## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11

Question 1.
Find the co-ordinates of the mid-point of the line segments joining the following pairs of points:
(i) (2, -3), ( -6, 7)
(ii) (5, -11), (4, 3)
(iii) (a + 3, 5b), (2a – 1, 3b + 4)
Solution:

Question 2.
The co-ordinates of two points A and B are (-3, 3) and (12, -7) respectively. P is a point on the line segment AB such that AP : PB = 2 : 3. Find the co-ordinates of P.
Solution:

Question 3.
P divides the distance between A (-2, 1) and B (1, 4) in the ratio of 2 : 1. Calculate the co-ordinates of the point P.
Solution:

Question 4.
(i) Find the co-ordinates of the points of trisection of the line segment joining the point (3, -3) and (6, 9).
(ii) The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, – 2) and $$\left( \frac { 5 }{ 3 } ,q \right)$$ respectively, find the values of p and q.
Solution:

Question 5.
(i) The line segment joining the points A (3, 2) and B (5, 1) is divided at the point P in the ratio 1 : 2 and it lies on the line 3x – 18y + k = 0. Find the value of k.
(ii) A point P divides the line segment joining the points A (3, -5) and B (-4, 8) such that $$\frac { AP }{ PB } =\frac { k }{ 1 }$$ If P lies on the line x + y = 0, then find the value of k.
Solution:

Question 6.
Find the coordinates of the point which is three-fourths of the way from A (3, 1) to B (-2, 5).
Solution:

Question 7.
Point P (3, -5) is reflected in P’ in the x-axis. Also, P on reflection in the y-axis is mapped as P”.
(i) Find the co-ordinates of P’ and P”.
(ii) Compute the distance P’ P”.
(iii) Find the middle point of the line segment P’ P”.
(iv) On which co-ordinate axis does the middle point of the line segment P P” lie?
Solution:

Question 8.
Use graph paper for this question. Take 1 cm = 1 unit on both axes. Plot the points A(3, 0) and B(0, 4).
(i) Write down the co-ordinates of A1, the reflection of A in the y-axis.
(ii) Write down the co-ordinates of B1, the reflection of B in the x-axis.
(iii) Assign the special name to the quadrilateral ABA1B1.
(iv) If C is the midpoint is AB. Write down the co-ordinates of the point C1, the reflection of C in the origin.
(v) Assign the special name to quadrilateral ABC1B1.
Solution:

Question 9.
The line segment joining A (-3, 1) and B (5, -4) is a diameter of a circle whose centre is C. find the co-ordinates of the point C. (1990)
Solution:

Question 10.
The mid-point of the line segment joining the points (3m, 6) and (-4, 3n) is (1, 2m – 1). Find the values of m and n.
Solution:

Question 11.
The co-ordinates of the mid-point of the line segment PQ are (1, -2). The co-ordinates of P are (-3, 2). Find the co-ordinates of Q.(1992)
Solution:

Question 12.
AB is a diameter of a circle with centre C (-2, 5). If point A is (3, -7). Find:
(i) the length of radius AC.
(ii) the coordinates of B.
Solution:

Question 13.
Find the reflection (image) of the point (5, -3) in the point (-1, 3).
Solution:

Question 14.
The line segment joining A $$\left( -1,\frac { 5 }{ 3 } \right)$$ the points B (a, 5) is divided in the ratio 1 : 3 at P, the point where the line segment AB intersects y-axis. Calculate
(i) the value of a
(ii) the co-ordinates of P. (1994)
Solution:

Question 15.
The point P (-4, 1) divides the line segment joining the points A (2, -2) and B in the ratio of 3 : 5. Find the point B.
Solution:

Question 16.
(i) In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7 ,6) ?
(ii) In what ratio does the point (-4, b) divide the line segment joining the points P (2, -2), Q (-14, 6) ? Hence find the value of b.
Solution:

Question 17.
The line segment joining A (2, 3) and B (6, -5) is intercepted by the x-axis at the point K. Write the ordinate of the point k. Hence, find the ratio in which K divides AB. Also, find the coordinates of the point K.
Solution:|

Question 18.
If A (-4, 3) and B (8, -6)
(i) find the length of AB.
(ii) in what ratio is the line joining AB, divided by the x-axis? (2008)
Solution:

Question 19.
(i) Calculate the ratio in which the line segment joining (3, 4) and(-2, 1) is divided by the y-axis.
(ii) In what ratio does the line x – y – 2 = 0 divide the line segment joining the points (3, -1) and (8, 9)?
Also, find the coordinates of the point of division.
Solution:

Question 20.
Given a line segment AB joining the points A (-4, 6) and B (8, -3). Find:
(i) the ratio in which AB is divided by the y-axis.
(ii) find the coordinates of the point of intersection.
(iii)the length of AB.
Solution:

Question 21.
(i) Write down the co-ordinates of the point P that divides the line joining A (-4, 1) and B (17, 10) in ratio 1 : 2.
(ii)Calculate the distance OP where O is the origin.
(iii)In what ratio does the y-axis divide the line AB?
Solution:

