ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test

Question 1.
A cylindrical container is to be made of tin sheet. The height of the container is 1 m and its diameter is 70 cm. If the container is open at the top and the tin sheet costs Rs 300 per m2, find the cost of the tin for making the container.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q1

Question 2.
A cylinder of the maximum volume is cut out from a wooden cuboid of length 30 cm and cross-section of square of side 14 cm. Find the volume of the cylinder and the volume of wood wasted.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q2

Question 3.
Find the volume and the total surface area of a cone having slant height 17 cm and base diameter 30 cm. Take π = 3.14.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q3

Question 4.
Find the volume of a cone given that its height is 8 cm and the area of base 156 cm2.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q4

Question 5.
The circumference of the edge of a hemispherical bowl is 132 cm. Find the capacity of the bowl.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q5

Question 6.
The volume of a hemisphere is \(2425 \frac { 1 }{ 2 } \) cm2. Find the curved surface area.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q6

Question 7.
A solid wooden toy is in the shape of a right circular cone mounted on a hemisphere. If the radius of the hemisphere is 4.2 cm and the total height of the toy is 10.2 cm, find the volume of the toy?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q7
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q7.1

Question 8.
A medicine capsule is in the shape of a cylinder of diameter 0.5 cm with two hemispheres stuck to each of its ends. The length of the entire capsule is 2 cm. Find the capacity of the capsule.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q8
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q8.1

Question 9.
A solid is in the form of a cylinder with hemispherical ends. The total height of the solid is 19 cm and the diameter of the cylinder is 7 cm. Find the volume and the total surface area of the solid.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q9
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q9.1

Question 10.
The radius and height of a right circular cone are in the ratio 5 : 12. If its volume is 2512 cm , find its slant height. (Take π = 3.14).
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q10

Question 11.
A cone and a cylinder are of the same height. If diameters of their bases are in the ratio 3 : 2, find the ratio of their volumes.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q11

Question 12.
A solid cone of base radius 9 cm and height 10 cm is lowered into a cylindrical jar of radius 10 cm, which contains water sufficient to submerge the cone completely. Find the rise in water level in the jar.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q12

Question 13.
An iron pillar has some part in the form of a right circular cylinder and the remaining in the form of a right circular cone. The radius of the base of each of cone and cylinder is 8 cm. The cylindrical part is 240 cm high and the conical part is 36 cm high. Find the weight of the pillar if one cu. cm of iron weighs 7.8 grams.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q13
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q13.1

Question 14.
A circus tent is made of canvas and is in the form of a right circular cylinder and a right circular cone above it. The diameter and height of the cylindrical part of the tent are 126 m and 5 m respectively. The total height of the tent is 21 m. Find the total cost of the tent if the canvas used costs Rs 36 per square metre.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q14
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q14.1

Question 15.
The entire surface of a solid cone of base radius 3 cm and height 4 cm is equal to the entire surface of a solid right circular cylinder of diameter 4 cm. Find the ratio of their
(i) curved surfaces
(ii) volumes.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q15
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q15.1

Question 16.
A cone is 8.4 cm high and the radius of its base is 2.1 cm. It is melted and recast into a sphere. Find the radius of the sphere.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q16

Question 17.
How many lead shots each of diameter 4.2 cm can be obtained from a solid rectangular lead piece with dimensions 66 cm, 42 cm and 21 cm.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q17

Question 18.
Find the least number of coins of diameter 2.5 cm and height 3 mm which are to be melted to form a solid cylinder of radius 3 cm and height 5 cm.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q18

Question 19.
A hemisphere of the lead of radius 8 cm is cast into a right circular cone of base radius 6 cm. Determine the height of the cone correct to 2 places of decimal.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q19

Question 20.
A vessel in the form of a hemispherical bowl is full of water. The contents are emptied into a cylinder. The internal radii of the bowl and cylinder are respectively 6 cm and 4 cm. Find the height of the water in the cylinder.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q20

Question 21.
The diameter of a metallic sphere is 42 cm. It is melted and drawn into a cylindrical wire of 28 cm diameter. Find the length of the wire.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q21

Question 22.
A sphere of diameter 6 cm is dropped into a right circular cylindrical vessel partly filled with water. The diameter of the cylindrical vessel is 12 cm. If the sphere is completely submerged in water, by how much will the level of water rise in the cylindrical vessel?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q22

Question 23.
A solid sphere of radius 6 cm is melted into a hollow cylinder of uniform thickness. If the external radius of the base of the cylinder is 5 cm and its height is 32 cm, find the uniform thickness of the cylinder.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q23

Question 24.
A solid is in the form of a right circular cone mounted on a hemisphere. The radius of the hemisphere is 3.5 cm and the height of the cone is 4 cm. The solid is placed in a cylindrical vessel, full of water, in such a Way that the whole solid is submerged in water. If the radius of the cylindrical vessel is 5 cm and its height is 10.5 cm, find the volume of water left in the cylindrical vessel.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q24
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Chapter Test Q24.1

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 1.
Find the length of the tangent drawn to a circle of radius 3 cm, from a point distant 5 cm from the centre.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

