1. in fig,a circle touches the side bc of triangle abc at p and touches ab and ac produced at q and r respectively if aq= 5cm then the perimeter of triangle abc=?
2. the value of 6+6+6+...................is
3.(xb-c)1bybc (xc-a)1byca(xa-b)1byab=
4.a sector is cut from a circle exceeds the diameter by 16.8 cm,then the length of arc is?
5.in an A.P 3rd term is 5 and the 7th term is 9 then 11th term is?
6.in a cyclic quadrilateral ABCD,cosA+cosB+cosC+cosD=
7.the value of log [(tan1)tan2)(tan3)...................(tan89)]is
8.median of xby 5,xby4,xby2,xby3is 8.if x>0,then thevalueof x is
9. the roots of the equation a(b-c)x2+b(c-a)x+c(a-b)=0 are
Answers
Answered by
15
see diagram.
AQ = AR = lengths of tangents from point A to the ex-circle of triangle ABC.
BQ = BP = lengths tangents from B onto the excircle.
CR = CP = lengths of tangents from C onto the excircle.
AQ = AB + BQ = AB + BP
AR = AQ = AC + CR = AC + CP
AQ + AR = 2 AQ = AB + BP + PC + CA = perimeter of triangle ABC
So perimeter = 2 * 5 cm = 10 cm
===================================
2) 6 + 6 + 6 + .... n terms = 6 * n
3) (xb - c) /bc + (xc -a )/ca + (xa - b) / ab
= [ ax b - ac + x bc - ab + xac - bc ] / abc
= (x - 1) ( ab + bc + ca) / abc
4)
case 1) perimeter of the sector exceeds the diameter by 16.8 cm
(2 R + arc length) - 2 R = 16.8 cm => arc length = 16.8 cm
case 2) arc length exceeds
R Ф - 2 R = 16.8 cm.
there are two parameters R & Ф. we need more information to solve this.
==================
5) T3 = 5 = a + 2 d
T7 = a + 8 d = 9
subtract equations, 6 d = 4 => d = 2/3
=> a = 11/3
=====================
6) In a cyclic quadrilateral, sum of opposite angles is 180 deg.
Cos A + Cos C = 2 Cos (A+C)/2 Cos (A-C)/2 = 0
As (A+C)/2 = 90 deg
similarly Cos B + Cos D = 0 as (B+D)/2 = 90 deg.
hence the answer is 0.
=======================
7) tan 1 tan2 tan 3 ...... tan 88 tan 89
= tan1 tan2 tan 3 .... tan 43 tan 44 tan 45 cot 44 cot 43 .... cot 3 cot 2 cot 1
= tan 1 cot 1 tan 2 cot2 tan3 cot3 .... tan43 cot43 tan44 cot44 * 1
= 1
Log 1 = 0
==============
8)
x / 5 , x/4 x/3 , x/2
median is the average of x/4 and x/3
so x/4 + x/3 = 2 * 8 = 16
7 x / 12 = 16
x = 12 * 16 / 7
======================
9)
D = discriminant = b² (c-a)² - 4 ac (b-c) (a-b)
= b² (c² + a² - 2 a c) - 4 a c ( a b - b² - a c + b c)
= b² c² + a² b² - 2 a c b² - 4 a² b c + 4 a c b² + 4 a² c² - 4 a b c²
= b² c² + a² b² + 2 a c b² - 4 a² b c + 4 a² c² - 4 a b c²
= b² (a +c)² - 4 a c (a b - a c + b c)
= (a+c)² [ b² - 4 abc ] + 4 a² c²
x = [ - b (c - a) + - √ D ] / [ 2 a (b-c) ]
AQ = AR = lengths of tangents from point A to the ex-circle of triangle ABC.
BQ = BP = lengths tangents from B onto the excircle.
CR = CP = lengths of tangents from C onto the excircle.
AQ = AB + BQ = AB + BP
AR = AQ = AC + CR = AC + CP
AQ + AR = 2 AQ = AB + BP + PC + CA = perimeter of triangle ABC
So perimeter = 2 * 5 cm = 10 cm
===================================
2) 6 + 6 + 6 + .... n terms = 6 * n
3) (xb - c) /bc + (xc -a )/ca + (xa - b) / ab
= [ ax b - ac + x bc - ab + xac - bc ] / abc
= (x - 1) ( ab + bc + ca) / abc
4)
case 1) perimeter of the sector exceeds the diameter by 16.8 cm
(2 R + arc length) - 2 R = 16.8 cm => arc length = 16.8 cm
case 2) arc length exceeds
R Ф - 2 R = 16.8 cm.
there are two parameters R & Ф. we need more information to solve this.
==================
5) T3 = 5 = a + 2 d
T7 = a + 8 d = 9
subtract equations, 6 d = 4 => d = 2/3
=> a = 11/3
=====================
6) In a cyclic quadrilateral, sum of opposite angles is 180 deg.
Cos A + Cos C = 2 Cos (A+C)/2 Cos (A-C)/2 = 0
As (A+C)/2 = 90 deg
similarly Cos B + Cos D = 0 as (B+D)/2 = 90 deg.
hence the answer is 0.
=======================
7) tan 1 tan2 tan 3 ...... tan 88 tan 89
= tan1 tan2 tan 3 .... tan 43 tan 44 tan 45 cot 44 cot 43 .... cot 3 cot 2 cot 1
= tan 1 cot 1 tan 2 cot2 tan3 cot3 .... tan43 cot43 tan44 cot44 * 1
= 1
Log 1 = 0
==============
8)
x / 5 , x/4 x/3 , x/2
median is the average of x/4 and x/3
so x/4 + x/3 = 2 * 8 = 16
7 x / 12 = 16
x = 12 * 16 / 7
======================
9)
D = discriminant = b² (c-a)² - 4 ac (b-c) (a-b)
= b² (c² + a² - 2 a c) - 4 a c ( a b - b² - a c + b c)
= b² c² + a² b² - 2 a c b² - 4 a² b c + 4 a c b² + 4 a² c² - 4 a b c²
= b² c² + a² b² + 2 a c b² - 4 a² b c + 4 a² c² - 4 a b c²
= b² (a +c)² - 4 a c (a b - a c + b c)
= (a+c)² [ b² - 4 abc ] + 4 a² c²
x = [ - b (c - a) + - √ D ] / [ 2 a (b-c) ]
Attachments:
Answered by
3
1. in fig,a circle touches the side bc of triangle abc at p and touches ab and ac produced at q and r respectively if aq= 5cm then the perimeter of triangle abc=?
2. the value of 6+6+6+...................is
3.(xb-c)1bybc (xc-a)1byca(xa-b)1byab=
4.a sector is cut from a circle exceeds the diameter by 16.8 cm,then the length of arc is?
5.in an A.P 3rd term is 5 and the 7th term is 9 then 11th term is?
6.in a cyclic quadrilateral ABCD,cosA+cosB+cosC+cosD=
7.the value of log [(tan1)tan2)(tan3)...................(tan89)]is
8.median of xby 5,xby4,xby2,xby3is 8.if x>0,then thevalueof x is
9. the roots of the equation a(b-c)x2+b(c-a)x+c(a-b)=0 are
2. the value of 6+6+6+...................is
3.(xb-c)1bybc (xc-a)1byca(xa-b)1byab=
4.a sector is cut from a circle exceeds the diameter by 16.8 cm,then the length of arc is?
5.in an A.P 3rd term is 5 and the 7th term is 9 then 11th term is?
6.in a cyclic quadrilateral ABCD,cosA+cosB+cosC+cosD=
7.the value of log [(tan1)tan2)(tan3)...................(tan89)]is
8.median of xby 5,xby4,xby2,xby3is 8.if x>0,then thevalueof x is
9. the roots of the equation a(b-c)x2+b(c-a)x+c(a-b)=0 are
Similar questions