PUC Science कक्षा ११ - Karnataka Board PUC Question Bank Solutions

Check the validity of the statement:

If a is the G.M. of 2 and \[\frac{1}{4}\] , find a.

 statement are true and false? In each case give a valid reason for saying so

 p : Each radius of a circle is a chord of the circle.

 statement are true and false? In each case give a valid reason for saying so

q : The centre of a circle bisects each chord of the circle.

 statement are true and false? In each case give a valid reason for saying so

r : Circle is a particular case of an ellipse.

 statement are true and false? In each case give a valid reason for saying so

 s : If x and y are integers such that x > y, then − x < − y.

 statement are true and false? In each case give a valid reason for saying so

 t :  \[\sqrt{11}\]  is a rational number. 

Determine whether the argument used to check the validity of the following statement is correct:
p : "If x² is irrational, then x is rational"
The statement is true because the number x² = π² is irrational, therefore x = π is irrational.

If AM and GM of two positive numbers a and b are 10 and 8 respectively, find the numbers.

If the A.M. of two positive numbers a and b (a > b) is twice their geometric mean. Prove that:

\[a : b = (2 + \sqrt{3}) : (2 - \sqrt{3}) .\]

If one A.M., A and two geometric means G₁ and G₂ inserted between any two positive numbers, show that \[\frac{G_1^2}{G_2} + \frac{G_2^2}{G_1} = 2A\]

\[\lim_{x \to 0} \frac{\sqrt{1 + x + x^2} - 1}{x}\]

\[\lim_{x \to 0} \frac{2x}{\sqrt{a + x} - \sqrt{a - x}}\] 

\[\lim_{x \to 0} \frac{\sqrt{a^2 + x^2} - a}{x^2}\] 

\[\lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{2x}\]

\[\lim_{x \to 2} \frac{\sqrt{3 - x} - 1}{2 - x}\]