STANTARD X or SSLC MATHEMATICS Question and Answers

5)Use a calculator to determine upto two decimal places, the perimeter and the area of the circle in the picture. Answer  BC = 3cm and BD = 6cm Join CD , thus we obtain a diameter CD ii. As   ∠ CBD = 90° ( ∵   angle in a semi circle is right angle) Therefore by Pythagoras theorem BC 2  + BD 2  = CD 2 3 2  + 6 2  = CD 2 CD 2  = 9 + 36 =45 CD = 3√5 Thus, diameter = CD = 3√5 Radius,   iii)Now change the position of a set of square , and draw another diameter to get the center  Thus, A is a centre. Hence the radius  Ee need to find perimeter of a circle? We know that, Perimeter of a circle = 2πr where r = radius We need to find area of a circle? We know that, Area of a circle=  π r 2  where r= radius 

4) In the picture, a circle is drawn with a line as diameter and a smaller circle with half the line as to diameter. Prove that any chord of the larger circle through the point where the circles meet is bisected by the small circle. Answer : ∠ A is common to both the triangles. AO � 1  = BO 1  (the radii of the big circle) ∠ ACO 1   =   ∠ ADB = 90° Hence, in ΔACO 1 ∼   ΔADB, ⇒   Since, the ratio is always equal, we can say that   any chord of the larger circle through the point where the circles meet is bisected by the small circle.

 3)If circles are drawn with each side of a triangle of sides 5 centimetres, 12 centimetres and 13 centimetres, as diameters, then with respect to each circle, where would be the third vertex? Answer: Let, ABC be a triangle with sides AB= 12cm BC = 5cm and AC = 13cm Case 1:   Let us draw a circle considering AB = 12cm as a diameter Then, point C lies outside the circle. Case 2:   Let us draw a circle considering BC = 5cm as a diameter Then point A lies outside the circle  Case 3: Let us draw a circle considering AC = 13cm as a diameter Then. Point B lies outside the circle 

2)For each diagonal of the quadrilateral shown, check whether the other two corners are inside, on or outside the circle with that diagonal as diameter. Ans   Case:1 As   ∠ A and   ∠ C are larger then 90° . Thus, point A and C lies inside the circle. (As seen above, point on a circle, such an angle is a right angle. Point outside the circle , such an angle is larger than 90° Point inside the circle, such an angle is smaller than 90° ) Case 2:   let us now draw a diagonal AC in quadrilateral ABCD Also, By angle sum property of quadrilateral ⇒   ∠ A +   ∠ B +   ∠ C +   ∠ D = 360° ⇒   105° +   ∠ B + 110° + 55° = 360° ⇒   ∠ B + 270° = 360° ⇒   ∠ B = 360° – 270° = 90° ● Draw a circle considering, AC as a diagonal. ● As   ∠ B is 90° . Thus, point B lies on the circle. As   ∠ D is smaller then 90°. Thus, point D lies outside the circle. (As seen above, point on a circle, such an angle is a right angle. Point outside the circle , such an angle is larger than 90° Point inside the circle, such a

1) Suppose we draw a circle with the bottom side of the triangles in the picture as diameter . Find out whether the top corner of each triangle is inside the circle, on the circle or outside the circle. Ans:  

5)  A tank contains 1000 litres of water and it flows out at the rate of 5 litres per second. How much water is there in the tank after each second? Write their numbers as a sequence. Ans   

( 4) Write down the sequence of natural numbers ending in 1 or 6 and describe it in two other ways. Ans: Sequence. 1,6,11,16,21,26 We can also say it as 1+5n  where n is from 0 to as much you want

  3)Write down the sequence of natural numbers leaving remainder 1 on division by 3 and the sequence of natural numbers leaving remainder 2 on division by 3. Ans:  The numbers that leave 1 as reminder when devided by 3 are 1,4,10.. (Sequence each with difference of 3 and starting from 1)  The no. That leave 2 as reminder when divided by 3 are  2,5,8,11,14,... (sequence each with difference of 3 and starting from 2) .

  2)  Look at these triangles made with dots. How many dots are there in each? Compute the no.of dots needed to make the next three triangles Ans:   3,6,10    3+3=6    6+4=10   10+5=15        The no of dots needed to make the next three triangles will be:       10+5  =15    15+6  =21    21+7  =28    15,21,27 dots  

1) Make the following number sequences, from the sequence of equilateral triangles, squares, regular pentagons and so on, of regular polygons:  No. Of sides.                               3,4,5,..... Sum of inner angles  One inner angle One outer angle Ans:      No.of sides :3,4,5       Sum of inner angles: Sum of inner angles of a regular polygon of n sides = n × [ (n - 2)180° ]/2  = (n - 2)180° ∴ Sum of interior angles of equilateral triangle (n = 3), square (n = 4), regular pentagon (n = 5), etc are: 180°, 360°, 540°, .......... Sum of outer angles of a regular polygon of n sides = n × 180° - (n - 2) × 180° = 2 × 180° =  360° which is free from "n". 1 interior angle One interior angle of triangle, square, regular pentagon etc are, 180°/3, 360°/4, 540°/5,....... = 60°, 90°, 180°,......... 1 exterior angle One exterior angle of triangle, square, regular pentagon etc are, 360°/3, 360°/4, 360°/5,....... = 120°, 90°, 72°,.........