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

The vector (cos α cos β) \[\hat{i}\] + (cos α sin β) \[\hat{j}\] + (sin α) \[\hat{k}\]  is a 

If the position vectors of P and Q are \[\hat{i} + 3 \hat{j} - 7 \hat{k} \text{ and } 5 \text{i} - 2 \hat{j} + 4 \hat{k}\] then the cosine of the angle between \[\vec{PQ}\] and y-axis is 

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then which of the following values of \[\vec{a} . \vec{b}\] is not possible? 

If the vectors `hati - 2xhatj + 3 yhatk and hati + 2xhatj - 3yhatk` are perpendicular, then the locus of (x, y) is ______.

The vector component of \[\vec{b}\] perpendicular to \[\vec{a}\] is 

What is the length of the longer diagonal of the parallelogram constructed on \[5 \vec{a} + 2 \vec{b} \text{ and } \vec{a} - 3 \vec{b}\] if it is given that \[\left| \vec{a} \right| = 2\sqrt{2}, \left| \vec{b} \right| = 3\] and the angle between \[\vec{a} \text{ and } \vec{b}\] is π/4? 

If \[\vec{a}\] is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λ \[\vec{a}\] is a unit vector if 

If θ is the angle between two vectors `veca` and `vecb` then, `veca * vecb` ≥ 0, only when

The values of x for which the angle between \[\vec{a} = 2 x^2 \hat{i} + 4x \hat{j} + \hat{k} , \vec{b} = 7 \hat{i} - 2 \hat{j} + x \hat{k}\]  is obtuse and the angle between \[\vec{b}\] and the z-axis is acute and less than \[\frac{\pi}{6}\]  are

If \[\vec{a} , \vec{b} , \vec{c}\] are any three mutually perpendicular vectors of equal magnitude a, then \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] is equal to 

If the vectors \[3 \hat{i} + \lambda \hat{j} + \hat{k} \text{ and } 2 \hat{i} - \hat{j} + 8 \hat{k}\] are perpendicular, then λ is equal to 

The projection of the vector \[\hat{i} + \hat{j} + \hat{k}\] along the vector of \[\hat{j}\] is

The vectors \[2 \hat{i} + 3 \hat{j} - 4 \hat{k}\] and \[a \hat{i} + \hat{b} j + c \hat{k}\] are perpendicular if 

If \[\left| \vec{a} \right| = \left| \vec{b} \right|, \text{ then } \left( \vec{a} + \vec{b} \right) \cdot \left( \vec{a} - \vec{b} \right) =\]

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors inclined at an angle θ, then the value of \[\left| \vec{a} - \vec{b} \right|\] 

If \[\vec{a} \text{ and } \vec{b}\] are unit vectors, then the greatest value of \[\sqrt{3}\left| \vec{a} + \vec{b} \right| + \left| \vec{a} - \vec{b} \right|\] 

If the angle between the vectors \[x \hat{i} + 3 \hat{j}- 7 \hat{k} \text{ and } x \hat{i} - x \hat{j} + 4 \hat{k}\] is acute, then x lies in the interval 

If \[\vec{a} \text{ and } \vec{b}\] are two unit vectors inclined at an angle θ, such that \[\left| \vec{a} + \vec{b} \right| < 1,\] then 

Let \[\vec{a} , \vec{b} , \vec{c}\] be three unit vectors, such that \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] =1 and \[\vec{a}\] is perpendicular to \[\vec{b}\]  If \[\vec{c}\] makes angles α and β with \[\vec{a} and \vec{b}\] respectively, then cos α + cos β =

The orthogonal projection of \[\vec{a} \text{ on } \vec{b}\] is