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

If \[\vec{a} = 5 \hat{i} - \hat{j} - 3 \hat{k} \text{ and } \vec{b} = \hat{i} + 3 \hat{j} - 5 \hat{k} ,\] then show that the vectors \[\vec{a} + \vec{b} \text{ and } \vec{a} - \vec{b} \] are orthogonal.

f(x) = x⁴ \[-\] 62x² + 120x + 9.

f(x) = x³\[-\] 6x² + 9x + 15

f(x) = (x - 1) (x + 2)².

`f(x) = 2/x - 2/x^2,  x>0`

f(x) = xe^x.

`f(x) = x/2+2/x, x>0 `.

`f(x) = (x+1) (x+2)^(1/3), x>=-2` .

`f(x)=xsqrt(32-x^2),  -5<=x<=5` .

f(x) = \[x^3 - 2a x^2 + a^2 x, a > 0, x \in R\] .

A unit vector \[\vec{a}\] makes angles \[\frac{\pi}{4}\text{ and }\frac{\pi}{3}\] with \[\hat{i}\] and \[\hat{j}\]  respectively and an acute angle θ with \[\hat{k}\] .  Find the angle θ and components of \[\vec{a}\] .

f(x) = \[x + \frac{a2}{x}, a > 0,\] , x ≠ 0 .

f(x) = \[x\sqrt{2 - x^2} - \sqrt{2} \leq x \leq \sqrt{2}\] .

If two vectors \[\vec{a} \text{ and } \vec{b}\] are such that \[\left| \vec{a} \right| = 2, \left| \vec{b} \right| = 1 \text{ and } \vec{a} \cdot \vec{b} = 1,\]  then find the value of \[\left( 3 \vec{a} - 5 \vec{b} \right) \cdot \left( 2 \vec{a} + 7 \vec{b} \right) .\] 

f(x) = \[x + \sqrt{1 - x}, x \leq 1\] .

f(x) = (x \[-\] 1) (x \[-\] 2)².

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 8\] 

`f(x)=xsqrt(1-x),  x<=1` .

If \[\vec{a}\] is a unit vector, then find \[\left| \vec{x} \right|\]  in each of the following. 

\[\left( \vec{x} - \vec{a} \right) \cdot \left( \vec{x} + \vec{a} \right) = 12\] 

f(x) = \[- (x - 1 )^3 (x + 1 )^2\] .