#### Chapters

Chapter 2: Banking (Recurring Deposit Account)

Chapter 3: Shares and Dividend

Chapter 4: Linear Inequations (In one variable)

Chapter 5: Quadratic Equations

Chapter 6: Solving (simple) Problems (Based on Quadratic Equations)

Chapter 7: Ratio and Proportion (Including Properties and Uses)

Chapter 8: Remainder and Factor Theorems

Chapter 9: Matrices

Chapter 10: Arithmetic Progression

Chapter 11: Geometric Progression

Chapter 12: Reflection

Chapter 13: Section and Mid-Point Formula

Chapter 14: Equation of a Line

Chapter 15: Similarity (With Applications to Maps and Models)

Chapter 16: Loci (Locus and Its Constructions)

Chapter 17: Circles

Chapter 18: Tangents and Intersecting Chords

Chapter 19: Constructions (Circles)

Chapter 20: Cylinder, Cone and Sphere

Chapter 21: Trigonometrical Identities

Chapter 22: Height and Distances

Chapter 23: Graphical Representation

Chapter 24: Measure of Central Tendency(Mean, Median, Quartiles and Mode)

Chapter 25: Probability

## Chapter 15: Similarity (With Applications to Maps and Models)

#### Exercise 15(A) [Pages 213 - 215]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 15 Similarity (With Applications to Maps and Models) Exercise 15(A) [Pages 213 - 215]

In the figure, given below, straight lines AB and CD intersect at P; and AC ‖ BD. Prove that: ΔAPC and ΔBPD are similar.

In the figure, given below, straight lines AB and CD intersect at P; and AC ‖ BD. Prove that :

If BD = 2.4 cm, AC = 3.6 cm, PD = 4.0 cm and PB = 3.2 cm; find the lengths of PA and PC.

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that : Δ APB is similar to Δ CPD.

In a trapezium ABCD, side AB is parallel to side DC; and the diagonals AC and BD intersect each other at point P. Prove that : PA x PD = PB x PC.

P is a point on side BC of a parallelogram ABCD. If DP produced meets AB produced at point L, prove that : DP : PL = DC : BL.

P is a point on side BC of a parallelogram ABCD. If DP produced meets AB produced at point L, prove that : DL : DP = AL : DC.

In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that : Δ AOB is similar to Δ COD

In quadrilateral ABCD, the diagonals AC and BD intersect each other at point O. If AO = 2CO and BO = 2DO; show that : OA x OD = OB x OC.

In Δ ABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that : CB : BA = CP : PA

In Δ ABC, angle ABC is equal to twice the angle ACB, and bisector of angle ABC meets the opposite side at point P. Show that : AB x BC = BP x CA

In ΔABC; BM ⊥ AC and CN ⊥ AB; show that:

`(AB)/(AC) = (BM)/(CN) = (AM)/(AN)`

In the given figure, DE ‖ BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm.

Write all possible pairs of similar triangles.

In the given figure, DE ‖ BC, AE = 15 cm, EC = 9 cm, NC = 6 cm and BN = 24 cm.

Find lengths of ME and DM.

In the given figure, AD = AE and AD^{2} = BD x EC.

Prove that: triangles ABD and CAE are similar.

In the given figure, AB ‖ DC, BO = 6 cm and DQ = 8 cm; find: BP x DO.

Angle BAC of triangle ABC is obtuse and AB = AC. P is a point in BC such that PC = 12 cm. PQ and PR are perpendiculars to sides AB and AC respectively. If PQ = 15 cm and PR = 9 cm; find the length of PB.

** True or False:**

Two similar polygons are necessarily congruent.

True

False

**True or False:**

Two congruent polygons are necessarily similar.

True

False

**True or False:**

all equiangular triangles are similar.

True

False

**True or False:**

all isosceles triangles are similar.

True

false

**True or False:**

Two isosceles – right triangles are similar

True

false

**True or False:**

Two isosceles triangles are similar, if an angle of one is congruent to the corresponding angle of the other.

True

False

**True or False:**

The diagonals of a trapezium, divide each other into proportional segments.

True

False

Given:

∠GHE = ∠DFE = 90 ,

DH = 8, DF = 12,

DG = 3x – 1 and DE = 4x + 2.

Find: the lengths of segments DG and DE

D is a point on the side BC of triangle ABC such that angle ADC is equal to angle BAC. Prove

that: `"CA"^2` = CB × CD

In the given figure, ∆ ABC and ∆ AMP are right angled at B and M respectively.

Given AC = 10 cm, AP = 15 cm and PM = 12 cm.

(i) Prove ∆ ABC ~ ∆ AMP

(ii) Find AB and BC

Given: RS and PT are altitudes of ΔPQR. Prove that:

(i) Δ PQT ~ ΔQRS

(ii) PQ × QS = RQ × QT

Given: ABCD is a rhombus, DRP and CBR are straight lines.

Prove that:

DP × CR = DC × PR

Given: FB = FD, AE ⊥ FD and FC ⊥ AD

Prove: `("FB")/("AD")=("BC")/("ED")`

In ∆PQR, ∠Q = 90° and QM is perpendicular to PR. Prove that:

(i) PQ^{2} = PM × PR

(ii)QR^{2} = PR × MR

(iii) PQ^{2} + QR^{2} = PR^{2}

In ∆ABC, ∠B = 90° and BD ⊥ AC.

(i) If CD = 10 cm and BD = 8 cm; find AD.

(ii) IF AC = 18 cm and AD = 6cm; find BD.

(iii) If AC = 9 cm and AB = 7cm; find AD.

In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10cm, L is a point on PR

such that RL: LP = 2: 3. QL produced meets RS at M and PS produced at N.

Find the lengths of PN and RM.

In quadrilateral ABCD, diagonals AC and BD intersect at point E such that

AE: EC = BE: ED

Show that ABCD is a trapezium.

In triangle ABC, AD is perpendicular to side BC and AD^{2} = BD × DC.

Show that angle BAC = 90°

In the given figure, AB ∥ EF ∥ DC ; AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.

(i) Name the three pairs of similar triangles.

(ii) Find the lengths of EC and EF.

In the given figure, QR is parallel to AB and DR is parallel to AB and DR is parallel to QB.

Prove that:

`PQ^2= PD × PA`

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting diagonal AC in L and AD produced in E. Prove that: EL = 2BL.

In the figure, given below, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.

(i) Calculate the ratio PQ : AC, giving reason for your answer.

(ii) In triangle ARC, ∠ARC = 90°. Given QS = 6cm, calculate the length of AR.

In the right-angled triangle QPR, PM is an altitude.

Given that QR = 8cm and MQ = 3.5 cm, calculate the value of PR.

In the figure, given below, the medians BD and CE of a triangle ABC meet at G. Prove that:

(i) Δ EGD ~ ΔCGB and

(ii) BG = 2 GD from (i) above.

#### Exercise 15(B) [Page 218]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 15 Similarity (With Applications to Maps and Models) Exercise 15(B) [Page 218]

In the following figure, point D divides AB in the ratio 3 : 5. Find : `(AE)/(EC)`

In the following figure, point D divides AB in the ratio 3 : 5. Find : `(AD)/(AB)`

In the following figure, point D divides AB in the ratio 3 : 5. Find : `(AE)/(AC)`

In the following figure, point D divides AB in the ratio 3 : 5. Find :

DE = 2.4 cm, find the length of BC.

In the following figure, point D divides AB in the ratio 3 : 5. Find :

BC = 4.8 cm, find the length of DE.

In the given figure, PQ ‖ AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find : `(CP)/(PA)`

In the given figure, PQ ‖ AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :

PQ

In the given figure, PQ ‖ AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. Find :

If AP = x, then the value of AC in terms of x.

A line PQ is drawn parallel to the side BC of Δ ABC which cuts side AB at P and side AC at Q. If AB = 9.0 cm, CA = 6.0 cm and AQ = 4.2 cm, find the length of AP.

In Δ ABC, D and E are the points on sides AB and AC respectively.

Find whether DE ‖ BC, if

AB = 9cm, AD = 4cm, AE = 6cm and EC = 7.5cm.

In Δ ABC, D and E are the points on sides AB and AC respectively.

Find whether DE ‖ BC, if AB = 6.3 cm, EC = 11.0 cm, AD =0.8 cm and EA = 1.6 cm.

In the given figure, Δ ABC ~ Δ ADE. If AE: EC = 4 : 7 and DE = 6.6 cm, find BC. If 'x' be the length of the perpendicular from A to DE, find the length of perpendicular from A to BC in terms of 'x'.

A line segment DE is drawn parallel to base BC of Δ ABC which cuts AB at point D and AC at point E. If AB = 5BD and EC = 3.2 cm, find the length of AE.

In the figure, given below, AB, CD and EF are parallel lines. Given AB = 7.5 cm, DC = y cm, EF = 4.5 cm, BC = x cm and CE = 3. cm, calculate the values of x and y.

In the figure, given below, PQR is a right-angle triangle right angled at Q. XY is parallel to QR, PQ = 6 cm, PY = 4 cm and PX : XQ = 1 : 2. Calculate the lengths of PR and QR.

In the following figure, M is mid-point of BC of a parallelogram ABCD. DM intersects the diagonal AC at P and AB produced at E.

Prove that : PE = 2 PD

The given figure shows a parallelogram ABCD. E is a point in AD and CE produced meets BA produced at point F. If AE = 4 cm, AF = 8 cm and AB = 12 cm, find the perimeter of the parallelogram ABCD.

#### Exercise 15(C) [Page 224]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 15 Similarity (With Applications to Maps and Models) Exercise 15(C) [Page 224]

The ratio between the corresponding sides of two similar triangles is 2 is to 5. Find the ratio

between the areas of these triangles.

Area of two similar triangles are 98 sq.cm and 128 sq.cm. Find the ratio between the lengths

of their corresponding sides.

A line PQ is drawn paral el to the base BC of ∆ABC which meets sides AB and AC at points P

and Q respectively. If AP`= 1/3`PB; find the value of:

1) `"Area of ΔABC"/"Area of ΔAPQ"`

2)`"Area of ΔAPQ"/"Area of trapeziumPBCQ"`

The perimeter of two similar triangles are 30 cm and 24 cm. If one side of the first triangle is 12 cm, determine the corresponding side of the second triangle.

In the given figure, AX : XB = 3: 5

(i) the length of BC, if the length of XY is 18 cm.

(ii) the ratio between the areas of trapezium XBCY and triangle ABC.

ABC is a triangle. PQ is a line segment intersecting AB in P and AC in Q such that PQ ∥ BC and divides triangle ABC into two parts equal in area. Find the value of ratio BP : AB.

In the given triangle PQR, LM is parallel to QR and PM : MR = 3: 4

Calculate the value of ratio:

1) `("PL")/("PQ") "and then" ("LM")/("QR")`

2) `"Area of Δ LMN"/"Area of Δ MNR"`

3) `"Area of Δ LQM"/"Area of Δ LQN"`

The given diagram shows two isosceles triangles which are similar also. In the given diagram,

PQ and BC are not parallel; PC = 4, AQ = 3, QB = 12, BC = 15 and AP = PQ

**Calculate:**(i) the length of AP,

(ii) the ratio of the areas of triangle APQ and triangle ABC.

In the figure, given below, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1:

2. DP produced meets AB produces at Q. Given the area of triangle CPQ = 20 cm^{2}.

Calculate:

(i) area of triangle CDP,

(ii) area of parallelogram ABCD.

In the given figure, BC is parallel to DE. Area of triangle ABC = 25 cm^{2}, Area of trapezium BCED = 24 cm^{2} and DE = 14cm.

Calculate the length of BC. Also, find the area of triangle BCD.

The given figure shows a trapezium in which AB is parallel to DC and diagonals AC and BD

intersect at point P. If AP : CP = 3: 5,

Find:

(1) ∆APB : ∆CPB (2 ) ∆DPC : ∆APB

(3) ∆ADP : ∆APB (iv) ∆APB : ∆ADB

In the given figure, ABC is a triangle. DE is parallel to BC and `("AD")/("DB")=3/2`

(i) Determine the ratios `("AD")/("AB") and ("DE")/("BC")`

(ii) Prove that ∆DEF is similar to ∆CBF Hence, find `("EF")/("FB").`

(iii) What is the ratio of the areas of ∆DEF and ∆BFC.

In the given figure, ∠B = ∠E, ∠ACD = ∠BCE, AB = 10.4cm and DE = 7.8 cm. Find the ratio

between areas of the ∆ABC and ∆ DEC

#### Exercise 15(D) [Page 229]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 15 Similarity (With Applications to Maps and Models) Exercise 15(D) [Page 229]

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C' Calculate : the length of AB, if A' B' = 6 cm.

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A' B' C' Calculate : the length of C' A' if CA = 4 cm.

A triangle LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate: the length of M' N', if MN = 8 cm.

A triangle LMN has been reduced by scale factor 0.8 to the triangle L' M' N'. Calculate: the length of LM, if L' M' = 5.4 cm.

A triangle ABC is enlarged, about the point O as centre of enlargement, and the scale factor is 3. Find : A' B', if AB = 4 cm.

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find : BC, if B' C' = 15 cm.

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find : OA, if OA' = 6 cm.

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find : OC', if OC = 21 cm

Also, state the value of : (a) `(OB')/(OB)` (b) `(C'A')/(CA)`

A model of an aeroplane is made to a scale of 1 : 400. Calculate :

the length, in cm, of the model; if the length of the aeroplane is 40 m

A model of an aeroplane is made to a scale of 1 : 400. Calculate :

the length, in m, of the aeroplane, if length of its model is 16 cm.

The dimensions of the model of a multistorey building are 1.2 m x 75 cm x 2 m. If the scale factor is 1 : 30; find the actual dimensions of the building.

On a map drawn to a scale of 1 : 2,50,000; a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and ∠ABC = 90°.

Calculate : the actual lengths of AB and BC in km.

On a map drawn to a scale of 1 : 2,50,000; a triangular plot of land has the following measurements : AB = 3 cm, BC = 4 cm and angle ABC = 90°.

Calculate : the area of the plot in sq. km.

A model of a ship is made to a scale 1 : 300.

(i) The length of the model of the ship is 2 m. Calculate the length of the ship.

(ii) The area of the deck of the ship is 180,000 m^{2}. Calculate the area of the deck of the model.

(iii). The volume of the model is 6.5 m^{3}. Calculate the volume of the ship.

#### Exercise 15(E) [Pages 229 - 232]

### Selina solutions for Concise Mathematics Class 10 ICSE Chapter 15 Similarity (With Applications to Maps and Models) Exercise 15(E) [Pages 229 - 232]

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

`(AY)/(YC)`

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

`(YC)/(AC)`

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm.

XY

In the following figure, ABCD to a trapezium with AB ‖ DC. If AB = 9 cm, DC = 18 cm, CF= 13.5,cm, AP = 6 cm and BE = 15 cm, Calculate: EC

In the following figure, ABCD to a trapezium with AB ‖ DC. If AB = 9 cm, DC = 18 cm, CF= 13.5,cm, AP = 6 cm and BE = 15 cm, Calculate: AF

In the following figure, ABCD to a trapezium with AB ‖ DC. If AB = 9 cm, DC = 18 cm, CF= 13.5,cm, AP = 6 cm and BE = 15 cm, Calculate: PE

In the following figure, AB, CD and EF are perpendicular to the straight line BDF.

If AB = x; CD = z unit and EF = y unit, prove that:

`1/"x" + 1/"y" = 1/"z"`

Triangle ABC is similar to triangle PQR. If AD and PM are corresponding medians of the two triangles, prove that: `(AB)/(PQ) = (AD)/(PM)`

Triangle ABC is similar to triangle PQR. If AD and PM are altitudes of two triangles, Prove that : `(AB)/(PQ) = (AD)/(PM)`

Triangle ABC is similar to triangle PQR. If bisector of angle BAC meets BC at point D and bisector of angle QPR meets QR at point M, prove that ` (AB)/(PQ)=(AD)/(PM)`

In the following figure, ∠AXY = ∠AYX. If `(BX)/(AX) = (CY)/(AY)` show that triangle ABC is isosceles.

In the given figure, lines l, m and n are such that ls ∥ m ∥ n. Prove that:

`(AB)/(BC)=(PQ)/(QR)`

In the following figure, DE || Ac and DC || AP. Prove that: `(BE)/(EC) = (BC)/(CP)`

In the figure given below, AB ‖ EF ‖ CD. If AB = 22.5 cm, EP = 7.5 cm, PC = 15 cm and DC = 27 cm. Calculate : EF

In the figure given below, AB ‖ EF ‖ CD. If AB = 22.5 cm, EP = 7.5 cm, PC = 15 cm and DC = 27 cm. Calculate : AC

In ΔABC, ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm.

Prove that ΔACD is similar to ΔBCA.

In the given triangle P, Q and R are the mid points of sides AB, BC and AC respectively. Prove that triangle PQR is similar to triangle ABC.

In the following figure, AD and CE are medians of ΔABC. DF is drawn parallel to CE. Prove that :

(i) EF = FB,

(ii) AG : GD = 2 : 1

The two similar triangles are equal in area. Prove that the triangles are congruent.

The ratio between the altitudes of two similar triangles is 3 : 5; write the ratio between their:

(i) medians (ii) perimeters (iii) areas

The ratio between the areas of two similar triangles is 16 : 25, Find the ratio between their:

(i) perimeters (ii) altitudes (iii) medians

The following figure shows a triangle PQR in which XY is parallel to QR. If PX : XQ = 1 : 3 and QR = 9 cm. find the length of XY.

Further, if the area of Δ PXY =x cm^{2 }; find, in terms of x the area of:

(i) triangle PQR (ii) trapezium XQRY

On a map, drawn to a scale of 1 : 20000, a rectangular plot of land ABCD has AB = 24cm and BC = 32 cm. Calculate:

(i) the diagonal distance of the plot in kilometer

(ii) the area of the plot in sq.km

The dimensions of the model of a multistoreyed building are 1 m by 60 cm by 1.20 m. if the scale factor is 1 : 50, find the actual dimensions of the building.

Also find:

(i) the floor area of a room of the building, if the floor area of the corresponding room in the model is 50 sq. cm

(ii) the space (volume) inside a room of the model, if the space inside the corresponding room of the building is `90 m^3`

In a triangle PQR, L and M are two points on the base QR, such that ∠LPQ = ∠QRP and ∠RPM

= ∠RQP. Prove that:

(i) `ΔPQL and ΔRMP`

(ii) `QL xx RM = P L × PM`

(iii) ` PQ^2 = QR × QL`

A triangle ABC with AB = 3 cm, BC = 6 cm and AC = 4 cm is enlarged to ΔDEF such that the longest side of ΔDEF = 9 cm. Find the scale factor and hence, the lengths of the other sides of ΔDEF.

Two isosceles triangles have equal vertical angles. Show that the triangles are similar.If the ratio between the areas of these two triangles is 16 : 25, find the ratio between their corresponding altitudes.

In ΔABC, AP : PB = 2 : 3. PO is parallel to BC and is extended to Q so that CQ is parallel to BA**.**

Find**:**

(i) area ΔAPO : area Δ ABC.

(ii) area ΔAPO : area Δ CQO.

The following figure shows a triangle ABC in which AD and BE are perpendiculars to BC and AC respectively.

Show that;

(i) ΔADC ~ ΔBEC

(ii) CA × CE = CB × CD

(iii) ΔABC ~ ΔDEC

(iv) CD × AB = CA × DE

In the give figure, ABC is a triangle with ∠EDB = ∠ACB. Prove that ΔABC ~ ΔEBD. If BE =6 cm, EC = 4cm, BD = 5cm and area of ΔBED = 9 cm2. Calculate the:

(i) length of AB]

(ii) area of Δ ABC

In the given figure, ABC is a right angled triangle with ∠BAC = 90°.

(i) Prove ΔADB ~ ΔCDA.

(ii) If BD = 18 cm and CD = 8cm, find AD.

(iii) Find the ratio of the area of ΔADB is to area of ΔCDA.

In the given figure, AB and DE are perpendiculars to BC.

Prove that : ΔABC ~ ΔDEC

In the given figure, AB and DE are perpendiculars to BC.

If AB = 6 cm, DE = 4 cm and AC = 15 cm. Calculate CD.

In the given figure, AB and DE are perpendiculars to BC.

Find the ratio of the area of a ΔABC : area of ΔDEC.

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :

ΔADE ~ ΔACB.

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :

If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.

ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that :

Find, area of ΔADE : area of quadrilateral BCED.

Given: AB || DE and BC || EF. Prove that:

`(AD)/(DG)=(CF)/(FG)`

` ∆"DFG" ~ ∆"ACG"`

PQR is a triangle. S is a point on the side QR of ΔPQR such that ∠PSR = ∠QPR. Given QP = 8 cm, PR = 6 cm and SR = 3 cm.

- i. ProveΔPQR∼ Δ
- Find the lengths of QR and PS.
- `(Area of DeltaPQR)/(area of Delta SPR)`

## Chapter 15: Similarity (With Applications to Maps and Models)

## Selina solutions for Concise Mathematics Class 10 ICSE chapter 15 - Similarity (With Applications to Maps and Models)

Selina solutions for Concise Mathematics Class 10 ICSE chapter 15 (Similarity (With Applications to Maps and Models)) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Concise Mathematics Class 10 ICSE solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Concise Mathematics Class 10 ICSE chapter 15 Similarity (With Applications to Maps and Models) are Similarity of Triangles, Similarity Triangle Theorem, Axioms of Similarity of Triangles, Areas of Similar Triangles Are Proportional to the Squares on Corresponding Sides.

Using Selina Class 10 solutions Similarity (With Applications to Maps and Models) exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Selina Solutions are important questions that can be asked in the final exam. Maximum students of CISCE Class 10 prefer Selina Textbook Solutions to score more in exam.

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