Q21 of 180 Page 1

Show that (2 + √3) is an irrational number.

Let’s assume that is a rational number.


We know that, rational number is represented in the form of where p and q are integers and q ≠ 0.


⇒ , where a and b are coprime and b ≠ 0


⇒


⇒


Here, a and b are integers, therefore must also be integer.


Right hand side of the above equation is integer, hence left side must also be integer.


But, above statement contradicts the fact that is irrational. This contradiction arises because of our wrong initial assumption that is a rational number.


Hence, is an irrational number.


More from this chapter

All 180 →
19

Consider the points P(6, 4) and Q(–5, –3). Draw QS perpendicular to the x-axis. Also draw a perpendicular PT from the point P on QS (extended) to meet y-axis at the point R.


(i) Coordinates of point T is


(a) (-5, 5)


(b) (-5, 4)


(c) (-4, 5)


(d) (4, -5)


(ii) Equation of line PT is


(a) x = 4


(b) x + y = 0


(c) y = 4


(d) x – y = 0


(iii) Distance between points P and Q is


(a) 14 units


(b) 15 units


(c) 13.5 units


(d) None of the above


(iv) If S be any point in the 4th quadrant such that PTQS is the rectangle, then coordinates of point S is


(a) (6, -3)


(b) (5, -4)


(c) (6, -4)


(d) None of the above


 20

A survey regarding the heights (in cm) of 50 girls of a class was conducted and the following data was obtained


(i) Mean height (in cm) of the girls is


(a) 146.8


(b) 147.8


(c) 148.8


(d) 149.8


(ii) The commutative frequency table is useful in determining the


(a) Mean


(b) Median


(c) Mode


(d) All of these


(iii) Lower limit of the median class is


(a) 130


(b) 140


(c) 150


(d) None of the above


(iv) Sum of lower limit of median class and modal class is


(a) 270


(b) 290


(c) 310


(d) None of these


 22

The line segment XY is parallel to side AC of ΔABC and it divides the triangle into two parts of equal area. Prove that AX : XB = (√2 – 1) : 1.

OR


In a trapezium ABCD, O is the point of intersection of AC and BD, AB || CD and AB = 2 × CD. If the area of ΔAOB is 84 cm2, find the area of ΔCOD.


 23

If a 1.5 m tall girl stands at a distance of 3 m from a lamp-post casts a shadow of length 4.5 m on the ground then find the height of the lamp-post.