@pyashraj thanks :)
@incognitos -sir how did u reach these values??sorry if it sounds like a lame question,but i need to know
@JoyNick said:how many integral values of a are there such that quadratic expression (x+a)(x+1991) + 1 can be factored as a product (x+b)(x+c) where b and c are integers
Given, (x+a)(x+1991) + 1 =0
=>x^2+x(1991+a)+(1991a+1)=0
Now, x = {-(1991+a)+/- _/(1991+a)^2-4(1991a+1)}/2
=>{-(1991+a)+/- _/(1991-a)^2-4)}/2
=>{-(1991+a)+/- _/(1991-a)^2-4)}/2
=>{-(1991+a)+/- _/(1989-a)(1993-a)}/2
=>(1989-a)*(1993-a) must be a perfect square.
Thus, k*(k+4) has to be a Perfect Square..or, (k+2)^2 - 2^2 has to be Perfect Square..
Now, No 2 squares will ever give a difference of 4..
Thus, there are only two values of a as 1989 and 1993..At which the D becomes zero.. :)
@falcao said:@incognitos -sir how did u reach these values??sorry if it sounds like a lame question,but i need to know
there is no fixed method but in most of the cases discriminant is the starting point....
@JoyNick said:how many integral values of a are there such that quadratic expression (x+a)(x+1991) + 1 can be factored as a product (x+b)(x+c) where b and c are integers
a^2 + (1991+a)x +1991a + 1 = y
now y havv integer roots when D perfect square
m^2 = (1991+a)^2 - 1991*4a - 4
for real roots, D>0
we can get range...
(a-1993)(a-1989) > 0
so a cud be out of 1990,1991,1992
out of these only 1991 will give us a prefect square but -4 so nhn chalega..,also at a=1993,1989 we get D=0..hence that will also suffice..hence 2 values...in total ?
@Crysis /\
@JoyNick -u can do it like this
(x+a)(x+1991)+1=(x+b)(x+c)
x^2+(1991+a)x+1991a+1=x^2+(b+c)x+bc
b+c=1991+a, bc=1991a+1
put the value of b in equation 1,
c^2-(1991+a)c+(1991a+1)=0
now for the values to be integral the discriminant should be 0
a^2-2*1991a+(1991)^2-4=0
(a-1991)^2-2^2=0
(a-1989)(a-1993)=0
so a=1989 and 1993
(x+a)(x+1991)+1=(x+b)(x+c)
x^2+(1991+a)x+1991a+1=x^2+(b+c)x+bc
b+c=1991+a, bc=1991a+1
put the value of b in equation 1,
c^2-(1991+a)c+(1991a+1)=0
now for the values to be integral the discriminant should be 0
a^2-2*1991a+(1991)^2-4=0
(a-1991)^2-2^2=0
(a-1989)(a-1993)=0
so a=1989 and 1993
Let N be a number in base 16,but contains only 3 zeros and 3 ones.Find the maximum number of zeros,when the number is denoted in base 2.
Karan sells a chair at a loss of 10%. If he had sold at a profit of 5%, he would have earned Rs. 75. What is the cost price of the chair??.
plz give explanation......
@gudda1122 said:Let N be a number in base 16,but contains only 3 zeros and 3 ones.Find the maximum number of zeros,when the number is denoted in base 2.
18 ????
let the number in base 16 be abcdef where a, b, c, d, e, f are 0 or 1.
so the number in decimal system is..... a.16^5 + b.16^4 + c.16^3 + d.16^2 + e.16^1 +f.16^0 .
this can be written as.... a.2^20 + b.2^16 + c.2^12 + d.2^8 + e.2^4 +f.2^0 .....
as a,b,c,d,e,f are either 0 or 1, the above written number is decimal representation a number in base 2 and digits a,b,c,d,e,f at 21st, 17th, 13th, 9th, 5th, 1st places and 0's at remaining places.
if we want to find maximum zeroes then 'a' has to be 1....we can assume b, c to be 1 and d,e,f to be 0.
so there are 3 zeroes between 21 and 17, 3 between 17 and 13, 12 zeroes after 13...total 18
let the number in base 16 be abcdef where a, b, c, d, e, f are 0 or 1.
so the number in decimal system is..... a.16^5 + b.16^4 + c.16^3 + d.16^2 + e.16^1 +f.16^0 .
this can be written as.... a.2^20 + b.2^16 + c.2^12 + d.2^8 + e.2^4 +f.2^0 .....
as a,b,c,d,e,f are either 0 or 1, the above written number is decimal representation a number in base 2 and digits a,b,c,d,e,f at 21st, 17th, 13th, 9th, 5th, 1st places and 0's at remaining places.
if we want to find maximum zeroes then 'a' has to be 1....we can assume b, c to be 1 and d,e,f to be 0.
so there are 3 zeroes between 21 and 17, 3 between 17 and 13, 12 zeroes after 13...total 18
@tmohan02 said:Karan sells a chair at a loss of 10%. If he had sold at a profit of 5%, he would have earned Rs. 75. What is the cost price of the chair??.plz give explanation......
c be the c.p
1.05 cp=cp+75
.05cp=75
cp=1500
i dont think first statement is needed
1.05 cp=cp+75
.05cp=75
cp=1500
i dont think first statement is needed
one side of a right angled triangle is 8 cm.What is the length of the hypotenuse if the area of the traingle is 24 sq cm. my doubt is if lets say it is given one side is 80 cm then how to find which side is it ?
Directions for Qs. 1 to 2: Refer to the following data and answer the following questions.
A B C
D
E F G
H
I
D
E F G
H
I
Each of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 is represented by a different letter in the figure.
A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.
A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.
1. Which of the digits does E represent?
(1) 9 (2) 4 (3) 7 (4) 1
(1) 9 (2) 4 (3) 7 (4) 1
2. Which of the digits does D represent?
(1) 8 (2) 7 (3) 3 (4) 7 or 8
(1) 8 (2) 7 (3) 3 (4) 7 or 8
@nole said:one side of a right angled triangle is 8 cm.What is the length of the hypotenuse if the area of the traingle is 24 sq cm. my doubt is if lets say it is given one side is 80 cm then how to find which side is it ?
1)we can assume that this 8 cm is one of the non-hyp side and say 8*b/2 = 24 =>b=6 =>hyp = rt(8^ + 6^2) = 10cm
2)we can assume that this 8 cm is the hyp and find the other sides(just to be sure if it is a meaningful assumption or not)
a^2 + b^2 = 64 -->(1)
and ab/2 = 24 ==> ab=48-->(2)
these 2 equations will lead to the imaginary values for a and b....
so i will go with the 1st assumption and hyp = 10cm
i didn't clearly understand ur doubt....y would u assume the side when that can be derived?
2)we can assume that this 8 cm is the hyp and find the other sides(just to be sure if it is a meaningful assumption or not)
a^2 + b^2 = 64 -->(1)
and ab/2 = 24 ==> ab=48-->(2)
these 2 equations will lead to the imaginary values for a and b....
so i will go with the 1st assumption and hyp = 10cm
i didn't clearly understand ur doubt....y would u assume the side when that can be derived?
@nole said:one side of a right angled triangle is 8 cm.What is the length of the hypotenuse if the area of the traingle is 24 sq cm. my doubt is if lets say it is given one side is 80 cm then how to find which side is it ?
It is not needed... the side can either be a base or height coz we have to find the hypotenuse...
so Area= 0.5*(8) (x)= 24
x= 6
Hypotenuse = 10
Incase of 80 its can either be the hypotenuse or one of the sides...
@shadowwarrior my doubt was in questions generally it is given " in a right angled triangle.one side is 'x' cm and area of triangle is 'y' cm. " i wanted to know how to find whether side given to us is base,hypotenuse or perpendicular ?
I think u said there are two cases one in which that side is hypotenuse and other in which either base or perpendicular.we just need to find which case is it .right ?
I think u said there are two cases one in which that side is hypotenuse and other in which either base or perpendicular.we just need to find which case is it .right ?
@nole said:@shadowwarrior my doubt was in questions generally it is given " in a right angled triangle.one side is 'x' cm and area of triangle is 'y' cm. " i wanted to know how to find whether side given to us is base,hypotenuse or perpendicular ?I think u said there are two cases one in which that side is hypotenuse and other in which either base or perpendicular.we just need to find which case is it .right ?
yes
@paridhi11890 said:Directions for Qs. 1 to 2: Refer to the following data and answer the following questions.A B C D E F G H IEach of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 is represented by a different letter in the figure.A + B + C, C + D + E, E + F + G and G + H + I is equal to 13.1. Which of the digits does E represent?(1) 9 (2) 4 (3) 7 (4) 12. Which of the digits does D represent?(1) 8 (2) 7 (3) 3 (4) 7 or 8
this seems weird:
i added all the given ones: A+B+D+F+H+2(C+E+G+I) = 13*4 = 52
and A+B+C+D+E+F+G+H+I = 1+2+..9 = 45
diffing these 2, C+E+G+I = 52-45 = 7
and there are no 4 integers in the set {1,2,...9} such that their sum is 7.....am i going wrong somewhere...?
i added all the given ones: A+B+D+F+H+2(C+E+G+I) = 13*4 = 52
and A+B+C+D+E+F+G+H+I = 1+2+..9 = 45
diffing these 2, C+E+G+I = 52-45 = 7
and there are no 4 integers in the set {1,2,...9} such that their sum is 7.....am i going wrong somewhere...?
@shadowwarrior said:this seems weird:i added all the given ones: A+B+D+F+H+2(C+E+G+I) = 13*4 = 52and A+B+C+D+E+F+G+H+I = 1+2+..9 = 45diffing these 2, C+E+G+I = 52-45 = 7and there are no 4 integers in the set {1,2,...9} such that their sum is 7.....am i going wrong somewhere...?
4+2+1=7 :)