Question 22.
Calculate the length of the median through the vertex A of the triangle ABC with vertices A (7, -3), B (5, 3) and C (3, -1)
Solution:

Question 23.
Three consecutive vertices of a parallelogram ABCD are A (1, 2), B (1, 0) and C (4, 0). Find the fourth vertex D.
Solution:

Question 24.
If the points A (-2, -1), B (1, 0), C (p, 3) and D (1, q) from a parallelogram ABCD, find the values of p and q.
Solution:

Question 25.
If two vertices of a parallelogram are (3, 2) (-1, 0) and its diagonals meet at (2, -5), find the other two vertices of the parallelogram.
Solution:

Question 26.
Prove that the points A (-5, 4), B (-1, -2) and C (5, 2) are the vertices of an isosceles right-angled triangle. Find the coordinates of D so that ABCD is a square.
Solution:

Question 27.
Find the third vertex of a triangle if its two vertices are (-1, 4) and (5, 2) and the midpoint of one side is (0, 3).
Solution:

Question 28.
Find the coordinates of the vertices of the triangle the middle points of whose sides are $$\left( 0,\frac { 1 }{ 2 } \right) ,\left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right) and\left( \frac { 1 }{ 2 } ,0 \right)$$
Solution:

Question 29.
Show by section formula that the points (3, -2), (5, 2) and (8, 8) are collinear.
Solution:

Question 30.
Find the value of p for which the points (-5, 1), (1, p) and (4, -2) are collinear.
Solution:

Question 31.
A (10, 5), B (6, -3) and C (2, 1) are the vertices of triangle ABC. L is the midpoint of AB, M is the mid-point of AC. Write down the co-ordinates of L and M. Show that LM = $$\\ \frac { 1 }{ 2 }$$ BC.
Solution:

Question 32.
A (2, 5), B (-1, 2) and C (5, 8) are the vertices of a triangle ABC. P and.Q are points on AB and AC respectively such that AP : PB = AQ : QC = 1 : 2.
(i) Find the co-ordinates of P and Q.
(ii) Show that PQ = $$\\ \frac { 1 }{ 3 }$$ BC.
Solution:

Question 33.
The mid-point of the line segment AB shown in the adjoining diagram is (4, -3). Write down die co-ordinates of A and B.

Solution:

Question 34.
Find the co-ordinates of the centroid of a triangle whose vertices are A (-1, 3), B(1, -1) and C (5, 1) (2006)
Solution:

Question 35.
Two vertices of a triangle are (3, -5) and (-7, 4). Find the third vertex given that the centroid is (2, -1).
Solution:

Question 36.
The vertices of a triangle are A (-5, 3), B (p, -1) and C (6, q). Find the values of p and q if the centroid of the triangle ABC is the point (1, -1).
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 10 Reflection Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 10 Reflection Chapter Test

Question 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:

Question 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:

Question 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:

Question 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:

Question 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:

Question 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 the 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. (2017)
Solution:

Question 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:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 10 Reflection MCQS

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 10 Reflection MCQS

Choose the correct answer from the given four options (1 to 7):

Question 1.
The reflection of the point P (-2, 3) in the x-axis is
(a) (2, 3)
(b) (2, -3)
(c) (-2, -3)
(d) (-2, 3)
Solution:

Question 2.
The reflection of the point P (-2, 3) in the y- axis is
(a) (2, 3)
(b) (2, -3)
(c) (-2, -3)
(d) (0, 3)
Solution:

Question 3.
If the image of the point P under reflection in the x-axis is (-3, 2), then the coordinates of the point P are
(a) (3, 2)
(b) (-3, -2)
(c) (3, -2)
(d) (-3, 0)
Solution:

Question 4.
The reflection of the point P (1, -2) in the line y = -1 is
(a) ( -3, -2)
(b) (1, – 4)
(c) (1 , 4)
(d) (1, 0)
Solution:

Question 5.
The reflection of the point A (4, -1) in the line x = 2 is
(a) (0, -1)
(b) (8, -1)
(c) (0, 1)
(d) none of these
Solution:

Question 6.
The reflection of the point (-3, 0) in the origin is the point
(a) (0, -3)
(b) (0, 3)
(c) (3, 0)
(d) none of these
Solution:

Question 7.
Which of the following points is invariant with respect to the line y = -2 ?
(a) (3, 2)
(b) (3, -2)
(c) (2, 3)
(d) (-2, 3)
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 10 Reflection Ex 10

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 10 Reflection Ex 10

Question 1.
Find the co-ordinates of the images of the following points under reflection in the x- axis:
(i) (2, -5)
(ii) $$-\frac { 3 }{ 2 } ,-\frac { 1 }{ 2 }$$
(iii) (-7, 0)
Solution:

Question 2.
Find the co-ordinates of the images of the following points under reflection in the y-axis:
(i) (2, -5)
(ii) $$-\frac { 3 }{ 2 } ,\frac { 1 }{ 2 }$$
(iii) (0, -7)
Solution:

Question 3.
Find the co-ordinates of the images of the following points under reflection in the origin:
(i) (2, -5)
(ii) $$\frac { -3 }{ 2 } ,\frac { -1 }{ 2 }$$
(iii) (0, 0)
Solution:

Question 4.
The image of a point P under reflection in the x-axis is (5, -2). Write down the coordinates of P.
Solution:

Question 5.
A point P is reflected in the x-axis. Co-ordinates of its image are (8, -6).
(i) Find the co-ordinates of P.
(ii) Find the co-ordinates of the image of P under reflection in the y-axis.
Solution:

Question 6.
A point P is reflected in the origin. Co-ordinates of its image are (2, -5). Find
(i) the co-ordinates of P.
(ii) the co-ordinates of the image of P in the x-axis.
Solution:

Question 7.
(i) The point P (2, 3) is reflected in the line x = 4 to the point P’. Find the co-ordinates of the point P’.
(ii) Find the image of the point P (1, -2) in the line x = -1.
Solution:

Question 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:

Question 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:

Question 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. (2000)
Solution:

Question 11.
Find the co-ordinates of the image of (3, 1) under reflection in x-axis followed by a reflection in the line x = 1.
Solution:

Question 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:

Question 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. (1998)
Solution:

Question 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”. (1997)
Solution:

Question 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:

Question 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:

Question 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:

Question 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:

Question 19.
The image of a point P on reflection in a line l is point P’. Describe the location of the line l.
Solution:

Question 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
(i) the x-axis
(ii) the y-axis and
(iii) the origin.
Solution:

Question 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:

Question 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:

Question 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. (2007)
Solution:

Question 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. (2001)
Solution:

Question 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:

Question 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’. (2009)
Solution:

Question 27.
The points A (2, 3), B (4, 5) and C (7, 2) are the vertices of ∆ABC. (2006)
(i) Write down the co-ordinates of A1, B1, C1 if ∆ A1B1C1 is the image of ∆ ABC when reflected in the origin.
(ii) Write down the co-ordinates of A2, B2, C2 if ∆ A2B2C2 is the image of ∆ ABC when reflected in the x-axis.
(iii) Assign the special name to the quadrilateral BCC2B2 and find its area.

Solution:

Question 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:

Question 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. (2005)
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Chapter Test

Question 1.
Write the first four terms of the A.P. when its first term is -5 and the common difference is -3.
Solution:

Question 2.
Verify that each of the following lists of numbers is an A.P., and the write its next three terms:
(i) $$0,\frac { 1 }{ 4 } ,\frac { 1 }{ 2 } ,\frac { 3 }{ 4 } ,…$$
(ii) $$5,\frac { 14 }{ 3 } ,\frac { 13 }{ 3 } ,4,…$$
Solution:

Question 3.
The nth term of an A.P. is 6n + 2. Find the common difference.
Solution:

Question 4.
Show that the list of numbers 9, 12, 15, 18, … form an A.P. Find its 16th term and the nth.
Solution:

Question 5.
Find the 6th term from the end of the A.P. 17, 14, 11, …, -40.
Solution:

Question 6.
If the 8th term of an A.P. is 31 and the 15th term is 16 more than its 11th term, then find the A.P.
Solution:

Question 7.
The 17th term of anA.P. is 5 more than twice its 8th term. If the 11th term of the A.P. is 43, then find the wth term.
Solution:

Question 8.
The 19th term of an A.P. is equal to three times its 6th term. If its 9th term is 19, find the A.P.
Solution:

Question 9.
If the 3rd and the 9th terms of an A.P. are 4 and -8 respectively, then which term of this A.P. is zero?
Solution:

Question 10.
Which term of the list of numbers 5, 2, -1, -4, … is -55?
Solution:

Question 11.
The 24th term of an A.P. is twice its 10th term. Show that its 72nd term is four times its 15th term.
Solution:

Question 12.
Which term of the list of numbers $$20,19\frac { 1 }{ 4 } ,18\frac { 1 }{ 2 } ,17\frac { 3 }{ 4 } ,..$$ is the first negative term?
Solution:

Question 13.
If the pth term of an A.P. is q and the qth term is p, show that its nth term is (p + q – n)
Solution:

Question 14.
How many three-digit numbers are divisible by 9?
Solution:

Question 15.
The sum of three numbers in A.P. is -3 and the product is 8. Find the numbers.
Solution:

Question 16.
The angles of a quadrilateral are in A.P. If the greatest angle is double of the smallest angle, find all the four angles.
Solution:

Question 17.
The nth term of an A.P. cannot be n² + n + 1. Justify your answer.
Solution:

Question 18.
Find the sum of first 20 terms of an A.P. whose nth term is 15 – 4n.
Solution:

Question 19.
Find the sum :
$$18+15\frac { 1 }{ 2 } +13+…+\left( -49\frac { 1 }{ 2 } \right)$$
Solution:

Question 20.
(i) How many terms of the A.P. -6,$$– \frac { 11 }{ 2 }$$ -5,… make the sum -25?
(ii) Solve the equation 2 + 5 + 8 + … + x = 155.
Solution:

Question 21.
If the third term of an A.P. is 5 and the ratio of its 6th term to the 10th term is 7 : 13, then find the sum of first 20 terms of this A.P.
Solution:

Question 22.
In an A.P., the first term is 2 and the last term is 29. If the sum of the terms is 155, then find the common difference of the A.P.
Solution:

Question 23.
The sum of the first 14 terms of an A.P. is 1505 and its first term is 10. Find its 25th term.
Solution:

Question 24.
Find the number of terms of the A.P. -12, -9, -6, …, 21. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.
Solution:

Question 25.
The sum of first n term of an A.P. is 3n² + 4n. Find the 25th term of this A.P.
Solution:

Question 26.
In an A.P., the sum of the first 10 terms is -150 and the sum of the next 10 terms is -550. Find A.P.
Solution:

Question 27.
The sum of first m terms of an A.P. is 4m² – m. If its nth term is 107, find the value of n. Also, find the 21 st term of this A.P.
Solution:

Question 28.
If the sum of first p, q and r terms of an A.P. are a, b and c respectively, prove that
$$\frac { a }{ p } (q-r)+\frac { b }{ q } (r-p)+\frac { c }{ r } (p-q)=0$$
Solution:

Question 29.
A sum of Rs 700 is to be used to give 7 cash prizes to students of a school for their overall academic performance. If each prize is Rs 20 less than its preceding prize, find the value of each of the prizes.
What is the importance of academic prise in students life? (Value Based)
Solution:

Question 30.
Find the geometric progression whose 4th term is 54 and 7th term is 1458.
Solution:

Question 31.
The fourth term of a G.P. is the square of its second term and the first term is -3. Find its 7th term.
Solution:

Question 32.
If the 4th, 10th and 16th terms of a G.P. are x, y, and z respectively, prove that x, y, and z are in G.P.
Solution:

Question 33.
The original cost of a machine is Rs 10000. If the annual depreciation is 10%, after how many years will it be valued at Rs 6561?
Solution:

Question 34.
How many terms of the G.P. $$3,\frac { 3 }{ 2 } ,\frac { 3 }{ 4 }$$,are needed to give the sum $$\\ \frac { 3069 }{ 512 }$$ ?
Solution:

Question 35.
Find the sum of first n terms of the series : 3 + 33 + 333 + …
Solution:

Question 36.
Find the sum of the series 7 + 7.7 + 7.77 + 7.777 + … to 50 terms.
Solution:

Question 37.
The inventor of the chessboard was a very clever man. He asked the king, h reward of one grain of wheat for the first square, 2 grains for the second, 4 grains for the third, and so on, doubling the number of the grains for each subsequent square. How many grains would have to be given?
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions MCQS

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions MCQS

Choose the correct answer from the given four options (1 to 33) :

Question 1.
The list of numbers – 10, – 6, – 2, 2, … is
(a) an A.P. with d = -16
(b) an A.P with d = 4
(c) an A.P with d = -4
(d) not an A.P
Solution:

Question 2.
The 10th term of the A.P. 5, 8, 11, 14, … is
(a) 32
(b) 35
(c) 38
(d) 185
Solution:

Question 3.
The 30th term of the A.P. 10, 7, 4, … is
(a) 87
(b) 77
(c) -77
(d) -87
Solution:

Question 4.
The 11th term of the A.P. -3, $$– \frac { 1 }{ 2 }$$, 2, … is
(a) 28
(b) 22
(c) -38
(d) -48
Solution:

Question 5.
The 4th term from the end of the A.P. -11, -8, -5, …, 49 is
(a) 37
(b) 40
(c) 43
(d) 58
Solution:

Question 6.
The 15th term from the last of the A.P. 7, 10, 13, …,130 is
(a) 49
(b) 85
(c) 88
(d) 110
Solution:

Question 7.
If the common difference of an A.P. is 5, then a18 – a13 is
(a) 5
(b) 20
(c) 25
(d) 30
Solution:

Question 8.
In an A.P., if a18 – a14 = 32 then the common difference is
(a) 8
(b) -8
(c) -4
(d) 4
Solution:

Question 9.
In an A.P., if d = -4, n = 7, an = 4, then a is
(a) 6
(d) 7
(c) 20
(d) 28
Solution:

Question 10.
In an A.P., if a = 3.5, d = 0, n = 101, then an will be
(a) 0
(b) 3.5
(c) 103.5
(d) 104.5
Solution:

Question 11.
In an A.P., if a = -7.2, d = 3.6, an = 7.2, then n is
(a) 1
(b) 3
(c) 4
(d) 5
Solution:

Question 12.
Which term of the A.P. 21, 42, 63, 84,… is 210?
(a) 9th
(b) 10th
(c) 11th
(d) 12th
Solution:

Question 13.
If the last term of the A.P. 5, 3, 1, -1,… is -41, then the A.P. consists of
(a) 46 terms
(b) 25 terms
(c) 24 terms
(d) 23 terms
Solution:

Question 14.
If k – 1, k + 1 and 2k + 3 are in A.P., then the value of k is
(a) – 2
(b) 0
(c) 2
(d) 4
Solution:

Question 15.
The 21st term of an A.P. whose first two terms are -3 and 4 is
(a) 17
(b) 137
(c) 143
(d) -143
Solution:

Question 16.
If the 2nd term of an A.P. is 13 and the 5th term is 25, then its 7th term is
(a) 30
(b) 33
(c) 37
(d) 38
Solution:

Question 17.
If the first term of an A.P. is -5 and the common difference is 2, then the sum of its first 6 terms is
(a) 0
(b) 5
(c) 6
(d) 15
Solution:

Question 18.
The sum of 25 terms of the A.P.$$-\frac { 2 }{ 3 } ,-\frac { 2 }{ 3 } ,-\frac { 2 }{ 3 }$$ is
(a) 0
(b) $$– \frac { 2 }{ 3 }$$
(c) $$– \frac { 50 }{ 3 }$$
(d) -50
Solution:

Question 19.
In an A.P., if a = 1, an = 20 and Sn = 399, then n is
(a) 19
(b) 21
(c) 38
(d) 42
Solution:

Question 20.
In an A.P., if a = -5, l = 21. and Sn = 200, then n is equal to
(a) 50
(b) 40
(c) 32
(d) 25
Solution:

Question 21.
In an A.P., if a = 3 and S8 = 192, then d is
(a) 8
(b) 7
(c) 6
(d) 4
Solution:

Question 22.
The sum of first five multiples of 3 is
(a) 45
(b) 55
(c) 65
(d) 75
Solution:

Question 23.
The number of two-digit numbers which are divisible by 3 is
(a) 33
(b) 31
(c) 30
(d) 29
Solution:

Question 24.
The number of multiples of 4 that lie between 10 and 250 is
(a) 62
(b) 60
(c) 59
(d) 55
Solution:

Question 25.
The sum of first 10 even whole numbers is
(a) 110
(b) 90
(c) 55
(d) 45
Solution:

Question 26.
The list of number $$\\ \frac { 1 }{ 9 }$$ , $$\\ \frac { 1 }{ 3 }$$, 1, – 3,… is a
(a) GP. with r = – 3
(b) G.P. with r = $$– \frac { 1 }{ 3 }$$
(c) GP. with r = 3
(d) not a G.P.
Solution:

Question 27.
The 11th of the G.P. $$\\ \frac { 1 }{ 8 }$$ , $$– \frac { 1 }{ 4 }$$ , 2, – 1, … is
(a) 64
(b) -64
(c) 128
(d) -128
Solution:

Question 28.
The 5th term from the end of the G.P. 2, 6, 18, …, 13122 is
(a) 162
(b) 486
(c) 54
(d) 1458
Solution:

Question 29.
If k, 2(k + 1), 3(k + 1) are three consecutive terms of a G.P., then the value of k is
(a) -1
(b) -4
(c) 1
(d) 4
Solution:

Question 30.
Which term of the G.P. 18, -12, 8, … is $$\\ \frac { 512 }{ 729 }$$ ?
(a) 12th
(b) 11th
(c) 10th
(d) 9th
Solution:

Question 31.
The sum of the first 8 terms of the series 1 + √3 + 3 + … is
Solution:

Question 32.
The sum of first 6 terms of the G.P. 1, $$– \frac { 2 }{ 3 }$$ ,$$\\ \frac { 4 }{ 9 }$$ ,… is
(a) $$– \frac { 133 }{ 243 }$$
(b) $$\\ \frac { 133 }{ 243 }$$
(c) $$\\ \frac { 793 }{ 1215 }$$
(d) none of these
Solution:

Question 33.
If the sum of the GP., 1, 4, 16, … is 341, then the number of terms in the GP. is
(a) 10
(b) 8
(c) 6
(d) 5
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.5

Question 1.
Find the sum of:
(i) 20 terms of the series 2 + 6 + 18 + …
(ii) 10 terms of series 1 + √3 + 3 + …
(iii) 6 terms of the GP. 1, $$– \frac { 2 }{ 3 }$$ , $$\\ \frac { 4 }{ 9 }$$, …
(iv) 20 terms of the GP. 0.15, 0.015, 0.0015,…
(v) 100 terms of the series 0.7 + 0.07 + 0.007 +…
(vi) 5 terms and n terms of the series $$1+\frac { 2 }{ 3 } +\frac { 4 }{ 9 } +…$$
(vii) n terms of the G.P. √7, √21, 3√7, …
(viii)n terms of the G.P. 1, -a, a², -a³, … (a ≠ -1)
(ix) n terms of the G.P. x3, x5 , x7, … (x ≠ ±1).
Solution:

Question 2.
Find the sum of the first 10 terms of the geometric series
√2 + √6 + √18 + ….
Solution:

Question 3.
Find the sum of the series 81 – 27 + 9….$$– \frac { 1 }{ 27 }$$
Solution:

Question 4.
The nth term of a G.P. is 128 and the sum of its n terms is 255. If its common ratio is 2, then find its first term.
Solution:

Question 5.
If the sum of the first six terms of any G.P. is equal to 9 times the sum of the first three terms, then find the common ratio of the G.P.
Solution:

Question 6.
A G.P. consists of an even number of terms. If the sum of all the terms is 3 times the sum of the odd terms, then find its common ratio.
Solution:

Question 7.
(i) How many terms of the G.P. 3, 32, 33, … are needed to give the sum 120?
(ii) How many terms of the G.P. 1, 4, 16, … must be taken to have their sum equal to 341?
Solution:

Question 8.
How many terms of the GP. 1, √2 > 2, 2 √2,… are required to give a sum of 1023( √2 + 1)?
Solution:

Question 9.
How many terms of the $$\frac { 2 }{ 9 } -\frac { 1 }{ 3 } +\frac { 1 }{ 2 } +…$$ will make the sum $$\\ \frac { 55 }{ 72 }$$ ?
Solution:

Question 10.
The 2nd and 5th terms of a geometric series are $$– \frac { 1 }{ 2 }$$ and sum $$\\ \frac { 1 }{ 16 }$$ respectively. Find the sum of the series upto 8 terms.
Solution:

Question 11.
The first term of a G.P. is 27 and 8th term is $$\\ \frac { 1 }{ 81 }$$ . Find the sum of its first 10 terms.
Solution:

Question 12.
Find the first term of the G.P. whose common ratio is 3, the last term is 486 and the sum of those terms is 728
Solution:

Question 13.
In a G.P. the first term is 7, the last term is 448, and the sum is 889. Find the common ratio.
Solution:

Question 14.
Find the third term of a G.P. whose common ratio is 3 and the sum of whose first seven terms is 2186.
Solution:

Question 15.
If the first term of a G.P. is 5 and the sum of first three terms is $$\\ \frac { 31 }{ 5 }$$, find the common ratio.
Solution:

Question 16.
The sum of first three terms of a GP. is to the sum of first six terms as 125 : 152. Find the common ratio of the GP.
Solution:

Question 17.
Find the sum of the products of the corresponding terms of the geometric progression 2, 4, 8, 16, 32 and 128, 32, 8, 2, $$\\ \frac { 1 }{ 2 }$$
Solution:

Question 18.
Evaluate $$\sum _{ n=1 }^{ 50 }{ \left( { 2 }^{ n }-1 \right) }$$
Solution:

Question 19.
Find the sum of n terms of a series whose mth term is 2m + 2m.
Solution:

Question 20.
Sum the series
x(x + y) + x2 (x2 + y2) + x3 (x3 + y3) … to n terms.
Solution:

Question 21.
Find the sum of the series
1 + (1 + x) + (1 + x + x2) + … to n terms, x ≠ 1.
Solution:

Question 22.
Find the sum of the following series to n terms:
(i) 7 + 77 + 777 + …
(ii) 8 + 88 + 888 + …
(iii) 0.5 + 0.55 + 0.555 + …
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.3

Question 1.
Find the sum of the following A.P.s:
(i) 2, 7, 12, … to 10 terms
(ii) $$\frac { 1 }{ 15 } ,\frac { 1 }{ 12 } ,\frac { 1 }{ 10 } ,…$$ t0 11 terms
Solution:

Question 2.
How many terms of the A.P. 27, 24, 21, …, should be taken so that their sum is zero?
Solution:

Question 3.
Find the sums given below :
(i) 34 + 32 + 30 + … + 10
(ii) -5 + ( -8) + ( -11) + … + ( -230)
Solution:

Question 4.
In an A.P. (with usual notations) :
(i) given a = 5, d = 3, an = 50, find n and Sn
(ii) given a = 7, a13 = 35, find d and S13
(iii) given d = 5, S9 = 75, find a and a9
(iv) given a = 8, an = 62, Sn = 210, find n and d
(v) given a = 3, n = 8, S = 192, find d.
Solution:

Question 5.
(i) The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.
(ii) The sum of the first 15 terms of an A.P. is 750 and its first term is 15. Find its 20th term.
Solution:

Question 6.
The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?
Solution:

Question 7.
Solve for x : 1 + 4 + 7 + 10 + … + x = 287.
Solution:

Question 8.
(i) How many terms of the A.P. 25, 22, 19, … are needed to give the sum 116? Also, find the last term.
(ii) How many terms of the A.P. 24, 21, 18, … must be taken so that the sum is 78? Explain the double answer.
Solution:

Question 9.
Find the sum of first 22 terms, of an A.P. in which d = 7 and a22 is 149.
Solution:

Question 10.
(i) Find the sum of the first 51 terms of the A.P. whose second and third terms are 14 and 18 respectively.
(ii) If the third term of an A.P. is 1 and 6th term is -11, find the sum of its first 32 terms.
Solution:

Question 11.
If the sum of the first 6 terms of an A.P. is 36 and that of the first 16 terms is 256, find the sum of the first 10 terms.
Solution:

Question 12.
Show that a1, a2, a3, … form an A.P. where an is defined as an = 3 + 4n. Also, find the sum of the first 15 terms.
Solution:

Question 13.
(i) If an = 3 – 4n, show that a1, a2, a3, … form an A.P. Also find S20.
(ii) Find the common difference of an A.P. whose first term is 5 and the sum of the first four terms is half the sum of the next four terms.
Solution:

Question 14.
The sum of first n terms of an A.P. whose first term is 8 and the common difference is 20 equal to the sum of first 2n terms of another A.P. whose first term is -30 and the common difference is 8. Find n.
Solution:

Question 15.
The sum of the first six terms of an arithmetic progression is 42. The ratio of the 10th term to the 30th term is $$\\ \frac { 1 }{ 3 }$$. Calculate the first and the thirteenth term.
Solution:

Question 16.
In an A.P., the sum of its first n terms is 6n – n². Find the 25th term.
Solution:

Question 17.
If the sum of first n terms of an A.P. is 4n – n², what is the first term (i. e. S1)? What is the sum of the first two terms? What is the second term? Also, find the 3rd term, the 10th term, and the nth terms?
Solution:

Question 18.
If Sn denotes the sum of first n terms of an A.P., prove that S30 = 3(S20 – S10).
Solution:

Question 19.
(i) Find the sum of the first 1000 positive integers.
(ii) Find the sum of first 15 multiples of 8.
Solution:

Question 20.
(i) Find the sum of all two digit natural numbers which are divisible by 4.
(ii) Find the sum of all natural numbers between 100 and 200 which are divisible by 4.
(iii) Find the sum of all multiples of 9 lying between 300 and 700.
(iv) Find the sum of all natural numbers less than 100 which are divisible by 6.
Solution:

Question 21.
(i) Find the sum of all two digit odd positive numbers.
(ii) Find the sum of all 3-digit natural numbers which are divisible by 7.
(iii) Find the sum of all two digit numbers which when divided by 7 yields 1 as the remainder.
Solution:

Question 22.
A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: Rs 200 for the first day, Rs 250 for the second day, Rs 300 for the third day, etc., the penalty for each succeeding day being Rs 50 more than for the preceding day. How much money the contractor has to pay a penalty if he has delayed the work for 30 days?
Solution:

Question 23.
Kanika was given her pocket money on 1st Jan 2016. She puts Rs 1 on Day 1, Rs 2 on Day 2, Rs 3 on Day 3, and continued on doing so till the end of the month, from this money into her piggy bank. She also spent Rs 204 of her pocket money and was found that at the end of the month she still has Rs 100 with her. How much money was her pocket money for the month?
Solution:

Question 24.
Yasmeen saves Rs 32 during the first month, Rs 36 in the second month and Rs 40 in the third month. If she continues to save in this manner, in how many months will she save Rs 2000?
Solution:

Question 25.
The students of a school decided to beautify the school on the Annual Day by fixing colourful flags on the straight passage of the school. They have 27 flags to be fixed at intervals of every 2 m. The flags are stored at the position of the middle most flag. Ruchi was given the responsibility of placing the flags. Ruchi kept her books where the flags were stored. She could carry only one flag at a time. How much distance did she cover in completing this job and returning back to collect her books? What is the maximum distance she traveled carrying a flag?
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.2

Question 1.
Find the A.P. whose nth term is 7 – 3K. Also, find the 20th term.
Solution:

Question 2.
Find the indicated terms in each of following A.P.s:
(i) 1, 6, 11, 16, …; a20
(ii) -4, -7, -10, -13, …, a25, an
Solution:

Question 3.
Find the nth term and the 12th term of the list of numbers: 5, 2, -1, -4, …
Solution:

Question 4.
Find the 8th term of the A.P. whose first term is 7 and the common difference is 3.
Solution:

Question 5.
(i) If the common difference of an A.P. is -3 and the 18th term is -5, then find its first term.
(ii) If the first term of an A.P. is -18 and its 10th term is zero, then find its common difference.
Solution:

Question 6.
Which term of the A.P.
(i) 3, 8, 13, 18, … is 78?
(ii) 7, 13, 19, … is 205 ?
(iii) 18, $$15 \frac { 1 }{ 2 }$$, 13, … is -47 ?
Solution:

Question 7.
(i) Check whether -150 is a term of the A.P. 11, 8, 5, 2, …
(ii) Find whether 55 is a term of the A.P. 7, 10, 13, … or not. If yes, find which term is it.
(iii) Is 0 a term of the A.P. 31, 28, 25,…? Justify your answer.
Solution:

Question 8.
(i) Find the 20th term from the last term of the A.P. 3, 8, 13, …, 253.
(ii) Find the 12th from the end of the A.P. -2, -4, -6, …; -100.
Solution:

Question 9.
Find the sum of the two middle most terms of the A.P.
$$-\frac { 4 }{ 3 } ,-1,-\frac { 2 }{ 3 } ,…,4\frac { 1 }{ 3 }$$
Solution:

Question 10.
Which term of the A.P. 53, 48, 43,… is the first negative term?
Solution:

Question 11.
Determine the A.P. whose fifth term is 19 and the difference of the eighth term from the thirteenth term is 20.
Solution:

Question 12.
Determine the A.P. whose third term is 16 and the 7th term exceeds the 5th term by 12
Solution:

Question 13.
Find the 20th term of the A.P. whose 7th term is 24 less than the 11th term, the first term is 12.
Solution:

Question 14.
Find the 31st term of an A.P. whose 11th term is 38 and 6th term is 73.
Solution:

Question 15.
If the seventh term of an A.P. is $$\\ \frac { 1 }{ 9 }$$ and its ninth term is $$\\ \frac { 1 }{ 7 }$$, find its 63rd term.
Solution:

Question 16.
(i) The 15th term of an A.P. is 3 more than twice its 7th term. If the 10th term of the A.P. is 41, find its nth term.
(ii) The sum of 5th and 7th terms of an A.P. is 52 and the 10th term is 46. Find A.P.
(iii) The sum of 2nd and 7th terms of an A.P. is 30. If its 15th term is 1 less than twice its 8th term, find the A.P.
Solution:

Question 17.
If the 8th term of an A.P. is zero, prove that its 38th term is triple of its 18th term.
Solution:

Question 18.
Which term of the A.P. 3, 10, 17,… will be 84 more than its 13th term?
Solution:

Question 19.
If the nth terms of the two A.P.s 9, 7, 5, … and 24, 21, 18, … are the same, find the value of n. Also, find that term
Solution:

Question 20.
(i) How many two digit numbers are divisible by 3?
(ii) Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.
(iii) How many numbers lie between 10 and 300, which when divided by 4 leave a remainder 3?
Solution:

Question 21.
If the numbers n – 2, 4n – 1 and 5n + 2 are in A.P., find the value of n.
Solution:

Question 22.
The sum of three numbers in A.P. is 3 and their product is -35. Find the numbers.
Solution:

Question 23.
The sum of three numbers in A.P. is 30 and the ratio of the first number to the third number is 3 : 7. Find the numbers.
Solution:

Question 24.
The sum of the first three terms of an A.P.is 33. If the product of the first and the third terms exceeds the second term by 29, find the A.P.
Solution:

Question 25.
A man starts repaying a loan as the first instalment of Rs 500. If he increases the instalment by Rs 25 every month, what, the amount will he pay in the 30th instalment?
Solution:

Question 26.
Ramkali saved Rs 5 in the first week of a year and then increased her savings by Rs 1.75. If in the right week, her weekly savings become Rs 20.75, find n.
Solution:

Question 27.
Justify whether it is true to say that the following are the nth terms of an A.P.
(i) 2n – 3
(ii) n² + 1
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 9 Arithmetic and Geometric Progressions Ex 9.1

Question 1.
For the following A.P.s, write the first term a and the common difference d:
(i) 3, 1, -1, – 3, …..
(ii) $$\frac { 1 }{ 3 } ,\frac { 5 }{ 3 } ,\frac { 9 }{ 3 } ,\frac { 13 }{ 3 } ,….$$
(iii) -3.2, -3, -2.8, -2.6, …
Solution:

Question 2.
Write first four terms of the A.P., when the first term a and the common difference d are given as follows:
(i) a = 10, d = 10
(ii) a = -2, d = 0
(iii) a = 4, d = -3
(iv) a = $$\\ \frac { 1 }{ 2 }$$, d = $$– \frac { 1 }{ 2 }$$
Solution:

Question 3.
Which of the following lists of numbers form an A.P.? If they form an A.P., find the common difference d and write the next three terms :
(i) 4, 10, 16, 22,…
(ii) -2, 2, -2, 2,…..
(iii) 2, 4, 8, 16,….
(iv) 2, $$\\ \frac { 5 }{ 2 }$$, 3, $$\\ \frac { 7 }{ 2 }$$,……
(v) -10, -6, -2, 2,….
(vi) 1², 3², 5², 7²,….
(vii) 1, 3, 9, 27,….
(viii) √2, √8, √18, √32,….
(ix) 3, 3 + √2, 3 + √2, 3 + 3√2,…..
(x) √3, √6, √9, √12,……
(xi) a, 2a, 3a, 4a,…….
(xii) a, 2a + 1, 3a + 2, 4a + 3,….
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths