Science (English Medium) Class 12 - CBSE Question Bank Solutions

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 \[f(x) = \cos^2 x + \sin x, x \in [0, \pi]\] .

Express the vector \[\vec{a} = 5 \text{i} - 2 \text{j} + 5 \text{k}\] as the sum of two vectors such that one is parallel to the vector \[\vec{b} = 3 \text{i} + \text{k}\]  and other is perpendicular to \[\vec{b}\]

Find the absolute maximum and minimum values of a function f given by `f(x) = 12 x^(4/3) - 6 x^(1/3) , x in [ - 1, 1]` .

Find the absolute maximum and minimum values of a function f given by f(x) = 2x³ − 15x² + 36x + 1 on the interval [1, 5].

If \[\vec{a} \text{ and } \vec{b}\] are two vectors of the same magnitude inclined at an angle of 30°, such that \[\vec{a} \cdot \vec{b} = 3, \text{ find } \left| \vec{a} \right|, \left| \vec{b} \right| .\] 

How should we choose two numbers, each greater than or equal to `-2, `whose sum______________ so that the sum of the first and the cube of the second is minimum?

Express \[2 \hat{i} - \hat{j} + 3 \hat{k}\] as the sum of a vector parallel and a vector perpendicular to \[2 \hat{i} + 4 \hat{j} - 2 \hat{k} .\]