SEE-SCANNER-2077-OPT-Maths

(d) If u.i  3 and  = 450, find the value of u Ans: 3

r r ur ur rr
3. (a) Find the angle between two vectors a and b if a  2 , b =12 and a.b=12 [SEE MODEL-2076] Ans: 600

rr rr Ans: 600
(b) If a = 4, b  5and a.b =10 then find the angle between a and b .

rr Ans: 450
(c) If a = 10, b  3and a.b =15 2 then find the angle between a and b

ur r ur r ur r
(d) If p.q =18 3 , p = 6 & q  6 and then find the angle between p and q . Ans: 300

(e) r = 5, r  3 and rr = 15 r r Ans: 600
If a b a.b then find the angle between a and b
2 Ans:: 450
rr Ans: 900
r rr r r r Ans: 900
4. (a) If a = 2i+ j and b= i+3j then find the angle between a and b .
rr
r rr r rr
(b) If a = -i + 2j and b = 4i + 2j , find the angle between a and b
rr
r rr r rr
(c) If a = 10i + 7j and b = 10i  7j , find the angle between a and b .

(d) r r are unit vectors so rr  1 , find the angle r and r Ans: 600
If a and b that ab between a b
2 Ans: 300
r r rr Ans:600
(e) Find the angle between i and a = 3i + j
r
5. (a) rrr r = 3, r r  7 , find the angle between a and r
If a+b+c = 0 , a b  5and c b

(b) rrr r = 6, r =7 and r  rr Ans:600
p q r 127 , find the angle between p & q
If p + q + r = 0 ,

rr r rr r rr Ans: 600
If a , b and c are unit vectors such that a  b  c  0 , calculate the angle between a and b .
(c) rr rr rr

6. (a) If a  2b and 5a  4b are perpendicular to each other and a and b are unit vectors, find the angle
rr
between a and b . Ans: 600
r r rr
(b) If 4a  5b and 2a  b are perpendicular to each other and a and b are unit vectors, find the angle
rr
between a and b . Ans: 600
rr r

(c) If 2a and a  b are perpendicular to each other and a and b are unit vectors, find the angle between a
r
Ans: 900
and b . r r r r rr

7. (a) Prove that the vectors a= 2i  5j and b = 15i+6j are perpendicular to each other.
r
(b) rr rr , prove that a and r are perpendicular to each other.
If a + b  ab b

r r rr rr

(c) If 4p + 3q  4p  3q , prove that p and q are perpendicular to each other.

r ur r r rr
(d) If (x + y)2  (x  y)2 , prove that x and y are perpendicular to each other.
rr
r ur r r
(e) If a and b are perpendicular to each other, prove that (a + b)2  (a  b)2

r rr r r r
(f) For what value of k, are the vectors a = 9i  k j and b = 2i + 6j are perpendicular to each other? Ans:3

(g) r =  5  and r  p  are perpendicular, find the value of p. Ans: 3
If a  3 b=  + 
  p 2 

8. (a) r r rr rr
If i and j
are the unit vectors along x-axis and y-axis respectively, prove that: i.i  1 and i.j = 0

(b) rr rr  2 cos θ
If a and b are unit vectors and  be the angle between them, prove that: a+b 2

VECTOR GEOMETRY

1. (a) If AM is the median of ABC. Prove that: uuuur 1 uuur uuur
AM = (AB +AC)
2

SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 51

(b) If the position vectors of A and B are 3i + 4j and 7i + 8j respectively, find the position vector of

the mid-point of the line joining A and B. Ans:5⃗i +6⃗j
r r rr
(c) If the position vectors of M and N are 7i + 2j and i + 4j . Find the position vector of a point P such that
uuur uur
MP = PN uur r r

Ans: OP  4i + 3j

(d) If M is the mid-point of the join of A (2, 5) and B (-6, -1), find the position vector of M. Ans: uuur  r + r
OM 2i 2j

rr r r
2. (a) The position vectors of A and B are i + 2j and 6i + 7 j . Find the position vector of a point P which
uur r r
divides AB internally in the rati0 2:3. Ans: OP  3i + 4j


(b) Point C divides the line AB internally in the ratio of 3:1. If the position vector of A and B are i - 3 j

and 2i + 5j respectively, find | AB | and the position vector of point C.

5  3
 
(c) Two points A and B have their position vectors are  2  and  6  respectively. Find the position vector
 

Ans: uur  3 r r
OP (3i -2 j)
of P which divides AB internally in the ratio 2:3. 5

(d) If the point P divides the line segment joining the points A (3, 1) and B (6, 4) in the ratio 2:1, find the
uur r r
Ans: OP  5i +3j
position vector of P.
r r rr
(e) The position vectors of A and B are i + j and 2i - j . Find the position vector of a point P which divides
uur r r
AB externally in the ratio 3:2. Ans: OP  4i -j

1 3
(f)    
Two points A and B have their position vectors are  2  and  2  respectively. Find the position vector

uur r r
Ans: OP  5i + 2j
of P which divides AB externally in the ratio 2:1.

ur r r
3. (a) In the given figure; find AP and express p in terms of a and b [SEE MODEL-2076]

uur r uuur r
In the given figure; OA  a and OB  b .
(b) uuur uuur r r
If AC 3AB a b
= , find uuur in terms of and
OC

(c) In the given figure; ABC = 900 uuur 2 uuur 2 uuur 2 P A
, prove that: AC = AB + BC p Q

(d) In the given figure, PA = 1 PQ , prove thata = 1 (3p + q ). a
4 4 q

(e) In the given figure, ABCD is a parallelogram. If AC and BD are B
uuur uuur uuur

diagonals, prove that: AC +BD = 2BC

(f) In ABC; D and E divide AB and AC in the ratio 1:2 respectively.

