The sum of two numbers is 15 . If the sum of their reciprocals is 3 10 , then find the numbers.

Given, sum of two numbers is 15 Let one of the number be x then another number is ( 15 - x ) . Sum of their reciprocal is 3 10 . ∴ 1 x + 1 ( 15 - x ) = 3 10 ⇒ 15 - x + x x ( 15 - x ) = 3 10 ⇒ 15 15 x - x 2 = 3 10 ⇒ 150 = 45 x - 3 x 2 ⇒ 3 x 2 - 45 x + 150 = 0 ⇒ x 2 - 15 x + 50 = 0 ⇒ x 2 - 10 x - 5 x + 50 = 0 ⇒ x ( x - 10 ) - 5 ( x - 10 ) = 0 ⇒ ( x - 10 ) ( x - 5 ) = 0 ⇒ x = 10 or x = 5 Hence, the numbers are 5 and 10 .

The sum of two numbers is 15 . If the sum of their reciprocals is 3 10 , then find the numbers.

Given, sum of two numbers is 15 Let one of the number be x then another number is ( 15 - x ) . Sum of their reciprocal is 3 10 . ∴ 1 x + 1 ( 15 - x ) = 3 10 ⇒ 15 - x + x x ( 15 - x ) = 3 10 ⇒ 15 15 x - x 2 = 3 10 ⇒ 150 = 45 x - 3 x 2 ⇒ 3 x 2 - 45 x + 150 = 0 ⇒ x 2 - 15 x + 50 = 0 ⇒ x 2 - 10 x - 5 x + 50 = 0 ⇒ x ( x - 10 ) - 5 ( x - 10 ) = 0 ⇒ ( x - 10 ) ( x - 5 ) = 0 ⇒ x = 10 or x = 5 Hence, the numbers are 5 and 10 .

Draw a regular hexagon of side 4 cm . Draw the circle about the hexagon.

Given, side of the hexagon is 4 cm . Steps of construction: a) Draw a circle of radius 4 cm with centre O b) Since the interior angle of regular hexagon is 120 ° , draw radii OA and OB such that ∠ AOB = 60 ° . c) Cut off arcs BC , CD , EF and each equal to arc AB on given circle. d) Join AB , BC , CD , DE , EF , FA to get required regular hexagon ABCDEF in a given circle. The circle is the required circumcircle, circumscribing the hexagon of side 4 cm .

If x is a negative integer, then find the solution set of 2 3 + 1 3 ( x + 1 ) > 0 .

Given, x is a negative integer. 2 3 + 1 3 ( x + 1 ) > 0 ⇒ x + 1 3 > - 2 3 ⇒ x + 1 > - 2 ⇒ x > - 3 As is x is negative integer so x < 0 and from above x > - 3 so allowed values for x are - 1 and - 2 Hence, solution is x = { - 1 , - 2 }

If B = 3 4 2 - 1 , then find a matrix A satisfying B 2 = B + 2 A .

Given, B = 3 4 2 - 1 The equation, B 2 = B + 2 A , putting the value of B , we get, 3 4 2 - 1 3 4 2 - 1 = 3 4 2 - 1 + 2 A Multiply the elements of matrices, 3 × 3 + 4 × 2 3 × 4 + 4 × - 1 2 × 3 + - 1 × 2 2 × 4 + 1 = 3 4 2 - 1 + 2 A ⇒ 17 8 4 9 - 3 4 2 - 1 = 2 A ⇒ 14 4 6 10 = 2 A ⇒ A = 7 2 3 5 Hence, value of A = 7 2 3 5