR P X [CSE Marking Scheme, 06] Q QX QY Y If equals are subtracted from equals, the remainders are also equal We have R PQ QX QR QY Q PX RY [CSE Marking Scheme, 06] 6. A C (given) X Y (given) If equals are subtracted from equals, then remains are also equal. A X C Y AX CY. Here OX XY Also Þ PX XZ OX PX XY XZ XY XZ Things equal to half of equals, are equal to one another. Two other aioms : Things coincide with one another are equal to one another. e.g., if A coincide with XY, such that A falls on X and falls on Y, then A XY The whole is greater than the part e.g., if mð mð + mð, then mð > mð and mð > mð. [CSE Marking Scheme, 06] WORKSHEET-. A system of aioms is called consistent, when it is impossible to deduce from these aioms, a statement that contradicts any aiom or previously proved statement. [CSE Marking Scheme, 0]. Since, A C AX + X Y + CY Since, X Y AX + X X Y + CY Y \ AX CY Aiom : If equals are subtracted from the equals, the remainders are equal. [CSE Marking Scheme, 0]. In a circle having centre at P, we have PR PQ radius In a circle having centre at Q, we have QR QP radius Euclid s first aiom : Things which are equal to the same thing are equal to one another. PR PQ QR.....() Adding () + (), we get...() + + AC DC Euclid s aiom used : If equals are added to equals, wholes are equal.. Given, AC AC [CSE Marking Scheme, 0] (As, ). P-0 M A T H E M A T I C S IX
41 TOPIC- Euclid s Postulates WORKSHEET-8. Only one line passes through two distinct points. A. Two planes intersect each other to form a straight line.. (i) Infinite, if they are collinear. (ii) Only one, if they are non-collinear. +. Let A be perpendicular to a line l and AP is any other line segment. In right AP, > P, ( 90 ) AP > A or A < AP.. Playfair s Aiom (statement) : For every line l and for every point P not lying on l, there eists a unique line m passing through P and parallel to l. It is equivalent to Euclid s fifth postulate. 6. A C D AC D (given) A + C C + CD A CD. [CSE Marking Scheme 0, 0]. Concermed, Caring. Things equal to same things are equal to one another. Rehman contributed ` 00. All right angles are, equal to one another (OR) Any postulate of Euclid can be stated. 8. Polygon : A simple closed figure made up of three or more line segments. Line Segment : Part of a line with two end points. Angle : A figure formed by two rays with a common initial point Right angle : Angle whose mcahre in 90 [CSE Marking Scheme, 06] WORKSHEET-9. Meeting place of two walls.. A surface is that which has length and breadth. y. A AE S O L U T I O N S Surface breadth ( y) length ( ) (E is the mid-point of A) CD DF (F is the mid-point of CD) Also, AE DF (Given) Therefore, A CD (Things which are double of the same thing are equal to one another). (i) Since it is true for things in any part of universe so this is a universal truth. (ii) If the sum of the cointerior angles made by a transversal intersect two straight lines at distinct points is less than 80, then the lines cannot be parallel.. (i) False : ecause infinitely many lines can pass through a single point. (ii) False : ecause only one line can pass through two distinct points. (iii) True : Two circles are equal if : (a) their circumferences are equal, or (b) their radii are equal. (iv) True : Things which are equal to the same thing are equal to one another (Euclid s aioms). 6. Their sales in July will also be equal as things which are double of the same things are equal to one another. Two other aioms are : (i) The whole is greater than the Part. (ii) Things which are halves of the same thing are equal to one another. P-
42 WORKSHEET-0. Dimension of Surface Length and readth (which is ).. Lines are parallel if they do not intersect on being etended. For eample : A or A Lines A and are parallel lines.. There are two undefined terms, line and point. They are consistent, because they deal with two different situations. (i) Says that given two points be A and, there is a point C, lying on the line which is in between them. (ii) Says that given A and, we can take C not lying on the line passing through A and. These Postulates do not follow from Euclid s postulates. However, (ii) follow from given postulate (i).. Euclid s aiom : If C be the mid-point of a line segment A, then AC A. AC A and AD AC AD A AD A. [CSE Marking Scheme, 0]. Given, AC C A C So, AC + AC AC + C (Equals are added to equals) AC A, ( AC + C coincides with A) AC A. 6. Here, OX XY, PX XZ XY (OX), XZ (PX) Also, OX PX, (Given) XY XZ, (ecause things which are double of the same things are equal to one another.). (i) If a straight line l falls on two straight lines m and n such that the sum of the interior angles on one side of l is equals to two right angles then by Euclid s fifth postulate the lines will not meet on this side of l. Net, we know that the sum of the interior angles on the other side of line l will also be two right angles. Therefore, they will not meet on the other side also. So, the lines m and n never meet and are therefore, parallel. (ii) Introduction to Euclid s Geometry. (iii) Universal truth. P- M A T H E M A T I C S IX
43 SECTION CHAPTER 6 LINES AND ANGLES TOPIC- Different Types of Angles WORKSHEET ( Straight line makes an angle of 80 ) Thus, COD Let the angle be, then Angle Complement of 90.. ( ) + ( + ) 90 Angles are 60 and 0.. Let the two supplementary angles are and, then Hence, the angles are and or and 08.. ÐPOR + ÐROQ [CSE Marking Scheme, 0] 6 \ ÐPOR, ÐROQ 08 [CSE Marking Scheme, 0, 0] 6. Let normals at A and meet at P. m D C As mirrors are perpendicular to each other there fore P OA and AP O. So P ^ PA, i.e, ÐPA 90 Therefore P Ð + Ð 90 O A...(i) (Angle sum property) Also Ð Ð and Ð Ð (Angle of incidence Angle of reflection) Therefore Ð + Ð 90...(ii) [form (i)] Adding () and (), we have Ð + Ð + Ð + Ð 80 i.e. ÐCA + ÐDA 80 Hence CA D [CSE Marking Scheme, 06] WORKSHEET-. Angles (0 a) and ( + a) are supplementary of each other, then 0 a + + a 80 a 80.. The complement of (90 a) 90 (90 a) (As sum of complementary angles is 90 ) a a.. Let the angle be, then y given condition, (90 ) Let, a and b Then, a + b a 8 6 S O L U T I O N S P-
44 b 8 c 80 b [CSE Marking Scheme, 0]. POC y (Vertically opp. angles) y + y + y 80 y 80 y. [CSE Marking Scheme, 0] 6. ÐCOF (vertically opposite angles) \ Þ 0 80 Þ 8 ÐAOC ÐOF 90 ÐDOE 6 [CSE Marking Scheme, 06] WORKSHEET-. Angle ( + a) and ( a) are supplement of each other, then + a + a 80 a Here, + 80 ( Straight line makes an angle of 80 ) Here, 0 + y + y 80 ( Straight line makes an angle of 80 ) y 0 y 8.. a + b 80 (Linear pair) a b 80 (Given) Adding, a 60 a 0 and b 80 a [CSE Marking Scheme, 0, 00, 0]. Let, AO and OC AO + OC AOC + AO 0 and OC. [CSE Marking Scheme, 0] 6. b + + b 80 [Linear pair] Þ b 0 Þ b \ b 8 [Vertically opposite anlge] a a 8 \ c 80 a c 80 8 c 96 c 8 [CSE Marking Scheme, 06] WORKSHEET Þ 0 So the angles are So two angles are 0 and 0. [CSE Marking Scheme, 06] q 0. (Vertically opposite angles). E C O Etend AO to E, AO + OE 80 (Linear pair)...() EOC + COD + DOA 80 (Adjacent angles)...() D A P- M A T H E M A T I C S IX
45 Adding () and (), we get AO + OE + EOC + COD + DOA AO + OC + COD + DOA 60. [CSE Marking Scheme, 0]. A C 8 O y z D. Given, COA 8 CO + OA 8 CO + 8 ( OA ) CO 8 Similarily, DO DOC + CO 8 DOC + DOC 8 0. [CSE Marking Scheme, 0] TOPIC- Transversal Line E (linear pair) 96 8 y + (VOA) y 8 96 z 8 (VOA) WORKSHEET-. m and n lines are parallel [CSE Marking Scheme, 06] z 0 (V.O.A.) Pair of alternate interior angles. [CSE Marking Scheme, 06]. of a right angle 90 0 (Sum of supplementary angles is 80 ) Supplement of [CSE Marking Scheme, 0] 6. Q P 6 S R A C l m. 0 (V.O.A.) y (Linear pair) y 0 y 0 ut and y are alternate angles. l m. [CSE Marking Scheme, 0, 00, 0] S O L U T I O N S (Corresponding interior angles) y 80 ( 0 ) 80 (0 0 ) y T Given : l m line t is a transversal intersecting them at P and Q rerpectively. To prove : PR QS Proof : Þ Ð Ð6 (Corresponding angles and l m) Ð Ð6 Þ Ð Ð \ PR QS [CSE Marking Scheme, 06] P-
46 WORKSHEET-6. y + y (Corresponding interior angles) (Vertically opposite angles) 0. ÐDS ÐCQ [Conresponding angles] ÐRD 80 ÐDS 80 [CSE Marking Scheme, 06]. 60 (corr S) + 80 WORKSHEET ( 60 ) Let, a, b Then, a h + a 80 (Corresponding eterior angles) h Let, l A CD (by construction) A O 0 I F 0 E C D O' then, OEF + OE 80 O EF + DOE 80 [Corresponding interior angles] OEF + O EF OEO z z + (Alternate interior angles) z + z DNM z A DC CD AD [alternate angles] + y AD y [angle sum property] DC z 80 [ + ] 0 [CSE Marking Scheme, 0] Alternative Method : A DC y (Corresponding interior angles) y 80 y 6 AD CD (Alternate angles) In CD, + y 0 + z 80 z 80 y z 6 z 0. CNM 80 DNM. ÐEFD 60 Þ + y 60 Þ y (Linear pair) 80. [CSE Marking Scheme, 0] Þ ÐEF + ÐEFD 80 [Sum of co-interior angles on same side of transversal is supphymentry] Þ ÐEF Þ ÐEF 0 Þ ÐPEF Þ ÐPEF 80 Now in DPEF Þ [Angle sum property] Þ [CSE Marking Scheme, 06] P-6 M A T H E M A T I C S IX
47 WORKSHEET-8. PQS + QSF 80 (Angles on the same side of transversal) PQS + RFE 80, as QSF EFR 60 + RFE 80 (Corresponding S) RFE 0.. Let, k, y k Then, + y k + k 80 (Angles on the same side of transversal) k 80 k 6 k 08 y k Thus, z 08. (Alternate interior angles). Ða (Linear pair) Ðb 0 (Since Vertically opposite angles are equal) This showes that Ða Ðb ut the are alternate interior angles, \ LM XY [CSE Marking Scheme, 06]. Let DAF CFA, FE, EF Since, AE DC D + 80 (Angles on the same side of transversal) (Alternate angles) Again, (Angle on the same side of transversal) (Vertically opp. angles) D C A 0 9 F In EF, + + EF 80 (Angle sum property) EF 80 EF 80 EF. [CSE Marking Scheme, 0] E TOPIC- Angle Sum Property of a Triangle WORKSHEET-9. Let, equal angles are and, then (Eterior angle is the sum of the two opposite interior angles). Eterior angle Sum of opposite two interior angle other angle ACD A + [Eterior angle is the sum of the two interior opposite angles] S O L U T I O N S ACD 0. ÐCD (Eterior Ð of DAC) ÐED + ÐED [CSE Marking Scheme, 06]. In AC, A + + C 80 (Angle sum prop. of D) A + C ( A + C) P-
48 6.. In AOC, A + C + AOC 80 A WORKSHEET-60 AOC 80. [CSE Marking Scheme, 0] C. We know that (Eterior angle is the sum of the two interior opposite angles) 0.. Given, A + 6 Given, + C 0 A C ut, A + + C 80 (Angle sum prop. of D) C 0 [CSE Marking Scheme, 0, 0] Alternative Method : We know that, A + + C 80...(i) (Angle Sum prop. of D) 6 + C 80 C Again by (i), A A 0 Again by (i), (Angle Sum prop. of D).. In AC, A + + C (Angle Sum prop. of D) A First prove the theorem Let A, & C are the s of Then, A + C + A + + C 80 ( sum prop. of ) C [CSE Marking Scheme, 0] (Linear pair) A a + b + (Eterior angle) +. [CSE Marking Scheme, 0] y z C To prove : Sum of all the angles of AC is 80. Construction : Draw a line l parallel to C. Proof : Since l C, we have y (Alternate angles are equal)... (i) Similarly, l C z (Alternate angles are equal)...(ii) l P-8 M A T H E M A T I C S IX
49 Also, sum of angles at a point A on line l is (linear pair) i.e., y + + z 80 (from (i) and (ii)) + y + z 80 A + + C 80 WORKSHEET-6. Since sum of all the eterior angles formed by producing the sides of a polygon is y + z 60.. Given, z + y z + y + y + z 80 (Angle sum property) z + z 80 z 60.. A 60 0 C C 80 ( ) AC is the smallest side Reason : Side opposite to smaller angle is shortest. [CSE Marking Scheme, 0] WORKSHEET-6. In AC, A + + C 80 (y given conditions) A + A + 6 A 80 9 A 80 A 0.. In AC, A + + C 80 Also, in DEF, D + E + F 80 A + + C + D + E + F Hence, k.. APQ 0 (Alternate angles) + y 8 (Eterior angle is the sum of the two opposite interior angles) 0 + y 8 y S O L U T I O N S Sum of all angles of a is Angles are 0, 60 and 0 respectively. Hence proved. [CSE Marking Scheme, 0]. Since A CD AC DCE (Corresponding angles) DCE CEF + CFE 0 + CFE (Eterior angle) CFE. [CSE Marking Scheme, 0]. y (Corr. angles) y PRQ 80 (8 + ) (Linear pair) ( ) (Angle sum property) [CSE Marking Scheme, 0]. A + C A + + C 80 A 80 A 90. [CSE Marking Scheme, 0] Alternative Method : Given, A + C A + + C 80 A + A 80 A 90. Hence, AC is a right angled triangle.. P Q A C P-9
50 Through verte A, draw PAQ C PA AC QAC AC Adding above equalities, (Alternate angles) (Alternate angles) PA + QAC AC + AC Adding AC to both sides, we get PA + QAC + AC AC + AC + AC 80 AC + AC + AC (Linear pair) [CSE Marking Scheme, 0] 6. P D y y C ACD AC CP DCP y Et APC + y ACP + y ACP APC. [CSE Marking Scheme, 0] A P-0 M A T H E M A T I C S IX
51 SECTION CHAPTER TRIANGLES TOPIC- Criteria for Congruence of Triangles WORKSHEET-6. ASA congruence : Two triangles are congruent, if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.. A A A, AE AFC 90 AE ACF, E CF AE AFC (ASA) A AC (c.p.c.t) [CSE Marking Scheme, 0] Alternative method : A C F E D Join AD. In AC and ACD, A AC (Given) D CD (Given) AD AD (Common) y using SSS Congruency Rule, AD ACD AD ACD (y c.p.c.t.). In AD and CD, A CD (Given) AD CD (Given) D D (Common) AD CD (y SAS) AD C. (y c.p.c.t.) [CSE Marking Scheme, 0]. In AE & ACF, A F E C S O L U T I O N S D C In CE and CF, EC FC 90 (Given) E CF (Given) C C (Common) CE CF (y RHS) C AC A (y c.p.c.t.) Similarly, AD AE AC C Therefore, A C AC Thus, AC is an equilateral triangle.. Given : D is the mid-point of side AC To Prove : D AC Const : Draw, DE C Proof : In AC, D is the mid point of AC & ED C y mid point theorem, AE E...(i) Now, ED C (y const) AED AC (Corresponding s) AED 90 AED DE 90...() Now, In ADE & DE, AE E (from eq....()) AED DE (from eq....()) ED ED (Common) ADE DE (y SAS rule) P-
CLASS IX : CHAPTER - 1 NUMBER SYSTEM
MCQ WORKSHEET-I CLASS IX : CHAPTER - 1 NUMBER SYSTEM 1. Rational number 3 40 is equal to: (a) 0.75 (b) 0.1 (c) 0.01 (d) 0.075. A rational number between 3 and 4 is: (a) 3 (b) 4 3 (c) 7 (d) 7 4 3. A rational
More informationClass-IX CBSE Latest Pattern Sample Paper {Mathematics}
Class-IX CBSE Latest Pattern Sample Paper {Mathematics} Term-I Examination (SA I) Time: 3hours Max. Marks: 90 General Instructions (i) All questions are compulsory. (ii) The question paper consists of
More informationCLASS IX MATHS CHAPTER REAL NUMBERS
Previous knowledge question Ques. Define natural numbers? CLASS IX MATHS CHAPTER REAL NUMBERS counting numbers are known as natural numbers. Thus,,3,4,. etc. are natural numbers. Ques. Define whole numbers?
More informationREAL NUMBERS. Rational Numbers P-1 TOPIC-1 SECTION. Solutions S O L U T I O N S CHAPTER. 7. x = 1 7
SETION HPTER REL NUMERS TOPI- Rational Numbers.. 58 0.058 (Decimal point is shifted three places 000 to the left) 5 6 S O L U T I O N S.0.... It is non-repeating, so it is not a rational number. 0.60....
More information5. Introduction to Euclid s Geometry
5. Introduction to Euclid s Geometry Multiple Choice Questions CBSE TREND SETTER PAPER _ 0 EXERCISE 5.. If the point P lies in between M and N and C is mid-point of MP, then : (A) MC + PN = MN (B) MP +
More informationSample Question Paper Mathematics First Term (SA - I) Class IX. Time: 3 to 3 ½ hours
Sample Question Paper Mathematics First Term (SA - I) Class IX Time: 3 to 3 ½ hours M.M.:90 General Instructions (i) All questions are compulsory. (ii) The question paper consists of 34 questions divided
More information9 th CBSE Mega Test - II
9 th CBSE Mega Test - II Time: 3 hours Max. Marks: 90 General Instructions All questions are compulsory. The question paper consists of 34 questions divided into four sections A, B, C and D. Section A
More informationSOLUTIONS SECTION A [1] = 27(27 15)(27 25)(27 14) = 27(12)(2)(13) = cm. = s(s a)(s b)(s c)
1. (A) 1 1 1 11 1 + 6 6 5 30 5 5 5 5 6 = 6 6 SOLUTIONS SECTION A. (B) Let the angles be x and 3x respectively x+3x = 180 o (sum of angles on same side of transversal is 180 o ) x=36 0 So, larger angle=3x
More informationClass : IX(CBSE) Worksheet - 1 Sub : Mathematics Topic : Number system
Class : IX(CBSE) Worksheet - Sub : Mathematics Topic : Number system I. Solve the following:. Insert rational numbers between. Epress 57 65 in the decimal form. 8 and.. Epress. as a fraction in the simplest
More informationPOLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL. A real number α is a root or zero of polynomial f(x) = x + a x + a x +...
POLYNOMIALS ML 5 ZEROS OR ROOTS OF A POLYNOMIAL n n 1 n an n 1 n 1 + 0 A real number α is a root or zero of polynomial f(x) = x + a x + a x +... + a x a, n n an n 1 n 1 0 = if f (α) = 0. i.e. α + a + a
More informationClass IX Chapter 5 Introduction to Euclid's Geometry Maths
Class IX Chapter 5 Introduction to Euclid's Geometry Maths Exercise 5.1 Question 1: Which of the following statements are true and which are false? Give reasons for your answers. (i) Only one line can
More informationANSWERS. CLASS: VIII TERM - 1 SUBJECT: Mathematics. Exercise: 1(A) Exercise: 1(B)
ANSWERS CLASS: VIII TERM - 1 SUBJECT: Mathematics TOPIC: 1. Rational Numbers Exercise: 1(A) 1. Fill in the blanks: (i) -21/24 (ii) -4/7 < -4/11 (iii)16/19 (iv)11/13 and -11/13 (v) 0 2. Answer True or False:
More informationCBSE Class IX Syllabus. Mathematics Class 9 Syllabus
Mathematics Class 9 Syllabus Course Structure First Term Units Unit Marks I Number System 17 II Algebra 25 III Geometry 37 IV Co-ordinate Geometry 6 V Mensuration 5 Total 90 Second Term Units Unit Marks
More informationSOLUTIONS SECTION A SECTION B
SOLUTIONS SECTION A 1. C (1). A (1) 3. B (1) 4. B (1) 5. C (1) 6. B (1) 7. A (1) 8. D (1) SECTION B 9. 3 3 + 7 = 3 3 7 3 3 7 3 3 + 7 6 3 7 = 7 7 6 3 7 3 3 7 0 10 = = 10. To find: (-1)³ + (7)³ + (5)³ Since
More informationQuestion 1: Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0?
Class IX - NCERT Maths Exercise (.) Question : Is zero a rational number? Can you write it in the form p, where p and q are integers and q 0? q Solution : Consider the definition of a rational number.
More informationExercise 5.1: Introduction To Euclid s Geometry
Exercise 5.1: Introduction To Euclid s Geometry Email: info@mywayteaching.com Q1. Which of the following statements are true and which are false? Give reasons for your answers. (i)only one line can pass
More informationTOPIC-1 Rational Numbers
TOPI- Rational Numbers Unit -I : Number System hapter - : Real Numbers Rational Number : number r is called a rational number, if it can be written in the form p/q, where p and q are integers and q 0,
More informationCBSE Sample Paper-05 (Solved) SUMMATIVE ASSESSMENT I MATHEMATICS Class IX. Time allowed: 3 hours Maximum Marks: 90. Section A.
CBSE Sample Paper-05 (Solved) SUMMATIVE ASSESSMENT I MATHEMATICS Class IX Time allowed: hours Maximum Marks: 90 General Instructions: a) All questions are compulsory. b) The question paper consists of
More informationQUESTION BANK ON STRAIGHT LINE AND CIRCLE
QUESTION BANK ON STRAIGHT LINE AND CIRCLE Select the correct alternative : (Only one is correct) Q. If the lines x + y + = 0 ; 4x + y + 4 = 0 and x + αy + β = 0, where α + β =, are concurrent then α =,
More informationMathematics Class 9 Syllabus. Course Structure. I Number System 17 II Algebra 25 III Geometry 37 IV Co-ordinate Geometry 6 V Mensuration 5 Total 90
Mathematics Class 9 Syllabus Course Structure First Term Units Unit Marks I Number System 17 II Algebra 25 III Geometry 37 IV Co-ordinate Geometry 6 V Mensuration 5 Total 90 Second Term Units Unit Marks
More information(D) (A) Q.3 To which of the following circles, the line y x + 3 = 0 is normal at the point ? 2 (A) 2
CIRCLE [STRAIGHT OBJECTIVE TYPE] Q. The line x y + = 0 is tangent to the circle at the point (, 5) and the centre of the circles lies on x y = 4. The radius of the circle is (A) 3 5 (B) 5 3 (C) 5 (D) 5
More informationSUMMATIVE ASSESSMENT I, IX / Class IX
I, 0 SUMMATIVE ASSESSMENT I, 0 0 MATHEMATICS / MATHEMATICS MATHEMATICS CLASS CLASS - IX - IX IX / Class IX MA-0 90 Time allowed : hours Maximum Marks : 90 (i) (ii) 8 6 0 0 (iii) 8 (iv) (v) General Instructions:
More informationSo, eqn. to the bisector containing (-1, 4) is = x + 27y = 0
Q.No. The bisector of the acute angle between the lines x - 4y + 7 = 0 and x + 5y - = 0, is: Option x + y - 9 = 0 Option x + 77y - 0 = 0 Option x - y + 9 = 0 Correct Answer L : x - 4y + 7 = 0 L :-x- 5y
More informationREVISED vide circular No.63 on
Circular no. 63 COURSE STRUCTURE (FIRST TERM) CLASS -IX First Term Marks: 90 REVISED vide circular No.63 on 22.09.2015 UNIT I: NUMBER SYSTEMS 1. REAL NUMBERS (18 Periods) 1. Review of representation of
More informationTERMWISE SYLLABUS SESSION CLASS-IX SUBJECT : MATHEMATICS. Course Structure. Schedule for Periodic Assessments and CASExam. of Session
TERMWISE SYLLABUS SESSION-2018-19 CLASS-IX SUBJECT : MATHEMATICS Course Structure Units Unit Name Marks I NUMBER SYSTEMS 08 II ALGEBRA 17 III COORDINATE GEOMETRY 04 IV GEOMETRY 28 V MENSURATION 13 VI STATISTICS
More informationTRIANGLES CHAPTER 7. (A) Main Concepts and Results. (B) Multiple Choice Questions
CHAPTER 7 TRIANGLES (A) Main Concepts and Results Triangles and their parts, Congruence of triangles, Congruence and correspondence of vertices, Criteria for Congruence of triangles: (i) SAS (ii) ASA (iii)
More informationCOURSE STRUCTURE CLASS IX Maths
COURSE STRUCTURE CLASS IX Maths Units Unit Name Marks I NUMBER SYSTEMS 08 II ALGEBRA 17 III COORDINATE GEOMETRY 04 IV GEOMETRY 28 V MENSURATION 13 VI STATISTICS & PROBABILITY 10 Total 80 UNIT I: NUMBER
More informationDefinitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg ( )
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg (2009-03-26) Logic Rule 0 No unstated assumptions may be used in a proof.
More informationCLASS-IX MATHEMATICS. For. Pre-Foundation Course CAREER POINT
CLASS-IX MATHEMATICS For Pre-Foundation Course CAREER POINT CONTENTS S. No. CHAPTERS PAGE NO. 0. Number System... 0 3 0. Polynomials... 39 53 03. Co-ordinate Geometry... 54 04. Introduction to Euclid's
More informationSTUDY MATERIAL SUBJECT: MATHEMATICS CLASS - IX
STUDY MATERIAL SUBJECT: MATHEMATICS CLASS - IX 2 INDEX PART - I SA - 1 1. Number System 2. Polynomials 3. Coordinate Geometry 4. Introduction to Euclid Geometry 5. Lines and Angles 6. Triangles 7. Heron's
More informationKENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION
KENDRIYA VIDYALAYA SANGATHAN, HYDERABAD REGION SAMPLE PAPER 02 FOR HALF YEARLY EXAM (2017-18) SUBJECT: MATHEMATICS(041) BLUE PRINT FOR HALF YEARLY EXAM: CLASS IX Chapter VSA (1 mark) SA I (2 marks) SA
More informationAbhilasha Classses. Class X (IX to X Moving) Date: MM 150 Mob no (Set-AAA) Sol: Sol: Sol: Sol:
Class X (IX to X Moving) Date: 0-6 MM 0 Mob no.- 97967 Student Name... School.. Roll No... Contact No....... If = y = 8 z and + + =, then the y z value of is (a) 7 6 (c) 7 8 [A] (b) 7 3 (d) none of these
More informationCBSE OSWAAL BOOKS LEARNING MADE SIMPLE. Published by : 1/11, Sahitya Kunj, M.G. Road, Agra , UP (India) Ph.: ,
OSWAAL BOOKS LEARNING MADE SIMPLE CBSE SOLVED PAPER 2018 MATHEMATICS CLASS 9 Published by : OSWAAL BOOKS 1/11, Sahitya Kunj, M.G. Road, Agra - 282002, UP (India) Ph.: 0562 2857671, 2527781 email: contact@oswaalbooks.com
More informationProperties of the Circle
9 Properties of the Circle TERMINOLOGY Arc: Part of a curve, most commonly a portion of the distance around the circumference of a circle Chord: A straight line joining two points on the circumference
More information6 CHAPTER. Triangles. A plane figure bounded by three line segments is called a triangle.
6 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius A plane figure bounded by three line segments is called a triangle We denote a triangle by the symbol In fig ABC has
More informationChapter 7. Geometric Inequalities
4. Let m S, then 3 2 m R. Since the angles are supplementary: 3 2580 4568 542 Therefore, m S 42 and m R 38. Part IV 5. Statements Reasons. ABC is not scalene.. Assumption. 2. ABC has at least 2. Definition
More informationTriangles. 3.In the following fig. AB = AC and BD = DC, then ADC = (A) 60 (B) 120 (C) 90 (D) none 4.In the Fig. given below, find Z.
Triangles 1.Two sides of a triangle are 7 cm and 10 cm. Which of the following length can be the length of the third side? (A) 19 cm. (B) 17 cm. (C) 23 cm. of these. 2.Can 80, 75 and 20 form a triangle?
More informationMATHEMATICS. QUESTION BANK for. Summative Assessment -I CLASS IX CHAPTER WISE COVERAGE IN THE FORM MCQ WORKSHEETS AND PRACTICE QUESTIONSS
MATHEMATICS QUESTION BANK for Summative Assessment -I CLASS IX 014 15 CHAPTER WISE COVERAGE IN THE FORM MCQ WORKSHEETS AND PRACTICE QUESTIONSS Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist
More informationQ.2 A, B and C are points in the xy plane such that A(1, 2) ; B (5, 6) and AC = 3BC. Then. (C) 1 1 or
STRAIGHT LINE [STRAIGHT OBJECTIVE TYPE] Q. A variable rectangle PQRS has its sides parallel to fied directions. Q and S lie respectivel on the lines = a, = a and P lies on the ais. Then the locus of R
More informationADARSHA VIDYALAYA HUNASHYAL P.B
ADARSHA VIDYALAYA HUNASHYAL P.B SUBJECT : MATHEMATICS MATHEMATICS FA 1 SL.N O CONTENTS TOPIC FA 1 PLYING WITH-NUMBERS ARITHMETIC FA - 1 2 SQUARE,SQUARE ROOTS CUBES AND CUBE ROOTS ARITHMETIC FA - 1 3 RATIONAL
More informationTRIPURA BOARD OF SECONDARY EDUCATION. SYLLABUS (effective from 2016) SUBJECT : MATHEMATICS (Class IX)
TRIPURA BOARD OF SECONDARY EDUCATION SYLLABUS (effective from 2016) SUBJECT : MATHEMATICS (Class IX) Total Page- 10 MATHEMATICS COURSE STRUCTURE Class IX HALF YEARLY One Paper Time: 3 Hours Marks: 90 Unit
More informationCOURSE STRUCTURE CLASS -IX
environment, observance of small family norms, removal of social barriers, elimination of gender biases; mathematical softwares. its beautiful structures and patterns, etc. COURSE STRUCTURE CLASS -IX Units
More informationTin Ka Ping Secondary School F.2 Mathematics Teaching Syllabus
Tin Ka Ping Secondary School 05-06 F. Mathematics Syllabus Chapter Rate and Time Guide. Rates. s A. Basic Concept of s B. s of Three Quantities Learn the concept of a rate. Learn the concepts of a ratio
More informationPaper: 02 Class-X-Math: Summative Assessment - I
1 P a g e Paper: 02 Class-X-Math: Summative Assessment - I Total marks of the paper: 90 Total time of the paper: 3.5 hrs Questions: 1] The relation connecting the measures of central tendencies is [Marks:1]
More information(1) then x y z 3xyz (1/2) (1/2) 8. Given, diameter of the pillar, d 50 cm m (1/2)
Sample Question Paper (Detailed Solutions) Mathematics lass th. p( x) x 6x. Let the angle x Then, supplement angle 80 x and complement angle ccording to the question, Supplement angle 0 x omplement angles
More information3. AD is a diameter of a circle and AB is a chord. If AD = 34 cm, AB = 30 cm, the distance of AB from the centre of the circle is:
Solved Paper 2 Class 9 th, Mathematics, SA 2 Time: 3hours Max. Marks 90 General Instructions 1. All questions are compulsory. 2. Draw neat labeled diagram wherever necessary to explain your answer. 3.
More informationSUMMATIVE ASSESSMENT-1 SAMPLE PAPER (SET-2) MATHEMATICS CLASS IX
SUMMATIVE ASSESSMENT-1 SAMPLE PAPER (SET-) MATHEMATICS CLASS IX Time: 3 to 3 1 hours Maximum Marks: 80 GENERAL INSTRUCTIONS: 1. All questions are compulsory.. The question paper is divided into four sections
More informationchapter 1 vector geometry solutions V Consider the parallelogram shown alongside. Which of the following statements are true?
chapter vector geometry solutions V. Exercise A. For the shape shown, find a single vector which is equal to a)!!! " AB + BC AC b)! AD!!! " + DB AB c)! AC + CD AD d)! BC + CD!!! " + DA BA e) CD!!! " "
More informationChapter (Circle) * Circle - circle is locus of such points which are at equidistant from a fixed point in
Chapter - 10 (Circle) Key Concept * Circle - circle is locus of such points which are at equidistant from a fixed point in a plane. * Concentric circle - Circle having same centre called concentric circle.
More informationPart (1) Second : Trigonometry. Tan
Part (1) Second : Trigonometry (1) Complete the following table : The angle Ratio 42 12 \ Sin 0.3214 Cas 0.5321 Tan 2.0625 (2) Complete the following : 1) 46 36 \ 24 \\ =. In degrees. 2) 44.125 = in degrees,
More informationTriangle Congruence and Similarity Review. Show all work for full credit. 5. In the drawing, what is the measure of angle y?
Triangle Congruence and Similarity Review Score Name: Date: Show all work for full credit. 1. In a plane, lines that never meet are called. 5. In the drawing, what is the measure of angle y? A. parallel
More information(1/2) a a (1/2) 6. Area of ABC will be 127cm because two congruent (1/2) 8. Given, the an gles of a tri an gle are 5( y 1) 180 x x (1/2) (1/2)
Sample Question Paper (etailed Solutions) Mathematics lass th. Given, a and b b a ( a b ) ( ) (/) ( 8 ) ( ). In the given figure, AB E EBA EBA 0 a a (/) [alternate interior angles] In ABE, EBA EAB AEB
More informationDESIGN OF THE QUESTION PAPER
SET-II DESIGN OF THE QUESTION PAPER MATHEMATICS CLASS IX Time : 3 Hours Maximum Marks : 80 The weightage or the distribution of marks over different dimensions of the question paper shall be as follows:
More informationCBSE Class IX Mathematics Term 1. Time: 3 hours Total Marks: 90. Section A
CBSE sample papers, Question papers, Notes for Class 6 to 1 CBSE Class IX Mathematics Term 1 Time: 3 hours Total Marks: 90 General Instructions: 1. All questions are compulsory.. The question paper consists
More informationCHAPTER 7 TRIANGLES. 7.1 Introduction. 7.2 Congruence of Triangles
CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a
More informationNumber Systems. Exercise 1.1. Question 1. Is zero a rational number? Can you write it in the form p q,
s Exercise. Question. Is zero a rational number? Can you write it in the form p q, where p and q are integers and q 0? Solution Yes, write 0 (where 0 and are integers and q which is not equal to zero).
More informationCBSE CLASS X MATH -SOLUTION Therefore, 0.6, 0.25 and 0.3 are greater than or equal to 0 and less than or equal to 1.
CBSE CLASS X MATH -SOLUTION 011 Q1 The probability of an event is always greater than or equal to zero and less than or equal to one. Here, 3 5 = 0.6 5% = 5 100 = 0.5 Therefore, 0.6, 0.5 and 0.3 are greater
More informationKENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32
KENDRIYA VIDYALAYA GACHIBOWLI, GPRA CAMPUS, HYD 32 SAMPLE PAPER 02 FOR PERIODIC TEST II EXAM (2018-19) SUBJECT: MATHEMATICS(041) BLUE PRINT FOR PERIODIC TEST II EXAM: CLASS IX Chapter VSA (1 mark) SA I
More informationYes zero is a rational number as it can be represented in the
1 REAL NUMBERS EXERCISE 1.1 Q: 1 Is zero a rational number? Can you write it in the form 0?, where p and q are integers and q Yes zero is a rational number as it can be represented in the form, where p
More information1 / 23
CBSE-XII-017 EXAMINATION CBSE-X-008 EXAMINATION MATHEMATICS Series: RLH/ Paper & Solution Code: 30//1 Time: 3 Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question
More informationCONGRUENCE OF TRIANGLES
Congruence of Triangles 11 CONGRUENCE OF TRIANGLES You might have observed that leaves of different trees have different shapes, but leaves of the same tree have almost the same shape. Although they may
More informationSimilarity of Triangle
Similarity of Triangle 95 17 Similarity of Triangle 17.1 INTRODUCTION Looking around you will see many objects which are of the same shape but of same or different sizes. For examples, leaves of a tree
More informationINTERNATIONAL INDIAN SCHOOL, RIYADH. 11cm. Find the surface area of the cuboid (240cm 2 )
INTERNATIONAL INDIAN SCHOOL, RIYADH CLASS: IX SUBJECT: MATHEMATICS 1. SURFACE AREAS AND VOLUMES 1. The diagonal of a cube is 12cm. Find its volume. 2. If the lateral surface area of a cube is 1600cm 2,
More informationProofs. by Bill Hanlon
Proofs by Bill Hanlon Future Reference To prove congruence, it is important that you remember not only your congruence theorems, but know your parallel line theorems, and theorems concerning triangles.
More informationFill in the blanks Chapter 10 Circles Exercise 10.1 Question 1: (i) The centre of a circle lies in of the circle. (exterior/ interior) (ii) A point, whose distance from the centre of a circle is greater
More informationRAO IIT ACADEMY / GANIT PRABHUTWA EXAMINATION / STD - VIII / QP + SOLUTIONS JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS GANIT PRABHUTWA EXAMINATION
JEE MEDICAL-UG BOARDS KVPY NTSE OLYMPIADS GANIT PRABHUTWA EXAMINATION Date : 0 - - 07 Std - VIII Total Marks : 00 Time : Hours Q. A) Choose the correct alternative and write it against the correct sub
More informationCCE RR. ( / English Version ) ( / New Syllabus ) ( / Regular Repeater )
CCE RR 1 81-E : 81-E Code No. : 81-E CCE RR Subject : MATHEMATICS ( / English Version ) ( / New Syllabus ) ( / Regular Repeater ) General Instructions : i) The Question-cum-Answer Booklet consists of objective
More informationANSWER KEY & SOLUTIONS
PRE-HALFYEARLY ASSESSMENT- [P-H-A MATHS SYLLABUS] ANSWER KEY & SOLUTIONS General Instructions:. The question paper comprises of four sections, A, B, C & D.. All questions are compulsory. 3. Section A Q
More informationLabel carefully each of the following:
Label carefully each of the following: Circle Geometry labelling activity radius arc diameter centre chord sector major segment tangent circumference minor segment Board of Studies 1 These are the terms
More informationMATHEMATICS. IMPORTANT FORMULAE AND CONCEPTS for. Summative Assessment -II. Revision CLASS X Prepared by
MATHEMATICS IMPORTANT FORMULAE AND CONCEPTS for Summative Assessment -II Revision CLASS X 06 7 Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya Vidyalaya GaCHiBOWli
More information(Question paper - With Answers) STD. X - MATHEMATICS. [Time Allowed : 2½ Hrs.] [Maximum Marks : 100]
GOVT SUPPLEMENTARY EXAM OCTOER - 06 (Question paper - With Answers) STD. X - MATHEMATICS [Time Allowed : ½ Hrs.] [Maimum Marks : 00] SECTION - I 8. The equation of a straight line passing through the point
More information1 / 23
CBSE-XII-07 EXAMINATION CBSE-X-009 EXAMINATION MATHEMATICS Series: HRL Paper & Solution Code: 0/ Time: Hrs. Max. Marks: 80 General Instuctions : (i) All questions are compulsory. (ii) The question paper
More informationMATHEMATICS QUESTION BANK. for CLASS IX CHAPTER WISE COVERAGE IN THE FORM MCQ WORKSHEETS AND PRACTICE QUESTIONSS.
MATHEMATICS QUESTION BANK for CLASS IX 2017 18 CHAPTER WISE COVERAGE IN THE FORM MCQ WORKSHEETS AND PRACTICE QUESTIONSS Prepared by M. S. KUMARSWAMY, TGT(MATHS) M. Sc. Gold Medallist (Elect.), B. Ed. Kendriya
More informationMATHEMATICS (IX-X) (CODE NO. 041) Session
MATHEMATICS (IX-X) (CODE NO. 041) Session 2018-19 The Syllabus in the subject of Mathematics has undergone changes from time to time in accordance with growth of the subject and emerging needs of the society.
More informationSUMMATIVE ASSESSMENT II SAMPLE PAPER I MATHEMATICS
SUMMATIVE ASSESSMENT II SAMPLE PAPER I MATHEMATICS Class: IX Time: 3-3 ½ hours M.Marks:80 General Instructions: 1. All questions are compulsory 2. The question paper consists of 34 questions divided into
More information( )( ) PR PQ = QR. Mathematics Class X TOPPER SAMPLE PAPER-1 SOLUTIONS. HCF x LCM = Product of the 2 numbers 126 x LCM = 252 x 378
Mathematics Class X TOPPER SAMPLE PAPER- SOLUTIONS Ans HCF x LCM Product of the numbers 6 x LCM 5 x 378 LCM 756 ( Mark) Ans The zeroes are, 4 p( x) x + x 4 x 3x 4 ( Mark) Ans3 For intersecting lines: a
More informationIt is known that the length of the tangents drawn from an external point to a circle is equal.
CBSE -MATHS-SET 1-2014 Q1. The first three terms of an AP are 3y-1, 3y+5 and 5y+1, respectively. We need to find the value of y. We know that if a, b and c are in AP, then: b a = c b 2b = a + c 2 (3y+5)
More informationGlossary. Glossary Hawkes Learning Systems. All rights reserved.