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

Question 2.
A point P is at a distance 13 cm from the centre C of a circle and PT is a tangent to the given circle. If PT = 12 cm, find the radius of the circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 3.
The tangent to a circle of radius 6 cm from an external point P, is of length 8 cm. Calculate the distance of P from the nearest point of the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 4.
Two concentric circles are of the radii 13 cm and 5 cm. Find the length of the chord of the outer circle which touches the inner circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 5.
Two circles of radii 5 cm and 2-8 cm touch each other. Find the distance between their centres if they touch:
(i) externally
(ii) internally.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 6.
(a) In figure (i) given below, triangle ABC is circumscribed, find x.
(b) In figure (ii) given below, quadrilateral ABCD is circumscribed, find x.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 7.
(a) In figure (i) given below, quadrilateral ABCD is circumscribed; find the perimeter of quadrilateral ABCD.
(b) In figure (ii) given below, quadrilateral ABCD is circumscribed and AD ⊥ DC ; find x if radius of incircle is 10 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 8.
(a) In the figure (i) given below, O is the centre of the circle and AB is a tangent at B. If AB = 15 cm and AC = 7.5 cm, find the radius of the circle.
(b) In the figure (ii) given below, from an external point P, tangents PA and PB are drawn to a circle. CE is a tangent to the circle at D. If AP = 15 cm, find the perimeter of the triangle PEC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 9.
(a) If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by
\(r= \frac { a+b-c }{ 2 } \)
(b) In the given figure, PB is a tangent to a circle with centre O at B. AB is a chord of length 24 cm at a distance of 5 cm from the centre. If the length of the tangent is 20 cm, find the length of OP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 10.
Three circles of radii 2 cm, 3 cm and 4 cm touch each other externally. Find the perimeter of the triangle obtained on joining the centres of these circles.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 11.
(a) In the figure (i) given below, the sides of the quadrilateral touch the circle. Prove that AB + CD = BC + DA.
(b) In the figure (ii) given below, ABC is triangle with AB = 10cm, BC = 8cm and AC = 6cm (not drawn to scale). Three circles are drawn touching each other with vertices A, B and C as their centres. Find the radii of the three circles
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 12.
(a) ln the figure (i) PQ = 24 cm, QR = 7 cm and ∠PQR = 90°. Find the radius of the inscribed circle ∆PQR
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, two concentric circles with centre O are of radii 5 cm and 3 cm. From an external point P, tangents PA and PB are drawn to these circles. If AP = 12cm, find BP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 13.
(a) In the figure (i) given below, AB = 8 cm and M is mid-point of AB. Semi-circles are drawn on AB, AM and MB as diameters. A circle with centre C touches all three semi-circles as shown, find its radius.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, equal circles with centres O and O’ touch each other at X. OO’ is produced to meet a circle O’ at A. AC is tangent to the circle whose centre is O. O’D is perpendicular to AC. Find the value of :
(i) \(\\ \frac { AO’ }{ AO } \)
(ii) \(\frac { area\quad of\quad \Delta ADO’ }{ area\quad of\quad \Delta ACO } \)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 14.
The length of the direct common tangent to two circles of radii 12 cm and 4 cm is 15 cm. Calculate the distance between their centres.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 15.
Calculate the length of a direct common tangent to two circles of radii 3 cm and 8 cm with their centres 13 cm apart.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 16.
In the given figure, AC is a transverse common tangent to two circles with centres P and Q and of radii 6 cm and 3 cm respectively. Given that AB = 8 cm, calculate PQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 17.
Two circles with centres A, B are of radii 6 cm and 3 cm respectively. If AB = 15 cm, find the length of a transverse common tangent to these circles.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 18.
(a) In the figure (i) given below, PA and PB are tangents at a points A and B respectively of a circle with centre O. Q and R are points on the circle. If ∠APB = 70°, find (i) ∠AOB (ii) ∠AQB (iii) ∠ARB
(b) In the figure (ii) given below, two circles touch internally at P from an external point Q on the common tangent at P, two tangents QA and QB are drawn to the two circles. Prove that QA = QB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 19.
In the given figure, AD is a diameter of a circle with centre O and AB is tangent at A. C is a point on the circle such that DC produced intersects the tangent af B. If ∠ABC = 50°, find ∠AOC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 20.
In the given figure, tangents PQ and PR are drawn from an external point P to a circle such that ∠RPQ = 30°. A chord RS is drawn parallel to the tangent PQ, Find ∠RQS
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 21.
(a) In the figure (i) given below, PQ is a tangent to the circle at A, DB is a diameter, ∠ADB = 30° and ∠CBD = 60°, calculate (i) ∠QAB (ii) ∠PAD (iii) ∠CDB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, ABCD is a cyclic quadrilateral. The tangent to the circle at B meets DC produced at F. If ∠EAB = 85° and ∠BFC = 50°, find ∠CAB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 22.
(a) In the figure (i) given below, O is the centre of the circle and SP is a tangent. If ∠SRT = 65°, find the value of x, y and z. (2015)
(b) In the figure (ii) given below, O is the centre of the circle. PS and PT are tangents and ∠SPT = 84°. Calculate the sizes of the angles TOS and TQS.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 23.
In the given figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO (ii) ∠AOR (iii) ∠APB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 24.
(a) In the figure (i) given below, O is the centre of the circle. The tangent at B and D meet at P. If AB is parallel to CD and ∠ ABC = 55°. find (i) ∠BOD (ii) ∠BPD
(b) In the figure (ii) given below. O is the centre of the circle. AB is a diameter, TPT’ is a tangent to the circle at P. If ∠BPT’ = 30°, calculate : (i) ∠APT (ii) ∠BOP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 25.
In the adjoining figure, ABCD is a cyclic quadrilateral.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
The line PQ is the tangent to the circle at A. If ∠CAQ : ∠CAP = 1 : 2, AB bisects ∠CAQ and AD bisects ∠CAP, then find the measure of the angles of the cyclic quadrilateral. Also prove that BD is a diameter of the circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 26.
In a triangle, ABC, the incircle (centre O) touches BC, CA and AB at P, Q and R respectively. Calculate (i) ∠QOR (ii) ∠QPR given that ∠A = 60°.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 27.
(a) In figure (0 given below, AB is a diameter. The tangent at C meets AB produced at Q, ∠CAB = 34°. Find
(i) ∠CBA (ii) ∠CQA (2006)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, AP and BP are tangents to the circle with centre O. Given ∠APB = 60°, calculate.
(i) ∠AOB (ii) ∠OAB (iii) ∠ACB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 28.
(a) In the figure (i) given below, O is the centre of the circumcircle of triangle XYZ. Tangents at X and Y intersect at T. Given ∠XTY = 80° and ∠XOZ = 140°, calculate the value of ∠ZXY. (1994)
(b) In the figure (ii) given below, O is the centre of the circle and PT is the tangent to the circle at P. Given ∠QPT = 30°, calculate (i) ∠PRQ (ii) ∠POQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 29.
Two chords AB, CD of a circle intersect internally at a point P. If
(i) AP = cm, PB = 4 cm and PD = 3 cm, find PC.
(ii) AB = 12 cm, AP = 2 cm, PC = 5 cm, find PD.
(iii) AP = 5 cm, PB = 6 cm and CD = 13 cm, find CP.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 30.
(a) In the figure (i) given below, PT is a tangent to the circle. Find TP if AT = 16 cm and AB = 12 cm.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure given below, diameter AB and chord CD of a circle meet at P. PT is a tangent to the circle at T. CD = 7.8 cm, PD = 5 cm, PB = 4 cm. Find (i) AB. (ii) the length of tangent PT.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 31.
PAB is secant and PT is tangent to a circle
(i) PT = 8 cm and PA = 5 cm, find the length of AB.
(ii) PA = 4.5 cm and AB = 13.5 cm, find the length of PT.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 32.
In the adjoining figure, CBA is secant and CD is tangent to the circle. If AB = 7 cm and BC = 9 cm, then
(i) Prove that ∆ACD ~ ∆DCB.
(ii) Find the length of CD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 33.
(a) In the figure (i) given below, PAB is secant and PT is tangent to a circle. If PA : AB = 1 : 3 and PT = 6 cm, find the length of PB.
(b) In the figure (ii) given below, ABC is an isosceles triangle in which AB = AC and Q is mid-point of AC. If APB is a secant, and AC is tangent to the circle at Q, prove that AB = 4 AP.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 34.
Two chords AB, CD of a circle intersect externally at a point P. If PA = PC, Prove that AB = CD.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 35.
(a) In the figure (i) given below, AT is tangent to a circle at A. If ∠BAT = 45° and ∠BAC = 65°, find ∠ABC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, A, B and C are three points on a circle. The tangent at C meets BA produced at T. Given that ∠ATC = 36° and ∠ACT = 48°, calculate the angle subtended by AB at the centre of the circle. (2001)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 36.
In the adjoining figure, ∆ABC is isosceles with AB = AC. Prove that the tangent at A to the circumcircle of ∆ABC is parallel to BC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 37.
If the sides of a rectangle touch a circle, prove that the rectangle is a square.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 38.
(a) In the figure (i) given below, two circles intersect at A, B. From a point P on one of these circles, two line segments PAC and PBD are drawn, intersecting the other circle at C and D respectively. Prove that CD is parallel to the tangent at P.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, two circles with centres C, C’ intersect at A, B and the point C lies on the circle with centre C’. PQ is a tangent to the circle with centre C’ at A. Prove that AC bisects ∠PAB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

