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

Two sides of a triangle have lengths 'a' and 'b' and the angle between them is \[\theta\]. What value of \[\theta\] will maximize the area of the triangle? Find the maximum area of the triangle also.  

If \[\vec{a,} \vec{b,} \vec{c}\]  are three non-coplanar vectors, such that \[\vec{d} \cdot \vec{a} = \vec{d} \cdot \vec{b} = \vec{d} \cdot \vec{c} = 0,\] then show that \[\vec{d}\] is the null vector.

A square piece of tin of side 18 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. What should be the side of the square to be cut off so that the volume of the box is maximum? Find this maximum volume.

If a vector \[\vec{a}\] is perpendicular to two non-collinear vectors \[\vec{b} \text{ and } \vec{c} , \text{ then show that } \vec{a}\] is perpendicular to every vector in the plane of \[\vec{b} \text{ and } \vec{c} .\] 

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?

If \[\vec{a} + \vec{b} + \vec{c} = \vec{0} ,\] show that the angle θ between the vectors \[\vec{b} \text{ and } \vec{c}\] is given by  \[\frac{\left| \vec{a} \right|^2 - \left| \vec{b} \right|^2 - \left| \vec{c} \right|^2}{2\left| \vec{b} \right| \left| \vec{c} \right|} .\]

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 ?