A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Acute triangle A triangle in which all three angles are acute Addends The
More informationCBSE MATHEMATICS (SET-2)_2019
CBSE 09 MATHEMATICS (SET-) (Solutions). OC AB (AB is tangent to the smaller circle) In OBC a b CB CB a b CB a b AB CB (Perpendicular from the centre bisects the chord) AB a b. In PQS PQ 4 (By Pythagoras
More informationClass IX Chapter 8 Quadrilaterals Maths
Class IX Chapter 8 Quadrilaterals Maths Exercise 8.1 Question 1: The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the quadrilateral. Answer: Let the common ratio between
More informationClass IX Chapter 8 Quadrilaterals Maths
1 Class IX Chapter 8 Quadrilaterals Maths Exercise 8.1 Question 1: The angles of quadrilateral are in the ratio 3: 5: 9: 13. Find all the angles of the quadrilateral. Let the common ratio between the angles
More informationGlossary. Glossary 981. Hawkes Learning Systems. All rights reserved.
A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Addends The numbers being added in an addition problem Addition principle
More informationMT EDUCARE LTD. SUMMATIVE ASSESSMENT Roll No. Code No. 31/1
CBSE - X MT EDUCARE LTD. SUMMATIVE ASSESSMENT - 03-4 Roll No. Code No. 3/ Series RLH Please check that this question paper contains 6 printed pages. Code number given on the right hand side of the question
More information5. Introduction to Euclid s Geometry
This is only a sample of the book. For complete book contact us... 5. Introduction to Euclid s Geometry EXERISE 5.1 One Mark Questions 1. How many lines can pass through two distinct points? Sol. Only
More informationGeometry 3 SIMILARITY & CONGRUENCY Congruency: When two figures have same shape and size, then they are said to be congruent figure. The phenomena between these two figures is said to be congruency. CONDITIONS
More informationDefinitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg Undefined Terms: Point, Line, Incident, Between, Congruent. Incidence Axioms:
More informationHonors Geometry Mid-Term Exam Review
Class: Date: Honors Geometry Mid-Term Exam Review Multiple Choice Identify the letter of the choice that best completes the statement or answers the question. 1. Classify the triangle by its sides. The
More informationClass IX Chapter 1 Number Sustems Maths
Class IX Chapter 1 Number Sustems Maths Exercise 1.1 Question Is zero a rational number? Can you write it in the form 0? and q, where p and q are integers Yes. Zero is a rational number as it can be represented
More informationSAMPLE QUESTION PAPER Class-X ( ) Mathematics. Time allowed: 3 Hours Max. Marks: 80
SAMPLE QUESTION PAPER Class-X (017 18) Mathematics Time allowed: 3 Hours Max. Marks: 80 General Instructions: (i) All questions are compulsory. (ii) The question paper consists of 30 questions divided
More informationSubject: General Mathematics
Subject: General Mathematics Written By Or Composed By:Sarfraz Talib Chapter No.1 Matrix A rectangular array of number arranged into rows and columns is called matrix OR The combination of rows and columns
More informationSUPPORT STUDY MATERIAL
SUPPORT STUDY MATERIAL IX Maths Support Material and VBQ How to use this study material? This study material contains gist of the topic/units along with the assignments for self assessment. Here are some
More informationCBSE CLASS-10 MARCH 2018
CBSE CLASS-10 MARCH 2018 MATHEMATICS Time : 2.30 hrs QUESTION & ANSWER Marks : 80 General Instructions : i. All questions are compulsory ii. This question paper consists of 30 questions divided into four
More informationDefinitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg ( )
Definitions, Axioms, Postulates, Propositions, and Theorems from Euclidean and Non-Euclidean Geometries by Marvin Jay Greenberg (2005-02-16) Logic Rules (Greenberg): Logic Rule 1 Allowable justifications.
More information1. B (27 9 ) = [3 3 ] = (3 ) = 3 2. D. = c d dy d = cy + c dy cy = d + c. y( d c) 3. D 4. C
HKDSE03 Mathematics (Compulsory Part) Paper Full Solution. B (7 9 ) [3 3 ] (3 ) 3 n + 3 3 ( n + ) 3 n + 5 3 6 n + 5. D y y + c d dy d cy + c dy cy d + c y( d c) c + d c + d y d c 3. D hl kl + hm km hn
More informationEuclidian Geometry Grade 10 to 12 (CAPS)
Euclidian Geometry Grade 10 to 12 (CAPS) Compiled by Marlene Malan marlene.mcubed@gmail.com Prepared by Marlene Malan CAPS DOCUMENT (Paper 2) Grade 10 Grade 11 Grade 12 (a) Revise basic results established
More informationQuestion 1: In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that ABC ABD. What can you say about BC and BD?
Class IX - NCERT Maths Exercise (7.1) Question 1: In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that ABC ABD. What can you say about BC and BD? Solution 1: In ABC and ABD,
More informationMT EDUCARE LTD. SUMMATIVE ASSESSMENT Roll No. Code No. 31/1
CBSE - X MT EDUCARE LTD. SUMMATIVE ASSESSMENT - 03-4 Roll No. Code No. 3/ Series RLH Please check that this question paper contains 6 printed pages. Code number given on the right hand side of the question
More informationEXERCISE 10.1 EXERCISE 10.2
NCERT Class 9 Solved Questions for Chapter: Circle 10 NCERT 10 Class CIRCLES 9 Solved Questions for Chapter: Circle EXERCISE 10.1 Q.1. Fill in the blanks : (i) The centre of a circle lies in of the circle.
More informationGeometry First Semester Exam Review
Geometry First Semester Exam Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Name three points that are collinear. a. points T, Q, and R c. points
More informationClass IX Chapter 7 Triangles Maths
Class IX Chapter 7 Triangles Maths 1: Exercise 7.1 Question In quadrilateral ACBD, AC = AD and AB bisects A (See the given figure). Show that ABC ABD. What can you say about BC and BD? In ABC and ABD,
More information