## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.1

Question 1.
State which pairs of triangles in the figure given below are similar. Write the similarity rule used and also write the pairs of similar triangles in symbolic form (all lengths of sides are in cm):

LearnCram.com also covered previous Year argumentative essay topics asked in ICSE board exams.

Solution:

Question 2.
It is given that ∆DEF ~ ∆RPQ. Is it true to say that ∠D = ∠R and ∠F = ∠P ? Why?
Solution:

Question 3.
If in two right triangles, one of the acute angles of one triangle is equal to an acute angle of the other triangle, can you say that the two triangles are similar? Why?
Solution:

ML Aggarwal Class 10 Solutions Similarity Question 4. In the given figure, BD and CE intersect each other at the point P. Is ∆PBC ~ ∆PDE? Give reasons for your answer.

Solution:

Question 5.
It is given that ∆ABC ~ ∆EDF such that AB = 5 cm, AC = 7 cm, DF = 15 cm and DE = 12 cm.
Find the lengths of the remaining sides of the triangles.

Solution:

Question 6.
(a) If ∆ABC ~ ∆DEF, AB = 4 cm, DE = 6 cm, EF = 9 cm and FD = 12 cm, then find the perimeter of ∆ABC.
(b) If ∆ABC ~ ∆PQR, Perimeter of ∆ABC = 32 cm, perimeter of ∆PQR = 48 cm and PR = 6 cm, then find the length of AC.
Solution:

ML Aggarwal Similarity Solutions Question 7.
Calculate the other sides of a triangle whose shortest side is 6 cm and which is similar to a triangle whose sides are 4 cm, 7 cm and 8 cm.
Solution:

Question 8.
(a) In the figure given below, AB || DE, AC = , 3 cm, CE = 7.5 cm and BD = 14 cm. Calculate CB and DC.

(b) In the figure (2) given below, CA || BD, the lines AB and CD meet at G.
(i) Prove that ∆ACO ~ ∆BDO.
(ii) If BD = 2.4 cm, OD = 4 cm, OB = 3.2 cm and AC = 3.6 cm, calculate OA and OC.

Solution:

ML Aggarwal Class 10 Similarity Question 9.
(a) In the figure
(i) given below, ∠P = ∠RTS.
Prove that ∆RPQ ~ ∆RTS.
(b) In the figure (ii) given below,
∠ADC = ∠BAC. Prove that CA² = DC × BC.

Solution:

Question 10.
(a) In the figure (1) given below, AP = 2PB and CP = 2PD.
(i) Prove that ∆ACP is similar to ∆BDP and AC || BD.
(ii) If AC = 4.5 cm, calculate the length of BD.

(b) In the figure (2) given below,
(i) Prove that ∆s ABC and AED are similar.
(ii) If AE = 3 cm, BD = 1 cm and AB = 6 cm, calculate AC.

(c) In figure (3) given below, ∠PQR = ∠PRS. Prove that triangles PQR and PRS are similar. If PR = 8 cm, PS = 4 cm, calculate PQ.

Solution:

Similarity Class 10 ICSE ML Aggarwal Solutions Question 11.
In the given figure, ABC is a triangle in which AB = AC. P is a point on the side BC such that PM ⊥ AB and PN ⊥ AC. Prove that BM × NP = CN × MP.

Solution:

Question 12.
Prove that the ratio of the perimeters of two similar triangles is the same as the ratio of their corresponding sides.
Solution:

Question 13.
In the given figure, ABCD is a trapezium in which AB || DC. The diagonals AC and BD intersect at O. Prove that $$\frac { AO }{ OC } =\frac { BO }{ OD }$$

Using the above result, find the value(s) of x if OA = 3x – 19, OB = x – 4, OC = x – 3 and OD = 4.
Solution:

ML Aggarwal Class 10 Similarity Solutions Question 14.
In ∆ABC, ∠A is acute. BD and CE are perpendicular on AC and AB respectively. Prove that AB x AE = AC x AD.
Solution:

Question 15.
In the given figure, DB ⊥ BC, DE ⊥ AB and AC ⊥ BC. Prove that $$\frac { BE }{ DE } =\frac { AC }{ BC }$$

Solution:

Question 16.
(a) In the figure (1) given below, E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. show that ∆ABE ~ ∆CFB.
(b) In the figure (2) given below, PQRS is a parallelogram; PQ = 16 cm, QR = 10 cm. L is a point on PR such that RL : LP = 2 : 3. QL produced meets RS at M and PS produced at N.
(i) Prove that triangle RLQ is similar to triangle PLN. Hence, find PN.
(ii) Name a triangle similar to triangle RLM. Evaluate RM.

Solution:

ML Aggarwal Class 9 Solutions ICSE Chapter 13 Question 17.
The altitude BN and CM of ∆ABC meet at H. Prove that
(i) CN . HM = BM . HN .
(ii) $$\frac { HC }{ HB } =\sqrt { \frac { CN.HN }{ BM.HM } }$$
(iii) ∆MHN ~ ∆BHC.
Solution:

Question 18.
In the given figure, CM and RN are respectively the medians of ∆ABC and ∆PQR. If ∆ABC ~ ∆PQR, prove that:
(i) ∆AMC ~ ∆PNR
(ii) $$\frac { CM }{ RN } =\frac { AB }{ PQ }$$
(iii) ∆CMB ~ ∆RNQ.

Solution:

Ex 13.1 Class 10 Question 19.
In the given figure, medians AD and BE of ∆ABC meet at the point G, and DF is drawn parallel to BE. Prove that
(i) EF = FC
(ii) AG : GD = 2 : 1

Solution:

Question 20.
(a) In the figure given below, AB, EF and CD are parallel lines. Given that AB =15 cm, EG = 5 cm, GC = 10 cm and DC = 18 cm. Calculate
(i) EF
(ii) AC.

(b) In the figure given below, AF, BE and CD are parallel lines. Given that AF = 7.5 cm, CD = 4.5 cm, ED = 3 cm, BE = x and AE = y. Find the values of x and y.

Solution:

Question 21.
In the given figure, ∠A = 90° and AD ⊥ BC If BD = 2 cm and CD = 8 cm, find AD.

Solution:

Question 22.
A 15 metres high tower casts a shadow of 24 metres long at a certain time and at the same time, a telephone pole casts a shadow 16 metres long. Find the height of the telephone pole.
Solution:

Question 23.
A street light bulb is fixed on a pole 6 m above the level of the street. If a woman of height casts a shadow of 3 m, find how far she is away from the base of the pole?
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 18 Trigonometric Identities Ex 18

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 18 Trigonometric Identities Ex 18

Question 1.
If A is an acute angle and sin A = $$\\ \frac { 3 }{ 5 }$$ find all other trigonometric ratios of angle A (using trigonometric identities).
Solution:

Trigonometric Identities Class 10 ML Aggarwal Question 2.
If A is an acute angle and sec A = $$\\ \frac { 17 }{ 8 }$$, find all other trigonometric ratios of angle A (using trigonometric identities).
Solution:

Question 3.
Express the ratios cos A, tan A and sec A in terms of sin A.
Solution:

Question 4.
If tan A = $$\frac { 1 }{ \sqrt { 3 } }$$, find all other trigonometric ratios of angle A.
Solution:

ML Aggarwal Trigonometry Solutions Question 5.
If 12 cosec θ = 13, find the value of $$\frac { 2sin\theta -3cos\theta }{ 4sin\theta -9cos\theta }$$
Solution:

Without using trigonometric tables, evaluate the following (6 to 10):

Question 6.
(i) cos² 26° + cos 64° sin 26° + $$\frac { tan{ 36 }^{ ° } }{ { cot54 }^{ ° } }$$
(ii) $$\frac { sec{ 17 }^{ ° } }{ { cosec73 }^{ ° } } +\frac { tan68^{ ° } }{ cot22^{ ° } }$$ + cos² 44° + cos² 46°
Solution:

Question 7.
(i) $$\frac { sin65^{ ° } }{ { cos25 }^{ ° } } +\frac { cos32^{ ° } }{ sin58^{ ° } }$$ – sin 28° sec 62° + cosec² 30° (2015)
(ii) $$\frac { sin29^{ ° } }{ { cosec61 }^{ ° } }$$ + 2 cot 8° cot 17° cot 45° cot 73° cot 82° – 3(sin² 38° + sin² 52°).
Solution:

ML Aggarwal Trigonometry Question 8.
(i) $$\frac { { sin }35^{ ° }{ cos55 }^{ ° }+{ cos35 }^{ ° }{ sin }55^{ ° } }{ { cosec }^{ 2 }{ 10 }^{ ° }-{ tan }^{ 2 }{ 80 }^{ ° } }$$
(ii) $${ sin }^{ 2 }{ 34 }^{ ° }+{ sin }^{ 2 }{ 56 }^{ ° }+2tan{ 18 }^{ ° }{ tan72 }^{ ° }-{ cot }^{ 2 }{ 30 }^{ ° }$$
Solution:

Question 9.
(i) $${ \left( \frac { { tan25 }^{ ° } }{ { cosec }65^{ ° } } \right) }^{ 2 }+{ \left( \frac { { cot25 }^{ ° } }{ { sec65 }^{ ° } } \right) }^{ 2 }+{ 2tan18 }^{ ° }{ tan }45^{ ° }{ tan72 }^{ ° }$$
(ii) $$\left( { cos }^{ 2 }25+{ cos }^{ 2 }65 \right) +cosec\theta sec\left( { 90 }^{ ° }-\theta \right) -cot\theta tan\left( { 90 }^{ ° }-\theta \right)$$
Solution:

Question 10.
(i) 2(sec² 35° – cot² 55°) – $$\frac { { cos28 }^{ ° }cosec{ 62 }^{ ° } }{ { tan18 }^{ ° }tan{ 36 }^{ ° }{ tan30 }^{ ° }{ tan54 }^{ ° }{ tan72 }^{ ° } }$$
(ii) $$\frac { { cosec }^{ 2 }(90-\theta )-{ tan }^{ 2 }\theta }{ 2({ cos }^{ 2 }{ 48 }^{ ° }+{ cos }^{ 2 }{ 42 }^{ ° }) } -\frac { { 2tan }^{ 2 }{ 30 }^{ ° }{ sec }^{ 2 }{ 52 }^{ ° }{ sin }^{ 2 }{ 38 }^{ ° } }{ { cosec }^{ 2 }{ 70 }^{ ° }-{ tan }^{ 2 }{ 20 }^{ ° } }$$
Solution:

ML Aggarwal Class 9 Chapter 18 Solutions Question 11.
Prove that following:
(i) cos θ sin (90° – θ) + sin θ cos (90° – θ) = 1
(ii) $$\frac { tan\theta }{ tan({ 90 }^{ ° }-\theta ) } +\frac { sin({ 90 }^{ ° }-\theta ) }{ cos\theta } ={ sec }^{ 2 }\theta$$
(iii) $$\frac { cos({ 90 }^{ ° }-\theta )cos\theta }{ tan\theta } +{ cos }^{ 2 }({ 90 }^{ ° }-\theta )=1$$
(iv) sin (90° – θ) cos (90° – θ) = $$\frac { tan\theta }{ { 1+tan }^{ 2 }\theta }$$
Solution:

Prove that following (12 to 30) identities, where the angles involved are acute angles for which the trigonometric ratios as defined:

Trigonometry ML Aggarwal Class 10 Question 12.
(i) (sec A + tan A) (1 – sin A) = cos A
(ii) (1 + tan2 A) (1 – sin A) (1 + sin A) = 1.
Solution:

Question 13.
(i) tan A + cot A = sec A cosec A
(ii) (1 – cos A)(1 + sec A) = tan A sin A.
Solution:

ML Aggarwal Class 10 Trigonometry Solutions Question 14.
(i) $$\frac { 1 }{ 1+cosA } +\frac { 1 }{ 1-cosA } =2{ cosec }^{ 2 }A$$
(ii) $$\frac { 1 }{ secA+tanA } +\frac { 1 }{ secA-tanA } =2{ sec }A$$
Solution:

Question 15.
(i) $$\frac { sinA }{ 1+cosA } =\frac { 1-cosA }{ sinA }$$
(ii) $$\frac { 1-{ tan }^{ 2 }A }{ { cot }^{ 2 }A-1 } ={ tan }^{ 2 }A$$
(iii) $$\frac { sinA }{ 1+cosA } =cosecA-cotA$$
Solution:

Trigonometric Identities Class 10 Question 16.
(i) $$\frac { secA-1 }{ secA+1 } =\frac { 1-cosA }{ 1+cosA }$$
(ii) $$\frac { { tan }^{ 2 }\theta }{ { (sec\theta -1) }^{ 2 } } =\frac { 1+cos\theta }{ 1-cos\theta }$$
(iii) $${ (1+tanA) }^{ 2 }+{ (1-tanA) }^{ 2 }=2{ sec }^{ 2 }A$$
(iv) $${ sec }^{ 2 }A+{ cosec }^{ 2 }A={ sec }^{ 2 }A{ .cosec }^{ 2 }A$$
Solution:

Question 17.
(i) $$\frac { 1+sinA }{ cosA } +\frac { cosA }{ 1+sinA } =2secA$$
(ii) $$\frac { tanA }{ secA-1 } +\frac { tanA }{ secA+1 } =2cosecA$$
Solution:

Question 18.
(i) $$\frac { cosecA }{ cosecA-1 } +\frac { cosecA }{ cosecA+1 } =2{ sec }^{ 2 }A$$
(ii) $$cotA-tanA=\frac { { 2cos }^{ 2 }A-1 }{ sinA-cosA }$$
(iii) $$\frac { cotA-1 }{ 2-{ sec }^{ 2 }A } =\frac { cotA }{ 1+tanA }$$
Solution:

ML Aggarwal Class 10 Solutions Question 19.
(i) $${ tan }^{ 2 }\theta -{ sin }^{ 2 }\theta ={ tan }^{ 2 }\theta { sin }^{ 2 }\theta$$
(ii) $$\frac { cos\theta }{ 1-tan\theta } -\frac { { sin }^{ 2 }\theta }{ cos\theta -sin\theta } =cos\theta +sin\theta$$
Solution:

Question 20.
(i) cosec4 θ – cosec2 θ = cot4 θ + cot2 θ
(ii) 2 sec2 θ – sec4 θ – 2 cosec2 θ + cosec4 θ = cot4 θ – tan4 θ.
Solution:

Question 21.
(i) $$\frac { 1+cos\theta -{ sin }^{ 2 }\theta }{ sin\theta (1+cos\theta ) } =cot\theta$$
(ii) $$\frac { { tan }^{ 3 }\theta -1 }{ tan\theta -1 } ={ sec }^{ 2 }\theta +tan\theta$$
Solution:

Question 22.
(i) $$\frac { 1+cosecA }{ cosecA } =\frac { { cos }^{ 2 }A }{ 1-sinA }$$
(ii) $$\sqrt { \frac { 1-cosA }{ 1+cosA } } =\frac { sinA }{ 1+cosA }$$
Solution:

Question 23.
(i) $$\sqrt { \frac { 1+sinA }{ 1-sinA } } =tanA+secA$$
(ii) $$\sqrt { \frac { 1-cosA }{ 1+cosA } } =cosecA-cotA$$
Solution:

Question 24.
(i) $$\sqrt { \frac { secA-1 }{ secA+1 } } +\sqrt { \frac { secA+1 }{ secA-1 } } =2cosecA$$
(ii) $$\frac { cotAcotA }{ 1-sinA } =1+cosecA$$
Solution:

Question 25.
(i) $$\frac { 1+tanA }{ sinA } +\frac { 1+cotA }{ cosA } =2(secA+cosecA)$$
(ii) $${ sec }^{ 4 }A-{ tan }^{ 4 }A=1+2{ tan }^{ 2 }A$$
Solution:

Question 26.
(i) cosec6 A – cot6 A = 3 cot2 A cosec2 A + 1
(ii) sec6 A – tan6 A = 1 + 3 tan2 A + 3 tan4 A
Solution:

Question 27.
(i) $$\frac { cot\theta -cosec\theta -1 }{ cot\theta -cosec\theta +1 } =\frac { 1+cos\theta }{ sin\theta }$$
(ii) $$\frac { sin\theta }{ cot\theta +cosec\theta } =2+\frac { sin\theta }{ cot\theta -cosec\theta }$$
Solution:

Question 28.
(i) (sinθ + cosθ)(secθ + cosecθ) = 2 + secθ cosecθ
(ii) (cosecA – sinA)(secA – cosA) sec2A = tanA
(iii) (cosecθ – sinθ)(secθ – cosθ)(tan θ + cotθ) = 1
Solution:

Question 29.
(i) $$\frac { { sin }^{ 3 }A+{ cos }^{ 3 }A }{ sinA+cosA } +\frac { { sin }^{ 3 }A-{ cos }^{ 3 }A }{ sinA-cosA } =2$$
(ii) $$\frac { { tan }^{ 2 }A }{ { 1+tan }^{ 2 }A } +\frac { cot^{ 2 }A }{ 1+{ cot }^{ 2 }A } =1$$
Solution:

Question 30.
(i) $$\frac { 1 }{ secA+tanA } -\frac { 1 }{ cosA } =\frac { 1 }{ cosA } -\frac { 1 }{ secA-tanA }$$
(ii) $${ (sinA+secA) }^{ 2 }+{ (cosA+cosecA) }^{ 2 }={ (1+secA\quad cosecA) }^{ 2 }$$
(iii) $$\frac { tanA+sinA }{ tanA-sinA } =\frac { secA+1 }{ secA-1 }$$
Solution:

Question 31.
If sin θ + cos θ = √2 sin (90° – θ), show that cot θ = √2 + 1
Solution:

Question 32.
If 7 sin2 θ + 3 cos2 θ = 4, 0° ≤ θ ≤ 90°, then find the value of θ.
Solution:

Question 33.
If sec θ + tan θ = m and sec θ – tan θ = n, prove that mn = 1.
Solution:

Question 34.
If x – a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 – y2 = a2 – b2.
Solution:

ICSE Maths Solutions For Class 10 Chapter 18 Question 35.
If x = h + a cos θ and y = k + a sin θ, prove that (x – h)2 + (y – k)2 = a2.
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 11 Understanding Symmetrical Shapes Ex 11.1

Question 1.
Why is it better to use a divider and a ruler than a ruler only, while measuring the length of a line segment?
Solution:

Question 2.
In the given figure, compare the line segments with the help of a divider and fill in the blanks by using the symbol >, = or <:
(i) AB ………. CD
(ii) BC ………. AB
(iii) AC ………. BD
(iv) CD ………. BD

Solution:

Question 3.
If A, B and C are collinear points such that AB = 6 cm, BC = 4 cm and AC = 10 cm, which one of them lies between the other two?
Solution:

Question 4.
In the given figure, verify the following by measurement:
(i) AB + BC = AC
(ii) AC – BC = AB

Solution:

Question 5.
In the given figure, verify by measurement that:
(i) AC + BD = AD + BC
(ii) AB + CD = AD – BC

Solution:

Question 6.
In the given figure, measure the lengths of the sides of the triangle ABC and verify:
(i) AB + BC > AC
(ii) BC + AC > AB
(iii) AC + AB > BC

Solution:

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 10 Basic Geometrical Concept Check Your Progress

Question 1.
(i) Name all the rays shown in the given figure whose initial point is A.
(ii) Is ray $$\overrightarrow{\mathrm{AB}}$$ different from ray $$\overrightarrow{\mathrm{AD}}$$ ?
(iii) Is ray $$\overrightarrow{\mathrm{CA}}$$ different from ray $$\overrightarrow{\mathrm{CE}}$$ ?
(iv) Is ray $$\overrightarrow{\mathrm{BA}}$$ different from ray $$\overrightarrow{\mathrm{CA}}$$ ?
(v) Is ray $$\overrightarrow{\mathrm{ED}}$$ different from ray $$\overrightarrow{\mathrm{DE}}$$ ?

Solution:

Question 2.
From the given figure, write

(i) all pairs of parallel lines.
(ii) all pairs of intersecting lines.
(iii) lines whose point of intersecting is E.
(iv) collinear points.
Solution:

Question 3.
In the given figure :

(a) Name;
(i) Parallel lines.
(ii) All pairs of intersecting lines.
(iii) concurrent lines.
(b) State wheather true or false:
(i) points A, B and D are collinear.
(ii) lines AB and ED interesect at C.
Solution:

Question 4.
In context of the given figure, state whether the following statements are true (T) or false (F):

(i) Point A is in the interior of ∠AOD.
(ii) Point B is in the interior of ∠AOC.
(iii) Point C is in the exterior of ∠AOB.
(iv) Point D is in the exterior of ∠AOC.
Solution:

Question 5.
How many angles are marked in the given figure? Name them?

Solution:

Question 6.
In context of the given figure, name
(i) all triangles
(ii) all triangles having point E as common vertex.

Solution:

Question 7.
In context of the given figure, answer the following questions:

(i) Is ABCDEFG a polygon?
(ii) How many sides does it have?
(iii) How many vertices does it have?
(iv) Are $$\overline{\mathrm{AB}}$$
and $$\overline{\mathrm{FE}}$$ adjacent sides?
(v) Is $$\overline{\mathrm{GF}}$$
a diagonal of the polygon?
(vi) Are $$\overline{\mathrm{AC}}, \overline{\mathrm{AD}} \text { and } \overline{\mathrm{AE}}$$ diagonals of the polygon?
(vii) Is point P in the interior of the polygon?
(viii)Is point A in the exterior of the polygon?
Solution:

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 10 Basic Geometrical Concept Objective Type Questions

Mental Maths
Question 1.
Fill in the blanks:
(i) There is exactly one line passing through ……….. distinct points in a plane.
(ii) Two different lines in a plane either ……….. at exactly one point or are parallel.
(iii) The curves which have different beginning and endpoints are called ……….. curves.
(iv) A curve which does not cross itself at any point is called a ……….. curve.
(v) A simple closed curve made up entirely of line segments is called a ………..
(vi) A line segment formed by joining two non- adjacent vertices of a polygon is called its ………..
(vii) A quadrilateral has ……….. diagonals.
(viii) A-lines segment has a ……….. length.
Solution:

Question 2.
Fill in the blanks with correct word(s) to make the statement true.
(i) Radius of a circle is one-half of its ………..
(ii) A radius of a circle is a line segment with one end point
at ……….. and the other end-point on
(iii) A chord of a circle is a line segment with its end points ………..
(iv) A diameter of a circle is a chord that ……….. the centre of the circle.
(v) All radii of a circle are ………..
Solution:

Question 3.
State whether the following statements are true (T) or false (F):
(i) The line segment AB is the shortest route from A to B.
(ii) A line cannot be drawn wholly on a sheet of paper..
(iii) A line segment is made of infinite (uncountable) number of points.
(iv) Two lines in a plane always intersect.
(v) Through a given point only one line can be drawn.
(vi) Two different lines can be drawn passing through two distinct points.
(vii) Every simple closed curve is a polygon.
(viii)Every polygon has atleast three sides.
(ix) A vertex of a quadrilateral lies in its interior.
(x) A line segment with its end-points lying on a circle is called a diameter of the circle.
(xi) Diameter is the longest chord of the circle.
(xii) The end-points of a diameter of a circle divide the circle into two points, each part is called a semi-circle.
(xiii) A diameter of a circle divides the circular region into two parts, each part is called a semi-circular region.
(xiv) The diameter’s of a circle are concurrent the centre of the circle is the point common to all diameters.
(xv) Every circle has unique centre and it lies inside the circle.
(xvi) Every circle has unique diameter.
Solution:

Multiple Choice Questions
Choose the correct answer from the given four options (4 to 20):
Question 4.
Which of the following has no end points?
(a) a line
(b) a ray
(c) a line segment
(d) none of these
Solution:

Question 5.
Which of the following has definite length?
(a) a line
(b) a ray
(c) a line segment
(d) none of these
Solution:

Question 6.
The number of points required to name a line if
(a) 1
(b) 2
(c) 3
(d) 4
Solution:

Question 7.
The number of lines that can be drawn through a given point is
(a) 1
(b) 2
(c) 3
(d) infinitely many
Solution:

Question 8.
The number of lines that can be drawn passing through two distinct points is
(a) 1
(b) 2
(c) 3
(d) infinitely many
Solution:

Question 9.
The maximum number of points of intersection of three lines drawn in a plane is
(a) 1
(b) 2
(c) 3
(d) 6
Solution:

Question 10.
The minimum number of points of intersection of three lines drawn in a plane is
(a) 0
(b) 1
(c) 2
(d) 3
Solution:

Question 11.
In the given figure, the number of line segment is
(a) 5
(b) 10
(c) 12
(d) 15
Solution:

Question 12.
In a polygon with 5 sides, the number of diagonals is
(a) 3
(b) 4
(c) 5
(d) 10
Solution:

Question 13.
The number of lines passing through 5 points such that no three of them are collinear are
(a) 10
(b) 5
(c) 8
(d) 20
Solution:

Question 14.
In context of the given figure, which of the following statement is correct ?
(a) B is not a point on segment $$\overline{\mathrm{AC}}$$
(b) B is the initial point of the ray $$\overrightarrow{\mathrm{AD}}$$
(c) D is a point on the ray $$\overrightarrow{\mathrm{CA}}$$
(d) C is a point on the ray $$\overrightarrow{\mathrm{BD}}$$

Solution:

Question 15.
The figure formed by two rays with same initial point is known as
(a) a line
(b) a line segment
(c) a ray
(d) an angle
Solution:

Question 16.
In the given figure, the number of angles is
(a) 3
(b) 4
(c) 5
(d) 6

Solution:

Question 17.
Which of the following statements is false?
(a) A triangle has three sides
(b) A triangle has three vertices
(c) A triangle has three angles
(d) A triangle has two diagonals
Solution:

Question 18.
Which of the following statements is false?
(a) A quadrilateral has four sides and four vertices
(b) A quadrilateral has four angles
(c) A quadrilateral has four diagonals
(d) A quadrilateral has two diagonals
Solution:

Question 19.
By joining any two points of a circle, we obtain its
(b) chord
(c) diameter
(d) circumference
Solution:

Question 20.
If the radius of a circle is 4 cm, then the length of its diameter is
(a) 2 cm
(b) 4 cm
(c) 8 cm
(d) 16 cm
Solution:

Higher Order Thinking Skills (Hots)
Question 1.
Can a sector and segment of a circle coincide? If so, name it.
Solution:

Question 2.
In the given figure, find:
(i) the number of triangles pointing up.
(ii) the total number of triangles.

Solution:

Question 3.
In the given figure, find the total number of squares.

Solution:

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 10 Basic Geometrical Concept Ex 10.4

Question 1.
In the given figure, identify:
(i) the centre of the circle
(iii) a diameter
(iv) a chord
(v) two points in the interior
(vi) a point in the exterior
(vii) a sector
(viii) a segment

Solution:

Question 2.
State whether the following statements are true (T) or false (F):
(i) Every diameter of a circle is also a chord.
(ii) Every chord of a circle is also a diameter.
(iii) Two diameters of a circle will necessarily intersect.
(iv) The centre of the circle is always in its interior.
Solution:

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 10 Basic Geometrical Concept Ex 10.3

Question 1.
Draw rough diagrams to illustrate the following:
(i) open simple curve
(ii) closed simple curve
(iii) open curve that is not simple
(iv) closed curve that is not simple.
Solution:

Question 2.
Consider the given figure and answer the following questions:
(i) Is it a curve?
(ii) Is it a closed curve?
(iii) Is it a polygon?

Solution:

Question 3.
Draw a rough sketch of a triangle ABC. Mark a point P in its interior and a point Q in its exterior. Is the point A in its exterior or in its interior?
Solution:

Question 4.
Draw a rough sketch of a quadrilateral PQRS. Draw its diagonals. Name them.
Solution:

Question 5.
In context of the given figure:
(i) Is it a simple closed curve?
(iii) Draw its diagonals and name them.
(iv) State which diagonal lies in the interior and which diagonal lies in the exterior of the quadrilateral.

Solution:

Question 6.
Draw a rough sketch of a quadrilateral KLMN. State,
(i) two pairs of opposite sides
(ii) two pairs of opposite angles
(iii) two pairs of adjacent sides
(iv) two pairs of adjacent angles.

Solution:

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 10 Basic Geometrical Concept Ex 10.2

Question 1.
How many angles are shown in the following figure? Name them.

Solution:

Question 2.
In the given figure, name the point(s)
(i) In the interior of ∠DOE
(ii) In the exterior of ∠EOF
(iii) On ∠EOF

Solution:

Question 3.
Draw rough diagrams of two angles such that they have
(i) One point in common.
(ii) Two points in common.
(iii) One ray in common.
Solution:

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 10 Basic Geometrical Concept Ex 10.1

Question 1.
How many lines can be drawn through a given point?
Solution:

Question 2.
How many lines can be drawn through two distinct given points?
Solution:

Question 3.
How many lines can be drawn through three collinear points?
Solution:

Question 4.
Mark three non-collinear points A, B and C in your notebook. Draw lines through these points taking two at a time and name these lines. How many such different lines can be drawn?
Solution:

Question 5.
Use the figure to name :
(i) Five point
(ii) Aline
(iii) Four rays
(iv) Five line segments

Solution:

Question 6.
Use the figure to name:

(i) Line containing point E.
(ii) Line passing through A.
(iii) Line on which point O lies.
(iv) Two pairs of intersecting lines.
Solution:

Question 7.
From the given figure, write

(i) collinear points
(ii) concurrent lines and their points of concurrence.
Solution:

Question 8.
In the given figure, write

(i) all pairs of parallel lines.
(ii) all pairs of intersecting lines,
(iii) concurrent lines.
(iv) collinear points.
Solution:

Question 9.
Count the number of line segments drawn in each of the following figures and name them:

Solution:

Question 10.
(i) Name all the rays shown in the following figure whose initial points are A, B and C respectively.

(ii) Is ray AB different from ray AD?
(iii) Is ray CA different from ray CE?
(iv) Is ray BA different from ray CA?
(v) Is ray ED different from ray DE?
Solution:

Question 11.
Consider the following figure of line $$\overleftrightarrow { MN }$$. Says whether following statements are true or false in context of the given figure.

(i) Q, M, O, N and P are points on the line
$$\overleftrightarrow { MN }$$.
(ii) M, O and N are points on a line segment
$$\overline{\mathrm{MN}}$$.
(iii) M and N are end points of line segment
$$\overline{\mathrm{MN}}$$ .
(iv) O and N are end points of line segment
$$\overline{\mathrm{OP}}$$.
(v) M is a point on the ray $$\overline{\mathrm{OP}}$$.
(vi) M is one of the end points of line segment
$$\overline{\mathrm{QO}}$$.
(vii) Ray $$\overrightarrow { OP }$$ is same as ray
$$\overrightarrow { OM }$$.
(viii)Ray $$\overrightarrow { OM }$$ is not opposite to ray
$$\overrightarrow { OP }$$.
(ix) Ray $$\overrightarrow { OP }$$ is different from ray
$$\overrightarrow { QP }$$.
(x) O is not an initial point of ray $$\overrightarrow { OP }$$.
(xi) N is the initial point of $$\overrightarrow { N }$$ and
$$\overrightarrow { NM }$$.
Solution:

## ML Aggarwal Class 6 Solutions for ICSE Maths Chapter 9 Algebra Check Your Progress

Question 1.
Look at the following matchstick pattern of polygons. Complete the table. Also write the general rule that gives the number of matchsticks.

Solution:

Question 2.
Write an algebraic expression for each of the following:
(i) If 1-metre cloth costs ₹ x, then what is the cost of 6-metre cloth?
(ii) If the cost of a notebook is ₹ x and the cost of a book is ₹ y, then what is the cost of 5 notebooks and 2 books?
(iii) The score of Ragni in Mathematics is 23 more than two-third of her score in English. If she scores x marks in English, what is her score in Mathematics?
(iv) If the length of a side of a regular pentagon is x cm, then what is the perimeter of the pentagon?
Solution:

Question 3.
When x = 4 and y = 2, find the value of:
(i) x + y
(ii) x – y
(iii) x2 + 2
(iv) x2 – 2xy + y2
Solution:

Question 4.
When a = 3 and b = -1, find the value of 2a3 – b4 + 3a2b3.
Solution:

Question 5.
When a = 3, b = 0, c = -2, find the values of:
(i) ab + 2bc + 3ca + 4abc
(ii) a3 + b3 + c3 – 3abc
Solution:

Question 6.
Solve the following linear equations:
(i) 2x – $$1 \frac{1}{2}=4 \frac{1}{2}$$
(ii) 3(y – 1) = 2(y + 1)
(ii) n – 3 = 5n + 21
(iv) $$\frac{1}{3}(7 x-1)=\frac{1}{4}$$
Solution: