SURDS (COMPETITION)

SURDS

Any number of the form p/q, where p and q are integers and q≠ 0 is called a rational number. Any real number which is not a rational numbers, of particular interest to us are SURDS  Amongst surds, we will specifically be looking at 'quadratic surds' :- shrds of the type a+√b and a+ √b + √c, where the terms involve only square roots and any higher roots. We do not need to go very deep into the area of surds - what is required is a basic understanding of some of the operation on surds.

If there is a surd of the form (a+√b), then a surd of the form ±(a - √b) is called conjugate of the surd (a+√b). The product of a surd and its conjugate will always be a rational number.

 Rationalisation of a surd

 When there is a surd of the 1/(a+√b), it is difficult to perform arithmetical operations on it. Hence, the denominator is converted into rational number thereby facilitating ease of handling the surd. This process of converting the denominator into a rational number without charging the value of the surd is called rationalisation.

        To convert the denominator of a surd into a rational number, multiply the denominator and the numerator simultaneously with the conjugate of the surds in the denominator so that the denominator gets converted to a rational number without changing the value of the fraction. That is, if there is a surd of the type a+√b in the denominator, then both the numerator and the denominator have to be multiplied with a surd of the form a- √b or a surd of the type (-a-√b) to convert the denominator into a rational number.

 If there is a surd of the form (a+√b +√c) in the denominator, then the process of multiplying the denominator with its conjugate surd has to be carried out TWICE to rationalise denominator.

SQUARE ROOT OF A SURD

If they are exists a square root of a surd of the type a+√b, then it will be of the form √x +√y. We can equate the square of  √x +√y to √a +√b and thus solve for x and y. Here, one point should be noted :- when there is an equation with rational and irrational terms, the rational part of the left hand side is equal to the irrational part on the right hand side and, the irrational part on the right hand side of the equation.

 However, for the problems which are expected in the competitive exams, there is no need for solving for the square root in such an elaborate manner. We will look at finding the square root of the surd in a much simpler manner. Here, first the given surd is written in the form of (√x +√y)² or (√x -√y)². Then the square root of the surd will be (√x +√y) or (√x -√y) respectively.

 Comparison of surds

Sometimes we will need to compare two or more surds to identify the largest one or to arrange the given surds in ascending/descending order. The surds given in such cases will be such that they will be close to each other and hence we will not be able to identify the largest one by taking the approximate square root of each of the terms. In such a case, the surds can be both be squared and the common rational part be subtracted. At thid stage, normally one will be able to make out the order of surds. If even at this stage, it is not possible to identify the larger of the two, then the number should be squared once more.

Some Identities Related to Real Numbers:

1) (√a)²= a

2) √(ab)= √a √b

3) √(a/b)= √a/√b

4) (√a+√b)(√a - √b)= a - b

5) (a+√b)(a - √b)= a² - b

6) (√a+√b)(√c + √d)= √(ac)+ √(ad) + √(bc) + √(bd)

7) (√a- √b)² = a - 2√(ab) + b

1) Express 5 as a biquadratic surd.

A) √5 B) ³√5 C) ⁴√5 D) 5

Express in simplest rationalisation factor of:

2) √32

A) 4 B) √2 C) 4 √2 D) 2√2 

3) ³√49 

A) 7. B) √7 C) 14 D) none

4) ⁴√25

A) 5 B) √5 C) 2√5 D) n

5) ⁵√(a²b³c⁴)

A) abc B) a²bc C) a²b²c² D) ab³c³

6) ⁵√486

A) ⁵√16 B)  3 C) √3 D) 6

7)  ³√1080 

8) ³√192

9) ⁴√1280

Arrange in ascending as well as descending order of magnitude 

10) √5 , ²√11, 2 ⁶√2

11) ⁴√3, ⁵√4, ¹⁰√12

12) 2√2, 2 ³√2 , 2 ⁶√5

Add:

13) (2√3 +5√5 - 7√7) and (3√5 - √3 + √7)

A) 3√3+ √5- 6√7 

B) √3+ 8√5- 6√7.

C)  3√3+ √5- √7 

D) √3+ √5- 6√7 

14) (4√3 + 7√2) and (√3 - 5√2)

A) 3√3+ √2        B) 3√5+ 8√5

C) 5√3+ 2√2.     D) √3+ √2

15) (√5 +2√3) and (2√5 - 5√3)

A) 3√5 - 3√3.     B) √5+ 6√5

C)  3√3+ √5       D) 2√3+ √5

16) (-√2/2 +2√5/3 + 6√7) and (√5/3 + 3√3/2 - √7)

A) √2+ √5 +5√7. 

B) √3+ 8√5- 6√7

C)  3√3+ √5- √7 

D) √3+ √5- 6√7 

Multiply:

17) 7√6 by 5√24

A) 20 B) 240 C) 420. D) 460

18) ³√32 by ³√250

A) 20. B) 240 C) 420 D) 460

19) ³√7 by √2

A) ⁶√392 B) ⁵√392 C) ⁴√392 D) n

20) 2√3 by  5√27

A) 9 B) 90. C) 900 D) 9000

21) 3√28 by 2√7

A) 8 B) 4 C) 48 D) 84.

22) 3√8 by 3√2

A) 63 B) 36. C) 3 D) 6

23) 4√12 by  7 √6

A) 16 √2  B) 18√2 C) 168√2. D) 6√2

24) ³√2 by ⁴√3

A) ¹²√42 B) 6√42 C) 4√42 D) ¹²√432

25) 2 ⁴√3 by 5 ⁴√81

A) 30 ⁴√3.          B) 15 ⁴√3

C) 5 ⁴√3             D) 8 ³√3

Divide:

26) ³√18 by  ³√9

A) ³√2 B) ³√3 C) ³√9 D) ³√4

27) ³√128 by ⁵√64

A) 2 ¹⁵√4. B) ¹⁵√4 C) ¹⁵√2 D) none

28) 12√15 by 4√3

A)  √5 B) 3√5. C) 2√3 D)  3√7

29) 4√28 by 3√7 

A) 2/3 B) 4/3 C) 8/3. D) 10/3

30) 21√384 by 8√96

A) 21/4. B) 22/3 C) 41/3 D) 1

31) ⁶√12 by 3 ³√2

A) ³√(1/3). B) √4 C) √3 D) 6

Simplify:

32) (√147 - √108 - √3) 

A) 1 B) 0 C) 2 D) none

33) 4√3 - 3√12 + 2√75

A) 8√3. B) √3 C) 2√3  D) 3√3

34) 3√45- √125 + √200 - √50

A) 4√5 B) 5√2 C) 4 √5 + 5√2 D) 4√5 - 5√2

35) 3√48 - 5/2 √(1/3)+ 4 √3 

A) 91√3/6. B) ) 3√3 C) √5 D) 6√7 

36) 2 ³√4 + 7 ³√32 - ³√500

A) 11 √4 B) 11³√4. C) 11⁴√4 D) n

37) 2 ³√54 + 3 ³√16 + 5 ⁴√128

A) 32 ³√2.        B) 30 ³√2

C) 32 ³√3        D) 3√2

38) √125 - 4√6 + √294 - 2 √(1/6)

A) 5√5 B) √3+ 8√5 C) 6√7 D) none

39) ⁴√81 - 8 ³√216 + 15 ⁵√32 + √225

A) 0 A) 1 C) 2 D) 3

40) (5 +√3)(7 +√5)

A) 35 + 7√3 +5√5 +√15.

B) 3√5 - √3 + √15

C) 3√3+ √5- 6√5 

D) √3+ 8√5- 6√5

41) (3 + √2)(4+ √3)

A) 12 + √2+ √3+ √6

B) 12+ 4√2 +3√3 + √6

C)  12+ 3√2 - 3√3 + √6

D) ) 3√3+ √5- 6√6

42)  (√5+ √2)(√3+ √2)

A) √15- √10 +√6+ 2.

B) √5- 6√10 +√6

C) 2√5 +5√5 - 7√6

D) 3√5 - √5 + √2

43)(√13+ √11)(√13- √11)

A) 1 B) 0 C) 2 D) none

44) (4+ √3)(4 - √3)

A) 0 B) 1 C) 3 D) 13

45) (√13 - √6)(√13+ √6)

A) 3 B) 4 C) 7.

46) (3√5 +3√7)(3√5 - 2√7) 

A) 12  B) 12 + 2√35. C) 1 D) 5

47) (3√5 + 5√2)²

A) 95 + √10    B) 95+ 30√10.

C) 95 - √19     D) 95 +√3

48) (√5 + √7)²

A)  12 + 2√35. B) 12+ 2 √5

C) 12- 2√7       D) 12 + 2√3

49) (4√3 - 3√5)²

A) 93 + 24√15     B) 90 - √15

C) 93 - 24√15      D) none

50) (2√5/3 + √2/2 + 6√12) + (√5/3 + 3√2/2 - √11)

A) √5+ √2+ 5 √11

B) √5/2 + 2√2+ √11

C)  √5+ √2+ 6√11

D) √5+ 2√2+ 5 √11

51) ³√7 x √5

A) ³√35 B) ⁶√35 C) ⁶√6125 D) ⁶√1225

52)  √18/6 x+ √18/3

A) 1 B) 1/12 C) 1/3 D) √2

53) √5x √7 x √15 x √21

A) √105 B) √210 C) 105 D) 210

54) (3+ √3)(3 - √3)

A) 18 B) 2√3 C) 6 D) 9

55) (3+ √5)²(3- √5)²

A) 15 B) 16 C) 4 D) 14

56) ³√250 ÷ ³√10

A) ³√25  B) 5 C) √5 D) ³√2500

57) 30/(√20+√5)

A) 10/3√5  B) 30/√5 C) 10/√5 D) 12√5

58) 6/(√12+√3)

A) 1/√3  B) 2/√3  C) 2√3 D) 6√3

Simplify

59) ³√24 - ³√192 + ³√81

60) (√5 +√3)²(4 - √15)

61) 3√48 - 4√75 + 5√192

62) ³√56 - ³√875 + ³√189

63) √(x³y) + √(xy³) + √(xyz²)

64) (1+√2 - √3)(2 + √2 + √6)

65)(√50+√32-√18)/(√75 -3√3 +√12)

66) Express 4 as a cubic surd

67) Express x²y as fifth order surd

68) Find the square of √(x+y)-√(x-y)

