ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection

1. Find the co-ordinates of the images of the following points under reflection in the x- axis:
(i) (2,-5)
(ii) (-3/2,-1/2)
(iii) (-7,0)
Solution:
We have given the points:
(i) (2,-5)
(ii) (-3/2,-1/2)
(iii) (-7,0)
So, 
The co-ordinates of the images of the points under reflection in the x-axis will be:
(i) Image of (2,-5) will be (2,5)
(ii) Image of (-3/2,-1/2) will be (-3/2,1/2)
(iii) Image of (-7,0) will be (-7,0)
Hence, the coordinates will be (2,5),(-3/2,1/2),(-7,0)
 
2. Find the co-ordinates of the images of the following points under reflection in the y-axis:
(i) (2,-5)
(ii) (-3/2,1/2)
(iii) (0,-7)
Solution:
We have given that:
(i) (2,-5)
(ii) (-3/2,1/2)
(iii) (0,-7)
so, The co-ordinates of the image of the points under reflection in the y-axis will be:
(i) Image of (2,-5) will be (-2,-5)
(ii) Image of (-3/2,1/2) will be (3/2,1/2)
(iii) Image of (0,-7) will be (0,-7)
Hence, the coordinates will be (-2,-5),(3/2,1/2),(0,-7)
 
3. Find the co-ordinates of the images of the following points under reflection in the origin:
(i) (2,-5)
(ii) (-3/2,-1/2)
(iii) (0,0)
Solution:
We have given that:
(i) (2,-5)
(ii) (-3/2,-1/2)
(iii) (0,0)
So, The co-ordinate of the image of the points under reflection in the y-axis will be:
(i) Image of (2,-5) will be (-2,5)
(ii) Image of (-3/2,-1/2) will be (3/2,1/2)
(iii) Image of (0,0) will be (0,0)
Hence, the coordinates will be (-2,5),(3/2,1/2),(0,0)
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-1
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-2
 
 
8. (i) The point P (2,4) on reflection in the line y = 1 is mapped onto P’ Find the co-ordinates of P’.
(ii) Find the image of the point P ( -3,-5) in the line y = -2.
Solution:
We have given that:
Point P(2,4) in the line y=1 mapped onto P'
(i) The steps are:
(a) Draw axis XOX’ and YOY’ and take 1 cm = 1 unit.
(b) Plot point P (2,4) on it.
(c) Draw a line y = 1, which is parallel to x-axis.
(d) From P, draw a perpendicular on y = 1 meeting it at Q.
(e) Produce PQ to P’ such that QP’ = PQ.
Therefore, P’ is the reflection of P whose co-ordinates are (2,-2).
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-3
ii) we have given that point P(-3,-5) on line y=-2
The steps are:
(a) Draw axis XOX’ and YOY’ and take 1 cm = 1 unit.
(b) Plot point P (-3,-5) on it.
(c) Draw a line y = -2 which is parallel to the x-axis.
(d) From P, draw a perpendicular on y = -2 which meets it at Q.
(e) Produce PQ to P’ such that QP’ = PQ.
Therefore, P’ is the image of P, whose co-ordinates are (-3,1).
Hence, the given values p=(-3,1) are not roots of the equation.
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-4
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-5
 
 
9. The point P ( -4,-5) on reflection in y-axis is mapped on P’. The point P’ on reflection in the origin is mapped on P”. Find the co-ordinates of P’ and P”. Write down a single transformation that maps P onto P”.
Solution:
We have given that:
Given, point P (-4,-5)
And, P’ is the image of point P in y-axis
Thus, the co-ordinates of P’ will be (4,-5)
Again,
P” is the image of P’ under reflection in origin.
Then, the co-ordinates of P’’ will be (-4,5).
The single transformation that maps P onto P” is the x-axis.
∴ the given values p=(-4,5) are not roots of the equation.
 
10. Write down the co-ordinates of the image of the point (3,-2) when:
(i) reflected in the x-axis
(ii) reflected in the y-axis
(iii) reflected in the x-axis followed by a reflection in the y-axis
(iv) reflected in the origin.
Solution:
We have given that:
The co-ordinates of the given point are (3,-2).
Then,
(i) Co-ordinates of the image reflected in x- axis will be (3,2)
(ii) Co-ordinates of the image reflected in y- axis will be (-3,-2)
(iii) Co-ordinates of the point reflected in x- axis followed by reflection in the y-axis will be (-3,2)
(iv) Co-ordinates of the point reflected in the origin will be (−3,2).
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-6
 
The steps are:
(i) Draw axis XOX’ and YOY’ taking 1 cm = 1 unit.
(ii) Plot a point P (3,1).
(iii) Draw a line x = 1, which is parallel to y-axis.
(iv) From P, draw a perpendicular on x-axis meeting it at Q.
(v) Produce PQ to P’ such that QP’ = PQ, then
P’ is the image of P is x-axis. Then co-ordinates of P’ will be (3,-1)
(vi) From P’, draw a perpendicular on x = 1 meeting it at R.
(vii) Produce P’R to P” such that RP” = P’R
Then, P” is the image of P’ in the line x = 1
∴, the co-ordinates of P” are (-1,-1)
∴, the given values p=(-1,-1) are not roots of the equation.
 
12. If P’ (-4,-3) is the image of a point P under reflection in the origin, find
(i) the co-ordinates of P.
(ii) the co-ordinates of the image of P under reflection in the line y = -2.
Solution:
We have given that:
(i) Reflection of P is P’ (-4,-3) in the origin
Then, the co-ordinates of P will be (4,3)
Now,
Draw a line y = -2, which is parallel to x-axis
(ii) From P, draw a perpendicular on y = -2 meetings it at Q
Produce PQ to P” such that QP” = PQ
Then, P” will the image of P in the line y = -2
Hence, the co-ordinates of P” will be (4,-7).
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-7
 
13. A point P (a,b) is reflected in the x-axis to P’ (2,-3), write down the values of a and b. P” is the image of P, when reflected in the y-axis. Write down the co-ordinates of P”. Find the co-ordinates of P”, when P is reflected in the line parallel to y-axis such that x = 4.
Solution:
We have given that:
P’ (2,-3) is the reflection of P (a,b) in the x-axis
Hence, the co-ordinates of P’ will be (a,– b) but P’ is (2,-3)
After comparing, we get a = 2,b = 3
Then, the co-ordinates of P will be (2,3)
And,
P” is the image of P when reflected in y-axis
Hence, the co-ordinate of P” will be ( – 2,3)
Now, draw a line x = 4, which is parallel to y-axis
As P’” is the image of P when it is reflected in the line x = 4,
So, P’” is its reflection.
Then, the co-ordinates of P”’ will be (6,3).
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-8
 
14. (i) Point P (a,b) is reflected in the x-axis to P’ (5,-2). Write down the values of a and b.
(ii) P” is the image of P when reflected in the y-axis. Write down the co-ordinates of P”.
(iii) Name a single transformation that maps P’ to P”.
Solution:
(i) Image of P (a,b) reflected in the x-axis to P’ (5,-2) (Given)
So, the co-ordinates of P will be (5,2)
Hence, a = 5 and b = 2
∴ the given values a=5 and b=2 are not roots of the equation.
(ii) P” is the image of P when reflected in the y-axis
∴ its co-ordinates will be (-5,-2).
(iii) The single transformation that maps P’ to P” is the origin.
 
15. Points A and B have co-ordinates (2,5) and (0,3). Find
(i) the image A’ of A under reflection in the x-axis.
(ii) the image B’ of B under reflection in the line AA’.
Solution:
We have given that:
Given, co-ordinates of A are (2,5) and of B are (0,3)
Then, 
(i) Co-ordinates of A’, the image of A reflected in the x-axis will be (2,-5)
(ii) Co-ordinates of B’, the image of B under reflection in the line AA’ will be (4,3).
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-9
 
16. Plot the points A (2,-3),B (-1,2) and C (0,-2) on the graph paper. Draw the triangle formed by reflecting these points in the x-axis. Are the two triangles congruent?
Solution:
We have given that:
The points A (2,-3),B (-1,2) and C(0,-2) has been plotted on the graph paper as shown and are joined to form a triangle ABC. 
Now, the co-ordinates of the images of A,B and C reflected in x-axis will be:
A’ (2,3),B’ (-1,-2),C’ (0,2) respectively.
And, these are joined to from another ∆ A’B’C’
∴ we can say that 
Yes, these two triangles are congruent.
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-10
 
17. The points (6,2),(3,-1) and (-2,4) are the vertices of a right-angled triangle. Check whether it remains a right-angled triangle after reflection in the y-axis.
Solution:
We have given that:
Let A (6,2),B (3,-1) and C (-2,4) be the points of a right-angled triangle
Then,
The co-ordinates of the images of A,B,C reflected in y-axis will be:
A’ (-6,2),B’ (-3,-1) and C’ (2,4).  
Then, by joining these points
We can see that, ∆A’B’C’ is also found a right-angled triangle.
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-11
18. The triangle ABC where A (1,2),B (4,8),C (6,8) is reflected in the x-axis to triangle A’ B’ C’. The triangle A’ B’ C’ is then reflected in.the origin to triangle A”B”C” Write down the co-ordinates of A”,B”,C”. Write down a single transformation that maps ABC onto A” B” C”.
Solution:
We have given that:
The co-ordinates of ∆ ABC are A (1,2) B (4,8),C (6,8)
These vertices are reflected in x- axis as A’,B’ and C’. Hence, their co-ordinates are A’ (1,-2),B’ (4,-8) and C’ (6, -8).
Now,
A’,B’ and C’ are again reflected in origins to form an ∆A”B”C”.
Hence, the co-ordinates will be A” (-1,2),B” (-4,8) and C” (-6,8)
and The single transformation that maps ABC onto A” B” C” is y-axis.
 
19. The image of a point P on reflection in a line l is point P’. Describe the location of the line l.
Solution:
Plot a point p in the 1st quadrant anywhere and reflect this point in the 4th quadrant or y axis. 
Hence, we observe that an angle of 90 degree is formed where the line l touches the pp'. 
Therefore, The line will be the right bisector of the line segment joining P and P’.
 
20. Given two points P and Q, and that (1) the image of P on reflection in the y-axis is the point Q and (2) the midpoint of PQ is invariant on reflection in x-axis. Locate:
(ii) the y-axis and
(iii) the origin.
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-12
Solution:
We have given that:
Given, Q is the image of P on reflection in y-axis and mid-point of PQ is invariant on reflection in x-axis
(i) x-axis will be the line joining the points P and Q.
(ii) The line perpendicular bisector of line segment PQ is the y-axis.
(iii) The origin will be the mid-point of line segment PQ.
 
21. The point (-3,0) on reflection in a line is mapped as (3,0) and the point (2,-3) on reflection in the same line is mapped as (-2,-3).
(i) Name the mirror line.
(ii) Write the co-ordinates of the image of (-3,-4) in the mirror line.
Solution:
We have given that:
The point (-3,0) is the image of point (3,0) and point (2,-3) is image of point (-2,-3) reflected on the same line.
Then, 
(i) Clearly, it’s seen that the mirror line will be y-axis.
(ii) The co-ordinates of the image of the point (-3,-4) reflected in the same line i.e. y-axis will be (3,-4).
 
22. A (-2,4) and B (-4,2) are reflected in the y-axis. If A’ and B’ are images of A and B respectively, find 
(i) the co-ordinates of A’ and B’.
(ii) Assign a special name to a quad. AA’B’B.
(iii) State whether AB’ = BA’.
Solution:
We have given that:
A (-2,4) and B (-4,2) are reflected in the y-axis as A’ and B’.
Then, 
(i) The co-ordinates of A’ are (2,4) and of B are (4,2).
(ii) The quadrilateral AA’B’B is an isosceles trapezium.
(iii) Yes, it is found out that AB’ = BA’
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-13
 
23. Use graph paper for this question.
(i) The point P (2,-4) is reflected about the line x = 0 to get the image Q. Find the co-ordinates of Q.
(ii) Point Q is reflected about the line y = 0 to get the image R. Find the co-ordinates of R.
(iii) Name the figure PQR.
(iv) Find the area of figure PQR.
Solution:
We have given that:
(i) As the point Q is the reflection of the point P (2,-4) in the line x = 0,
Thus, the co-ordinates of Q are (2,4). 
(ii) As R is the reflection of Q (2,4) about the line y = 0, 
Thus, the co-ordinates of R are (– 2,4). 
(iii) Figure PQR is the right-angled triangle PQR. 
(iv) Area of ∆ PQR = ½ × QR × PQ 
= ½ × 4 × 8  
= 16 sq. units.
Hence, the area of ∆PQR is 16 sq.units
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-14
 
 
24. Use graph paper for this question. The point P (5,3) was reflected in the origin to get the image P’.
(i) Write down the co-ordinates of P’.
(ii) If M is the foot of the perpendicular from P to the x-axis, find the co-ordinates of M.
(iii) If N is the foot of the perpendicular from P’ to the x-axis, find the co-ordinates of N.
(iv) Name the figure PMP’N.
(v) Find the area of the figure PMP’N.
Solution:
We have given that:
Given, P’ is the image of point P (5,3) reflected in the origin.
(i) Co-ordinates of P’ will be (-5,-3). 
(ii) M is the foot of the perpendicular from P to the x-axis. 
Hence, the co-ordinates of M will be (5,0) 
Hence, the given values M=(5,0) are not roots of the equation.
(iii) N is the foot of the perpendicular from P’ to x-axis. 
Hence, the co-ordinates of N will be (-5,0). 
Hence, the given values N=(-5,0) are not roots of the equation.
(iv) By joining the points, the figure PMP’N is a parallelogram. 
(v) Area of the parallelogram = 2 x area of ∆ MPN 
= 2 × ½ × MN × PM 
= MN × PM = 10 × 3 
= 30 sq. units.
Hence, the area of parallelogram is 30 sq.units
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-15
 
25. Using a graph paper, plot the points A (6,4) and B (0,4).
(i) Reflect A and B in the origin to get the images A’ and B’.
(ii) Write the co-ordinates of A’ and B’.
(iii) State the geometrical name for the figure ABA’B’.
(iv) Find its perimeter.
Solution:
We have given that:
Points A (6,4) and B (0,4) are plotted on a graph paper.
(i) A and B are reflected in the origin to get images A’ and B’
(ii) Hence,
The co-ordinates of A’ are (-6,-4)
The co-ordinates of B’ are (0,-4)
(iii) The geometrical name for ABA’B’ is parallelogram
(iv) From the figure in graph paper, we see that
Length of AB = A’B’ = 6 units
And, BB’ = 8 units
In ∆ ABB’, by Pythagoras theorem
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-16
 
26. Use graph paper to answer this question
(i) Plot the points A (4,6) and B (1,2).
(ii) If A’ is the image of A when reflected in x-axis, write the co-ordinates of A’.
(iii) If B’ is the image of B when B is reflected in the line AA’, write the co-ordinates of B’.
(iv) Give the geometrical name for the figure ABA’B’.
Solution:
We have given that:
(i) Plotting the points A (4,6) and B (1,2) on the given graph. 
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-17
(ii) The co-ordinates of the image of A when reflected in axis are A’(4,-6) 
(iii) The co-ordinates of the image of B when reflected in the line AA’ are B’ = (7,2) 
(iv) It’s seen that in the quadrilateral ABA’B’, we have
AB = AB’ and A’B = A’B’ 
Thus, ABA’B’ is a kite.
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-18
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-19
 
28. The point P (3,4) is reflected to P’ in the x-axis and O’ is the image of O (origin) in the line PP’. Find:
(i) the co-ordinates of P’ and O’,
(ii) the length of segments PP’ and OO’.
(iii) the perimeter of the quadrilateral POP’O’.
Solution:
We have given that:
P’ is the image of P (3,4) reflected in x- axis and O’ is the image of O the origin in the line P’P. 
(i) Hence, co-ordinates of P’ are (3,-4) and co-ordinates of O’ reflected in PP’ are (6,0) 
(ii) Length of PP’ = 8 units and OO’ = 6 units 
(iii) Perimeter of POP’O’ is (4 x OP) units.
Let Q be the point of intersection of diagonals OO’ and PP’.
So, OQ = 3 units and OP = 4 units
Hence,
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-20
 
29. Use a graph paper for this question. (Take 10 small divisions = 1 unit on both axes). P and Q have co-ordinates (0,5) and (-2,4).
(i) P is invariant when reflected in an axis. Name the axis.
(ii) Find the image of Q on reflection in the axis found in (i).
(iii) (0,k) on reflection in the origin is invariant. Write the value of k.
(iv) Write the co-ordinates of the image of Q, obtained by reflecting it in the origin followed by a reflection in x-axis.
Solution:
We have given, 
Two points P (0,5) and Q (-2,4)
(i) As the abscissa of P is 0. It is invariant when is reflected in y-axis. 
(ii) Let Q’ be the image of Q on reflection in y-axis. 
Thus, the co-ordinate of Q’ will be (2,4) 
(iii) (0,k) on reflection in the origin is invariant. 
So, the co-ordinates of image will be (0,0)
Hence, k = 0 
(iv) The reflection of Q in the origin is the point Q” and its co-ordinates will be (2,– 4) and reflection of Q” (2,– 4) in x-axis is (2,4) which is the point Q’.
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-21
 
 
Chapter Test

1. The point P (4,– 7) on reflection in x-axis is mapped onto P’. Then P’ on reflection in the y-axis is mapped onto P”. Find the co-ordinates of P’ and P”. Write down a single transformation that maps P onto P”.
Solution:
We have given that:
P’ is the image of P (4,-7) reflected in x-axis
Then, the co-ordinates of P’ are (4,7) 
Again P” is the image of P’ reflected in y-axis
Hence, the co-ordinates of P” are (-4,7)
Therefore, single transformation that maps P and P” is in the origin.
Hence, the given values p=(-4,7) are not roots of the equation.
 
2. The point P (a,b) is first reflected in the origin and then reflected in the y-axis to P’. If P’ has co-ordinates (3,– 4), evaluate a,b
Solution:
We have given that:
The co-ordinates of image of P (a,b) reflected in origin are (-a,-b). 
Again, the co-ordinates of P’ which is image of the above point (-a,-b) reflected in the y-axis are (a,-b).
But the co-ordinates of P’ are (3,-4) (Given)
Therefore, a = 3 and -b = -4 ⇒ b = 4
 
3. A point P (a,b) becomes (– 2,c) after reflection in the x-axis, and P becomes (d,5) after reflection in the origin. Find the values of a,b,c and d.
Solution:
As we have given that:
Given, point P (a,b) and the image of P (a,b) after reflected in the x-axis be (a,-b)
But, we have already given as (-2,c)
Thus, a = -2,c = -b 
Then,
If P is reflected in the origin, then its co-ordinates will be (-a,-b)
But, as it is given that (d,5)
Thus,
-b = 5 ⇒ b = -5, 
d = -a = -(-2) = 2,
c = -b = -(-5) = 5 
Thus,
a = -2,b = -5,c = 5 and d = 2
 
4. A (4,– 1),B (0,7) and C ( – 2,5) are the vertices of a triangle. ∆ ABC is reflected in the y-axis and then reflected in the origin. Find the co-ordinates of the final images of the vertices.
Solution:
We have given that:
Given, A (4,-1),B (0,7) and C (-2,5) are the vertices of ∆ABC. 
∆ABC after reflecting in y-axis, the co-ordinates of points will be A’ (-4,-1),B’ (0,7),C’ (2,5).
Again, when ∆A’B’C’ reflecting in origin:
Hence, The co-ordinates of the images of the vertices will be A” (4,1),B” (0,-7),C” (-2,-5).
 
5. The points A (4,– 11),B (5,3),C (2,15), and D (1,1) are the vertices of a parallelogram. If the parallelogram is reflected in the y-axis and then in the origin, find the co-ordinates of the final images. Check whether it remains a parallelogram. Write down a single transformation that brings the above change.
Solution:
We have given that:
Given, points A (4,-11),B (5,3),C (2,15) and D (1,1) are the vertices of a parallelogram. 
After reflecting in y-axis, the images of these points will be 
A’ ( -4,11),B’ (-5,3),C (-2,15) and D’ (-1,1). 
Again, reflecting these points in origin, the image of these points will be A” (4,-11),B” (5,-3),C” (2,-15) and D” (0,-1).
Yes, the reflection of a single transformation is in the x-axis.
 
6. Use a graph paper for this question (take 2 cm = 1 unit on both x and y axes). 
(i) Plot the following points: A (0,4),B (2,3),C (1,1) and D (2,0). 
(ii) Reflect points B,C,D on y-axis and write down their coordinates. Name the images as B’,C’,D’ respectively. 
(iii) Join points A,B,C,D,D’,C’,B’ and A in order, so as to form a closed figure. Write down the equation of line of symmetry of the figure formed.
Solution:
We have given that, 
(i) On graph: A (0,4),B (2,3),C (1,1) and D (2,0) 
(ii) Point after reflection on y-axis are B’ = (-2,3),C’ = (-1,1) and D’ = (-2,0)
(iii) The points A,B,C,D,D’,C’,B’ and A in order to form a closed figure.
Hence, the equation of the line of symmetry is x = 0
 
ML Aggarwal Solutions Class 10 Maths Chapter 10 Reflection-22
 
7. The triangle OAB is reflected in the origin O to triangle OA’B’.A’ and B’ have coordinates ( – 3,– 4) and (0,– 5) respectively.
(i) Find the co-ordinates of A and B. 
(ii) Draw a diagram to represent the given information. 
(iii) What kind of figure is the quadrilateral ABA’B’?
(iv) Find the coordinates of A”, the reflection of A in the origin followed by reflection in the y-axis.
(v) Find the co-ordinates of B”, the reflection of B in the x-axis followed by reflection in the origin.
Solution:
We have given that:
∆ OAB is reflected in the origin O to ∆ OA’B’,
And the co-ordinates of A’ = (-3,-4) and B’ = (0,-5).
(i) Hence, the co-ordinates of A will be (3,4) and of B will be (0,5). 
(ii) The diagram representing the given information has been drawn here. 
(iii) The figure in the diagram is a rectangle. 
(iv) The co-ordinates of B’, the reflection of B is the x-axis is (0,-5) and co-ordinates of B”, the reflection in origin of the point (0,-5) will be (0,5). 
(v) The co-ordinates of the points, the reflection of A in the origin are (-3,-4) and coordinates of A”, the reflected in y-axis of the point (-3,– 4) are (3,-4).
 
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ML Aggarwal Solutions Class 10 Maths Chapter 22 Probability
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NCERT Exemplar Solutions Class 10 Maths Arithmetic Progression
NCERT Exemplar Solutions Class 10 Maths Circles
NCERT Exemplar Solutions Class 10 Maths Construction
NCERT Exemplar Solutions Class 10 Maths Coordinate Geometry
NCERT Exemplar Solutions Class 10 Maths Linear Equations
NCERT Exemplar Solutions Class 10 Maths Polynomials
NCERT Exemplar Solutions Class 10 Maths Quadratic Equation
NCERT Exemplar Solutions Class 10 Maths Real Numbers
NCERT Exemplar Solutions Class 10 Maths Surface Area and Volume
NCERT Exemplar Solutions Class 10 Maths Triangles
NCERT Exemplar Solutions Class 10 Maths Trigonometry