Question 39.
(a) In the figure (i) given below, AB is a chord of the circle with centre O, BT is tangent to the circle. If ∠OAB = 32°, find the values of x and y.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
(b) In the figure (ii) given below, O and O’ are centres of two circles touching each other externally at the point P. The common tangent at P meets a direct common tangent AB at M. Prove that:
(i) M bisects AB (ii) ∠APB = 90°.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.3

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 1.
If O is the centre of the circle, find the value of x in each of the following figures (using the given information):
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

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

Question 2.
(a) In the figure (i) given below, O is the centre of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
(b) In the figure (ii) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 3.
(a) In the figure, (i) given below, if ∠DBC = 58° and BD is a diameter of the circle, calculate:
(i) ∠BDC (ii) ∠BEC (iii) ∠BAC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
(b) In the figure (if) given below, AB is parallel to DC, ∠BCE = 80° and ∠BAC = 25°. Find:
(i) ∠CAD (ii) ∠CBD (iii) ∠ADC (2008)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 4.
(a) In the figure given below, ABCD is a cyclic quadrilateral. If ∠ADC = 80° and ∠ACD = 52°, find the values of ∠ABC and ∠CBD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
(b) In the figure given below, O is the centre of the circle. ∠AOE =150°, ∠DAO = 51°. Calculate the sizes of ∠BEC and ∠EBC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 5.
(a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC.
(b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find:
(i)∠BAD (ii) DBCD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 6.
(a) In the figure given below, O is the centre of the circle. If ∠BAD = 30°, find the values of p, q and r.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
(b) In the figure given below, two circles intersect at points P and Q. If ∠A = 80° and ∠D = 84°, calculate
(i) ∠QBC (ii) ∠BCP
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 7.
(a) In the figure given below, PQ is a diameter. Chord SR is parallel to PQ.Given ∠PQR = 58°, calculate (i) ∠RPQ (ii) ∠STP
(T is a point on the minor arc SP)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
(b) In the figure given below, if ∠ACE = 43° and ∠CAF = 62°, find the values of a, b and c (2007)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 8.
(a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB.
(b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values of x, y and z.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 9.
(a) In the figure (i) given below, ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If ∠ADE = 70° and ∠OBA = 45°, calculate
(i) ∠OCA (ii) ∠BAC
(b) In figure (ii) given below, ABF is a straight line and BE || DC. If ∠DAB = 92° and ∠EBF = 20°, find :
(i) ∠BCD (ii) ∠ADC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 10.
(a) In the figure (ii) given below, PQRS is a cyclic quadrilateral in which PQ = QR and RS is produced to T. If ∠QPR = 52°, calculate ∠PST.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
(b) In the figure (ii) given below, O is the centre of the circle. If ∠OAD = 50°, find the values of x and y.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 11.
(a) In the figure (i) given below, O is the centre of the circle. If ∠COD = 40° and ∠CBE = 100°, then find :
(i) ∠ADC
(ii) ∠DAC
(iii) ∠ODA
(iv) ∠OCA.
(b) In the figure (ii) given below, O is the centre of the circle. If ∠BAD = 75° and BC = CD, find :
(i) ∠BOD
(ii) ∠BCD
(iii) ∠BOC
(iv) ∠OBD (2009)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 12.
In the given figure, O is the centre and AOE is the diameter of the semicircle ABCDE. If AB = BC and ∠AEC = 50°, find :
(i) ∠CBE
(ii) ∠CDE
(iii) ∠AOB.
Prove that OB is parallel to EC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 13.
(a) In the figure (i) given below, ED and BC are two parallel chords of the circle and ABE, ACD are two st. lines. Prove that AED is an isosceles triangle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
(b) In the figure (ii) given below, SP is the bisector of ∠RPT and PQRS is a cyclic quadrilateral. Prove that SQ = RS.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 14.
In the given figure, ABC is an isosceles triangle in which AB = AC and circle passing through B and C intersect sides AB and AC at points D and E. Prove that DE || BC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 15.
(a) Prove that a cyclic parallelogram is a rectangle.
(b) Prove that a cyclic rhombus is a square.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

Question 16.
In the given figure, chords AB and CD of the circle are produced to meet at O. Prove that triangles ODB and OAC are similar. Given that CD = 2 cm, DO = 6 cm and BO = 3 cm, area of quad. CABD
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 1.
Draw a straight line AB of length 8 cm. Draw the locus of all points which are equidistant from A and B. Prove your statement.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 2.
A point P is allowed to travel in space. State the locus of P so that it always remains at a constant distance from a fixed point C.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 3.
Draw a line segment AB of length 7 cm. Construct the locus of a point P such that area of triangle PAB is 14 cm².
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 4.
Draw a line segment AB of length 12 cm. Mark M, the mid-point of AB. Draw and describe the locus of a point which is
(i) at a distance of 3 cm from AB.
(ii) at a distance of 5 cm from the point M. Mark the points P, Q, R, S which satisfy both the above conditions. What kind of quadrilateral is PQRS? Compute the area of the quadrilateral PQRS.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 5.
AB and CD are two intersecting lines. Find the position of a point which is at a distance of 2 cm from AB and 1.6 cm from CD.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 6.
Two straight lines PQ and PK cross each other at P at an angle of 75°. S is a stone on the road PQ, 800 m from P towards Q. By drawing a figure to scale 1 cm = 100 m, locate the position of a flagstaff X, which is equidistant from P and S, and is also equidistant from the road.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 7.
Construct a rhombus PQRS whose diagonals PR, QS are 8 cm and 6 cm respectively. Find by construction a point X equidistant from PQ, PS and equidistant from R, S. Measure XR.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 8.
Without using set square or protractor, construct the parallelogram ABCD in which AB = 5.1 cm. the diagonal AC = 5.6 cm and the diagonal BD = 7 cm. Locate the point P on DC, which is equidistant from AB and BC.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

