The area of the curve enclosed by is :
(A) 16
(B) 12
(C) 8
(D) 4
Area of is equivalent to area of
So it is a rhombus
Hence area will be,
In a rectangle , point
lies on
such that
and point
lies on
such that
2. Lines
and
intersect
at
and
respectively. If
, are relatively prime positive integers, then the minimum value of
is :
(i)
.(ii)
A regular polygon has 100 sides each of length. A another regular polygon has 200 sided each of length 2. When the area of the larger polygon is divided by the area of the smaller polygon, the quotient is closest to the integer
(A) 2
(B) 4
(C) 8
(D) 16
In a rectangle
, points
are taken on the sides
respectively such that the lengths
and
are integers and
is rectangle. The largest possible area of PQRS is
Area of
is a point inside an equilateral triangle
. The perpendicular distance
to the sides of the triangle are in the ratio
. If
where
are co-prime positive integers, then
equals
Let
Similarly
Area of quadrilateral
Area of
In and
. Point
lies on
and
bisects
. Point
lies on
and
bisects
. If the bisectors intersect at
, then the ratio
bisect
Length of
so
and also bisect
Ans.
In quadrilateral and
. If
is an integer then
Difference of two side is less than third side
in
also in
so
A box of dimension units is used to keep smaller cuboidal boxes so that no space is left between the boxes. If the box is packed with 100 such smaller boxes of the same size, then dimension of the smaller box is
Dimension of bigger box
No. of smaller box
Volume of smaller box
Small rectangular sheets of length units and breadth
units are available. These sheets are assembled and pasted in a big cardboard sheet, edge to edge and made into a square. The minimum number of such sheet required is
Area of square
No. of rectangle
Therefore
is a square.
is one fourth of the way from
to
and
is one fourth of the way from
to C.
is the centre of the square. Side of the square is
. Then the area of the shaded region in the figure in
is
Draw perpendicular from
to
\&
on
.
So required area Area of
Area of
Area of quadrilateral
.
Therefore
is a rectangle with
and
are midpoints of
and
respectively and
is the mid-point of
. The ratio of the area of
to area of
is
Let length of rectangle breadth of rectangle
So area of trapezium
In quadrilateral and
. If
is an integer then
Extend to
such that
So by SAS rule
CPCT
So of
of
add on both sides
So (1)
In
In the given figure, is a right angled triangle with
are points on
,
respectively such that
and
. Then,
(in degree).
Let
So
So
The area of a sector and the length of the arc of the sector are equal in numerical value. Then the radius of the circle is
Area of sector = length of are of sector
In , the medians through
and
are perpendicular. Then
is equal to
Let
In
Equation (i) + (ii)
In a quadrilateral . Then
(A) is a cyclic quadrilateral
(B) ABCD has an in-circle
(C) ABCD has both circum-circle and in-circle
(D) It has neither a circum-circle nor an in-circle
For incircle
Option B is correct
In a rhombus of side length 5 , the length of one of the diagonals is at least 6 , and the length of the other diagonal is at most 6 . What is the maximum value of the sum of the diagonals ?
(A)
(B) 14
(C)
(D) 12
Let diagonal are and
We have to find ?
By option (B)
The number of acute angled triangles whose vertices are chosen from the vertices of a rectangular box is
(A) 6
(B) 8
(C) 12
(D) 24
From each surface diagonals there are 4 triangles are possible which are equilateral of side
each.
But in this manner each triangle is counted thrice therefore
Option (B).
Circles A, B and C are externally tangent to each other and internally tangent to circle D. Circles A and are congruent. Circle
has radius 1 unit and passes through the centre of circle D. Then the radius of circle
is units.
In
In CAN
In shows below,
is a point on
and
a point on
such that
is a point in the interior of
is a point on
and
is a point on
such that
= BG. Also,
. The measure of
in degree is
As
Therefore
The area of the curve enclosed by is :
(A) 16
(B) 12
(C) 8
(D) 4
Area of is equivalent to area of
So it is a rhombus
Hence area will be,
In a rectangle , point
lies on
such that
and point
lies on
such that
2. Lines
and
intersect
at
and
respectively. If
, are relatively prime positive integers, then the minimum value of
is :
(i)
.(ii)
A regular polygon has 100 sides each of length. A another regular polygon has 200 sided each of length 2. When the area of the larger polygon is divided by the area of the smaller polygon, the quotient is closest to the integer
(A) 2
(B) 4
(C) 8
(D) 16
In a rectangle
, points
are taken on the sides
respectively such that the lengths
and
are integers and
is rectangle. The largest possible area of PQRS is
Area of
is a point inside an equilateral triangle
. The perpendicular distance
to the sides of the triangle are in the ratio
. If
where
are co-prime positive integers, then
equals
Let
Similarly
Area of quadrilateral
Area of
In and
. Point
lies on
and
bisects
. Point
lies on
and
bisects
. If the bisectors intersect at
, then the ratio
bisect
Length of
so
and also bisect
Ans.
In quadrilateral and
. If
is an integer then
Difference of two side is less than third side
in
also in
so
A box of dimension units is used to keep smaller cuboidal boxes so that no space is left between the boxes. If the box is packed with 100 such smaller boxes of the same size, then dimension of the smaller box is
Dimension of bigger box
No. of smaller box
Volume of smaller box
Small rectangular sheets of length units and breadth
units are available. These sheets are assembled and pasted in a big cardboard sheet, edge to edge and made into a square. The minimum number of such sheet required is
Area of square
No. of rectangle
Therefore
is a square.
is one fourth of the way from
to
and
is one fourth of the way from
to C.
is the centre of the square. Side of the square is
. Then the area of the shaded region in the figure in
is
Draw perpendicular from
to
\&
on
.
So required area Area of
Area of
Area of quadrilateral
.
Therefore
is a rectangle with
and
are midpoints of
and
respectively and
is the mid-point of
. The ratio of the area of
to area of
is
Let length of rectangle breadth of rectangle
So area of trapezium
In quadrilateral and
. If
is an integer then
Extend to
such that
So by SAS rule
CPCT
So of
of
add on both sides
So (1)
In
In the given figure, is a right angled triangle with
are points on
,
respectively such that
and
. Then,
(in degree).
Let
So
So
The area of a sector and the length of the arc of the sector are equal in numerical value. Then the radius of the circle is
Area of sector = length of are of sector
In , the medians through
and
are perpendicular. Then
is equal to
Let
In
Equation (i) + (ii)
In a quadrilateral . Then
(A) is a cyclic quadrilateral
(B) ABCD has an in-circle
(C) ABCD has both circum-circle and in-circle
(D) It has neither a circum-circle nor an in-circle
For incircle
Option B is correct
In a rhombus of side length 5 , the length of one of the diagonals is at least 6 , and the length of the other diagonal is at most 6 . What is the maximum value of the sum of the diagonals ?
(A)
(B) 14
(C)
(D) 12
Let diagonal are and
We have to find ?
By option (B)
The number of acute angled triangles whose vertices are chosen from the vertices of a rectangular box is
(A) 6
(B) 8
(C) 12
(D) 24
From each surface diagonals there are 4 triangles are possible which are equilateral of side
each.
But in this manner each triangle is counted thrice therefore
Option (B).
Circles A, B and C are externally tangent to each other and internally tangent to circle D. Circles A and are congruent. Circle
has radius 1 unit and passes through the centre of circle D. Then the radius of circle
is units.
In
In CAN
In shows below,
is a point on
and
a point on
such that
is a point in the interior of
is a point on
and
is a point on
such that
= BG. Also,
. The measure of
in degree is
As
Therefore