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1

### JEE Main 2015 (Offline)

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $$(0, 0)$$ $$(0, 41)$$ and $$(41, 0)$$ is :
A
820
B
780
C
901
D
861

## Explanation

The number of integral points lie inside the triangle are

1. If x = 1, then y may be 1, 2, 3, ....., 39

2. If x = 2, then y may be 1, 2, 3, ....., 38

3. If x = 3, then y may be 1, 2, 3, ....., 37

$$\vdots$$

39. If x = 39, then the value of y is 1.

Hence, the number of interior points are

$$1 + 2 + 3 + .... + 39 = {{39 \times 40} \over 2} = 780$$

2

### JEE Main 2014 (Offline)

Let $$a, b, c$$ and $$d$$ be non-zero numbers. If the point of intersection of the lines $$4ax + 2ay + c = 0$$ and $$5bx + 2by + d = 0$$ lies in the fourth quadrant and is equidistant from the two axes then
A
$$3bc - 2ad = 0$$
B
$$3bc + 2ad = 0$$
C
$$2bc - 3ad = 0$$
D
$$2bc + 3ad = 0$$

## Explanation

Since the point of intersection lies on fourth quadrant and equidistant from the two axes,

i.e., let the point be (k, $$-$$k) and this point satisfies the two equations of the given lines.

$$\therefore$$ 4ak $$-$$ 2ak + c = 0 ......... (1)

and 5bk $$-$$ 2bk + d = 0 ..... (2)

From (1) we get, $$k = {{ - c} \over {2a}}$$

Putting the value of k in (2) we get,

$$5b\left( { - {c \over {2a}}} \right) - 2b\left( { - {c \over {2a}}} \right) + d = 0$$

or, $$- {{5bc} \over {2a}} + {{2bc} \over {2a}} + d = 0$$ or, $$- {{3bc} \over {2a}} + d = 0$$

or, $$- 3bc + 2ad = 0$$ or, $$3bc - 2ad = 0$$

3

### JEE Main 2014 (Offline)

Let $$PS$$ be the median of the triangle with vertices $$P(2, 2)$$, $$Q(6, -1)$$ and $$R(7, 3)$$. The equation of the line passing through $$(1, -1)$$ band parallel to PS is:
A
$$4x + 7y + 3 = 0$$
B
$$2x - 9y - 11 = 0$$
C
$$4x - 7y - 11 = 0$$
D
$$2x + 9y + 7 = 0$$

## Explanation

Let $$P,Q,R,$$ be the vertices of $$\Delta PQR$$

Since $$PS$$ is the median, $$S$$ is mid-point of $$QR$$

So, $$S = \left( {{{7 + 6} \over 2},{{3 - 1} \over 2}} \right) = \left( {{{13} \over 2},1} \right)$$

Now, slope of $$PS$$ $$= {{2 - 1} \over {2 - {{13} \over 2}}} = - {2 \over 9}$$

Since, required line is parallel to $$PS$$ therefore slope of required line $$=$$ slope of $$PS$$

Now, equation of line passing through $$(1, -1)$$ and having slope $$- {2 \over 9}$$ is

$$y - \left( { - 1} \right) = - {2 \over 9}\left( {x - 1} \right)$$

$$9y + 9 = - 2x + 2$$

$$\Rightarrow 2x + 9y + 7 = 0$$
4

### JEE Main 2013 (Offline)

The $$x$$-coordinate of the incentre of the triangle that has the coordinates of mid points of its sides as $$(0, 1) (1, 1)$$ and $$(1, 0)$$ is :
A
$$2 + \sqrt 2$$
B
$$2 - \sqrt 2$$
C
$$1 + \sqrt 2$$
D
$$1 - \sqrt 2$$

## Explanation

From the figure, we have

$$a = 2,b = 2\sqrt 2 ,c = 2$$

$${x_1} = 0,\,{x^2} = 0,\,{x_3} = 2$$

Now, $$x$$-co-ordinate of incenter is given as

$${{a{x_1} + b{x_2} + c{x_3}} \over {a + b + c}}$$

$$\Rightarrow x$$-coordinate of incentre

$$= {{2 \times 0 + 2\sqrt 2 .0 + 2.2} \over {2 + 2 + 2\sqrt 2 }}$$

$$=$$ $${2 \over {2 + \sqrt 2 }} = 2 - \sqrt 2$$

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