Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions for Mathematics

A tank with rectangular base and rectangular sides, open at the top, is to the constructed so that its depth is 2 m and volume is 8 m³. If building of tank cost 70 per square metre for the base and Rs 45 per square metre for sides, what is the cost of least expensive tank?

Let \[\vec{u,} \vec{v} \text{ and } \vec{w}\]  be vectors such \[\vec{u} + \vec{v} + \vec{w} = \vec{0} .\] If \[\left| \vec{u} \right| = 3, \left| \vec{v} \right| = 4 \text{ and } \left| \vec{w} \right| = 5,\] then find \[\vec{u} \cdot \vec{v} + \vec{v} \cdot \vec{w} + \vec{w} \cdot \vec{u} .\]

A window in the form of a rectangle is surmounted by a semi-circular opening. The total perimeter of the window is 10 m. Find the dimension of the rectangular of the window to admit maximum light through the whole opening.

A large window has the shape of a rectangle surmounted by an equilateral triangle. If the perimeter of the window is 12 metres find the dimensions of the rectangle will produce the largest area of the window.

Show that the height of the cylinder of maximum volume that can be inscribed a sphere of radius R is \[\frac{2R}{\sqrt{3}} .\]

Let \[\vec{a} = x^2 \hat{i} + 2 \hat{j} - 2 \hat{k} , \vec{b} = \hat{i} - \hat{j} + \hat{k} \text{ and } \vec{c} = x^2 \hat{i} + 5 \hat{j} - 4 \hat{k}\] be three vectors. Find the values of x for which the angle between \[\vec{a} \text{ and } \vec{b}\ \]  is acute and the angle between \[\vec{b} \text{ and } \vec{c}\] is obtuse.

A rectangle is inscribed in a semi-circle of radius r with one of its sides on diameter of semi-circle. Find the dimension of the rectangle so that its area is maximum. Find also the area ?

Prove that a conical tent of given capacity will require the least amount of  canavas when the height is \[\sqrt{2}\] times the radius of the base.

Find the values of x and y if the vectors \[\vec{a} = 3 \hat{i} + x \hat{j} - \hat{k} \text{ and } \vec{b} = 2 \hat{i} + \hat{j} + y \hat{k}\] are mutually perpendicular vectors of equal magnitude. 

If \[\vec{a}\] \[\vec{b}\]  are two vectors such that \[\left| \vec{a} + \vec{b} \right| = \left| \vec{b} \right|\] then prove that \[\vec{a} + 2 \vec{b}\] is perpendicular to \[\vec{a}\] 

An isosceles triangle of vertical angle 2 \[\theta\] is inscribed in a circle of radius a. Show that the area of the triangle is maximum when \[\theta\] = \[\frac{\pi}{6}\] .

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is \[6\sqrt{3}\]r. 

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible when revolved about one of its sides ?

Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm ?

A closed cylinder has volume 2156 cm³. What will be the radius of its base so that its total surface area is minimum ?

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius \[5\sqrt{3 cm} \text { is }500 \pi  {cm}^3 .\]

Show that among all positive numbers x and y with x² + y² =r², the sum x+y is largest when x=y=r \[\sqrt{2}\] .

Find the point on the curve y² = 4x which is nearest to the point (2,\[-\] 8).