Maths i b Guess Paper

Maths IB Section A

I) very short answer type question i) Answer all questions

II) Each question carries two marks 1) If the area of the triangle formed by the straight lines x = 0 and Y = 0 and 3x+4y = a (a> 0 ) is 6. Find the value of a. 2) Find the equation of the straight line passing through (-4, 5) and cutting off equal non zero intercepts on the coordinate axes. 3) Find the fourth vertex of the parallelogram whose consecutive vertices are (2, 4, -1) (3, 6, -1) and (4, 5, 1) 4) If (3, 2, -1), (4, 1, 1) and (6, 2, 5) are three vertices and (4, 2, 2) is the centroid of a tetrahedron, find the fourth vertex.

5) Compute lim + 1 - 6) Compute lim 7) If Y = 7 + 3x (x > 0 ) then find

8) Differentiate f(x) w.r.to g(x) if f(x) = and g(x) =

9) Find the approximate value of 82 10) Find the value of C in Rolls theorem for the function f(x) = + 4 on (-3, 3)

Section B

11) A (2, 3); B (-3, 4) are two given points. Find the equation of locus of P so that the area of P so that the area of PAB is 8.5 Sq. units. 12) When the axes are rotated through an angle of 45, the transformed equation of a curve is 17-16xy+17 = 225. Find the original equation of the curve. 13) Find the equation of the line perpendicular to the line 3x+4y+6 = 0 and making an intercept 4 on axis.

14) If f(x) given by f(x) = 12 < 1 is continuous function on R then find the values of K.

15) Find the derivative of cos by using first principle w.r.to x. 16) The distance time formula for the motion of a particle along a straight line S = - 9 + 24t 18 then find when and where the velocity is zero.

17) Show that the curves 6 -5x+2y = 0 and 4+8 = 3 touch each other at

,

Section C

III) Long answer type question i) Answer any five questions ii) Each question carries seven marks. 18) Find the ortho centre of the triangle with the vertices (-2, -1), (6, 1) and (2, 5). 19) Show that the two pairs of lines 6-5xy-6 = 0 and 6 -5xy-6 +x+5y-1 = 0

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20) Find the angle between the lines joining the origin to the points of intersection of the curve +2xy+ +2 + 2 5 = 0 and the line 3x y +1 = 0 21) If a ray makes the angles , , , with four diagonals of a cube then find cos + cos + cos + cos 22) If Y = x + + log(+ + ) then prove that

= 2 +

23) If the tangent at any point P on the curve . = (mn 0) meets the coordinate axes in A, B then show that AP:BP is a constant. 24) A window is in the shape of a rectangle surmounted by a semicircle. If the perimeter of the wondow is 20ft, find the maximum area.

Solutions

I 1) 3x+4y-a = 0

area of triangle

|| = 6

|| = 6 = 6 64 a = 12 2) Point (-4, 5)

Each line

+

= 1

+

= 1

(-4, 5) -4 + 5 = 9 x + y = 9 a = 1 line x + y = 1

3) (2, 4, -1) (3, 6, -1) (4, 5, 1) Fourth vertex (- + , + , + ) = (2-3+64, 4-6+5, -1+1+1) = (3, 3, 1) 4) (3, 2,-1) (4, 1, 1) (6, 2, 5) three vertices centroid = (4, 2, 2) G =

,

,

(4, 2, 2) =

,

,

4 =

2 =

2 =

= 3 = 3 = 3 (3, 3, 3)

5) lim + 1 -





Rationalize lim

lim

lim

X =

lim

=

6) lim lim

1 + + 1 lim

lim

1 + + 1

1 1 + 0 + 1 =2

7) y = 7

= 7. (2x + 3)

8) f(x) = g(x) = f () = g() =

= 2 .

= ()

() =

= 2 .

9) 82 X = 81 = 1 f(x) = f () =

=

1 =

= 0.055

82 = f(x + ) = f(x) + 81 + 0.055 82 = 9.055 10) f(x) = + 4 f () = () ()

() = (9+4) (9-





II)

11) A (2, 3) B (-3, 4) P (x, y)

PAB =

|(3 4) + 2( ) 3( 3)| | 5 + 17| = 17

X + 5y -17 = 17 (x + 5y 34) (x + 5y) = 0 12) = 45 17 - 16 y + 17 = 225 (1) X = cos 45 + sin 45 =

Y = -x sin 45 + cos 45 = -

(1) 17

- 16

+ 17

= 225

13) Equation of the line 3x+4y+6 = 0 (1)

Slope =

Slope of the line r to (1) =

Y = mx + c

m =

c = -4

y =

- 4

4x 3y -4 = 0

14) f(x) = 12 < 1 lim

f(x) = 2 lim

f(x) = lim

k k = 2 k 2 = 0 (k 2) (k + 1) = 0 k = 2 K = -1 15) f(x) = cos f () = lim

() ()

lim

()





lim

-2 lim

. sin

-2 . a . sin

= -a sin 16) S = - 9 + 24t 18

V =

= 3 + 18t +24 = 0

- 6t +8 = 0 (t 2) (t 4) = 0 T = 2 t = 4 T = 2 S = 8-36+48-18 = 56-54 = 2

Section C III 18) A (-2, -1), B 96, -1), C (2, 5) are the vertices of

Slope of BC =

=

=

AD is to BC

Slope of AD =

Eq. of AD is

Y + 1 =

(x + 2)

2x-3y+1 = 0 (1) Slope of BE is

Y + 1 = -

(x -6)

2x + 3y 9 = 0 (2)

By solving (1) and (2)

3 -9 2 3

-3 1 2 -3

A

B C D

0 E

(-2, -1)

(6, -1) (2, -5)





=

=

=

=

X =

= 2 y =

=

Coordinates of ortho centre 0 are (2,

)

19) H 6-5xy-6 = (3x+2y)(2x-3y) and

S 6 -5xy-6 +x+5y-1 = (3x+2y - 1)(2x-3y + 1) Clearly, H = 0 represents a pair of St. lines and S = 0 also represents a pair of lines. Further the lines represents by H = 0 are parallel to the lines represented by S = o. therefore, the four lines form a rectangle.

But the distance of each lines 3x + 2y 1 = 0. 2x 3y + 1` = 0 from origin

. Hence the rectangle is square.

20)

Eqn. of the curve is +2xy+ + 2 + 2 5 = 0 (1) Eqn of AB is 3x y +1 = 0 Y 3x = 1 (2) Homogenizing (1) with the help of (2) combined equation of OA, OB is +2xy+ + 2. 1 + 2. 1 5(1)= 0 +2xy+ + 2( 3) + 2( 3) 5( 3) = 0 +2xy+ + 2 6 + 2 6 5( + 9 6) = 0 -5 -2xy + 3 -5 - 45 + 30xy = 0 25 - 14xy + = 0 Suppose Q is the angle b/w OA and AB cos = ||

() = ||() =

=

=

= cos

B

A





21) let each side of the cube be of length a. let one of the vertices of the cube be the origin 0 and the co-ordinate axes be along the 3 edges , and passing through the origin. The co-ordinates of the vertices of the cube w.r.t the frame of reference OABC are as shown in figure. The diagonals of the cube are , , , and (a, a, a), (a, a, -a), (-a, a, a) and (a, -a, a) are direction ratios of these diagonals.

20x - 2 +

f(x) = 20x - 2

f () = 0 20 4x - (( + 4)x = 20 X =

f () = -4 - < 0

f(x) is max when x =

=

=

Max area = 2xy +

=

+

()

= ()

=()() = Sq feet.

Let the direction cosinesof the given ray be (l, m, n).. if this ray makes the angles , , , with the four diagnols of the cube, then cos = ||

|| = ly cos = ||

cos = ||

cos = ||

cos + cos + cos + cos = { | + + | + | + | + | + + | + | + |}

=

[4( + + )] =

Since + + = 1

B (0, 4, 0) D (a, a, 0)

P (a, a, a)

A (a, a, 0) F (a, 0, a)

0 (0, 0, 0)

C (0,0, a)

E (a, a, a)

Z





X

Y

B

A 0

P(x1, y1)

22) Given Y = x + + log + +

=

2 + + 1 +

1 + 2

=

+ + + =

+ + + =

+ + = + + + =2 + 23) Eq. of the curve = Differentiating w.r.t x

+ = 0

n

= - .

=

..

= -

Slope of the tgt at P(x1, y1) =

Eqn of the tgt at P is

y y1 =

(x x1)

nx1y nx1y1 = -my1x + mx1y1 ny1x + nx1y = mx1y1+nx1y1 = (m+n) x1y1

() + () = 1

+

= 1 OA =

, OB =

Co ordinates of A are

, 0 and B are 0,

Let P divide AB in the ratio K:l Coordinates of P are

()

, () = (, )

=

=

(1)

= 1

=

(2)





x

y

x

Dividing (1) by (2)

=

=

P divides AB in the ratio n:m i.e. AP:PB = n:m = constant. 24) Perimeter = 2x + 2y +.x = 20 2y = 20 2x

= 2x 10

+





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