Set-3 Sec-b

CBSE X Mathematics 2009 Solution (SET 3)

CBSE X 2009 Mathematics Section B Question Number 11 to 15 carry 2 marks each.

11.

If the points A (4, 3) and B (x, 5) are on the circle with the centre O (2, 3), find the value of x. Solution: If the points A (4, 3) and B (x, 5) are on the circle with the centre O (2, 3), then OA and OB will be the radii of the circle. OA = OB OA2 = OB2 (4 – 2)2 + (3 – 3)2 = (x – 2)2 + (5 – 3)2 [Using distance formula] 2 2 2 2 2 + 0 = (x – 2) + 2 0 = (x – 2)2 x–2=0 x=2 Thus, the required value of x is 2.

12.

Which term of the A.P. 4, 12, 20, 28, … will be 120 more than its 21st term? Solution: The given A.P. is 4, 12, 20, 28, … Here, first term = a = 4 Common difference = d = a2 – a = 12 – 4 = 8 a21 = a + (21 – 1) d [an = a + (n – 1) d] = 4 + 20 × 8 = 4 + 160 = 164 120 more than the 21st term = 120 + 164 = 284 Let the nth term of the given A.P. be 284. an = a + (n – 1) d 284 = 4 + (n – 1) 8 280 = (n – 1) 8 280 35 n–1= 8 n = 35 + 1 = 36 Thus, the 36th term of the given A.P. is 120 more than its 21st term.

CBSE X Mathematics 2009 Solution (SET 3)

13.

15 (2 2sin )(1 sin ) , then evaluate 8 (1 cos )(2 2cos ) OR

If cot

Find the value of tan 60°, geometrically.

Solution: The given expression is

(2 2sin )(1 sin ) . (1 cos )(2 2cos )

(2 2sin )(1 sin ) (1 cos )(2 2 cos ) 2(1 sin )(1 sin ) 2(1 cos )(1 cos ) 2(12 sin 2 ) 2(12 cos 2 ) 2 cos 2 2sin 2 cot 2 15 8 225 64

[(a b)(a b) (a 2 b 2 )] sin 2

cos 2

cot

15 8

2

Thus, the value of the given expression is

1

225 . 64

OR Consider an equilateral ABC. Then, ABC = BCA = CAB = 60° and AB = BC = CA = 2a (say) Now, draw AD BC.

Comparing ABD and ACD, we have:

CBSE X Mathematics 2009 Solution (SET 3) ADB = ADC = 90 AB = AC AD = AD ABD ACD BD = DC BC 2a BD a 2 2

[By construction] [Sides of equilateral triangle] [Common] [By RHS congruency axiom] [CPCT]

On applying Pythagoras’ Theorem in ABD, we get:

AB2

AD 2

(2a ) 2

BD 2

AD 2

4a 2

AD 2

AD 2

3a 2

AD

3a

a2 a2

Opposite side Adjacent side

tan 60º = tan B

AD BD

3a a

3

Thus, the value of tan 60º is 3 .

14.

Find all the zeroes of the polynomial x3 + 3x2 – 2x – 6, if two of its zeroes are 2 and 2 . Solution: The given polynomial is p( x)

x3 3x 2 2 x 6 .

It is given that 2 and 2 are the two zeroes of p(x). 2) and ( x 2) are the factors of p(x). Thus, ( x This means, ( x

2)( x

We can divide p( x)

x

2

2

2)

x 3 x 3x 2 2 x 6 2x 3x 2

6

2

6

3x

0

2 is also a factor of p(x).

x3 3x 2 2 x 6 by x2 – 2 as

3

x3

x2

CBSE X Mathematics 2009 Solution (SET 3) p(x) = (x2 – 2) (x + 3) As (x + 3) is a factor of the polynomial p(x), so x = –3 is a zero of the polynomial. Thus, –3 is the third zero of the given polynomial. Thus, the three zeroes of p( x)

15.

x3 3x 2 2 x 6 are –3,

2 , and 2 .

In Figure 2, ∆ABD is a right triangle, right-angled at A and AC that AB2 = BC . BD.

BD. Prove

Solution: In ACB and DAB, we have C DAB Each is 90 C = DBA [Common angle] ACB DAB [By AA similarity criterion] It is known that if two triangles are similar, then the corresponding sides are proportional. AB BC BD AB AB2 BC BD Hence, proved.