69) Find the cube of 2√3 - 3√2

B) Rationalise the denominator

70) 3/√2                                   3√2/2

71) 3/(2+√3).                        3(2- √3)

72) 2/(√5 - √3)                     (√5+√3)

73) 3/(4√5 - 2).           3(2√5 +1)/38

74) 5/(4√5 - 5√3).           (4√5+5√3)

75) (3√2+2√3)/(3√2 -2√3).  (5+2√6)

76)√3/(√2+√3 -√5).    (2+√6+√10)/4

77) 1/(1 + √2 - √3).       (√2+2+√6)/4

78) 12/(3 +√5 + 2√2).         (1+√5+√2-√10)

79) (√3 - √2)/(√3 + √2).            5-2√6

C) Simplify 

80) 1/(√6 -√5) - 3/(√7 - √2) - 4/(√6 + √2).                                 2√2+ √5+ √7

81) √5/(√3 +√2) - 3√3/(√2 + √5) + 2√2/(√5 +√3).                           0

82) If x= √{(√5 +1)/(√5 -1)}, Show that x² - x -1=0

83) If x= √2 +√3, show that x⁻² = 5 - 2√6

84) If a= √3/ 2, show that √(1+a) +√(1- a) = 2a

85) If x = 4+ 2√3, find the value of (√x - 2/√x)

86) If x = 4 + √15, Show that √x + 1/√x = √10

E) Find the value of

87) √[2 +√{2 + √(2 + ....... up to ∞)}]

88) √[³√{b √(a ³√b ..... up to ∞)}]

89) √[6 + √{6 +√(6+ .....up to ∞)}]

EXERCISE -1

1) Simplify: 1/(4-√5) - 1/(4+ √5)

A) 2 B) √5 C) 2√5 D) 2√5/11 E) n

2) Rationalise the denominator: 1/(1+ (√6 - √7).

A) (√6+ 6+ √42)/12

B) (6+ √6+ √42)/12

C) (√6- 6+ √42)/12

D) (√6- 6- √42)/12.  E) n

3) Find the value of √{62+ √480}

A) √60+√2 B) √60- √2 C) 60+√2 D) √60+ 2 E) n

4) Which of the surds is greater? √3+ √23 and √6+ √19.

A) √6+ √19 > √3+ √23 

B) √3+ √23 > √6+ √19

C) both are equal

D) not equable

5) Which of the following is the conjugate of the surd √7- 2 ?

A) √7+ 2 B) -2 -√7 C) either A or B D) n

6) simplify: 3/(√5+ √2) + 1)(√6+ √5)

A) √6-√2 B) √6+ √2 C) √5+ √6 D) √2- √6 E) n

7) Find the positive square root of √180 - √125

A)√5 B) 5¹⁾⁴ C) 65¹⁾⁴ D)180¹⁾⁴ -125¹⁾⁴ E) n

8) Find the positive square root of 27- 10√2.

A) 5- √2 B) √200- √27 C) 6- √2 D)  4- √5  E) n

9) 9/(6²⁾³ - 18¹⁾³ + 3²⁾³) =

A) (6¹⁾³ + 3¹⁾³) B) (1/3)(6¹⁾³+3¹⁾³) C) (6¹⁾³ - 3¹⁾³) D) (1/9)(6¹⁾³+ 3¹⁾³) E) n

10) Which of the following is a rationalising factor of 10¹⁾³ - 9¹⁾³ ?

A) 10²⁾³ + 9²⁾³) B) 10²⁾³ - 9²⁾³) C) 10²⁾³ + 90¹⁾³ + 9² D) 19²⁾³ - 9¹⁾³ + 9²⁾³ E) n

11) Which of the following is the rationalising factor of 12 + 12√6 ?

A) -12 - 12√6 B) -12 - 12√6 C) 12 - 12√6 D) either B or C E) n

12) Find the square root of √98 + √96

A) ⁴√2(2+ √3) B) √2(√2 + √3) C) √2(2+ √3) D) ⁴√2(√2 + √3) E) n

13) Arrange the following in ascending order a= √2 + √11, b= √6 + √7, c= √3 +  √10 and d= √5+ √8

A) abcd B) adbc C) acdb D) acbd E) n

14) Arrange the following in descending order a= √20 + √2, b= √24 + √6, c= √22 + 2 and d= √26+ √8

A) dcba B) dcab C) dbca D) dbca E) n

15) a= √3 + √23, b= √6 + √19, is greater than b ?

A) true B) false C) cannot say 

16) Arrange the following in descending order a= √13 + √11, b= √15 + √9, c= √18 +  √6 and d= √7+ √17

A) abdc B) dcab C) adcb D) acdb E) n

17) Arrange the following in ascending order p= √26 - √23, q= √18 - √15, r= √11 -  √8 and s= √24 - √21

A) rqsp B) psrq C) pqrs D) psqr E) n

18) Simplify: = [a/(√b - √c) + a/(√b + √c)]²

A) 2a²c/(b- c)² B) (b- c)²/4a²b  C) 4a²b/(b- c)²  D) 2a²/(b²+c² - a²)  E) 2ac/(b- c)² F) n

19) If 2√2 + √3= x, what is the value of (11+ 4√6)/(2√2 - √3) in terms of x ?

A) x²/√2 B) x³ C) x³/8 D) x E) x³/5

20) If √[x {√x (√x.......∞)= 11ˣ, what is the value of (ˣ√x) ?

A) ³√11 B) 1 C) √11 D) 11¹⁾ˣ E) 11

21) If 1/(√x + √(x+1)) + 1/(√(x+1) + √(x+2)) + 1/(√(x +2)+ √(x+3))+ 1/(√(x +98)+ √(x+99)) = 9, find which of the following is a positive value of x.

A) 2 B) 1 C) 4 D) 3 E) 5

22) What is the cube root of (3√3)(2√2)+ 7√7 + 3√3)(√2)(√7)(√3 √2+ √7) ?

A)√3 + 7√2 B) 6+ √7 C) √6 + 7√7 D) √6 + √7 E)  7 + √6

23) Find the value of  √(16 + 2√55).

A) 1+ √15 B) √10 + √5 +5 C) 1+ 2√5 + √15 D) √12 + √5 E) √11+ 3√5

24) Simplify: √{(a+b+c) + 2√(ac+ bc)}.

A) √a + √b +√c B) √(a+b) + √c D) √(ab+ bc) D) √(abc) E) √(ac+ bc)

25) Arrange a, b, c, d in ascending order if a= √13 + √9, b= √19 + √3, c= √17 +  √5 and d= √12+ √10

A) b,c, a,d B) d,b,a,c C) d,b,c,a D) d, c, b, a E) d,a,c,b

26) if a= 4(b-1) and b≠ 2, then simplify: √{b - √a)/(b+ √a)} + {b+ √a)/(b -  √a)} - 4/{(b- √a)(b +  √a)} 

A) √b B) 2 C) b/(b-2) D) (b-2)¹⁾² E) 4

27) Simplify: (a- b)/{³√a² + ³√(ab) + ³√b²} - (a+ b)/{³√a² - ³√(ab) + ³√b²}.

A) -2b¹⁾³ B) -2(ab)¹⁾³ C) a¹⁾³ - b¹⁾³ D) (a/b)¹⁾³ E) (b/a)¹⁾³

28) If √[x+ √{x²+ √(x⁴+√(x⁸.......∞).

A) √x{(1+√5)/2}  B) {(3+ 5√2)/√x}  C) x)(1+√x) D) (√2+√3)/x³⁾²   E) (√3+√5)/x³⁾²

29) If A/a = B/b = C/c = D/d, what is the value of √(Aa) + √(BB) + √(Cc) + √(Dd), given that (a+ b+c+ d)≠ 0 ?

A) 1 B) (A+B+C+D)/(a+b+c+ d)

C) =(√A+ √B+ √C+ √D)(a+b+c+d)

D) √{(a+b+c+d)(A+B+C+D)

E) √{(A+B+C+D)/(a+b+c+d)}

30) √(2x²+9) + √(2x²- 9)= 9+ 3√7 and √(2x²+9) - √(2x²- 9)= √a- √b, find the values of a and b

A) 18, 36 B) 81, 63 C) 27, 96 D) 34, 28 E) 63, 18

31) Find the square root of 23+ 4√10 - 10√2- 8√5.

A) √5 + √10 - √8 B) √10 + √8 - √5 C) √8 + √5 + √10 D) √5 + √8 - √10 E) √10 - √5 - √8 

32) a, b, c and d are rational numbers such that 16/2+ √5 + √13)= a+ b √5 + c √13 + d√65. Find the value of abcd 

A)21 B) 25 C) 28 D) 35 E) 42

33) What is the mean proportional of 2 - √3 and 26 - 15 √3 ?

A) 7  - 4√3 B) 12- 7√3 C)!13- 5√3 D) 24√3 - 14√3 E) 12 - 4 √3

34) Find the square root of [1+ 1/(√2 +1) + 1/(√3 + √2) + 1/(√4+ √3)  + 1/(√324 + √323)]

A) 3√2 B) 1/√2 C) 2 √3 D) (√5 -2)/2 E) 1/√3

35) If a= √6 + √8, what is 2 √2(26+ 15√3) in terms of a ?

A) √a³ B) √a⁵  C) a⁴ D) a³ E) a²

36) solve: √x + √{x - √(1- x)}= 1.

A) 1 B) 16/25 C) 4/5 D) 0 E) 5/4

37) value of √(7 - 3√5)

A) √10 - 2√3 B) (3-√5)/√2 C) (√3 - √7)/2√2 D) (√5 + √2)/3 E) 5 - √5)/2

38) Which of the following is a rationalising factor of (⁸√3 - ⁸√2)(⁴√3 + ⁴√2)(√3 - √2) ?

A) (⁴√3 - ⁴√2( B) (⁸√3 + ⁸√2) C) (1- √3 - √2) D) (√3 - √2)(⁴√3  - ⁴√2) E) (√3 - √2)

39) if a= 1/(2+√3), b= 1/(2- √3), what is the value of 7b² + 11ab  - 7a²

A) -14 + 21√3 B) 11 + 56√3 C) 49 + 8 √3 D) √3 + √11 E) 11 - √3

40) The arithmetic mean of two surds is 5+ 9√2, and one of the surds is 1+ 12√2. What is the square root of the other surd ?

A) 6 - 21√2 B) 4- 3√2 C) √3(√2+ 1) D) √2(2 - √3) E) 2(√3 - √2)

41) If x= 3 - √5 then the value of √1x/{√2 + √(3x- 2)}

A) 1/(4 - 2√5) B) √3 - 2√5 C) 1/√5 D) 1/(√2- 3 √5) E) 1/(√2 + 3√5)

42) if x= 5 - √21, find the value of √x/{√32 - 2x) - √21}

A) (√7 - √3)/2 B) (√7 - √3)/√3 C) (√7 - √3)/√2 E) (√7- √3)/3 E) √7 - √3

43) which of the following in ascending order if a= √12 + 2, b= √3+ 4, c=  √6+ √8 and d= √2 + √24 ?

A) cbda B) cbad C) cadb D) cabd E) cdab

44) If (x - y)[1/(√x + √y)  + 1/(√x - √y)= 12, find the value of y

A) 36 B) 25 C) 49 D) 64  E) can not be determined

45) If x= 3+ √5, then find the value of x³ - 9x² + 22x. 

A) -15 B) 12 C) 42 D) 45 E) 48

EXERCISE -2

1) If p= (√6 + √3)/(√6 - √3) and q= (√6 - √3)/(√6 + √3), then find the value of p+ q

A) √6 - √3 B) √6 + √3 C) 6 D) 3 E) 4

2) If x= 3+√5, find the value of x² - 1/x².

A) (3- √5)/4 B) 12√5 C) 4 D) (105 + 51 √5)/8 E) (135+ 63 √5)/8

3) If √{31 + 4√21}=  √x + √y and x > y, then the values of x and y respectively are:

A) 27,4 B) 30,1 C) 24,7 D) 21,10 E) 28,3 

4) The positive square root of 41 + 24 √2 is

A) 20+ 2√3 B) 21- 12√2 C) 4√2 - 3 D) 6√2 - 2 E) n

5) Given that x is a rational number and x² - 1= 0, the value of x is

A) (-1- √-3)/2 B) -1 C) (1 + √-3)/2 D) 1 E) n

6) The surd 1/(2+ √3 + √5) is equal to

A) {4+(4√3 - 8)}/16 

B)  {4- (4√3 + 18)}√5/16

C) {4+(4√3 + 8)}√5/16   

D) {4+(4√3 + 18)}√5/16  E( n

7) If √(x + 1/x)= 3, find the value of √(x - 1/x) if x≠ 0

A) 1/3 B) 2/3 C) 1 D) ⁴√79 E) ⁴√77

8) If a, b, c,d are rational √b, √c, √d are irrational √{a+ √b}= √d + c, and a > √b, then find the value of √{a - √b}.

A) c - √d B) - c + √d C) |c - √d| D) √{c - √d{ E) none

9) if a and b are natural numbers and √a, √b are irrational then √(ab) would always be

I. rational if the GCD (a,b) is 1.

II. irrational if the GCD (a,b) is 1.

III. rational if the LCM (a,b) is ab.

IV. irrational if the LCM (a,b) is ab.

A) I and III B) II and IV C) I and IV D) II and III E) III and IV

10) which of the following is the greatest

a) √10 +√7 b) √11 - √8 c)√13 - √10 d) √16 - √14 e) √20 - √17

11)  if a=⁶√(5 + 2√6) and b= ⁶√(5- 2√6), find the value of (a+ b)(a²+ b²+ 2ab -3)= 

A)  2√3 B) 2√2  C) 2√6  D) 10 E) 2√5

12) Evaluate √[5+√{5 - √(5+ √5....

A) (√13 - 1)/2 B) (√17- 1)/2 C) (√17  + 1)/2 D) √17 E) (√13+1)/2

13) if x= 12√6/(√2 + √3), then find the value of (x+ 6√2)/(x - 6√2) + (x + 6 √3)/(x  - 6√3)

A) 2 B) 3 C) 2√3 D) 3√3 E) n

14) which of the following is the Greatest:

a) ₂ √3 +√5 +√6+ √7 

b) ₃√2 +√5 +√6+ √7

c) ₅√2 +√3 +√6+ √7 

d) ₆√2 +√3 +√5+ √7

e) ₇√2 +√3 +√5+ √6

15) simplify: 1/(4+√2) - 1/(4- √2)

A) 2 B)  √6 C) 2/√7 D) -2/√7 E) n

16) Rationalise the denominator 1/(4- √13)

A) (4+√5)/3 B) (4 +√13)/3 C) (4+ √7)/3 D) n

17) Rationalise the denominator 1/(4 + √6 - √10)

A) (4+√5)/3 B) (4 +√13)/3 C) (4+ √7)/3 D) n

CIRCLE - XI

1) Find the equation of the circle with::

a) Centre (2,4), radius 5.         x² + y² - 4x - 8y - 5= 0

b) centre (-3, -2) radius=6.          x² + y² + 6x + 4y - 23= 0

c) centre (a, a), radius= a√2.      x² + y² + 2ax - 2ay = 0

d) centre (acos k, a sin k) , radius= √(a² - b²).                x² + y² - 2ax cos k - 2ay sin k = 0

e) centre (-a, -b) , radius= √(a² - b²).

          x² + y² +2ax + 2by + 2b²= 0 

f) centre at the origin and radius 4.          x² + y² - 16 = 0

2) Find the centre and radius of each of the following:

a) (x -3)² + (y -1)²= 9.            (3,1), 3

b) (x - 1/2)² + (y+ 1/3)²= 1/16.       (1/2, -1/3), 1/4

c) (x + 5)² + (y - 3)²= 20.        (-5,3), 2 √5

d) x ² + (y -1)²= 2.              (0,1), √2

e) x² + y² - 4x + 6y= 5.      (2,-3); 3√2

f) x² + y² - x + 2y= 3. (1/2,-); √17/2

3)a) Find the equation of the circle whose Centre is (2,-5) and which passes through the point (3,2).    x² + y² - 4x + 10y - 21= 0

b) Find the equation of the circle whose Centre is (1,2) and which passes through the point (4,6).    x² + y² - 2x - 4y - 20= 0

4)a) Find the equation of the circle radius 5 cm, whose centre lies on the y axis and which passes through the point (3,2).          (x² + y²- 12y+11) or (x² + y² + 4y - 21= 0)

b) Find the equation of the circle whose centre lies on the positive direction of y axis at a distance 6 from the origin and whose radius is 4.                         x² + y²- 12y+20= 0

5)a) Find the equation of the circle whose centre is (2, -3) and which passes through the intersection of the line 3x + 2y = 11 and 2x + 3y = 4.                      x² + y² - 4x + 6y - 3= 0

b) Find the equation of the circle whose centre is (2, 3) and which passes through the intersection of the line 3x - 2y = 1 and 4x + y = 27.       x² + y² - 4x - 6y - 12= 0

c) find the equation of the circle passing through the point (-1,3) and having its centre at the point of intersection of the lines x - 2y = 4 and 2x + 5y- 1 = 0.            x² + y² - 4x + 2y - 20= 0

d) find the equation of the circle passing through the point of intersection of the lines x + 3y = 0 and 2x - 7y = 0 and whose centre is the point of intersection of the lines x - 2y + 4= and x + y + 1 = 0.          x² + y² + 4x - 2y = 0

6)a) If two diameter of a circle lie along the lines x - y = 9 and x - 2y = 7 and the area of the circle is 38.5 sq.cm, find the equation of the circle.                  4x² + 4y² - 88x - 16y +451= 0

b) If the equation of two diameters of a circle 2x + y= 6 and 3x + 2y = 4 and the radius is 10, find the equation of the circle.            x² + y² - 16x + 20y + 64= 0   

7) find the equation of the circle, the co-ordinate of the end points of one of whose diameters are:

a) A(3,2), B(2,5).                x² + y² - 5x - 7y + 16= 0 

b) A(5,-3), B(2,-4).                   x² + y² - 7x + 7y + 22= 0

c) A(- 2, -3) , B(-3,5).               x² + y² + 5x - 2y - 9= 0 

d) A(p,q), B(r,s).                   (x - p)(x - r)+ (y - q)(y - s) = 0

e) A(1,3) , B(4,5). Find also the equation of the perpendicular diameter.                    x² + y² - 5x - 8y +19= 0; 6x + 4y = 31

8) The sides of a rectangle are given by the equation x= -2, x = 4, y =- 2 and y = 5. Find the equation of circle drawn on the diagonal of this rectangle and as its diameter.       x² + y² - 2x - 3y + 18= 0

9) Find the equation of a circle:

a) Which touches both the at a distance of 6 units from the origin.        x² + y² - 12x - 12y + 36= 0

b) Which touches x-axis at a distance of 5 units from the origin and radius 6 units.        x² + y² - 10x - 12y + 25= 0

c) Which touches both the axes and passes through the point (2,1).        x² + y² - 2x - 2y + 1= 0, x² + y² - 10x - 10y +25= 0

d) passing through the origin, radius and ordinate of the centre is -15.              x² + y² ±16x + 30y = 0

e) touches the axes and whose centre lies on x - 2y = 3.      x² + y² +6x + 6y+ 9 = 0 or x² + y² - 2x + 2y +1 = 0

f) whose centre at the point (3,4) and touches the straight line on 5x + 12y = 1.                  169(x² + y² - 6x - 8y)+ 381 = 0 

g) radius 4 units touches the coordinates axes in the first quadrant. Find the equation of its images with respect to the line mirrors x= 0 and y= 0.           x² + y² + 8x - 8y+ 16 = 0 ; x² + y² - 8x + 8y+ 16 = 0

h) touching y-axis at (0,3) and making an intercept of 8 units on the x-axis.                  x² + y² ± 10x - 6y+ 9 = 0 

i) touching x-axis whose centre is (0, -4).                         x² + y² + 8y = 0 

j) whose centre is at (3,4) and which touches the line 5x +12y = 1.         169(x² + y² -6x - 8y)+ 381 = 0 

k) Which passes through the origin and cuts off intercept of length 'a', each from positive direction of the axes.                    x² + y² - ax - ay = 0 

l) centre is (a, b) and which passes through the origin.         x² + y² - 2ax - 2by = 0 

m) pass through the origin and cuts off intercept equal to 3, 4 from x-axis.                    x² + y² - 3x - 4y = 0 

n) pass through the origin and cuts off intercept equal to 2a, 2b from y-axis.                 x² + y² - 2ax - 2by = 0 

o) touch the axis of x at a distance of 4 from the origin and cut off an intercept of 6 from the axis of y.      x² + y² ± 8x ± 10y +16 = 0

p) centre in the first quadrant touches the y-axis at the point (0,2) and passes through the point (1,0). Find the equation of the circle.       x² + y² - 5x - 4y +4 = 0

q) centre (1,0), which passes through the point (3, 3/2). Find also the equation of the equation which touches the given circle externally at P.                 4x² + 4y² - 8x - 21 = 0, 4x² + 4y² - 40x - 24y +111 = 0

10) Find the equation of the circle whose centre is (3, -1) and which cut-off an intercept of length 6 from the line 2x - 5y +18= 0.          x² + y² - 6x + 2y - 28 = 0 

11) If the line 2x - y +1 = 0 touches the circle at the point (2,5) and the centres of the circle lies on the line x+ y - 9 = 0. Find the equation of the circle.               (x - 6)² + (y - 3)² = 20

INTERPOLATION

               INTERPOLATION

1)X:  5     12    17   22    27   32    37

   Y: 1.1   2.8   5.1  8.8   12 15.2 17.3

a) Find x, when y= 12.8 ?

A) 22.85 B) 25.22 C) 28.25 D) 28.52

b) Find y when X= 19.3 ? 

A) 6.082 B) 6.802 C) 6.028 D) 6.208

2) X:1    3    5    7     9    11    13    15

    Y: 1  15  65 175 369 671 786 989

a) Find x when y= 100 ?

A) 14.321 B)14.213 C)14.123 D)14.231

b) find x when y=900 

A)5.6436 B)5.6653 C)5.6463 D)5.6364

c) Find y when xis 10 ?

A) 550 B) 520 C)580 D) 540

d) find y when X is 4 ?

A) 40 B) 42 C) 38 D) 44

3) What is the value ∆⁵y₀ in terms of y₀,y₁ ,y₂ ?

A) y₅ - 6y₄ + 10y₃ - 10y₂ + 5y₁ - y₀

B) y₅ - 5y₄ + 10y₃ - 10y₂ + 5y₁ - y₀

C)  y₅ - 5y₄ + 10y₃ - 12y₂ + 5y₁ - y₀

D)  y₅ - 5y₄ + 10y₃ - 10y₂ + 7y₁ - y₀

4) Given y₀= (10-x)(15+x), y₁= (17-x)(6+x), y₂ = (8+x)(9-x), y₃ = 100, obtain a value of x such that the second difference of y are constant.

A) 1/2  √1065 - 25/2, -1/2  √1065 - 25/2

B) 1/2  √1055 - 35/2, -1/2  √1065 - 35/2

C) 1/2  √1065 - 35/2, -1/2  √1065 - 35/2

D) 1/2  √1065 - 35/2, -1/2  √1065 - 35/2

5) Estimate y₂ from the table

X: 1        2        3        4          5

Y: 5        ?       17      29       151

A) 71/5 B) 71/4 C) 71/3 D) 71/2

6) Estimate y₁₂ from the table

X:    3      6       9         12          15

Yₓ: 15    47     79         ?          243

A) 136 B) 163 C) 613 D) 316

7) Using Lagrange's is interpolation formula, find the value of y at x=0 given some set of values (-2,5) (1,7) (3,11) (7,34) ?

A) 1078/180 B) 1708/180 C) 1087/180 D) 1087/108

8) Using Lagrange's interpolation find f(7) from the table

X:   4        5            6          8

Y: 3.11   2.96     2.85     2.70

A) 2.7675 B) 2.6775 C) 2.5675 D) 2.4675

9) Given f(x), 

f(0)=10, f(1)+f(2)=100, f(3)+ f(4) +f(5)= 650 find f(4)

A) 220 B) 200 C) 210 D) 240

10) find f(x) for x=24.

U:      20      25    30      35          40

F(u):30.5  34.5   40    47.75   59.25

A) 33.95 B)33.59 C) 39.35 D) 35.39

11) estimate y when x= 0.35

u:     0    0.1      0.2       0.3        0.4

f(u): 1  1.095  1.179   1.251  1.310

A) 1.228 B) 1.822 C) 1.283 D) 1.802

12) The nᵗʰ order forward difference is

A) ∆ⁿ f(x+h) - ∆ⁿ⁻¹f(x)

B) ∆ⁿ⁻¹ f(x) - ∆ⁿ⁻¹f(x+ h).

C) ∆ⁿ⁺¹ f(x+h) - ∆ⁿ f(x).

D) ∆ⁿ⁺¹ f(x+h) - ∆ⁿ⁻¹ f(x).

13) The nᵗʰ order backward difference is

A) ∆ⁿ⁻¹ f(x) - ∆ⁿ⁻¹f(x- h)

B) ∆ⁿ⁻¹ f(x) - ∆ⁿ⁻¹f(x+ h).

C) ∆ⁿ⁺¹ f(x+h) - ∆ⁿ f(x).

D) ∆ⁿ ⁻¹ f(x) - ∆ⁿ⁻¹ f(x - h). 

14) The value of ∆²x³ is

A) 6hx² + 6h³    B) 6h²x + 6h³

C) 6h²x - 6h³     D) 6hx + 3h³ 

15) The value of ∆eˣ is

A) eˣ⁺ʰ - eˣ          B) eˣ -  eˣ⁻ʰ

C) eˣ⁺ʰ + eˣ         D) eˣ⁻ʰ - eˣ

16) If f(x) be a polynomial of mᵗʰ degree then     

A) νᵐ⁻¹ f(x) = 0

B) ∆ ᵐ⁺¹ f(x)

C) ∆ᵐ⁻¹ f(x) = constant

D) ν ᵐ⁺¹ f(x) = constant

17) The value of ∆ⁿ [axⁿ + bxⁿ⁻²] with h= 1 is

A) a(n-2)! B) b(n-2)! C) b n! D) a n!

18) The value of ∆ᵐ xᵐ is

A) hᵐ⁻¹ m! B) hᵐ m! C) hᵐ /m! D) hᵐ (m+2)!

19) If the nᵗʰ order difference of a tabulated function f(x) are constant, the value of independent variables are taken at equal intervals, then 

A) f(x) is a polynomial of degree n

B) f(x) is a polynomial of degree n-1

C) f(x) is zero D) f(x) is constant

20) if f(x) is given at the points x=x , x , x ,....., Xₙ = b of the interval [a, b], where x₀ < x₁ > ....< xₙ , then the problem of finding f(x) at a points lying in any of the sub-intervals [xᵢ₋ ₁ , x], i= 1,2,.....n is known as

A) interpolation     B) extrapolation C) estimation D) none

21) Newton forward interpolation formula is used for

A) centre difference

B) unequal interval C) equal interval  D) none

22) Newton backward interpolation formula is used for computing f(x) for values of x lying at the

A) middle of the table

B) end of the table

C) beginning of the table

D) none

23) Find the value of x for which y= 40:

x: 10     12    15    20  

Y: 25     32    35    45

A) 19.65 B) 19.56 C) 15.96 D) 16.59

24) Find f(5) for the following data : f(3)=4, f(4)=13, f(6)=43.

A) 24 B) 25 C) 26 D) 27

25) find ∆³ f(2) from the following data of f(x): 

f(2)=9, f(4)=63, f(6) =211, f(8)=506.

A) 50 B) 51 C) 52 D) 53

26) Assuming f(x) to be a 3rd degree polynomial of x, find f(x) :

X:          0           1          2           3

Y=f(x): 1            2         11        34

A) x³+ x² - x +2.    B) x³+ x² - 2x + 1

C) x³+ 2x² - x +1   D) x³+ x² - x + 1

27) Find f(x) given that f(0) = -3, f(1)=6, f(2)=8, f(3)=12.

A) 126 B) 127 C) 128 D) 129

28) Find the value a and b 

X:    10     15       20     25       30     35

Y: 19.97 21.51    a    23.52  24.65   b

A) 21.58, 26.29  B) 22.58, 26.92 C) 22.58, 26.29 D) 22.58, 25.29

29) Calculate f(5) 

16) x:    2         4         7            9

      Y:   10       26       65        101

A) 37 B) 39 C) 38  D) 40

30) Find the polynomial function f(x) from the following values: f(3)=-1, f(4)=5, f(5)=15.

A) 2x² - 8x +5 B) 2x² - 7x +5 C) 2x² - 8x +6  D) x² - 8x +5

31) Assuming f(x) to be a 3rd degree polynomial of x, find f(2):

 X:. 0       1         3       4

 Y:  5       6        50   105

A) 17  B) 18 C) 19  D) 20

32) Estimate the value of m from the following table:

  z:    0     1     2     3       4

 f(z); 1     3     9     m     81

A) 30 B) 31 C) 32 D) 33

33) The population of a certain country is as given below:

year: 1971  1981  1991  2001  2012

Pop:    46        66      81     93     101

Estimate the population for the year 2005.

A) 96.48ml B) 96.84ml C) 96.68ml D) 96.86ml

34) Find the value of U₄ of a function Uₓ, given U₁=10, U₂=15, U₅=42.

A) 35 B) 13 C) 33 D) 31

STRAIGHT LINE - X

                  EXERCISE--1

                  -------------------

1) Find the slope of a line whose inclination is:

a) 30°.                                       1/√3

b) 90°.                             not defined

2) Find the inclination of a line whose slope is

a) √3.                                             60°

b) 1/√3.                                         30°

c) 1.                                               45°  

3) Find the slope of a line which passes through the points:

a) (0,0) and (4, -2).                    -1/2

b) (0,-3) and (2, 1).                        2

c) (2,5) and (-4, -4).                     3/2

d) (-2,3) and (4, -6).                    -3/2

4) Show that the following points are collinear:

a) (1,5),(3,14),(-1,-4).

b) (3,-4),(1,2),(2,1)

c) (3,-2),(-1,1),(-5,4)

d) (4,7),(-2,-5),(2,3)

5) If the slope of the line joining the points A(x,2) and B (6, -8) is -5/4, find the value of x.                         -2

6) Find y if the slope of the line joining (-8,11),(2,y) is -4/3.        -7/3

7) Show that the line through the points (5,6) and (2,3) is parallel to the line through the points (9,-2) and (6,-5).     

8) Find the value of x so that the line through (3,x) and (2,7) is parallel to the line through (-1,4) and (0,6).                                          9 

9) Show that the line through the point (-2,6) and (4,8) is perpendicular to the line through the points (3,-3) and (5,-9).   

10) If A(2,-5), B(-2,5), C(x,3) and D(1,1) be four points such that a AB and CD are perpendicular to each other, find the value of x.                6

11) Without using Pythagora's theorem, show that the points

a) A(1,2), B(4,5) and C(6,3) 

b) A(0,4), B(1,2) and C(3,3) are the vertices of a right-angled triangle.

12) Using slopes, find the value of x for which the points A(5,1), B(1,-1) and C(x,4) are collinear.                11

13) Find the value of x for which the points (x, -1),(2,1) and (4,5) are collinear.                                           1

14) Using slopes, show that point

a) A(-4,-1), B(-2,-4), C(4,0) and D(2,3)

b) (-4,-1),(-2,-4),(4,0),(2,3) 

 are taken in order, are the vertices of a rectangle.  

15) Using slopes prove that the points A(-2,-1), B(1,0), C(4,3) and D(1,2) are the vertices of a parallelogram.

16) If the points A(a,0), B(0,b) and P(x,y) are collinear, using slopes, prove that x/a + y/b = 1.

17) If three A(h,0), P(a,b) and B(0,k) lie on a line, show that: a/h + b/k = 1.

18) A line passes through the points A(4,-6) and B(-2,-5). Show that the line AB makes an obtuse angle with the x-axis.

19) The vertices of a quadrilateral are A(-4,2), B(2,6), C(8,5) and D(9,-7). Using slopes, show that the midpoints of the sides of the quadrilateral ABCD form a parallelogram.

20) Find the slope of the line which makes an angle 30° with the positive direction of the y-axis, measured anticlockwise.             -√3

21) 

22) Show that the points A(0,6), B(2,1) and C(7,3) are three corners of a square ABCD. find

a) the slope of the diagonal BD.  7/3

b) the co-ordinates of the fourth vertex D.                                       (5,8) 

23) A(1,1), B(7,3) and C(3,6) are the vertices of a ∆ABC. If D is the midpoint of the BC and AL perpendicular to BC. find the Slope of 

A) AD.                                            7/8

B) AL.                                            4/3

               EXERCISE -2

                 ----------------

1) Find the equation of a line parallel to the x-axis at a distance of 

a) 4 units above it.                  y-4= 0

b) 5 units below it.                  y+5= 0

2) Find the equation of a line parallel to the y-axis at a distance of 

a) 6 units to its right.             x- 6= 0

b) 3 units to the left.             x+ 3= 0

3)a) Find the equation of a line parallel to the x-axis and having intercept -3 on y-axis.              y+3= 0

b) Find the equation of the line parallel to x-axis and having intercept -2 on y-axis.                y= -2

c) Find the equation of the line parallel to x-axis and passing through (3,-5).                              y=-5

d) Find the equation of the line perpendicular to x-axis and having intercept -2 on x-axis.                x= -2

4) Find the equation of a horizontal line passing through the point (4,-2).                                        y+2 = 0

5)a) Find the equation of a vertical line passing through the point (-5, 6).                                              x+5= 0

b) Find the equation of the straight lines which pass through (4,3) and are respectively parallel and perpendicular to the x-axis.   y=3, x=4

6)a) Find the equation of a line which is equidistant from the lines x= -2 and x= 6.                           x= 2

b) Find the equation of a line equidistant from the lines y= 10 and y= -2.                                             y= 4

7) Find the equation of a line which is equidistant from the line y= 8 and y= -2.                                            y= 3

8) Find equation of a line

a) whose slope is 4 and which passes through the point (5,-7).    4x - y- 27= 0 

b) Whose slope is -3 and which passes through the point (-2,3). 3x+ y+3= 0

9) Find the equation of a line whose inclination with the x-axis is 30° and which passes through the point (0,5).                       x - √3 y + 5√3 = 0

10) find the equation of a line which cuts off intercept 5 on the x-axis and makes an angle 60° with the positive direction of x-axis.      √3 x - y -5√3= 0

11) Find the equation of the line passing through the point P(4,-5) and parallel to the line joining the points A(3,7) and B (-2,4).

12) find the equation of the line passing through the point P(-3,5) and perpendicular to the line passing through the points A(2,5) and B (-3,6).                  5x - y+20= 0 

13) find the slope and the equation of the line passing through the points

A) (3,-2) and (-5,-7).              5/8, 5x- 8y-31= 0

B) (-1,1) and (2,-4).               -5/3, 5x+ 3y+2= 0

C) (5,3) and (-5,-3).          3/5, 3x- 5y= 0

D) (a,b) and (-a,b).                   0, y= b

14) Find the angle which the line joining the points (1,√3) and (√2, √6) makes with the x-axis.           60°

15) Prove that the points A(1,4),.B(3,-2) and C (4,5) are collinear. also find the equation of the line on which these points lie.          3x+y= 7

16) If A(0, 0) B(2,4) and C(6,4) are the vertices of a ∆ABC, find the equations of its sides.         y= 4, 2x-3y= 0, 2x -y= 0.

17) If A(-1,6), B(-3,-9) and C(5,-8) are the vertices of a ∆ ABC, find the equation of its medians.     29x+4y+5= 0, 8x- 5y-21, 13x+ 14y+ 47= 0.

18) Find the equation of the perpendicular bisector of the line segment whose end points are A(10,4) and B(-4,9).          28x - 10y -19= 0

 

19) Find the equations of the altitudes of a ∆ABC, Whose vertices are A(2,-2), B(1,1) and C(-1,0).       2x+y -2= 0, 3x- 2y- 1= 0, x- 3y -1= 0

20) If A(4,3), B(0,0) and C (2,3) are the vertices of a ∆ABC, find the equation of the bisector of angle A. x- 3y+5= 0

21) The midpoints of the sides BC, CA and AB of a ∆ ABC are D(2,1), E(-5,7) and F(-5,-5) respectively. Find the equations of the sides of if 1423 are the third season of ∆ABC.          x- 2= 0, 6x - 7y +79= 0, 6x+ 7y +65= 0

22) If A(1,4), B(2,-3) and C(-1,-2) are the vertices of a∆ ABC, find the equation of

a) the median through A..       13x- y -9= 0

b) the altitude through A.          3x-y+1= 0

c) the perpendicular bisector of BC.       3x- y - 4= 0

             

                  EXERCISE -3

                    ---------------

1)Find the equation of the line which makes an angle of 30° with the positive direction of the x-axis and cuts off an intercept of 4 units with the negative direction of the y-axis.                           x - √3y - 4√3= 0

2) Find the equation of a straight line :

a) with slope 2 and y-intercept 3.     y= 2x+3

b) with slope -1/3 and y-intercept - 4.             x+ 3y+12= 0

c) with slop -2 and intersecting x- axis at a distance of 3 units to the left of the origin.    2x+ y+6= 0

d) Slope= 3, y-intercept=5.         3x-y+5= 0

e) Slope=-1, y-intercept= 4.     x+y-4= 0

f) Slope= -2/5, y-intercept= -3.     2x+ 5y+15= 0

3) Find the equation of the line cutting off an intercept -2 from the y-axis and inclined to the axes.     x- y - 2= 0, or x+ y +2= 0

5) Find the equation of a line that has y-intercept -4 and is parallel to the line joining (2,-5) and (1,2).     7x + y+4= 0

6) Find the equation of the line through the point (-1,5) and making an intercept of -2 on the y-axis.  7x+ y + 2= 0

7) find the equation of a line which is perpendicular to the line joining (4,2) and (3,5) and cuts off an intercept of length 3 on y- axis.       x - 3y+9= 0

8) find the equation of the perpendicular to the line segment joining (4,3) and (-1,1) if cuts off an intercept -3 from y-axis.     5x+ 2y+ 6= 0

9) find the equation of the straight line intersecting y-axis at a distance of 2 units above the origin and making an angle 30° with the positive direction of the x-axis.     x - √3 y+ 2√3= 0

10) Find the equation of the line which is parallel to the line 2x - 3y = 8 and whose y-intercept is 5 units.       2x- 3y +15= 0

11) Find the equation of the line which is perpendicular to the line x - 2y = -5 and passing through (0,3).       2x + y -3= 0     

12) Find the equation of the line passing through the point (2,3) and perpendicular to the line 4x +3y= 10.                         3x- 4y +6= 0

13) Find the equation of the line passing through the point (2,4) and perpendicular to the x-axis.      x = 2

14) Find the equation of the line that has x-intercept -3 and which is perpendicular to the line 3x +5y= 4.       5x- 3y + 15= 0

15) Find the equation of the line passing through the midpoint point of the line joining the point (6,4) and (4, -2) and perpendicular to the line 3x +2y= 8.              2x- 3y -7= 0

16) Find the equation of the line whose y-intercept is -3 and which is perpendicular to the line joining the points (-2,3) and (4,-5).             3x- 4y -12= 0

17) Find the equation of the line passing through (-3,5) and perpendicular to the line through the points (2,5) and (-3,6).    5x - y+20= 0

18) A line perpendicular to the line segment joining the points (1,0) and (2,3) divides it in the ratio 1:2. Find the equation of the line.    3x+ 9y-13= 0

MISCELLANEOUS-1 (A)

-------------------------------

1) Find the gradient of the line joining the pair of points:

a) (0,3),(4,5).                                1/2

b) (0,-2),(-2,4).                                -3

c) (1/2,3/2),(5/2,7/2).                    1

d) (√3+1,2),(√3+3,4).                      1

2) Find the Slope of a line through each of the pair of the following points:

a) (2,3) and (3,4).                             1

b) (2,-1) and (4,1).                            1

c) (-3,-2) and (-2,-1).                         1

d) (-5,-3) and (4,3).                       2/3

3) Find the inclination of the line joining the pair of points:

a) (1,2),(2,3).                                 45°

b) (-1,-3),(3,1).                               45°

4) If A(4,-3), B(6,5) and C(5,1) are three points, find the slope of AB and BC. Hence show that the points are collinear.                                 4, 4

5) Show that the following points are collinear:

a) A(5,-2), B(4,-1) and C(1,2). Hence find the inclination of the line AC.    

6)a) If (-5,a),(3,6) and (7,8) are collinear, find a.                                2

b) If A(2,a), B(3,-1) and C(4, -5) are collinear, find a.                               3

c) Find a, if the points (-2,3),(3,4),(a,5) are collinear.                            8

d) If the points (a,1),(1,2) and (0, b+1) are collinear, Show that 1/a + 1/b = 1.

7)a) If 2y - p²x= 3 and 2y - 4px +1= 0 are parallel, find the value of p.       4

b) If y - 2x = 3 and 2y = px +8 are parallel, find the value of p.             4

c) If 3(k -1)y - 6x = 2 and 4y - 8x +10= 0 are parallel, find the value of k.                                                       2

8)a) If (p+1)x + y= 3, and 3y - (p- 1)x= 4 are perpendicular, Find the value of p.                                       ±2

b) If 2my - 3x= 4 and 3my + 8x =10 are perpendicular to each other, find m.                                                      ±2

c) If y+ (2p+1)x +3= 0 and 8y- (2p -1)x= 5 are perpendicular, find the value of p.                                   ±3/2

9) State the slope (m) and y-intercept (c) of the line 2y = 4x - 3.             2, -3/3

10) Find the slope of a line

a) parallel 

b) perpendicular to to 2y= 3x+1.

     3/2, -2/3

MISCELLANEOUS-- 1(B)

-----------------------------

1) State the slope(m) and y-intercept(c) of the line:

a) y= x+1.                                       1,1

b) 2y= 4x+1.                                2,1/2

c) 3y= 6x -2.                              2, -2/3

d) 2y= - 3x -4.                            -3/1, 2

2) Find the Slope of a line which is parallel to:

a) 2y= x+2.                                     1/2,

b) 3y/4= 2x -2.                               8/3

3) Find the Slope of a line which is perpendicular to:

a) 3y= x+5.                                       -3

b) 2y/3= x -3.                              -2/3

4)a) Find the equation of the line y-intercept of 3, and a slope of 2.    y= 2x+ 3.      

b) Find the equation of the line y-intercept of 7, and a slope of 2.    y= 2x+ 7.      

c) Find the equation of the line with y-intercept of 4 and a slope of -3.        y= - 3x +4.      

d) Find the equation of the line with y-intercept of 1/2 and a slope of 2.       2y= 4x+1.     

e) Find the equation of the line with y-intercept of 5, and a slope of -2/3.     3 y= - 2x+15.      

f) Given, y-intercept of 2, and an inclination of 45° with the positive direction of the x-axis.        y= x+ 2.      

g) Find the equation of the line having y-intercept of 4 and which is equally inclined to the axes, in the second quadrant.                  y= x+ 4

5)a) Find the equation of a line through (0,3) and having slope = 4.     y= 4x +3

b) Find the equation of a line through (1,2) and having slope = 3.     y= 3x - 1

6)a) Find the equation of the line through (0,2) and parallel to y= 3x +2.                    y= - x+ 3.      

b) Find the equation of the line through (4,0) and parallel to 3y= 6x +2.                                         y= 2x - 8     

c) Find the equation of the line through (0,3) and parallel to 2y= x - 2.                                              2y= x -6

7) Find the equation of the line through (0,2) and perpendicular to

a) y= 2x+3.                    2y= - x+4.     

 

b) y= x/3 +2.                        y= - 3x+2

c) Find the equation of the line through (0,3) and perpendicular to 2y= x + 1.                      y= - 2x+ 3.      

8)a) Find the equation of the line through (0,2) and (-2,0).        y= x+2

b) Find the equation of the line through (3,0) and (0,3).         y= -x+3

c) Find the equation of the line through (2,1) and (4,3).         y= x -1

9)a) Find the equation of the line through (2,3) and (3,4) and y-intercept of 5 units.                y= x+ 5

b) Find the slope of the line joining the points (3,4) and (0,16). Hence or otherwise, write down the equation of this line.                  -4, y= - 4x+16

10) The equation of a line is y= 3x -5. Write down the slope of this line and intercept made by it on the y-axis. Hence or otherwise, write down the equation of a line which is parallel to this line and which passes through the point (0,5).      3, -5, y= 3x+5 

11) A(2,1), B(5,3), C(-1,3) are the vertices of the triangle ABC. Find

a) equation of the median AD. 2= x

b) the equation of the altitude BE.   2y= 3x -9     

c) the equation of the altitude CF.   2y= -3 x+ 3      

12) find the equation of the line through (-4,8) and parallel to x-axis.        y= 8

13) find the equation of the line through (3,5) and perpendicular to the axis.                                         y= 5

14) A(1,3) and C(6,8) are the opposite vertices of a square ABCD. Find the equation of the diagonal BD.                                     x +y = 9

15) Find the equation of the perpendicular bisector of the line segment joining (3,2) and (7,6).     x+ y = 9

16) A(-1,4) and B(5, -2) are two points. Find the equation of the perpendicular bisector of AB.     y + x -3= 0 

17) A straight line cuts off, on the axes of co-ordinates positive intercepts whose sum is 7. If the line passes through the point (-3,8), find its equation.          4x +3y= 12

18) A straight line cuts off, on the axes, positive intercepts whose sum is 7. If the line passes through the point (-8, 9), find its equation.    3x +4y= 12

19) The coordinates of the vertex A of a square ABCD are (1,2) and the equation of the diagonal BD is x + 2y= 10. Find the equation of the diagonal and the coordinates of the centre of the square.      y- 2x= 0; (2,4)

20) A, B are the points (0,6) and (10,0). O is origin, OM is a median and OP an altitude of triangle AOB. Find the equation of OM and OP.          5y- 3x= 0; 3y- 5x= 0

21) Find the equation of the line passing through the point (3,4) such that the portion between the axes is divided by P in the ratio 2:3.       y + 2x= 10

22) Find the equation of a line, which has the y-intercept of 5 and is parallel to the line 4x - 6y= 9. Find the coordinates of the point, where it cuts the x axis.              3y- 2x= 15, (-7.5,0)

              EXERCISE -4

               ----------------

1) Find the equation of the line which cuts off intercepts 

a) - 3 and 5                 5x - 3y +15= 0

b) -2, and 3.                3x - 2y +6= 0

c) -k/m and k.             mx - y + k= 0

d) 4 and -6 

on the x- axis and y- axis respectively.                3x - 2y -12= 0

2) Determine the x-intercept and y-intercept of the following:

a) 3x+ 5y -15= 0.                         5, 3

b) x - y - 7= 0.                             7, -7

3) Find the equation of the line that cuts off equal intercepts on the co-ordinate axes and passes through the point 

a) (4,7).                            x + y -11= 0

b) (2,3).           x+y-5= 0 or x- y+ 1= 0

4)a) Find the equation of the line which passes through the point (3,-5) and cut off intercept on the axes which are equal in magnitude but opposite in sign.        x - y -8= 0

b) Find the equation of the line which passes through the point (5, 6) and cut off intercept on the axes which are equal in magnitude but opposite in sign.        x - y +1 = 0

5) Find the equation of the line which makes an intercept of 2a on the x-axis and 3a on the y-axis. Given that the line passes through the point (14,9), find the numerical value of a.             3x+ 3y- 6a= 0, 4

6) Find the equation of the line passing through the point (2,2) and cutting off intercepts on the axes, whose sum is 9.     x + 2y -6= 0 or 2x +y -6 = 0

7) A straight line passing through (2,3) and the portion of the line intercepted between the axes is bisected at this point. Find the equation.                  3x + 2y -12= 0

8) Show that the three points (5,1),(1,-1) and (11,4) lie on a straight line. Further find

a) its intercepts on the axes.     3

b) the length of the portion of the line intercepted between the axes.  -3/2

c) the slope of the line.               1/2

9) Find the equation of the line  passing through the point (22,-6) and whose intercept on the x-axis exceeds the intercept on the y-axis by 5.     6x + 11y -66= 0 or x +2y - 10 = 0

10) Find the equation of the line whose portion intercepted between the axes is bisected at the point (3, -2).       2x - 3y - 12= 0 

11)a)  Find the equation of the line whose portion intercepted between the coordinates axes is divided at the point (5,6) in the ratio 3:1.     2x + 5y - 40= 0 

b) Find the equation of the line whose portion intercepted between the coordinates axes is divided at the point (3, -2) in the ratio 4: 3     3x + 4y - 1= 0 

12) A straight line passes through the point (-5,2) and the portion of the line intercepted between the axes is divided at this point in the ratio 2:3. Find the equation of the line.          3x - 5y + 25= 0

13)a) If the straight line x/a + y/b = 1 passes through the points (8,-9) and (12,-15), find the value of a and b.             2, 3

b) If the straight line x/a - y/b = 1 passes through the points (8, 6) and cuts off a triangle of area 12 units from the axes of coordinates. Find the equation of the straight line.      x/4 - y/6 = 1 and x/8 - y/3 = - 1 

                  EXERCISE- 5

                 ---------------------

1) Find the equation of the line for which:

a) p= 3 and φ= 45°.        x+y -3√2= 0

b) p= 5 and φ= 135°.      x-y +5√2= 0

c) p= 8 and φ= 150°.   √3 x-y +16= 0

d) p= 3 and φ= 225°.     x+y +3√2= 0

e) p= 2 and φ= 300°.      x- √3y -4= 0

f) p= 4 and φ= 180°.        x+ 4= 0

2) The length of the perpendicular segment from the origin to a line is 2 units and the inclination of this perpendicular is φ such that sin φ= 1/3 and φ is acute. Find the equation of the line.         2√2 x + y - 6= 0

3) Find the equation of the line which is at a distance of 3 units from the origin such that tan φ= 5/12, where φ is the acute angle which this perpendicular makes with the positive direction of x-axis.    12 x + 5y - 39= 0

               EXERCISE-6

               -----------------

1) Reduce the equation to slope intercept form, and find from it the slop and y+intercept:

A) 2x - 3y - 5= 0.         y= 2x/3 - 5/3, 2/3 and -5/3

B) 5x + 7y - 35= 0.     y= -5x/7 + 5, -5/7 and 5   

C) y+ 5= 0.        y= 0.x - 5, 0 and -5

d) 

2) Reduce the equation to intercept form. Hence, find the length of the portion of the line intercepted between the axes.

A) 3x - 4y+12= 0.       x/-4 + y/3 = 1, 5 units

B) 5x - 12y= 60.         x/12 + y/-5= 1, 13 units.

3) Find the inclination of the line:

a) x + √3 y +6= 0.                        150°

b) 3x + 3 y +8= 0.                       135°

c) √3 x - y - 4= 0.                           60°

4)a) Reduce the equation x+ y - √2= 0 to the normal form x cos φ + y sin φ = p, and hence find the value of φ, p.             x cos 45°+ y sin 45°= 1, φ= 45°, p= 1

b)  Reduce the equation x+ √3 y - 4= 0 to the normal form x cos φ + y sin φ = p, and hence find the value of φ, p.             x cos 60°+ y sin 60°= 2, φ= 60°, p= 2

c) Reduce the following equation to the normal form and find p and φ in each case:

i) x+ √3 y - 4= 0.                        2, π/3

ii) x+ y + √2 = 0.                      1, 225°

iii) x - y + 2 √2= 0.                   2, 135°

iv) x - 4= 0.                                   3, 0

v) y - 2 = 0.                                 2, π/2

5) Reduce each of the following equations to the normal form:

a) x+ y -2 = 0.       x cos 45°+ y sin 45°= √2

b) x+ y + √2 = 0.     x cos 225°+ y sin 225°= 1

c) x+ 5 = 0.       x cos 180°+ y sin 180°= 5

d) 2y - 3 = 0.       x cos 90°+ y sin 90°= 3/2

e) 4x+ 3y - 9 = 0.       x cos φ + y sin φ= p, where cos φ=4/5, sin φ=3/5 and p= 9/5

6) Reduce the equation √3 x + y +2= 0 to

a) slope-intercept form and find slope and y-intercept.           -√3, - 2

b) intercept form and find intercept on the axes.                         -2/√3, - 2

c) The normal form and find p and φ.                                             1, 210°

7) Put the equation x/a + y/b = 1 to the slope intercept form and find the slope and y-intercept.    -b/a, b

8) The perpendicular distance of a line from the origin is 5 units and its slope is - 1. Find the equation of the line.                               x + y - 5 √2= 0

9) Reduce the lines 3x - 4y +4= 0 and 3x + 4y -5 = 0 to the normal form and hence find which line is nearer to the origin.     3x - 4y +4= 0

10) Show that the origin is equidistant from the lines 4x + 3y + 10= 0; 5x - 12y + 26= 0 and 7x + 24y = 50.                

11) Find the values of φ and p, if the equation x cos φ + y sin φ = p is the normal form of the line √3 x + y +2= 0.                                             210°, 1

                  EXERCISE --7

                   -----------------

1) Write down the slopes of the following lines:

a) 2x+ 3y +1= 0.                           -2/3

b) 7x - 5y +8= 0.                            7/5

c) -11x - 6y = 0.                        -11/6

d) xx₁ + yy₁= a².                       -x₁/y₁

e) 3x + 4y - 2(x + x₁) - 5(y + y₁) + 2= 0.               1

2) Find the value of k such that the line (k - 2)x + (k + 3)y - 5 = 0 is

a) parallel to the line 2x - y +7= 0.   -4/3

b) perpendicular to it.                7

3) prove that the lines 

a) 3x + 4y - 7= 0 and 28x - 21y + 50= 0 are mutually perpendicular.    

b) px + qy - r = 0 and - 4px - 4qy + 5r = 0 are parallel.   

4)  find the slope of the line which is perpendicular to the line 7x + 11y - 2 = 0.         11/7 

5) Show that (2,-1) and (1,1) are on opposite sides of 3x + 4y = 6.

6) The sides of a Triangles are given by the equations 3x + 4y = 10, 4x - 3y = 5 and 7x + y+10=0; show that the origin lies within the triangle.

7) find by calculation whether the points (13,8), (26, -4) lie in the same, adjacent, or opposite angles formed by the straight lines 5x + 6y - 112=0, and 10x + 11y - 217=0.       Opposite

      

            EXERCISE --8

            ---------------------

1) Find the point of intersection of the following pairs of lines:

a) 2x - y+3= 0, x + y - 5= 0.     (2/13, 13/3)

b) 3x - 5y+ 5= 0, 2x + 3y - 22= 0.     (5,4)

c) 2x - 3y -7 = 0, 3x - 4y - 13= 0.     (11, 5)

d) bx + ay = ab, ax + by = ab.       (2ab/(a+b), ab/(a+b))

2) Find the coordinates of the vertices of a triangle, the equations of whose sides are:

a) x + y- 4= 0, 2x - y +3= 0, x -3 y+2= 0.     (1/3, 11/3),(-7/5,1/5),(5/2,3/2)

b) 

3) Find area of the triangle formed by the lines

a) y= 0, x = 2, x +2 y = 3.                 0

b) x + y- 6= 0, x - 3y -2= 0, 5x -3 y+2= 0.                                      12 sq. units

4) Find the equations of the medians of a triangle, the equations of whose sides are: 3x + 2y +6= 0, 2x - 5y +4 = 0, x -3 y - 6 = 0.      41x - 112y- 70= 0, 16x - 59y -120 = 0, 25x -53 y+ 50 = 0.       

5) Prove that the lines y= √3 x +1, y = 4, y = - √3 x +2 form an equilateral triangle.

6) Classify the following pairs of the lines are coincident, parallel or intersecting:

a) 2x + y- 1 = 0, 3x + 2y +5 = 0,   intersecting

b) x - y = 0, 3x - 3y + 5= 0.   Parallel

c) 3x + 2y- 4= 0, 6x + 4y - 8= 0.   Coincident

7) Find the equation of the line joining the points (3,5) to the point of intersection of the lines 4x + y- 1= 0, 7x - 3y - 35= 0.    12x - y- 31= 0

8) Find the equation of the line passing through the point of intersection of the lines 4x - 7y- 3= 0 and 2x - 3y +1 = 0 that has equal intercepts on the axes.    x + y +13 = 0

9) Show that the area of the triangle formed by the lines y = m ₁x, y= m₂x and y= c is equal to c²/4 (√33 + √11), m₁ , m₂ are the roots of the equation of x² (√3 + 2)x + √3 - 1= 0.      x+ y+13= 0

10) If the straight line x/a + y/b = 1 passes through the point of intersection of the lines x+ y - 3= 0 and  2x - 3y -1= 0 and is parallel to x- y- 6= 0, find a and b.              1, -1

b) Find the ortho-centre of the triangle whose angular points are (0,0),(2, -1),(-1,3).                     (-4,-3)

11) a) Find the orthocentre of the triangle is equations of whose sides are x+ y -1 = 0,  2x+ 3y = 6 and 4x- y+4 = 0.          (19/7, 18/7)

12) Three sides AB, BC, CA of a triangle ABC are 5x - 3y+ 2= 0, x - 3 y - 2= 0 and x+ y - 6 = 0 respectively. Find the equation of the altitude through the vertex A.    3x+ y -10= 0

13) Find the coordinates of the orthocentre of the triangle whose vertices are (-1,3),(2,-1) and (0, 0).   (-4,-3)

14) Find the coordinates of the incentre and centroid of the triangle whose sides have the equations 3x - 4 y = 0,  5x+ 12y = 0 and y -15= 0.  (-1, 8),(-16/3, 15)

15) Prove that the lines √3 x+ y= 0, x+ √3y = 0, √3 x + y = 1 and x+ √3 y= 1 form a rhombus.

16) The vertices of a triangle are A(0,5), B(-1, -2) and C(11,7). Write down the equations of BC and the perpendicular from A to BC and hence find the coordinates of the foot of the perpendicular.     3x -4y- 5= 0, 4x + 3y - 15; = 0, (3,1)

17) Find the equation of the line passing through the point of intersection of the two lines x + 2y + 3= 0, 3x + 4y +7= 0 and parallel to the straight line y - x = 8.    x- y= 0

              EXERCISE--9

             ---------------------

1) Prove that the following sets of three lines are concurrent:

a) 15x -18y +1= 0, 12x +10y -3= 0, 6x + 66y -11= 0.  

b) 3x - 5y - 11= 0, 5x + 3y -7 = 0, x -3 2y= 0.                             

c) x/a + y/b = 1, x/b +y/a = 2 , x = y.

d) (b+c) x + ay+  2= 0, (c+a)x + by +1= 0, (a+b) x - c y+ 1= 0                                

2)a)  For what value of K are the three lines  2x - 5y +3 = 0, 5x - 9y + K = 0, x - 2 y+ 1= 0 concurrent ?      4

b) For what value of m are the three lines x - y +1 = 0, 2(x+1) = y, y = mx + 3 concurrent.                               3 

c) For what value of m are the three lines 3x - 4y -13 = 0, 8x - 11y-33 = 0 and ,  2x - 3y + m= 0 are concurrent.                                      -7

3) If the three lines ax² + a²y +1 = 0, bx + b²y + 2 = 0, cx + c² y+ 1= 0 are concurrent, show that atleast two of three constants a, b, c are equal.

4) If a, b, c are in AP., Prove that the lines ax + 2y +2 = 0, bx + 3y + 2 = 0, cx + 4 y+ 1= 0 are concurrent.

5) Prove that the lines 5x + 3y -7 = 0, 3x - 4y = 10, x + 2 y = 0 meets in a point.

6) Show that the lines lx + my +n = 0, mx + ny + l = 0, x + ly+ m= 0 are concurrent if l+ m + n = 0.

7) Show that the lines x - y -6 = 0, 4x - 3y -20 = 0, 6x +5 y+ 8= 0 are concurrent. Also , find their common point of intersection.   (2,-4)

                EXERCISE-10

              ----------------------

1) a) Find the equation of a line passing through the point (2,3) and parallel to the line 3x - 4y +5= 0.    3x - 4y + 6= 0

b) Find the equation of a line passing through the point (4,5) and  is

i) parallel.                       3x - 3y - 2= 0

ii) perpendicular to the line 3x - 2y +5= 0.                           2x + y -20= 0

c) Find the equation of a line passing through the point (4,3) and is parallel to the line 3x - 4y +5= 0.       3x - 4y = 0

2)a) Find the equation of line passing through (3,-2) and perpendicular to the line x - 3y +5= 0.                                     3x +y -7=0 

b) Find the equation of a line passing through the point (4,3) and perpendicular to the line 3x - 4y +5= 0.                                     4x + 3y =  25

3) find the equation of the perpendicular bisector of the line joining the point (1,3) and (3,1).        x=y  

4) find the equation of the altitude of a ∆ ABC whose vertices are A(1,4), B(-3,2) and C(-5,-3).       2x + 5y -12= 0,  6x + 7y +4= 0, 2x +y +13 = 0

5) find the equation of a line which is perpendicular to the line √3 x - y +5= 0 and which cuts off intercept of 4 units with the negative direction of y-axis.            x + √3y + 4√3=0

6)  find the equation of a line perpendicular to the line √3 x -y +5 = 0 and at a distance of 3 units from the origin.      x + √3 y ± 6 =0 

7) Find the equation of the straight line through the point (a,b) and perpendicular to the line lx + my + n = 0.                      m(x -a) = l(y - b)= 0

8) Find the equation of the straight line perpendicular to 2x - 3y = 5 and cutting off an intercept 1 on the positive direction of x-axis.    8x + 2y - 3= 0

9)a)  Find the equation of the straight line perpendicular to 5x - 2y = 8 and which passes through the midpoint of the line segment (2,3) and (4,5).                 2x + 5y - 26 = 0

b) Find the equation of a line passing through the point of intersection of the straight line -x + y +7= 0, 2x + y - 2= 0.         4x+ 3y= 0

10) Find the equation of the line which has y-intercept equal to 4/3 and is perpendicular to 3x - 4y + 11= 0.                           4x + 3y - 4= 0

11)  Find the equation of the right bisector of the line segment joining the points on (a, b) and (a₁, b₁).   2x(a₁ - a) + 2y ((b₁ - b)+ (a² + b²) - (a₁² + b₁²) = 0

12) Find the image of the point (2,1) with respect to the line mirror x + y -  5= 0.                                         (4,3)

13) If the image of the point (2,1) with respect to the line mirror be (5,2), find the equation of the mirror.                         3x + y - 12= 0  

14) Find the equation oto the straight line parallel to 3x - 4y +6= 0 and passing through the middle point of the join of points (2,3) and (4, -1).                          3x - 4y - 5= 0

15) Prove that the line 2x - 3y +1 = 0, x + y - 4= 0, 2x - 3y - 2= 0 and x + y - 4= 0 form a parallelogram.

16) find the equation of a line drawn perpendicular to the line x/y + y/6 = 1 through the point where it meets the y-axis.    3x - 3y + 18= 0

17) The perpendicular from the origin to the line y = mx + c meets it at the point (-1,2). Find the values of m and c.                           1/2, 5/2

18) Find the equation of the right bisector of the line segment joining the points (3,4) and (-1,2).   2x + y - 5= 0 

19) The line through (h, 3) and (4,1) intersects the line 7x - 9y - 19= 0 at a right angle. Find the value of h.    22/9

20) find the image of the point (3,8) with respect to the lines  x + 3y - 7= 0 assuming the line to be a plane mirror.                                       (-1,-4)

21) find the coordinates of the foot of the perpendicular from the point (-1,3) to the line 3x - 4y - 16= 0.     (68/25, -49/25)

22) Find the projection of the point (1,0) on the line joining points (-1,2) and (5,4).                           (1/5,12/5)

23) find the equation of the straight line which cuts off intercepts on x-axis twice that on y-axis and is at a unit distance from the origin.   x + 2y ± √5 = 0.

24) The equation of perpendicular bisectors of the sides AB and AC of a triangle ABC are x - y + 5= 0 and x + 2y = 0 respectively. If the point A is (1,-2), find the equation of the line BC.                       14x + 23y - 40= 0

              EXERCISE -- 11

                --------------------

1) Find the distance of the point:

a) (3, -5) from 3x - 4y = 27.         2/5

b) (-2, 3) from  12x - 5y = 13.          4

c) (-4, 3) from 4(x +5)= 3(y -6).     13/5

d) (2, 3) from y = 4.                         1

e) (0, 0) from h(x+ h) + k(y +k) = 0.          √(h² + k²)

f) (4, ) from the line joining the points (4,1) and (2,3).              1/√2

2) Find the length of the perpendicular from the origin to each of the following lines:

a) 7x + 24y= 50.                      2 units

b) 4x + 3y= 9.                       9/5 units

c) x= 4.                                     4 units

d) the two points (a cos k, a sin k) and (a cos m, a sin m).                  a cos{(k -m)/2}

e) (cos k, sin k) and (cos m, sin m).   Cos {(k - m)/2}

3) Prove that the product of the lengths of particular drawn from the points A(√(a² - b²),0) and B(- √(a² - b²), to the line x/a cos k +y/b sink = 1 is b².

3) a) Find the values of k for which the length of the perpendicular from the point (4,1) on the line 3x - 4y +k = 0 is 2 units.                            2, -18

b) If p is the length of the perpendicular from the origin to the line x/a + y/b = 1, then prove 1/p² = 1/a² + 1/b².

c) If p and p' be the perpendicular from the origin upon the straight lines x sec k + y cosec k = a and x cos k - y sin k = a cos 2k. Prove that 4p² + p'² = a².

d) If the length of the perpendicular from the point (1,1) to line ax - by + c= 0 be unity, show that 1/c + 1/a - 1/b = c/2ab.

e) Show that the product of perpendiculars on the line x/a cos k + y/b sin k=1 from the points (±√(a² - b²),0) is b²

4) Show that the length of perpendicular from the point (7,0) to the line 5x + 12y= 9 is double the length of perpendicular to it from the point (2,1).    

5) The points A(2, 3), B(4,-1) and C(-1,2) are the vertices of ∆ABC. find the length of perpendicular from C on AB and hence find the area of ∆ABC.                    7/√5, 7

6) a) What are the points on the x-axis whose perpendicular distance from the line x/3 + y/4 =1 is 4 units.    (8,0) and (-2,0)

b) What are the points on the y-axis whose perpendicular distance from the line 4x - 3y=12 is 3 units.    (0,1) and (0,-9)

c) What are the points on the x-axis whose perpendicular distance from the line x/a + y/b =1 is a units.    [a/b (b ± √(a²+ b²) ,0) 

7) Find all the points on the line x + y= 4 that lie at a unit distance from the line 4x + 3y= 10.    (3,1),(-7,11)

8) The perpendicular distance of a line from the origin is 5 units and its slope is -1. find the equation of the line.   x + y= - 5√2 or x + y= 5 √2

9) Find the distance of the point of intersection of the lines 2x + 3y= 21 and 3x - 4y +11=0 from the line  8x + 6y +5= 0.                                59/10

10) Find the length of the perpendicular from the point (4,-7) to the line joining the origin and the point of intersection of the lines  2x - 3y +14= 0 and 5x + 4y= 7.             1

11) Find the distance of the point (1,2) from the straight line with Slope 5 and passing through the point of intersection of x+ 2y = 5 and x - 3y = 7.                132/√650

                  EXERCISE--12

                   --------------------

1) Determine the distance between the following pair of parallel lines:

a) 4x - 3y= -5 and 4x - 3y +7= 0.  2/5

b) 8x + 15y= 36 & 8x + 15y= -32.    4 

c) y= mx +c and y= mx +d.                              |d - c|/√(1+ m²)

d) p(x +y)+ q= 0 and p(x +y) - r = 0.           |q + r|/√(2p) 

2) The equation of two sides of a  square are 5x - 152y= 65 & 5x - 12y + 26=0. Find the area of the square.                                          49 

3) a) Find the equation of two straight lines which are parallel to x + 7y +2= 0 & at units distance from the point (2,-1).       x + 7y + 6 ± 5√2=0 

b) Find the equation of straight lines are parallel to 3x - 4y - 5= 0 at a units distance from it.       3x - 4y=0 or 3x - 4y= 10 

4) Prove that the lines 2x + 3y= 19 & 2x + 3y +7=0 are equidistant from the line 2x + 3y= 6.            

5) Find the equation of the line midway between the parallel lines 9x + 6y= 7 & 3x + 2y= 6.        18x + 12y +11= 0

6) Prove that the line 12x - 5y= 3 is mid parallel to the lines 12x - 5y= 7 and 12x - 5y= 13.

7) A vertex of a square is at the origin and its one side lies along the line 3x - 4y= 10. find the area of the square.                             4 sq.units

          EXERCISE --13

           ---------------------

1) Show that the area of the parallelogram formed by the lines 2x - 3y + a=0,  3x - 2y - a = 0, 2x - 3y + 3a =0and 3x - 2y - 2a= 0 is  2a²/5 square units.

2) prove that the area of the parallelogram formed by the x cos k + y sin k = p, x cos k + y sin k = q, x cos m + y sin m = r and x cos m + y sin m = d is ±(p + q)(r - s) cosec(k - m).

3) Prove that the four straight lines x/a + y/b =1,  x/b + y/a  = 2, x/a +y/b  =2 and x/a + y/a - 2=0 form a rhombus. find its area.       a²b²/|b² - a²| 

4) Show that the four lines ax ± b ± c = 0 encloses a rhombus whose area is 2c²/ab.

5) Prove that area of the parallelogram formed by the lines a₁x + b₁y + c₁ =0,  a₁x + b₁y + d₁= 0, a₂x + b₂y + c₂ =0,  a₂x +  b₂y + d₂ = 0 is |{(d₁ - c₁)(d₂ - c₂)}/(a₁ b₂ - a₂ b₁)| sq. units. Deduce the condition for these lines to form a rhombus.   P₁ P₂/sin k, (P₁ P₂are the distance between the pairs of parallel lines and k is the angle between two adjacent sides, for rhombus P₁ = P₂)

6) Prove that the area of the parallelogram formed by the lines 3x - 4y + a=0, 3x - 4y + 3a = 0, 4x - 3y - a =0 and 4x - 3y - 2a= 0 is 2a²/7 units.

7)  show that the diagonals of the parallelogram whose sides are lx + my + n =0,  lx + my + n' = 0, mx + ly + n =0and mx + ly + n' = 0  include an angle π/2.     (Use P₁ = P₂)