Prove that: uuur = 1 uuur
DE BC
3

HIGH ABILITY (HA) BASED PROBLEMS

1. (a) Prove by vector method that the median of an isosceles triangle is perpendicular to the base.
(b) In the given isosceles PQR, PQ = PR and PT is the median drawn from the vertex P upon QR. Prove
by vector method that PT  QR.
(c) Prove by vector method that the line segment joining the midpoint of a chord to the centre of the corcle
is perpendicular to the chord.

2. (a) Prove by vector method that the mid-point of hypotenuse of a right angled triangle is equidistance from
its every vertex.

(b) In the given right angled XYZ, XYZ = 900 and M is the mid-point of
hypotenuse XZ. Prove by vector method that MX = MY = MZ.

SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 52

3. (a) Prove by vector method that the angle in a semi-circle is right angle.
(b) In the given figure; O is the centre of semi-circle, AB is the diameter and C be any
point on the circumference. Prove by vector method that ACB = 900

4. (a) Prove by vector method that the diagonals of a rectangle are equal to each other.
(b) In the given figure; PR and QS are the diagonals of the rectangle PQRS.
Prove by vector method that PR = QS.

5. (a) By using vector method, prove that the quadrilateral formed by joining the midpoints of adjacent sides

of a quadrilateral is a parallelogram. [SEE MODEL-2076]

(b) Prove by vector method that the diagonals of a parallelogram bisect to each other.

(c) Prove by vector method that the line joining the mid-points of any two sides of a triangle is parallel and

half of the third side.

6. (a) Prove by vector method that the diagonals of a rhombus bisect to each other perpendicularly.

(b) In the given figure; PR and QS are its diagonals of a rhombus PQRS. Prove by

vector method that PR  QS.

7. (a) Prove by vector method that the position vector of centroid of a triangle is one-third of sum of position

vectors of its vertices.
uuur r uuur r
(b) If OA  a,OB  b and the point P divides the line segment AB internally in the ratio m1 : m2, prove that:

uur rr
OP
= m1 b + m2 a

m1+ m2
uuur r uuur r
(b) If OA  a,OB  b and the point P divides the line segment AB externally in the ratio m : n, then prove

that: uur = r  r
OP mb na

mn

UNIT-7 TRANSFORMATION

SEE SPECIFICATION GRID

Contents K U A HA Total Marks

1. Combined transformation 14=4 15=5 1+4+5 = 10

2. Inversion transformnation 11=1
3. Matrix transformation

A. COMBINED TRANSFORMATION

KNOWLEDGE (K) BASED PROBLEMS

1. (a) What do you mean by combined transformation?
(b)
What will be the combined transformation if a reflection is followed by another reflection and the axes
(c)
of reflections do intersect at a point?
2. (a)
What will be the combined transformation if a reflection is followed by another reflection and the axes
(b)
(c) of reflections being parallel?
(d)
(e) What is the single transformation when the reflection about x-axis is followed by the reflection about y-
(f)
axis?
3. (a)
(b) If r1represents the reflection in y = x and r2 represents the reflection in y = -x, what does r2or1 represent?
What is the single transformation when the reflection about x-axis is followed by the reflection in y = x?
4. (a)
If R1denotes the reflection on y-axis and R2 denotes the reflection in y = -x, what does R1oR2 denote?
If r1and r2 represent the reflections about the lines x =1and y = 2respectively, what does r2or1 represent?
If r1and r2 represent the reflections about the line x = -2 and y = 1 respectively, what does r1or2
represent?

What will be the image of a point P (x, y) if it is first reflected in x = h and then reflected in y = 0?

What will be the final image of a point P (x, y) if it is first reflected in y = k and then reflected in x = 0?

If T1  a and T2   c  , find T2oT1. (b) If T1   x1  and T2   x2  , find T1oT2.
   d   y1   y2 
 b       

SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 53

5. (a) What will be single transformation when a rotation through 1 about origin is followed by another

rotation through 2 about origin?
(b) What will be single transformation when a rotation through 600 about origin is followed by another

rotation through 300 about origin?
(c) What will be single transformation when a rotation through - 900 about origin is followed by another

rotation through 2700 about origin?

6. (a) What will be single transformation when an enlargement with scale factor k1 about origin is followed by
another enlargement with scale factor k1 about origin?

(b) If an enlargement E1 [(a, b), h] is followed by another enlargement E2 [(a, b), k], what is the single
enlargement E2oE1?

(c) If an enlargement E1 [(1, 2), 3] is followed by another enlargement E2 [(1, 2), 2], what is the single
enlargement E2oE1?

7. (a) What will be the image of a point P (x, y) if it is first reflected in x-axis and then rotated through +900

about origin?
(b) What will be the image of a point P (x, y) if it is first rotated through +900 about origin and then

reflected about y-axis?

8. (a) What will be the image of a point (x, y) if it is first reflected in y = x and then enlarged by E [O, k]?

(b) What will be the image of a point (x, y) if it is first reflected in y + x = 0 and then enlarged by E [O, k]?

m
 
9. (a) What will be the final image of (x, y) when it is reflected in x – p = 0 and then translated by  n  ?

a
 
(b) What will be the final image of (x, y) when it is rotated by R [+900; (0, 0)] and then translated by  b  ?

APPLICATION (A) BASED PROBLEMS

1. a) A (-1, 1), B (4, 2) and C (5, 6) are the vertices of a ABC. Find the coordinates of the vertices of image
b)
of ABC under the reflection on x-axis followed by the reflection on y-axis and draw the triangles on
2. a)
the same graph paper. Ans: A’ (-2, -3), B’ (-4, -5), C’ (-1, -4
b)
B (2, 1), U (4, 5) and Y (-1, 4) are the vertices of BUY.  Find the coordinates of the vertices of image
3. a)
b) of ABC under the reflection on about y = x followed by the reflection on x = 2 and draw the triangles

4. a) on the same graph paper. Ans: A’ (3, 1), B’ (-1, 4), C’ (0, -1)
b)
The vertices of ABC are A (2, 0), B (3, 1) and C (1, 1). ABC is translated by  2 the vector. The
 
 3 

image so obtained is reflected on the line x + y = 0, find the coordinates of the images under combined

transformation and draw all the triangles on the same graph paper.
Ans: A’ (4, -3), B’ (5, -2), C’(3, -2);A’’(3, -4), B’’(2, -5), C’’(2, -3)

Translate the parallelogram OABC having vertices O (0, 0), A (2, 0), B (3, 1) and C (1, 1) by translation

vector 0 . Reflect the image so formed on the line x = 3. Represent the object and image on the same
 
 2 

graph. Ans: O’ (0, 2), A’ (2, 2), B’ (3, 4), C’ (1, 4); O’’ (6, 2), A’’ (4, 2), B’’ (3, 4), C’’ (5, 4)

A (2, 5), B (-1, 3) and C (4, 1) are the vertices of a ABC. Find the coordinates of the vertices of image

of ABC under the rotation of positive 900 about origin followed by enlargement E [(0, 0); 2].

Represent the object and the images on the same graph paper.

Ans: A’ (-5, 2), B’(-3, -1), C’(-1, 4);A’’(-10, 4), B’’(-6, -2), C’’(-2, 8)

I (4, -2), L (2, 1) and U (5, 2) are the vertices of ILU. ILU is transformed by a single transformation
obtained by a rotation [(0, 0); 1800] and on the same direction a rotation [(0, 0); 900]. Find the

coordinates of the images of ILU and draw both the triangles on the same graph paper.
Ans: I’ (-4, 2), L’ (-2, -1), U’ (-5, -2); I’’ (-2, -4), L’’ (1, -2), U’’ (2, -5)

Triangle PQR having vertices P (2, 1), Q (4, 3) and R (0, 4) is reflected on the y-axis. The image so

formed is enlarged by E [(0, 0); 2]. Write the vertices of the images thus obtained and present the PQR
and its images on the same graph paper. Ans: P’ (-2, 1), Q’(-4, 3), R’(0, 4); P’’(-4, 2), Q’’(-8, 6), R’’(0, 8)

The vertices of ABC are A (2, 3), B (-2, 1) and C (0, 2). If E1 [(0, 0); 2] and E2 [(0, 2); 2], then find the
coordinates of the images of ABC under enlargement E20E1. Draw both triangles on the same graph.

SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 54

Ans: A’ (4, 6), B’ (-4, 2), C’ (0, 4); A’’ (8, 10), B’’ (-8, 2), C’’ (0, 6)

HIGHER ABILITY (HA) BASED PROBLEMS

1. a) The coordinates of vertices of a quadrilateral ABCD are A (1, 1), B (2,3), C (4, 2) and D(3,−2). Rotate
this quadrilateral about origin through180°. Reflect this image of quadrilateral about y = −x. Write the

name of transformation which denotes the combined transformation of above two transformations.
Ans: y = x, A’ (1, 1), B’ (3, 2), C’ (2, 4) and D’ (-2, 3)

b) State the single transformation equivalent to combination of reflections on the x-axis and y-axis

respectively. Using this single transformation, find the coordinates of the vertices of the image of PQR

with vertices P (4, 3), Q (1, 1) and R (5, -1). Also draw the object and the image on the same graph

paper. Ans: R [1800, (0, 0)]; P’ (-4, -3), Q’ (-1, -1), R’ (-5,-1)

c) State the single transformation equivalent to combination of reflections on the y-axis and y = x

respectively. Using this single transformation, find the coordinates of the vertices of the image of ABC

with vertices A (2, 3), B (3, -4) and C (1, -2). Also draw the object and the image on the same graph

paper. Ans: R [-900, (0, 0)]; A’ (3, -2), B’ (-4, -3), C’ (-2,-1)

d) ABC with vertices A (1, 2), B (4, -1) and C (2, 5) is reflected successively in the line y = x and x-axis.

Find by stating coordinates and graphically represent images under these transformations. State also the

single transformation given by the combination of these transformations.
Ans: A’ (2, 1), B’ (-1, 4), C’ (5, 2); A’’ (2, -1), B’’ (-1, -4), C’’ (5, -2); R [-900, (0, 0)]

2. a) A triangle with vertices A (1, 2), B (4, -1) and C (2, 5) is reflected successively in the lines x = 5 and y

= -2. Find by stating coordinates and graphically represent images under these transformations. State

also the single transformation given by the combination of these transformations.
Ans: A’ (9, 2), B’ (6, -1), C’ (8, 5); A’’ (9, -6), B’’ (6, -3), C’’ (8, -9); R [-1800, (5,-2)]

b) A triangle with vertices A (1, 2), B (4, -1) and C (2, 5) is reflected successively in the lines x = -1 and y

= 2. Find by stating coordinates and graphically represent images under these transformations. State

also the single transformation given by the combination of these transformations.
Ans: A’ (-3, 2), B’ (-6, -1), C’ (-4, 5); A’’ (-3, 2), B’’ (-6, 5), C’’ (-4, -1); R [-1800, (-1, 2)]

B. INVERSION TRANSFORMATION AND INVERSION CIRCLE

KNOWLEDGE (N) BASED PROBLEMS

1. (a) What do you mean by inversion transformation? (b) Define inversion circle.

(c) Write down a property of inversion transformation. (d) Write the definition of inversion centre.

2. (a) In an inversion transformation, if P’ is the image of the P and r is the radius of inversion circle with

centre O, write the relation of OP, OP’ and r. [SEE MODEL-2076]

(b) Under what condition, the point on the inversion transformation is invariant?

(c) If a point P is inside the inversion circle, where does its inverse lie?

(d) If a point P is outside the inversion circle, where does its inverse lie?
(e) In an inversion transformation, P’ is the image of the P and r is the radius of inversion circle with centre

O. If OP and OP’ are reciprocal to each other; find the length of radius (r).
3. (a) If P’ (x’, y’) is the inversion point of P (x, y) in the circle with centre origin and radius r, write down the

formula of finding x’.
(b) If P’ (x’, y’) is the inversion point of P (x, y) in the circle with centre origin and radius r, write down the

formula of y’.

(c) Write down the coordinates of inversion point of (x, y) with respect to the circle with centre at origin and

4. (a) radius r.
(b) If P’ (x’, y’) is the inversion point of P (x, y) in the circle with centre (h, k) and radius r, write down the
formula of finding x’.
If P’ (x’, y’) is the inversion point of P (x, y) in the circle with centre (h, k) and radius r, write down the
formula of finding y’.

(c) Write down the coordinates of the inversion of a point P (x, y) with respect to the circle having centre at

(h, k) and radius r.
5. (a) In a circle with centre O, P’ is the inversion point of P. If OP × OP’ = 36 inch2, find the radius of the

inversion circle.

SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 55

(b) In a circle with centre O, P’ is the inversion point of P. If OP × OP’ = 100 cm2, find the radius of the

inversion circle.
(c) In a circle with centre O, P’ is the inversion point of P. If OP × OP’ = 144 cm2, find the radius of the

inversion circle.
6. (a) In a circle with centre O, P’ is the inversion point of P. If the radius (r) = 2 cm, what is the value of OP

× OP’.

(b) In a circle with centre O, P’ is the inversion point of P. If the radius (r) = 3 cm, what is the value of OP
× OP’.

(c) In a circle with centre O, P’ is the inversion point of P. If the radius (r) = 3 cm, what is the value of OP
7. (a) × OP’.
In a circle with centre O, P’ is the inversion point of P. If the radius (r) = 4 cm and OP = 2cm, what is
the length of OP’?

(b) In a circle with centre O, P’ is the inversion point of P. If the radius (r) = 6 cm and OP’ = 9cm, what is

the length of OP?
(c) In a unit circle with centre O, P’ is the inversion point of P. If OP = 0.5 units, what is the length of OP’?
(d) In a unit circle with centre O, P’ is the inversion point of P. If OP = 0. 25 units, what is the length of

OP’?

APPLICATION (A) OR HIGH ABILITY (HA) BASED PROBLEMS

1. (a) Find the inverse of the point (2, 0) with respect to the circle x2 + y2 = 9. Ans: (18, 0)

(b) Find the inverse of the point (4, 4) with respect to the circle x2 + y2 = 64. Ans: (8, 8)

(c) Find the inverse of the point (3, 4) with respect to the circle x2 + y2 = 100. Ans: (12, 16)
2. (a)
Find the inverse of the point (6, 7) with respect to the circle x2 + y2 – 4x – 6y = 51. Ans: (10, 11)
(b)
(c) Find the inverse of the point (3, 4) with respect to the circle x2 + y2 – 4x – 6y = 23. Ans: (20, 21)
(d)
(e) Find the inverse of the point (4, 5) with respect to the circle x2 + y2 – 4x – 6y = 3. Ans: (6, 7)
3. (a)
Find the inverse of the point (3, 2) with respect to the circle x2 + y2 – 4x – 6y + 9 = 0. Ans: (4, 1)

Find the inverse of the point (6, 3) with respect to the circle x2 + y2 – 4x – 6y = 3. Ans: (6, 7)

If the inverse of the point A (3, 4) with respect to the circle having origin as the centre is A’ (12, k), find

the vslure of k. Ans: 16

(b) Find the inverse of the point P (4, 0) with respect to the circle having origin as the centre is P’ (1, m),

find the value of m. Ans: 0

(c) Find the inverse of the point (6, 7) with respect to the circle having centre (2, 3) is P’ (10, a), find the

value of a. Ans: 11

(d) Find the inverse of the point (8, 6) with respect to the circle having centre (5, 6) is P’ (k, 6), find the

value of k. Ans: 17

C. MATRIX TRANSFORMATION

KNOWLEDGE (N) BASED PROBLEMS

1. (a) To what transformation is the matrix 10 01 associated?

(b) Which type of transformation does  0 1  represent?
1 0

(c) Which type of transformation does 0 -1  represent?
 0 
 -1 

(d) To what transformation is the matrix 10 01 associated?

(e) What transformation does the matrix  0 1  denote?
1 0

(f) Which type of transformation does  1 01 represent?
0

SEE SCANNER OF OPTIONAL MATHEMATICS FOR SEE-2077 BY: TARA BDR MAGAR Page 56

(g) To what transformation is the matrix  2 02 associated?
0

(h) What transformation is represented by  3 0  ?
0 3

2. (a) Which transformation matrix is associated to reflection on x-axis?
(b) Which transformation matrix is associated to reflection on y-axis?
(c) Which transformation matrix is associated to reflection in y = x?
(d) Write the matrix that represents the rotation through +900 about origin.
(e) Write the matrix that represents the rotation through 1800 about origin.
(f) Write the matrix is representing to the enlargement [k ;O] .
(g) Write the matrix is representing to the enlargement with scale factor 4 and centre at origin.

3. (a) If a point (a, b) is transformed into (b, -a) by a 2×2 matrix, find the matrix.

(b) Find a2×2 matrix which transforms a point A (-3, 1) into A’ (3, 1)
(c) Find a2×2 matrix from the equations x’ = 2x + y and y’ = 3x – y.
(d) Find a2×2 transformation matrix from the equations x’ = y and y’ = - x.
4. a) Find a 2×1 matrix which transforms a point (a, b) in to the point (a + 2, b – 3).

(b) Find a 2×1 matrix which maps a point (x, y) onto the point(x + 2, y + 1).

APPLICATION (A) OR HIGH ABILITY (HA) BASED PROBLEMS

1. a) Find the 22 matrix which transforms the unit square into the parallelogram  0 3 4 11 . Ans:  3 11
0 0 1 0

b) Find the 22 matrix which transforms the unit square into the parallelogram  0 3 5 2 .Ans: 13 12
 0 1 2 
 1 

c) Find the22 matrix which transforms the unit square into a parallelogram  0 2 5 3 . Ans:  2 3
d)  0 1 0 1  1 1
2. a)  

Find the22 matrix that transforms 0 2 1 3 into  0 2 1 3 . Ans:  1 0
 4 3 1  0 4 3 1  1
 0   0


A square  0 a 3 2 is transformed by the matrix 1 2 to get a parallelogram  6 3 7 10  .
 b 1 c  1  6 1 1 6 
 d  2   


Find the values of a, b, c and d. Ans: a = 1, b = 2, c = 3 , d = 4

b) If the matrix a b  transforms a rectangle  0 2 2 0 into a rectangle 0 2 2 0  , find
d   0 0 1   1 1 
 c   1   0 0 


the values of a, b, c and d. Ans: a = 1, b = 0, c = 0 , d = -1

3. a) A square ABCD with vertices A (0, 3), B (1, 1), C (3, 2) and D (2, 4) is mapped on to  ABCD by a
2 2 transformation matrix so that the vertices of  ABCD are A (6, -6), B (3, -1), C (7, –1) and D

(10, –6). Find the2 2 matrix. 1 2
Ans: 1 2

b)  ABC with vertices A (3, 6), B (4, 2) and C (2, 2) is mapped on to aABC’ by a 22 transformation
matrix so that the vertices of ABC’ are A (-6, 3), B (-2, 4) and C (-2, 2). Find the 22 matrix

. Ans: 0 1
 0 
 1

c) Find a2 2 transformation matrix which maps PQR with vertices P (-3, 5), Q (7, 4) and R (6, -2) into

P1Q1R1 with vertices P1 (5, 3), Q (4, -7) and R (-2, -6) Ans:  0 1
 0 
 1

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d) A square ABCD with vertices A (0, 0), B (1, 0), C (1, 1) and D (0, 1) is mapped on to  ABCD by a

22 transformation matrix so that the vertices of  ABCD are A (0, 0), B (3, 0), C (4, 1) and D (1,

1). Find the 22 matrix. Ans: 3 1
 
 0 1 

e) A square PQRS with vertices P (0, 3), Q (1, 1), R (3, 2) and S (2, 4) is mapped on to the  PQRS by a

22 transformation matrix so that the vertices of  PQRS are P (3, 0), Q (1, 1), C (2, 3) and S (4,

2). Find the 22 matrix. Ans: 0 1
 
 1 0 

4. a) Find the coordinates of vertices of image A’B’C’ which is transformed by the transformation of

ABC having vertices A (3, 1), B (-2, -1) and C (4, 2) under the matrix 2 0
 2 
 0

Ans: A’ (6, 2), B’ (-4, -2),C’ (8, 4)

b) A square WXYZ has the vertices W (0, 3), X (1, 1), Y (3, 2) and Z (2, 4). Transform the given square

0 1
WXYZ under the matrix 1 0 and find the coordinates of vertices of its image.

Ans: W’ (-3, 0), X’ (-1, 1) ,Y’ (-2, 3) and Z’ (-4, 2)

5. a) A (4, 1), B (-2, 3) and C (-3, 5) are the vertices of ABC. Find the vertices of image triangle under
rotation through +900 about origin by using matrix method. Ans: A’ (-1, 4), B’ (-3, -2) and C’ (-5, -3)

b) X (-2, 3), Y (1, 4) and Z (0, -2) are the vertices of XYZ. Find the vertices of image triangle under
reflection on x-axis by using matrix method. Ans: X’ (-2, -3), Y’ (1, -4) and Z’ (0, 2)

c) An enlargement with centre O has the scale factor 3. Write the matrix to represent this enlargement and

hence use it to find the image of ABC with vertices A (2, 1), B (1, 3) and C (2, 4).

Ans: 3 0 ; A’ (6, 3), B’ (3, 9) and C’ (6, 12)
 
 0 3 

d) An enlargement with centre O has the scale factor 2. Write the matrix to represent this enlargement and

hence use it to find the image of ABC with vertices A (0, 1), B (2, 1) and C (-3, 5)

Ans:  2 02 ; A’ (0, 2), B’ (4, 2) and C’ (-6, 12)
0

e) By using matrix method, show that a reflection about y = x followed by a reflection in y- axis is

equivalent to the rotation through 900 about the origin. Ans: R 900; (0, 0)

UNIT-8 STATISTICS

SEE SPECIFICATION GRID K U A HA Total Marks
Contents
12=2 24=8 2+8 = 10
1. Quartile deviation
2. Maen deviation
3. Standard deviation

A. QUARTILE DEVIATION

UNDERSTANDING (U) BASED PROBLEMS

1. (a) What is measure of dispersion? What are the methods of measure of dispersion?
(b) Write any two differences between measure of dispersion and its coefficient.
(c) Define first quartile and third quartile of a data.
(d) Define quartile deviation and coefficient of quartile deviation.
(e) Write any two differences between quartile deviation and its coefficient.
(g) Write down the formula to calculate quartile deviation and its coefficient.

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(h) Write any two merits of quartile deviation.
(i)
(j) Write any two demerits of quartile deviation.
2. (a)
(b) Write any two differences between the first and third quartiles.
(c)
(d) In a continuous series, if the first quartile (Q1) is 10 and the third quartile (Q3) is 30, find the quartile
3. (a)
(b) deviation and its coefficient. Ans:10, 0.25
(c)
(d) In a continuous series, the first quartile (Q1) is 25 and the third quartile (Q3) is 100, find the quartile
4. (a)
(b) deviation and its coefficient. Ans: 37.5, 0.5
(c)
(d) In a continuous series, the lower and the upper quartiles are 18 and 48 respectively, calculate the
5. (a)
(b) quartile deviation and its coefficient. Ans: 15, 0.4545
(c)
(d) If 20 and 80 are the lower and the upper quartiles of a continuous series, find the quartile deviation and

its coefficient. Ans: 30, 0.6

In a continuous series, the third quartile is two times the first quartile. If the first quartile is 24, find the

third quartile and quartile deviation. Ans:12, 0.33

In a continuous series, the third quartile is three times the first quartile. If the third quartile is 60, find

the first quartile and quartile deviation. Ans:20, 0.5

In a continuous series, the third quartile is twice the first quartile. If the median of the data is 30, find

the quartile deviation. Ans:10, 0.5

In a continuous series, the third quartile is treble the first quartile. If the median of the data is 44, find

the quartile deviation. Ans:22, 0.5

In a grouped data, if the first quartile and the quartile deviation are 15 and 10 respectively, find the

coefficient quartile deviation.

In a grouped data, if the first quartile and the quartile deviation are 24.5 and 65 respectively, find the

coefficient quartile deviation.

In a grouped data, if the third quartile and the semi-interquartile range deviation are 80 and 30

respectively, find the coefficient quartile deviation.

In a grouped data, if the third quartile and the semi- interquartile range deviation are 100 and 20

respectively, find the coefficient quartile deviation.

In a grouped data, the quartile deviation and its coefficient are 5 and 0.2 respectively then find the

lower quartile.

In a continuous series, the quartile deviation and its coefficient are 30 and 0.6 respectively then find

the lower quartile.

In a continuous series, the quartile deviation and its coefficient are 25 and 0.5 respectively then find

the upper quartile.

In a continuous series, the quartile deviation and its coefficient are 18 and 0.6 respectively then find

the upper quartile.

APPLICATION (A) BASED PROBLEMS

1. Calculat e the quartile deviantion and its coeffient of the data given below. Ans:
OR, find the inter-quartile range and its coeffient of the data given below.

(a) Marks obtained 0-10 10-20 20-30 30-40 40-50 Ans: 8.625, 0.12
No. of students 2 1 2 3 5

6.44, 0.11
(b) Ans: 26.31 years, 0.42

Age in years 5-15 15-25 25-35 35-45 45-55
No. of people 12 15 10 11 14

Ans: 6.68 years, 0.18
(c) Weight in kg 0-10 10-20 20-30 30-40 40-50 50-60 Ans: 30 kg, 0.55

No. of girls 5 3 4 4 5 2

(d) Wage/hr (Rs) 40-60 60-80 80-100 100-120 120-140 140-160 Ans: Rs 27.74 , 0.24

No. of workers 6 9 13 12 15 20

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B. MEAN DEVIATION

UNDERSTANDING (U) BASED PROBLEMS

1. (a) Define mean deviation on the basis of mean and write its formula.
( b) Define mean deviation on the basis of median and write its formula.
(c) Define coefficient of mean deviation
Write any two merits of mean deviation.
2. (a) Write any two demerits of mean deviation.
(b) Which central tendency can be used to calculate mean deviation? Which method is considered as the
(c) best? Write with reason.

3. (a) In a grouped data fm = 200, fm - x = 480 and N = 40 then calculate the mean deviation and its

(b) coefficient. Ans: 12, 3

(c) In a grouped data fm = 480, fm - x = 200 and N = 40 then calculate the mean deviation and its

(d) coefficient. Ans: 5, 0.4167

4. (a) In a grouped data fm = 7800, fm - x = 5070 and N = 65 then calculate the mean deviation and

(b) its coefficient. Ans: 12, 0.65

(c) In a grouped data fm = 9240, fm - x = 2100 and N = 84 then calculate the mean deviation and

(d) its coefficient. Ans: 25, 0.2273
5. (a)
In a continuous data if fm = 600, fm - x = 70 and MD from mean = 14 find the mean.
(b) Ans: 120

(c) In a continuous data if fm = 2000, fm - x = 600 and MD from mean = 6 find the mean.
(d) Ans: 20

In a grouped data, if fm = 1350, mean ( x ) = 27 and fm - x = 560, find the mean deviation from

mean. Ans: 11.20

In a continuous data if fm = 600, N = 30 and fm - x = 120, find the coefficient of mean

deviation. Ans: 0.2

In a continuous data if fm - Mdn= 544 , N = 40 and median (Mdn) = 37 calculate the mean

deviation and its coefficient from median. Ans: 13.6, 0.37

In grouped data, if fm - Mdn= 560 , N = 50 and median (Mdn) = 27 calculate the mean deviation

and its coefficient from median. Ans: 11.2, 0.41

In a continuous data if fm - Mdn= 680 , N = 60 and median (Mdn) = 35 calculate the mean

deviation and its coefficient from median. Ans: 11.33, 0.32

In a continuous data if fm - Mdn= 448 , N = 100 and median (Mdn) = 24 calculate the mean

deviation and its coefficient from median. Ans: 4.48, 0.187

APPLICATION (A) BASED PROBLEMS

1. Find the mean deviation from mean and its coefficient of the given data.

(a) Marks obtained 0-10 10-20 20-30 30-40 40-50 [SEE MODEL-2076]
Ans: 10.3, 0.3678
No. of students 2 3 6 5 4

(b) Marks obtained 0-10 10-20 20-30 30-40 40-50 Ans: 11.2, 0.41
No. of students 5 8 15 16 6

(c) Height in cm 4-8 8-12 12-16 16-20 20-24 24-28 Ans: 10.1, 0.367
No. of plants 33 2 7 4 1

2. Find the mean deviation and its coefficient from median of the data given alongside.
(a)

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Marks obtained 0-10 10-20 20-30 30-40 40-50 50-60 Ans: 12.8, 0.43

No. of students 5 8 12 10 9 6

Ans: 6.2, 0.32 Ans: 9.36, 0.36
(b) Weight (in kg) 0-10 10-20 20-30 30-40 40-50

No. of people 5 8 15 10 6

C. STANDARD DEVIATION

UNDERSTANDING (U) BASED PROBLEMS

1. (a) Define standard deviation.
(b)
(c) What is coefficient of standard deviation?
(d)
(e) What do you mean by coefficient of variation?

2. (a) Write a merit and a demerit of a standard deviation.
(b)
(c) What does more and less value of coefficient of variation signify?
(d)
(e) Write the formula for calculating the standard deviation by exact mean method.

3. (a) Write the formula to find the standard deviation by assumed mean method.
(b)
Write the formula for calculating the standard deviation by direct method.
4. (a)
(b) Write the formula to find the coefficient of standard deviation.
(c)
(d) Write the formula to find the coefficient of variation.

5. (a) Write any two differences between coefficient of standard deviation and coefficient of variation.
(b)
(c) Write any two differences between mean deviation and standard deviation.

6. (a) If the standard deviation of set of data is 0.25, find its variance. [SEE MODEL-2076] Ans:0.0625

(b) If the standard deviation of set of data is 0.5, find its variance. Ans:0.25

(c) If the standard deviation of set of data is 0.35, find its variance. Ans:0.1225Ans: 9.36, 0.36

(d) If the standard deviation of set of data is , find its variance. Ans:0.25

7. (a) If the variance of set of data is 25, find its standard deviation. Ans:5

(b) If the variance of set of data is 12.25, find its standard deviation. Ans: 3.5

(c) If the variance of set of data is 0.144, find its standard deviation. Ans: 0.12

(d) In a continuous series, if N = 50, mean ( x ) = 20 and f(m - x )2 = 1250, find the standard deviation

8. (a) and its coefficient. Ans: 5, 0.25

(b) In a grouped data, if N = 7, mean ( x ) = 44 and f(m - x )2 = 432, find the standard deviation and its

(c) coefficient. Ans: 7.8, 0.18

In a continuous series, if N = 100, fm = 1200 and f(m - x )2 = 848, find the standard deviation and

its coefficient. Ans: 2.912, 0.247

In a grouped data, if N = 50, fm = 1550 and f (m - x )2 = 8400, find the standard deviation and its

coefficient. Ans: 12.96, 0.418 Ans: 9.36, 0.36

In a grouped data, if N = 20, fm = 480 and fm2 = 12400, find the standard deviation and its

coefficient. Ans: 6.633, 0.276

In a grouped data, N = 100, fm = 3100 and fm2 = 112800 , find the standard deviation and its

coefficient. Ans: 12.92, 0.416

In a grouped data, if N = 50, fm = 1000 and fm2 = 40000, find the standard deviation and its

coefficient. Ans: 20, 0.5

In a grouped data, if N = 50, fm = 1750 and fm2 = 61700, find the standard deviation and its

coefficient. Ans: 3, 0.0857

In a grouped data, if N = 30, fd = 90 and fd2 = 5700 where d = A –m , find the standard deviation .

Ans: 13.453

In a grouped data, if N = 50, fd = 200 and fd2 = 1250 where d = A –m , find the standard deviation

. Ans: 3
In a grouped data, if N = 100, fd = -600 and fd2 = 5200 where d = A –m , find the standard

deviation . Ans: 4

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(d) In a grouped data, if N = 40, fd = -120 and fd2 = 2000 where d = A –m , find the standard deviation
Ans: 6.403

APPLICATION (A) BASED PROBLEMS

1. Find the standard deviation, coefficient of standard deviation and coeficent of variation of the given data.

(a) Age in years 0-4 4-8 8-12 12-16 16-20 20-24 [SEE MODEL-2076]

No of students 7 7 10 15 7 6 Ans: 6.03, 0.5042, 50.42%

(b) Marks obtained 0-10 10-20 20-30 30-40 40-50 50-60 Ans: 12.96, 0.418, 41.8%

No of students 8 12 20 40 12 8

(c) Ans: 36.60, 63.17%
Age in years 0-4 4-8 8-12 12-16 16-20 20-24

No of students 7 7 10 15 7 6

SEE MODEL QUESTION SET

Optional-I (Mathematics)

Time: 3 hours Maximum Marks: 100

Candidates are required to answer in their own words as far as practicable. Credit shall be given to

originality in expression, creativity and neatness in hand, not to rote learning.

Attempt all the questions.

Group-A (10  1 = 10)

1. (a) Write down the period of sine function.

(b) State remainder theorem.

2. (a) Determine the continuity or discontinuity of the graph of

function f (x) at x = 1.

(b) Define singular matrix.

3. (a) Write down the condition of coincident of pair of straight lines represented by the
equation ax2 + 2hxy + by2 = 0.

(b) Identify and name the conic section from the figure given alongside.

4. (a) Write down the formula of cos2 in terms of tan.

(b) Express 2sinA.sinB in terms of sum or difference of cosines.
rr

5. (a) If a and b are two vectors and θ be angle between them then what will be the value of

cos θ?

(b) In an inversion transformation, if P is the image of the P and r is the radius of inversion

circle with centre O, write the relation of OP, OP and r.

Group-B (13  2 = 26)

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6. (a) If (x + 3) is a factor of x3 – (k – 1) x2 + kx + 54, find the value of k.

(b) Find the values of x, y and z from the given GP: 1 , x, y, z, 2

8

(c) Find the vertex of the parabola y = x2 – 2x – 3.

7. (a)  1 0 , find the determinant of AAT.
(b) If A =  
 2 4 
8. (a)
Find the value of x from the linear equations 3x – y = 7 and x + 2y = 4 by using

Cramer’s rule.

Find the angle between the pair of straight lines represented by 2x2 + 7xy + 3y2 = 0.

(b) Find the equation of a circle whose centre is (4, 5) and y-axis is tangent to it.

9. (a) Prove that: sinA - cosA +1 = tan A
sinA + cosA +1 2

(b) Find the value of cos1300 + cos1100 + cos100

(c) Solve: sin2 = sin [00    900]

r rr

10. (a) Find the angle between a = 3i + j and X-axis.

rr r r

(b) The position vectors of A and B are i + 2j and 6i + 7 j . Find the position vector of a point

P which divides AB internally in the ratio 2:3.

(c) In a continuous series, the lower quartile is 12 and the upper quartile exceeds twice the

lower quartile by 4. Find the quartile deviation and its coefficient.

Group-C (11 × 4 = 44)

11. If f = {(x, 3x – 11): xR}, g =  x, 2x - 3  : x  R  and ff (x) = g –1 (x), find the value of x.
 5  


12. If the third and eleventh terms of an arithmetic series are 8 and -8 respectively, find the

sum of the first seven terms of the series.

13. For a real valued function f (x) = 3x +5.

(i) What are the values of f (x) at x = 1.9, 1.99 and 1.999, 2.1 and 2.01 and 2.001?

(ii) Is this function continuous at x = 2?

14. Solve by matrix method: 3 x + y = 7 , x  y = 1

2 33

15. A (3, 5) and C (7, 9) are the opposite vertices of a rhombus ABCD, find the equation of the

diagonal BD.

 πc  3πc  5πc  7πc 
16. Calculate the value of 1+ cos  1+ cos 1+ cos 1+ cos 
 8  8  8  8 

17. If A + B + C = 1800, prove that sinA.cosA +sinB.cosB+sinC.cosC = 2sinA.sinB.sinC

18. The angle of depression and elevation of the top of a building 25m high from the top and
bottom of a cliff are found to be 600 and 300 respectively, find the height of the cliff.

19. Find the22 matrix which transforms the unit square into a parallelogram  0 2 5 3
 0 1 0 1 .


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20. Calculate the mean deviation and its coefficient of the data from median.

Height (in cm) 0-10 10-20 20-30 30-40 40-50
15 10 6
No. of plants 5 8

21. The age of 30 boys of a school is given below.

Age (in years) 0-4 4-8 8-12 12-16 16-20

No. of boys 4 6 10 6 4

(i) What is the average age of the boys?

(ii) What is the standard deviation of the data?

Group-D (4 × 5 = 20)

22. Write one importance of linear programming. Optimize the objective function P = 3x + 2y
subject to the following constraints: x + y ≤ 5, x + 4y ≤ 8, x ≥ 0, y ≥ 0

23. A cow tighten to a stake by a rope moves through the points (2, -2), (5, 7) and (6, 6) in a

path always keeping the rope tight. Find the coordinates of the point at which the stake is

fixed and the equation representing the locus of cow.

24. In a rhombus BEST, prove by vector method that the diagonals BS and ET bisect to each

other at a right angle.
25. Rotate the quadrilateral ABCD with vertices A (1, 1), B (2, 3), C (4, 2) and D (3, −2) about

origin through180°. Then reflect the image of quadrilateral so obtained about the line y = −

x. Find the coordinates of the vertices of images quadrilaterals and represent them in the

same graph paper. Also, write the name of transformation which denotes the combined

transformation of above two transformations.

The End

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