Question 9.
By using ruler and compass only, construct a quadrilateral ABCD in which AB = 6.5 cm, AD = 4 cm and ∠DAB = 75°. C is equidistant from the sides AB and AD, also C is equidistant from the points A and B.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 1.
A point moves such that its distance from a fixed line AB is always the same. What is the relation between AB and the path travelled by P?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 2.
A point P moves so that its perpendicular distance from two given lines AB and CD are equal. State the locus of the point P.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 3.
P is a fixed point and a point Q moves such that the distance PQ is constant, what is the locus of the path traced out by the point Q?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 4.
(i) AB is a fixed line. State the locus of the point P so that ∠APB = 90°.
(ii) A, B are fixed points. State the locus of the point P so that ∠APB = 60°.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 5.
Draw and describe the locus in each of the following cases :
(i) The locus of points at a distance 2.5 cm from a fixed line.
(ii) The locus of vertices of all isosceles triangles having a common base.
(iii) The locus of points inside a circle and equidistant from two fixed points on the circle.
(iv) The locus of centres of all circles passing through two fixed points.
(v) The locus of a point in rhombus ABCD which is equidistant from AB and AD. (1998)
(vi) The locus of a point in the rhombus ABCD which is equidistant from points A and C.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 6.
Describe completely the locus of points in each of the following cases :
(i) mid-point of radii of a circle.
(ii) centre of a ball, rolling along a straight line on a level floor.
(iii) the point in a plane equidistant from a given line.
(iv) the point in a plane, at a constant distance of 5 cm from a fixed point (in the plane).
(v) centre of a circle of varying radius and touching two arms of ∠ADC.
(vi) centre of a circle of varying radius and touching a fixed circle, centre O, at a fixed point A on it.
(vii) centre of a circle of radius 2 cm and touching a fixed circle of radius 3 cm with centre 0.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 7.
Using a ruler and compasses construct:
(i) a triangle ABC in which AB = 5.5 cm, BC = 3.4 cm and CA = 4.9 cm.
(ii) the locus of points equidistant from A and C.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 8.
Construct triangle ABC, with AB = 7 cm, BC = 8 cm and ∠ABC = 60°. Locate by construction the point P such that:
(i) P is equidistant from B and C and
(ii) P is equidistant from AB and BC
(iii) Measure and record the length of PB. (2000)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 9.
A straight line AB is 8 cm long. Locate by construction the locus of a point which is:
(i) Equidistant from A and B.
(ii) Always 4 cm from the line AB.
(iii) Mark two points X and Y, which are 4 cm from AB and equidistant from A and B. Name the figure AXBY.(2008)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 10.
Use ruler and compasses only for this question.
(i) Construct ∆ABC, where AB = 3.5 cm, BC = 6 cm and ∠ABC = 60°.
(ii) Construct the locus of points inside the triangle which are equidistant from BA and BC.
(iii) Construct the locus of points inside the triangle which are equidistant from B and C.
(iv) Mark the point P which is equidistant from AB, BC and also equidistant from B and C. Measure arid record the length of PB. (2010)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 11.
Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 105°. Hence:
(i) Construct the locus of points equidistant from BA and BC.
(ii) Construct the locus of points equidistant from B and C.
(iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 12.
In the given diagram, A, B and C are fixed collinear points; D is a fixed point outside the line: Locate
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
(i) the point P on AB such that CP = DP.
(ii) the points Q such that CQ = DQ = 3 cm. How many such points are possible?
(iii) the points R on AB such that DR = 4 cm. How many such points are possible?
(iv) the points S such that CS = DS and S is 4 cm away from the line CD. How many such points are possible?
(v) Are the points P, Q, R collinear?
(vi) Are the points P, Q, S collinear?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 13.
Points A, B and C represent the position of three towers such that AB = 60 m, BC = 73 m and CA = 52 m. Taking a scale of 10 m to 1 cm, make an accurate drawing of ∆ABC. Find by drawing, the location of a point which is equidistant from A, B and C, and its actual distance from any of the towers.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 14.
Draw two intersecting lines to include an angle of 30°. Use ruler and compasses to locate points which are equidistant from these lines and also 2 cm away from their point of intersection. How many such points exist? (1990)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 15.
Without using set square or protractor, construct the quadrilateral ABCD in which ∠BAD = 45°, AD = AB = 6 cm, BC = 3.6 cm and CD = 5 cm.
(i) Measure ∠BCD.
(ii) Locate the point P on BD which is equidistant from BC and CD. (1992)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 16.
Without using set square or protractor, construct rhombus ABCD with sides of length 4 cm and diagonal AC of length 5 cm. Measure ∠ABC. Find the point R on AD such that RB = RC. Measure the length of AR. (1990)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 17.
Without using set-squares or protractor construct :
(i) Triangle ABC, in which AB = 5.5 cm, BC = 3.2 cm and CA = 4.8 cm.
(ii) Draw the locus of a point which moves so that it is always 2.5 cm from B.
(iii) Draw the locus of a point which moves so that it is equidistant from the sides BC and CA.
(iv) Mark the point of intersection of the loci with the letter P and measure PC. (1994)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 18.
By using ruler and compasses only, construct an isosceles triangle ABC in which BC = 5 cm, AB = AC and ∠BAC = 90°. Locate the point P such that :
(i) P is equidistant from the sides BC and AC.
(ii) P is equidistant from the points B and C.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 19.
Using ruler and compasses only, construct a quadrilateral ABCD in which AB = 6 cm, BC = 5 cm, ∠B = 60°, AD = 5 cm and D is equidistant from AB and BC. Measure CD. (1983)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 20.
Construct an isosceles triangle ABC such that AB = 6 cm, BC = AC = 4 cm. Bisect ∠C internally and mark a point P on this bisector such that CP = 5 cm. Find the points Q and R which are 5 cm from P and also 5 cm from the line AB (2001)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 21.
Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of length 6 cm and 5 cm respectively.
(i) Construct the locus of points, inside the circle, that is equidistant from A and C. Prove your construction.
(ii) Construct the locus of points, inside the circle, that is equidistant from AB and AC, (1995)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

Question 22.
Ruler and compasses only may be used in this question. All construction lines and arcs must be clearly shown, and be of sufficient length and clarity to permit assessment.
(i) Construct a triangle ABC, in which BC = 6 cm, AB = 9 cm. and ∠ABC = 60°.
(ii) Construct the locus of all points, inside ∆ABC, which are equidistant from B and C.
(iii) Construct the locus of the vertices of the triangles with BC as a base, which is equal in area to ∆ABC.
(iv) Mark the point Q, in your construction, which would make ∆QBC equal in area to ∆ABC, and isosceles.
(v) Measure and record the length of CQ. (1998)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 14 Locus Ex 14

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Take π = \(\\ \frac { 22 }{ 7 } \) unless stated otherwise.

Question 1.
Find the total surface area of a solid cylinder of radius 5 cm and height 10 cm. Leave your answer in terms of π.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 2.
An electric geyser is cylindrical in shape, having a diameter of 35 cm and height 1.2 m. Neglecting the thickness of its walls, calculate
(i) its outer lateral surface area,
(ii) it’s capacity in litres.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 3.
A school provides milk to the students daily in cylindrical glasses of diameter 7 cm. If the glass is filled with milk up to a height of 12 cm, find how many litres of milk is needed to serve 1600 students.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 4.
In the given figure, a rectangular tin foil of size 22 cm by 16 cm is wrapped around to form a cylinder of height 16 cm. Find the volume of the cylinder.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 5.
(i) How many cubic metres of the soil must be dug out to make a well 20 metres deep and 2 metres in diameter?
(ii) If the inner curved surface of the well in part (i) above is to be plastered at the rate of Rs 50 per m2, find the cost of plastering.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 6.
A road roller (in the shape of a cylinder) has a diameter 0.7 m and its width is 1.2 m. Find the least number of revolutions that the roller must make in order to level a playground of size 120 m by 44 m.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 7.
If the volume of a cylinder of height 7 cm is 448 π cm3, find its lateral surface area and total surface area.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 8.
A wooden pole is 7 m high and 20 cm in diameter. Find its weight if the wood weighs 225 kg per m3.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 9.
The area of the curved surface of a cylinder is 4400 cm2, and the circumference of its base is 110 cm. Find
(i) the height of the cylinder.
(ii) the volume of the cylinder.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 10.
A cylinder has a diameter of 20 cm. The area of the curved surface is 1000 cm2. Find
(i) the height of the cylinder correct to one decimal place.
(ii) the volume of the cylinder correct to one decimal place. (Take π = 3.14)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 11.
The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used up when writing 310 words on an average. How many words would use up a bottle of ink containing one-fifth of a litre?
Answer correct to the nearest.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 12.
Find the ratio between the total surface area of a cylinder to its curved surface area given that its height and radius are 7.5 cm and 3.5 cm.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 13.
The radius of the base of a right circular cylinder is halved and the height is doubled. What is the ratio of the volume of the new cylinder to that of the original cylinder?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 14.
(i) The sum of the radius and the height of a cylinder is 37 cm and the total surface area of the cylinder is 1628 cm2. Find the height and the volume of the cylinder.
(ii) The total surface area of a cylinder is 352 cm2. If its height is 10 cm, then find the diameter of the base.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 15.
The ratio between the curved surface and the total surface of a cylinder is 1 : 2. Find the volume of the cylinder, given that its total surface area is 616 cm2.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 16.
Two cylindrical jars contain the same amount of milk. If their diameters are in the ratio 3 : 4, find the ratio of their heights.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 17.
A rectangular sheet of tin foil of size 30 cm x 18 cm can be rolled to form a cylinder in two ways along the length and along with breadth. Find the ratio of volumes of the two cylinders thus formed.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 18.
A cylindrical tube open at both ends is made of metal. The internal diameter of the tube is 11.2 cm and its length is 21 cm. The metal thickness is 0.4 cm. Calculate the volume of the metal.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 19.
The given figure shows a metal pipe 77 cm long. The inner diameter of a cross-section is 4 cm and the outer one is 4.4 cm. Find its
(i) inner curved surface area
(ii) outer curved surface area
(iii) total surface area.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 20.
A lead pencil consists of a cylinder of wood with a solid cylinder of graphite filled in the interior. The diameter of the pencil is 7 mm and the diameter of the graphite is 1 mm. If the length of the pencil is 14 cm, find the volume of the wood and that of the graphite.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 21.
A soft drink is available in two packs
(i) a tin can with a rectangular base of length 5 cm and width 4 cm, having a height of 15 cm and
(ii) a plastic cylinder with a circular base of diameter 7 cm and height 10 cm. Which container has greater capacity and by how much?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

Question 22.
A cylindrical roller made of iron is 2 m long. Its inner diameter is 35 cm and the thickness is 7 cm all around. Find the weight of the roller in kg, if 1 cm³ of iron weighs 8 g.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.1

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 1.
Draw a circle of radius 3 cm. Mark its centre as C and mark a point P such that CP = 7 cm. Using a ruler and compasses only, Construct two tangents from P to the circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 2.
Draw a line AQ = 7 cm. Mark a point P on AQ such that AP = 4 cm. Using a ruler and compasses only, construct:
(i) a circle with AP as diameter.
(ii) two tangents to the above circle from the point Q.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 3.
Using ruler and compasses only, construct a triangle ABC having given c = 6 cm, b = 1 cm and ∠A = 30°. Measure side a. Draw carefully the circumcircle of the triangle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 4.
Using ruler and compasses only, construct an equilateral triangle of height 4 cm and draw its circumcircle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 5.
Using a ruler and compasses only:
(i) Construct a triangle ABC with the following data: BC = 7 cm, AB = 5 cm and ∠ABC = 45°.
(ii) Draw the inscribed circle to ∆ABC drawn in part (i).
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 6.
Draw a triangle ABC, given that BC = 4 cm, ∠C = 75° and that radius of the circumcircle of ∆ABC is 3 cm.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 7.
Draw a regular hexagon of side 3.5 cm construct its circumcircle and measure its radius.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

Question 8.
Construct a triangle ABC with the following data: AB = 5 cm, BC = 6 cm and ∠ABC = 90°.
(i) Find a point P which is equidistant from B and C and is 5 cm from A. How many such points are there?
(ii) Construct a circle touching the sides AB and BC, and whose centre is equidistant from B and C.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 1.
Draw an equilateral triangle of side 4 cm. Draw its circumcircle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 2.
Using a ruler and a pair of compasses only, construct:
(i) a triangle ABC given AB = 4 cm, BC = 6 cm and ∠ABC = 90°.
(ii) a circle which passes through the points A, B and C and mark its centre as O. (2008)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 3.
Construct a triangle with sides 3 cm, 4 cm and 5 cm. Draw its circumcircle and measure its radius.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 4.
Using a ruler and compasses only:
(i) Construe a triangle ABC with the following data:
Base AB = 6 cm, AC = 5.2 cm and ∠CAB = 60°.
(ii) In the same diagram, draw a circle which passes through the points A, B and C. and mark its centre O.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 5.
Using ruler and compasses only, draw an equilateral triangle of side 5 cm and draw its inscribed circle. Measure the radius of the circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 6.
(i) Conduct a triangle ABC with BC = 6.4 cm, CA = 5.8 cm and ∠ ABC = 60°. Draw its incircle. Measure and record the radius of the incircle.
(ii) Construct a ∆ABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm. Construct the incircle of the triangle. Measure and record the radius of the incircle. (2014)
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 7.
The bisectors of angles A and B of a scalene triangle ABC meet at O.
(i) What is the point O called?
(ii) OR and OQ is drawn a perpendicular to AB and CA respectively. What is the relation between OR and OQ?
(iii) What is the relation between ∠ACO and ∠BCO?
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 8.
Using ruler and compasses only, construct a triangle ABC in which BC = 4 cm, ∠ACB = 45° and the perpendicular from A on BC is 2.5 cm. Draw the circumcircle of triangle ABC and measure its radius.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 9.
Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

Question 10.
Draw a regular hexagon of side 4 cm and construct its incircle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.2

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

Question 1.
Use a ruler and compass only in this question.
(i) Draw a circle, centre O and radius 4 cm.
(ii) Mark a point P such that OP = 7 cm.
Construct the two tangents to the circle from P. Measure and record the length of one of the tangents.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

Question 2.
Draw a line AB = 6 cm. Construct a circle with AB as diameter. Mark a point P at a distance of 5 cm from the mid-point of AB. Construct two tangents from P to the circle with AB as diameter. Measure the length of each tangent
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

Question 3.
Construct a tangent to a circle of radius 4cm from a point on the concentric circle of radius 6 cm and measure its length. Also, verify the measurement by actual calculation.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

Question 4.
Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

Question 5.
Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 16 Constructions Ex 16.1

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 1.
(a) In the figure (i) given below, triangle ABC is equilateral. Find ∠BDC and ∠BEC.
(b) In the figure (ii) given below, AB is a diameter of a circle with centre O. OD is perpendicular to AB and C is a point on the arc DB. Find ∠BAD and ∠ACD
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 2.
(a) In the figure given below, AB is a diameter of the circle. If AE = BE and ∠ADC = 118°, find
(i) ∠BDC (ii) ∠CAE.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
(b) In the figure given below, AB is the diameter of the semi-circle ABCDE with centre O. If AE = ED and ∠BCD = 140°, find ∠AED and ∠EBD. Also, Prove that OE is parallel to BD.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 3.
(a) In the figure (i) given below, O is the centre of the circle. Prove that ∠AOC = 2 (∠ACB + ∠BAC).
(b) In the figure (ii) given below, O is the centre of the circle. Prove that x + y = z.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 4.
(a) In the figure (i) given below, AB is the diameter of a circle. If DC is parallel to AB and ∠CAB = 25°, find:
(i)∠ADC
(ii) ∠DAC.
(b) In the figure (ii) given below, the centre O of the smaller circle lies on the circumference of the bigger circle. If ∠APB = 70° and ∠BCD = 60°, find:
(i) ∠AOB
(ii) ∠ACB
(iii) ∠ABD
(iv) ∠ADB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 5.
(a) In the figure (i) given below, ABCD is a cyclic quadrilateral. If AB = CD, Prove that AD = BC.
(b) In the figure (ii) given below, ABC is an isosceles triangle with AB = AC. If ∠ABC = 50°, find ∠BDC and ∠BEC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 6.
A point P is 13 cm from the centre of a circle. The length of the tangent drawn from P to the circle is 12 cm. Find the distance of P from the nearest point of the circle.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 7.
Two circles touch each other internally. Prove that the tangents drawn to the two circles from any point on the common tangent are equal in length.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 8.
From a point outside a circle, with centre O, tangents PA and PB are drawn. Prove that
(i) ∠AOP = ∠BOP.
(ii) OP is the perpendicular bisector of the chord AB.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 9.
(a) The figure given below shows two circles with centres A, B and a transverse common tangent to these circles meet the straight line AB in C. Prove that:
AP : BQ = PC : CQ.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
(b) In the figure (ii) given below, PQ is a tangent to the circle with centre O and AB is a diameter of the circle. If QA is parallel to PO, prove that PB is tangent to the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 10.
In the figure given below, two circles with centres A and B touch externally. PM is a tangent to the circle with centre A and QN is a tangent to the circle with centre B. If PM = 15 cm, QN = 12 cm, PA = 17 cm and QB = 13 cm, then find the distance between the centres A and B of the circles.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 11.
Two chords AB, CD of a circle intersect externally at a point P. If PB = 7 cm, AB = 9 cm and PD = 6 cm, find CD.
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 12.
(a) In the figure (i) given below, chord AB and diameter CD of a circle with centre O meet at P. PT is tangent to the circle at T. If AP = 16 cm, AB = 12 cm and DP = 2 cm, find the length of PT and the radius of the circle
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
(b) In the figure (ii) given below, chord AB and diameter CD of a circle meet at P. If AB = 8 cm, BP = 6 cm and PD = 4 cm, find the radius of the circle. Also, find the length of the tangent drawn from P to the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 13.
In the figure given below, chord AB and diameter PQ of a circle with centre O meet at X. If BX = 5 cm, OX = 10 cm and.the radius of the circle is 6 cm, compute the length of AB. Also, find the length of tangent drawn from X to the circle.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 14.
(a) In the figure (i) given below, ∠CBP = 40°, ∠CPB = q° and ∠DAB = p°. Obtain an equation connecting p and q. If AC and BD meet at Q so that ∠AQD = 2 q° and the points C, P, B and Q are concyclic, find the values of p and q.
(b) In the figure (ii) given below, AC is a diameter of the circle with centre O. If CD || BE, ∠AOB = 130° and ∠ACE = 20°, find:
(i)∠BEC
(ii) ∠ACB
(iii) ∠BCD
(iv) ∠CED.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 15.
(a) In the figure (i) given below, APC, AQB and BPD are straight lines.
(i) Prove that ∠ADB + ∠ACB = 180°.
(ii) If a circle can be drawn through A, B, C and D, Prove that it has AB as a diameter
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
(b) In the figure (ii) given below, AQB is a straight line. Sides AC and BC of ∆ABC cut the circles at E and D respectively. Prove that the points C, E, P and D are concyclic.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 16.
(a) In the figure (i) given below, chords AB, BC and CD of a circle with centre O are equal. If ∠BCD = 120°, find
(i) ∠BDC
(ii) ∠BEC
(iii) ∠AEC
(iv) ∠AOB.
Hence Prove that AOAB is equilateral.
(b) In the figure (ii) given below, AB is a diameter of a circle with centre O. The chord BC of the circle is parallel to the radius OD and the lines OC and BD meet at E. Prove that
(i) ∠CED = 3 ∠CBD
(ii) CD = DA.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

Question 17.
(a) In the adjoining figure, (i) given below AB and XY are diameters of a circle with centre O. If ∠APX = 30°, find
(i) ∠AOX
(ii) ∠APY
(iii) ∠BPY
(iv) ∠OAX.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
(b) In the figure (ii) given below, AP and BP are tangents to the circle with centre O. If ∠CBP = 25° and ∠CAP = 40°, find :
(i) ∠ADB
(ii) ∠AOB
(iii) ∠ACB
(iv) ∠APB.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
Solution:
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths