Tangents PA and PB are drawn from an external point P to two concentric circle with centre O and radii 8 cm and 5 cm respectively, as shown in Fig., If AP = 15 cm, then find the length of BP.
In fig., an isosceles triangle ABC, with AB =AB, circumscribes a circle. Prove that the point of contact P bisects the base BC.
OR
In fig., the chord AB of the larger of the two concentric circles, with centre O, touches the smaller circle at C. Prove that AC = CB.
In fig., OABC is a square of side 7 cm. If OAPC is a quadrant of a circle with centre O, then find the area of the shaded region.
Find the sum of all three digit natural numbers, which are multiples of 7.
First three-digit number that is divisible by 7 = 105
Next number = 105 + 7 = 112
Therefore the series is 105, 112, 119,........
The maximun possible three digit number is 999.
When we divide by 7, the remainder will be 5.
Clearly, 999 - 5 = 994 is the maximum possible three-digit number divisible by 7.
The series is as follows:
105, 112, 119,.........994.
Here a = 105, and d = 7
Let 994 be the nth term of this A.P.
an = a + ( n-1 ) d
994 = 105 + ( n-1 ) x 7
( n-1 ) x 7 = 889
( n-1 ) = 127
n = 128
So, there are 128 terms in the A.P.
A point P divides the line segment joining the points A(3,-5) and B(-4, 8) such that . If P lies on the line x + y = 0, then find the value of K.
If the vertices of a triangle are (1, -3), (4, p) and (-9, 7) and its area is 15 sq. units, find the value (s) of p.
Prove that the parallelogram circumscribing a circle is a rhombus.
OR
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base diameter is hollowed out. Find the total surface area of the remaining solid.
OR
A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, then find the radius and slant height of the heap.