PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Find \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 12 \text{ and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

Show that \[\frac{\log x}{x}\] has a maximum value at x = e ?

Find  \[\left| \vec{a} \right| \text{ and } \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 8 \text{ and } \left| \vec{a} \right| = 8\left| \vec{b} \right|\]

Find \[\left| \vec{a} \right| and \left| \vec{b} \right|\] if 

\[\left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) = 3\text{  and } \left| \vec{a} \right| = 2\left| \vec{b} \right|\]

Find the maximum and minimum values of the function f(x) = \[\frac{4}{x + 2} + x .\]

Find \[\left| \vec{a} - \vec{b} \right|\] if 

\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 5 \text{ and } \vec{a} \cdot \vec{b} = 8\]

Find \[\left| \vec{a} - \vec{b} \right|\]  

\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 4 \text{ and } \vec{a} \cdot \vec{b} = 1\] 

Find the maximum and minimum values of y = tan \[x - 2x\] .

Find \[\left| \vec{a} - \vec{b} \right|\] if  

\[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 4\]

If f(x) = x³ + ax² + bx + c has a maximum at x = \[-\] 1 and minimum at x = 3. Determine a, b and c ?

Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\] if 

\[\left| \vec{a} \right| = \sqrt{3}, \left| \vec{b} \right| = 2 \text{ and } \vec{a} \cdot \vec{b} = \sqrt{6}\] 

Prove that f(x) = sinx + \[\sqrt{3}\] cosx has maximum value at x = \[\frac{\pi}{6}\] ?

Find the angle between two vectors \[\vec{a} \text{ and } \vec{b}\]  

\[\left| \vec{a} \right| = 3, \left| \vec{b} \right| = 3 \text{ and } \vec{a} \cdot \vec{b} = 1\]

f(x) = 4x \[-\] \[\frac{x^2}{2}\] in [ \[-\] 2,4,5] .

f(x) = (x \[-\] 1)² + 3 in [ \[-\] 3,1] ?

`f(x) = 3x^4 - 8x^3 + 12x^2- 48x + 25 " in "[0,3]` .

f(x) = (x \[-\] 2) \[\sqrt{x - 1} \text { in  }[1, 9]\] .

Find the maximum value of 2x³\[-\] 24x + 107 in the interval [1,3]. Find the maximum value of the same function in [ \[-\] 3, \[-\] 1].

Find the absolute maximum and minimum values of the function of given by \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .