English Medium कक्षा १० - CBSE Question Bank Solutions for Mathematics

The values of k for which the quadratic equation  \[16 x^2 + 4kx + 9 = 0\]  has real and equal roots are

Find the number of metallic circular discs with a 1.5 cm base diameter and of height  0.2 cm to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

How many spherical lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead piece with dimension  6cm \[\times\] 42cm \[\times\] 21 cm.

A solid cuboid of iron with dimensions 53 cm ⨯ 40 cm ⨯ 15 cm is melted and recast into a cylindrical pipe. The outer and inner diameters of pipe are 8 cm and 7 cm respectively. Find the length of pipe.

The largest cone is curved out from one face of solid cube of side 21 cm. Find the volume of the remaining solid. 

Three solid spheres of radii 3, 4 and 5 cm respectively are melted and converted into a single solid sphere. Find the radius of this sphere.

The inner and outer radii of a hollow cylinder are 15 cm and 20 cm, respectively. The cylinder is melted and recast into a solid cylinder of the same height. Find the radius of the base of new cylinder.

A cylindrical bucket 28 cm in diameter and 72 cm high is full of water. The water is emptied into a rectangular tank 66 cm long and 28 cm wide. Find the height of the water level in the tank.

A solid metallic sphere of diameter 28 cm is melted and recast into a number of smaller cones, each of diameter  4 \[\frac{2}{3}\] cm and height 3 cm. Find the number of cones so formed.

Show that the points (−4, −1), (−2, −4) (4, 0) and (2, 3) are the vertices points of a rectangle.

 Prove hat the points A (2, 3) B(−2,2) C(−1,−2), and D(3, −1) are the vertices of a square ABCD.

A solid is composed of a cylinder with hemispherical ends. If the length of the whole solid is 108 cm and the diameter of the cylinder is 36 cm, find the cost of polishing the surface at the rate of 7 paise per cm² .

If the point P(x, 3) is equidistant from the point A(7, −1) and B(6, 8), then find the value of x and find the distance AP.   

If A(3, y) is equidistant from points P(8, −3) and Q(7, 6), find the value of y and find the distance AQ. 

If (0, −3) and (0, 3) are the two vertices of an equilateral triangle, find the coordinates of its third vertex.    

Show that ΔABC, where A(–2, 0), B(2, 0), C(0, 2) and ΔPQR where P(–4, 0), Q(4, 0), R(0, 2) are similar triangles.

If P ( 9a -2  , - b) divides the line segment joining A (3a + 1 , - 3 ) and B (8a, 5) in the ratio 3 : 1 , find the values of a and b .

If r₁ and r₂ be the radii of two solid metallic spheres and if they are melted into one solid sphere, prove that the radius of the new sphere is \[\left( r_1^3 + r_2^3 \right)^\frac{1}{3}\].

 If (a,b) is the mid-point of the line segment joining the points A (10, - 6) , B (k,4) and a - 2b = 18 , find the value of k and the distance AB.