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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:

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:

Question 3.
Construct a triangle with sides 3 cm, 4 cm and 5 cm. Draw its circumcircle and measure its radius.
Solution:

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:

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:

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:

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:

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:

Question 9.
Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.
Solution:

Question 10.
Draw a regular hexagon of side 4 cm and construct its incircle.
Solution:

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

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:

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:

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:

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:

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

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

Solution:

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.

(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.

Solution:

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.

Solution:

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.

Solution:

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.

Solution:

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:

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:

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:

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.

(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.

Solution:

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.

Solution:

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:

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

(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.

Solution:

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.

Solution:

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.

Solution:

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

(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.

Solution:

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.

Solution:

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.

(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.

Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 13 Similarity Ex 13.2

Question 1.
(a) In the figure (i) given below if DE || BG, AD = 3 cm, BD = 4 cm and BC = 5 cm. Find (i) AE : EC (ii) DE.
(b) In the figure (ii) given below, PQ || AC, AP = 4 cm, PB = 6 cm and BC = 8 cm. Find CQ and BQ.
(c) In the figure (iii) given below, if XY || QR, PX = 1 cm, QX = 3 cm, YR = 4.5 cm and QR = 9 cm, find PY and XY.

Solution:

Question 2.
In the given figure, DE || BC.
(i) If AD = x, DB = x – 2, AE = x + 2 and EC = x – 1, find the value of x.
(ii) If DB = x – 3, AB = 2x, EC = x – 2 and AC = 2x + 3, find the value of x.

Solution:

Question 3.
E and F are points on the sides PQ and PR respectively of a ∆PQR. For each of the following cases, state whether EF || QR:
(i) PE = 3.9 cm, EQ = 3 cm, PF = 8 cm and RF = 9 cm.
(ii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm.
Solution:

Question 4.
A and B are respectively the points on the sides PQ and PR of a triangle PQR such that PQ = 12.5 cm, PA = 5 cm, BR = 6 cm and PB = 4 cm. Is AB || QR? Give reasons for your answer.
Solution:

Question 5.
(a) In figure (i) given below, DE || BC and BD = CE. Prove that ABC is an isosceles triangle.
(b) In figure (ii) given below, AB || DE and BD || EF. Prove that DC² = CF × AC.
Solution:

Question 6.
(a) In the figure (i) given below, CD || LA and DE || AC. Find the length of CL if BE = 4 cm and EC = 2 cm.
(b) In the given figure, ∠D = ∠E and \(\frac { AD }{ BD } =\frac { AE }{ EC } \). Prove that BAC is an isosceles triangle.

Solution:

Question 7.
In the figure given below, A, B and C are points on OP, OQ and OR respectively such that AB || PQ and AC || PR. show that BC || QR.

Solution:

Question 8.
ABCD is a trapezium in which AB || DC and its diagonals intersect each other at O. Using Basic Proportionality theorem, prove that \(\frac { AO }{ BO } =\frac { CO }{ DO } \)
Solution:

Question 9.
(a) In the figure (1) given below, AB || CR and LM || QR.
(i) Prove that \(\frac { BM }{ MC } =\frac { AL }{ LQ } \)
(ii) Calculate LM : QR, given that BM : MC = 1 : 2.
(b) In the figure (2) given below AD is bisector of ∠BAC. If AB = 6 cm, AC = 4 cm and BD = 3cm, find BC

Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

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

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

Choose the correct answer from the given four options (1 to 32):

Question 1.
In a cylinder, if the radius is halved and height is doubled then the volume will be
(a) same
(b) doubled
(c) halved
(d) four times
Solution:

Question 2.
In a cylinder, if the radius is doubled &nd height is halved then its curved surface area will be
(a) halved
(b) doubled
(c) same
(d) four times
Solution:

Question 3.
If a well of diameter 8 m has been dug to the depth of 14 m, then the volume of the earth dug out is
(a) 352 m³
(b) 704 m³
(c) 1408 m³
(d) 2816 m³
Solution:

Question 4.
If two cylinders of the same lateral surface have their radii in the ratio 4 : 9, then the ratio of their heights is
(a) 2 : 3
(b) 3 : 2
(c) 4 : 9
(d) 9 : 4
Solution:

Question 5.
The radii of two cylinders are in the ratio 2 : 3 and their heights are in the ratio 5 : 3. The ratio of their volumes is
(a) 10 : 17
(b) 20 : 27
(c) 17 : 27
(d) 20 : 37
Solution:

Question 6.
The total surface area of a cone whose radius is \(\\ \frac { r }{ 2 } \) and slant height 2l is
(a) 2πr (l + r)
(b) \(\pi r\left( l+\frac { r }{ 4 } \right) \)
(c) πr(l + r)
(d) 2πrl
Solution:

Question 7.
If the diameter of the base of cone is 10 cm and its height is 12 cm, then its curved surface area is
(a) 60π cm²
(b) 65π cm²
(c) 90π cm²
(d) 120π cm²
Solution:

Question 8.
If the diameter of the base of a cone is 12 cm and height is 20 cm, then its volume is ,
(a) 240π cm³
(b) 480π cm³
(c) 720π cm³
(d) 960π cm³
Solution:

Question 9.
If the radius of a sphere is 2r, then its volume will be
(a) \(\frac { 4 }{ 3 } \pi { r }^{ 3 } \)
(b) \(4\pi { r }^{ 3 }\)
(c) \(\frac { 8\pi { r }^{ 3 } }{ 3 } \)
(d) \(\frac { 32\pi { r }^{ 3 } }{ 3 } \)
Solution:

Question 10.
If the diameter of a sphere is 16 cm, then its surface area is
(a) 64π cm²
(b) 256π cm²
(c) 192π cm²
(d) 256 cm²
Solution:

Question 11.
If the radius of a hemisphere is 5 cm, then its volume is
(a) \(\frac { 250 }{ 3 } \pi { \quad cm }^{ 3 }\)
(b) \(\frac { 500 }{ 3 } \pi { \quad cm }^{ 3 } \)
(c) \(75\pi { \quad cm }^{ 3 } \)
(d) \(\frac { 125 }{ 3 } \pi { \quad cm }^{ 3 } \)
Solution:

Question 12.
If the ratio of the diameters of the two spheres is 3 : 5, then the ratio of their surface areas is
(a) 3 : 5
(b) 5 : 3
(c) 27 : 125
(d) 9 : 25
Solution:

Question 13.
The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. The ratio of the surface areas of the balloon in the two cases is
(a) 1 : 4
(b) 1 : 3
(c) 2 : 3
(d) 2 : 1
Solution:

Question 14.
The shape of a gilli, in the the game of gilli- danda, is a combination of

(a) two cylinders
(b) a cone and a cylinders
(c) two cones and a cylinder
(d) two cylinders and a cone
Solution:

Question 15.
If two solid hemisphere of same base radius r are joined together along with their bases, then the curved surface of this new solid is
(a) 4πr²
(b) 6πr²
(c) 3πr²
(d) 8πr²
Solution:

Question 16.
During the conversion of a solid from one shape to another, the volume of the new shape will
(a) increase
(b) decrease
(c) remain unaltered
(d) be doubled
Solution:

Question 17.
If a solid of one shape is converted to another, then the surface area of the new solid
(a) remains same
(b) increases
(c) decreases
(d) can’t say
Solution:

Question 18.
If a marble of radius 2.1 cm is put into a cylindrical cup full of water of radius 5 cm and height 6 cm, then the volume of water that flows out of the cylindrical cup is
(a) 38.8 cm³
(b) 55.4 cm³
(c) 19.4 cm³
(d) 471.4 cm³
Solution:

Question 19.
The volume of the largest right circular cone that can be carved out from a cube of edge 4.2 cm is
(a) 9.7 cm³
(b) 77.6 cm³
(c) 58.2 cm³
(d) 19.4 cm³
Solution:

Question 20.
The volume of the greatest sphere cut off from a circular cylindrical wood of base radius 1 cm and height 6 cm is
(a) 288 π cm³
(b) \(\frac { 4 }{ 3 } \pi \) cm³
(c) 6 π cm³
(d) 4 π cm³
Solution:

Question 21.
The volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is
(a) 3 : 4
(b) 4 : 3
(c) 9 : 16
(d) 16 : 9
Solution:

Question 22.
If a cone, a hemisphere and a cylinder have equal bases and have same height, then the ratio of their volumes is
(a) 1 : 3 : 2
(b) 2 : 3 : 1
(c) 2 : 1 : 3
(d) 1 : 2 : 3
Solution:

Question 23.
If a sphere and a cube have equal surface areas, then the ratio of the diameter of the sphere to the edge of the cube is
(a) 1 : 2
(b) 2 : 1
(c) √π : √6
(d) √6 : √π
Solution:

Question 24.
A solid piece of iron in the form of a cuboid of dimensions 49 cm x 33 cm x 24 cm is moulded to form a sphere. The radius of the sphere is
(a) 21 cm
(b) 23 cm
(c) 25 cm
(d) 19 cm
Solution:

Question 25.
If a solid right circular cone of height 24 cm and base radius 6 cm is melted and recast in the shape of a sphere, then the radius of the sphere is
(a) 4 cm
(b) 6 cm
(c) 8 cm
(d) 12 cm
Solution:

Question 26.
If a solid circular cylinder of iron whose diameter is 15 cm and height 10 cm is melted and recasted into a sphere, then the radius of the sphere is
(a) 15 cm
(b) 10 cm
(c) 7.5 cm
(d) 5 cm
Solution:

Question 27.
The number of balls of radius 1 cm that can be made from a sphere of radius 10 cm is
(a) 100
(b) 1000
(c) 10000
(d) 100000
Solution:

Question 28.
A metallic spherical shell of internal and external diameters 4 cm and 8 cm, respectively is melted and recast into the form of a cone of base diameter 8 cm. The height of the cone is
(a) 12 cm
(b) 14 cm
(c) 15 cm
(d) 18 cm
Solution:

Question 29.
A cubical icecream brick of edge 22 cm is to be distributed among some children by filling ice cream cones of radius 2 cm and height 7 cm up to its brim. The number of children who will get the ice cream cones is
(a) 163
(b) 263
(c) 363
(d) 463
Solution:

Question 30.
Twelve solid spheres of the same size are made by melting a solid metallic cylinder of base diameter 2 cm and height 16 cm. The diameter of each sphere is
(a) 4 cm
(b) 3 cm
(b) 2 cm
(d) 6 cm
Solution:

Question 31.
A hollow cube of internal edge 22 cm is filled with spherical marbles of diameter 0.5 cm and it is assumed that \(\\ \frac { 1 }{ 8 } \) space of the cube remains unfilled. Then the number of marbles that the cube can accommodate is
(a) 142296
(b) 142396
(c) 142496
(d) 142596
Solution:

Question 32.
In the given figure, the bottom of the glass has a hemispherical raised portion. If the glass is filled with orange juice, the quantity of juice which a person will get is
(a) 135 π cm³
(b) 117 π cm³
(c) 99 π cm³
(d) 36 π cm³

Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Chapter Test

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Chapter Test

Question 1.
Find the equation of a line whose inclination is 60° and the y-intercept is -4.
Solution:

Question 2.
Write down the gradient and the intercept on the y-axis of the line 3y + 2x = 12.
Solution:

Question 3.
If the equation of a line is y – √3x + 1, find its inclination.
Solution:

Question 4.
If the line y = mx + c passes through the points (2, -4) and (-3, 1), determine the values of m and c.
Solution:

Question 5.
If the point (1, 4), (3, -2) and (p, – 5) lie on a straight line, find the value of p.
Solution:

Question 6.
Find the inclination of the line joining the points P (4, 0) and Q (7, 3).
Solution:

Question 7.
Find the equation of the line passing through the point of intersection of the lines 2x + y = 5 and x – 2y = 5 and having y-intercept equal to \(– \frac { 3 }{ 7 } \)
Solution:

Question 8.
If point A is reflected in the y-axis, the co-ordinates of its image A₁, are (4, -3),
(i) Find the co-ordinates of A
(ii) Find the co-ordinates of A₂, A₃ the images of the points A, A₁, Respectively under reflection in the line x = -2
Solution:

Question 9.
If the lines \(\frac { x }{ 3 } +\frac { y }{ 4 } =7 \) and 3x + ky = 11 are perpendicular to each other, find the value of k.
Solution:

Question 10.
Write down the equation of a line parallel to x – 2y + 8 = 0 and passing through the point (1, 2).
Solution:

Question 11.
Write down the equation of the line passing through (-3, 2) and perpendicular to the line 3y = 5 – x.
Solution:

Question 12.
Find the equation of the line perpendicular to the line joining the points A (1, 2) and B (6, 7) and passing through the point which divides the line segment AB in the ratio 3 : 2.
Solution:

Question 13.
The points A (7, 3) and C (0, -4) are two opposite vertices of a rhombus ABCD. Find the equation of the diagonal BD.
Solution:

Question 14.
A straight line passes through P (2, 1) and cuts the axes in points A, B. If BP : PA = 3 : 1, find:

(i) the co-ordinates of A and B
(ii) the equation of the line AB
Solution:

Question 15.
A straight line makes on the co-ordinates axes positive intercepts whose sum is 7. If the line passes through the point (-3, 8), find its equation.
Solution:

Question 16.
If the coordinates of the vertex A of a square ABCD are (3, -2) and the equestion of diagonal BD is 3x – 7y + 6 = 0, find the equation of the diagonal AC. Also, find the co-ordinates of the centre of the square.
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line MCQS

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line MCQS

Choose the correct answer from the given four options (1 to 13):

Question 1.
The slope of a line parallel to y-axis is
(a) 0
(b) 1
(c) -1
(d) not defined
Solution:

Question 2.
The slope of a line which makes an angle of 30° with the positive direction of x-axis is
(a) 1
(b) \(\frac { 1 }{ \sqrt { 3 } } \)
(c) √3
(d) \(– \frac { 1 }{ \sqrt { 3 } } \)
Solution:

Question 3.
The slope of the line passing through the points (0, -4) and (-6, 2) is
(a) 0
(b) 1
(c) -1
(d) 6
Solution:

Question 4.
The slope of the line passing through the points (3, -2) and (-7, -2) is
(a) 0
(b) 1
(c) \(– \frac { 1 }{ 10 } \)
(d) not defined
Solution:

Question 5.
The slope of the line passing through the points (3, -2) and (3, -4) is
(a) -2
(b) 0
(c) 1
(d) not defined
Solution:

Question 6.
The inclination of the line y = √3x – 5 is
(a) 30°
(b) 60°
(c) 45°
(d) 0°
Solution:

Question 7.
If the slope of the line passing through the points (2, 5) and (k, 3) is 2, then the value of k is
(a) -2
(b) -1
(c) 1
(d) 2
Solution:

Question 8.
The slope of a line parallel to the line passing through the points (0, 6) and (7, 3) is
(a) \(– \frac { 1 }{ 5 } \)
(b) \(\\ \frac { 1 }{ 5 } \)
(c) -5
(d) 5
Solution:

Question 9.
The slope of a line perpendicular to the line passing through the points (2, 5) and (-3, 6) is
(a) \(– \frac { 1 }{ 5 } \)
(b) \(\\ \frac { 1 }{ 5 } \)
(c) -5
(d) 5
Solution:

Question 10.
The slope of a line parallel to the line 2x + 3y – 7 = 0 is
(a) \(– \frac { 2 }{ 3 } \)
(b) \(\\ \frac { 2 }{ 3 } \)
(c) \(– \frac { 3 }{ 2 } \)
(d) \(\\ \frac { 3 }{ 2 } \)
Solution:

Question 11.
The slope of a line perpendicular to the line 3x = 4y + 11 is
(a) \(\\ \frac { 3 }{ 4 } \)
(b) \(– \frac { 3 }{ 4 } \)
(c) \(\\ \frac { 4 }{ 3 } \)
(d) \(– \frac { 4 }{ 3 } \)
Solution:

Question 12.
If the lines 2x + 3y = 5 and kx – 6y = 7 are parallel, then the value of k is
(a) 4
(b) -4
(c) \(\\ \frac { 1 }{ 4 } \)
(d) \(– \frac { 1 }{ 4 } \)
Solution:

Question 13.
If the line 3x – 4y + 7 = 0 and 2x + ky + 5 = 0 are perpendicular to each other, then the value of k is
(a) \(\\ \frac { 3 }{ 2 } \)
(b) \(– \frac { 3 }{ 2 } \)
(c) \(\\ \frac { 2 }{ 3 } \)
(d) \(– \frac { 2 }{ 3 } \)
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 12 Equation of a Straight Line Ex 12.2

Question 1.
State which one of the following is true:
The straight lines y = 3x – 5 and 2y = 4x + 7 are
(i) parallel
(ii) perpendicular
(iii) neither parallel nor perpendicular.
Solution:

Question 2.
If 6x + 5y – 7 = 0 and 2px + 5y + 1 = 0 are parallel lines, find the value of p.
Solution:

Question 3.
Lines 2x – by + 5 = 0 and ax + 3y = 2 are parallel. Find the relation connecting a and b. (1991)
Solution:

Question 4.
Given that the line \(\\ \frac { y }{ 2 } \) = x – p and the line ax + 5 = 3y are parallel, find the value of a. (1992)
Solution:

Question 5.
If the lines y = 3x + 7 and 2y + px = 3 perpendicular to each other, find the value of p. (2006)
Solution:

Question 6.
Find the value of k for which the lines kx – 5y + 4 = 0 and 4x – 2y + 5 = 0 are perpendicular to each other. (2003)
Solution:

Question 7.
If the lines 3x + by + 5 = 0 and ax – 5y + 7 = 0 are perpendicular to each other, find the relation connecting a and b.
Solution:

Question 8.
Is the line through (-2, 3) and (4, 1) perpendicular to the line 3x = y + 1 ?
Does the line 3x = y + 1 bisect the join of (-2, 3) and (4, 1). (1993)
Solution:

Question 9.
The line through A (-2, 3) and B (4, b) is perpendicular to the line 2x – 4y = 5. Find the value of b.
Solution:

Question 10.
If the lines 3x + y = 4, x – ay + 7 = 0 and bx + 2y + 5 = 0 form three consecutive sides of a rectangle, find the value of a and b.
Solution:

Question 11.
Find the equation of a line, which has the y-intercept 4, and is parallel to the line 2x – 3y – 7 = 0. Find the coordinates of the point where it cuts the x-axis. (1998)
Solution:

Question 12.
Find the equation of a straight line perpendicular to the line 2x + 5y + 7 = 0 and with y-intercept -3 units.
Solution:

Question 13.
Find the equation of a st. line perpendicular to the line 3x – 4y + 12 = 0 and having same y-intercept as 2x – y + 5 = 0.
Solution:

Question 14.
Find the equation of the line which is parallel to 3x – 2y = -4 and passes through the point (0, 3). (1990)
Solution:

Question 15.
Find the equation of the line passing through (0, 4) and parallel to the line 3x + 5y + 15 = 0. (1999)
Solution:

Question 16.
The equation of a line is y = 3x – 5. Write down the slope of this line and the intercept made by it on the y-axis. Hence or otherwise, write down the equation of a line which is parallel to the line and which passes through the point (0, 5).
Solution:

Question 17.
Write down the equation of the line perpendicular to 3x + 8y = 12 and passing through the point (-1, -2).
Solution:

Question 18.
(i) The line 4x – 3y + 12 = 0 meets the x-axis at A. Write down the co-ordinates of A.
(ii) Determine the equation of the line passing through A and perpendicular to 4x – 3y + 12 = 0. (1993)
Solution:

Question 19.
Find the equation of the line that is parallel to 2x + 5y – 7 = 0 and passes through the mid-point of the line segment joining the points (2, 7) and (-4, 1).
Solution:

Question 20.
Find the equation of the line that is perpendicular to 3x + 2y – 8 = 0 and passes through the mid-point of the line segment joining the points (5, -2), (2, 2).
Solution:

Question 21.
Find the equation of a straight line passing through the intersection of 2x + 5y – 4 = 0 with x-axis and parallel to the line 3x – 7y + 8 = 0.
Solution:

Question 22.
The equation of a line is 3x + 4y – 7 = 0. Find
(i) the slope of the line. .
(ii) the equation of a line perpendicular to the given line and passing through the intersection of the lines x – y + 2 = 0 and 3x + y – 10 = 0. (2010)
Solution:

Question 23.
Find the equation of the line perpendicular from the point (1, -2) on the line 4x – 3y – 5 = 0. Also, find the co-ordinates of the foot of the perpendicular.
Solution:

Question 24.
Prove that the line through (0, 0) and (2, 3) is parallel to the line through (2, -2) and (6, 4).
Solution:

Question 25.
Prove that the line through,(-2, 6) and (4, 8) is perpendicular to the line through (8, 12) and (4, 24).
Solution:

Question 26.
Show that the triangle formed by the points A (1, 3), B (3, -1) and C (-5, -5) is a right-angled triangle by using slopes.
Solution:

Question 27.
Find the equation of the line through the point (-1, 3) and parallel to the line joining the points (0, -2) and (4, 5).
Solution:

Question 28.
A (-1, 3), B (4, 2), C (3, -2) are the vertices of a triangle.
(i) Find the coordinates of the centroid G of the triangle.
(ii) Find the equation of the line through G and parallel to AC
Solution:

Question 29.
The line through P (5, 3) intersects y-axis at Q.
(i) Write the slope of the line.
(ii) Write the equation of the line.
(iii) Find the coordinates of Q.

Solution:

Question 30.
In the adjoining diagram, write down
(i) the co-ordinates of the points A, B and C.
(ii) the equation of the line through A parallel to BC. (2005)

Solution:

Question 31.
Find the equation of the line through (0, -3) and perpendicular to the line joining the points (-3, 2) and (9, 1).
Solution:

Question 32.
The vertices of a triangle are A (10, 4), B (4, -9) and C (-2, -1). Find the equation of the altitude through A. The perpendicular drawn from a vertex of a triangle to the opposite side is called altitude.
Solution:

Question 33.
A (2, -4), B (3, 3) and C (-1, 5) are the vertices of triangle ABC. Find the equation of:
(i) the median of the triangle through A
(ii) the altitude of the triangle through B
Solution:

Question 34.
Find the equation of the right bisector of the line segment joining the points (1, 2) and (5, -6).
Solution:

Question 35.
Points A and B have coordinates (7, -3) and (1, 9) respectively. Find
(i) the slope of AB.
(ii) the equation of the perpendicular bisector of the line segment AB.
(iii) the value of ‘p’ if (-2, p) lies on it.
Solution:

Question 36.
The points B (1, 3) and D (6, 8) are two opposite vertices of a square ABCD. Find the equation of the diagonal AC.
Solution:

Question 37.
ABCD is a rhombus. The co-ordinates of A and C are (3, 6) and (-1, 2) respectively. Write down the equation of BD. (2000)
Solution:

Question 38.
Find the equation of the line passing through the intersection of the lines 4x + 3y = 1 and 5x + 4y = 2 and
(i) parallel to the line x + 2y – 5 = 0
(ii) perpendicular to the x-axis.
Solution:

Question 39.
(i) Write down the co-ordinates of the point P that divides the line joining A (-4, 1) and B (17, 10) in the ratio 1 : 2.
(ii) Calculate the distance OP where 0 is the origin
(iii) In what ratio does the y-axis divide the line AB?
Solution:

Question 40.
Find the image of the point (1, 2) in the line x – 2y – 7 = 0
Solution:

Question 41.
If the line x – 4y – 6 = 0 is the perpendicular bisector of the line segment PQ and the co-ordinates of P are (1, 3), find the co-ordinates of Q.
Solution:

Question 42.
OABC is a square, O is the origin and the points A and B are (3, 0) and (p, q). If OABC lies in the first quadrant, find the values of p and q. Also, write down the equations of AB and BC.
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.5

Question 1.
The diameter of a metallic sphere is 6 cm. The sphere is melted and drawn into a wire of uniform cross-section. If the length of the wire is 36 m, find its radius.
Solution:

Question 2.
The radius of a sphere is 9 cm. It is melted and drawn into a wire of diameter 2 mm. Find the length of the wire in metres.
Solution:

Question 3.
A solid metallic hemisphere of radius 8 cm is melted and recast into a right circular cone of base radius 6 cm. Determine the height of the cone.
Solution:

Question 4.
A rectangular water tank of base 11 m x 6 m contains water up to a height of 5 m. if the water in the tank is transferred to a cylindrical tank of radius 3.5 m, find the height of the water level in the tank.
Solution:

Question 5.
The rainwater from a roof of dimensions 22 m x 20 m drains into a cylindrical vessel having a diameter of base 2 m and height; 3.5 m. If the rainwater collected from the roof just fill the cylindrical vessel, then find the rainfall in cm.
Solution:

Question 6.
The volume of a cone is the same as that of the cylinder whose height is 9 cm and diameter 40 cm. Find the radius of the base of the cone if its height is 108 cm.
Solution:

Question 7.
Eight metallic spheres, each of radius 2 cm, are melted and cast into a single sphere. Calculate the radius of the new (single) sphere.
Solution:

Question 8.
A metallic disc, in the shape of a right circular cylinder, is of height 2.5 mm and base radius 12 cm. Metallic disc is melted and made into a sphere. Calculate the radius of the sphere.
Solution:

Question 9.
Two spheres of the same metal weigh 1 kg and 7 kg. The radius of the smaller sphere is 3 cm. The two spheres are melted to form a single big sphere. Find the diameter of the big sphere.
Solution:

Question 10.
A hollow copper pipe of inner diameter 6 cm and outer diameter 10 cm is melted and changed into a solid circular cylinder of the same height as that of the pipe. Find the diameter of the solid cylinder.
Solution:

Question 11.
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 4 cm and height is 72 cm, find the uniform thickness of the cylinder.
Solution:

Question 12.
A hollow metallic cylindrical tube has an internal radius of 3 cm and height 21 cm. The thickness of the metal of the tube is \(\\ \frac { 1 }{ 2 } \) cm. The tube is melted and cast into a right circular cone of height 7 cm. Find the radius of the cone correct to one decimal place.
Solution:

Question 13.
A hollow sphere of internal and external diameters 4 cm and 8 cm respectively, is melted into a cone of base diameter 8 cm. Find the height of the cone. (2002)
Solution:

Question 14.
A well with inner diameter 6 m is dug 22 m deep. Soil taken out of it has been spread evenly all round it to a width of 5 m to form an embankment. Find the height of the embankment.
Solution:

Question 15.
A cylindrical can of internal diameter 21 cm contains water. A solid sphere whose diameter is 10.5 cm is lowered into the cylindrical can. The sphere is completely immersed in water. Calculate the rise in water level, assuming that no water overflows.
Solution:

Question 16.
There is water to a height of 14 cm in a cylindrical glass jar of radius 8 cm. Inside the water, there is a sphere of diameter 12 cm completely immersed. By what height will the water go down when the sphere is removed?
Solution:

Question 17.
A vessel in the form of an inverted cone is filled with water to the brim. Its height is 20 cm and diameter is 16.8 cm. Two equal solid cones are dropped in it so that they are fully submerged. As a result, one-third of the water in the original cone overflows. What is the volume of each of the solid cone submerged? (2002)
Solution:

Question 18.
A solid metallic circular cylinder of radius 14 cm and height 12 cm is melted and recast into small cubes of edge 2 cm. How many such cubes can be made from the solid cylinder?
Solution:

Question 19.
How many shots each having diameter 3 cm can be made from a cuboidal lead solid of dimensions 9 cm × 11 cm × 12 cm?
Solution:

Question 20.
How many spherical lead shots of diameter 4 cm can be made out of a solid cube of lead whose edge measures 44 cm?
Solution:

Question 21.
Find the number of metallic circular discs with 1.5 cm base diameter and height 0.2 cm to be melted to form a circular cylinder of height 10 cm and diameter 4.5 cm.
Solution:

Question 22.
A solid metal cylinder of radius 14 cm and height 21 cm is melted down and recast into spheres of radius 3.5 cm. Calculate the number of spheres that can be made.
Solution:

Question 23.
A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. Find the number of cones thus obtained. (2005)
Solution:

Question 24.
A certain number of metallic cones each of radius 2 cm and height 3 cm are melted and recast in a solid sphere of radius 6 cm. Find the number of cones. (2016)
Solution:

Question 25.
A vessel is in the form of an inverted cone. Its height is 11 cm and the radius of its top, which is open, is 2.5 cm. It is filled with water up to the rim. When some lead shots, each of which is a sphere of radius 0.25 cm, are dropped into the vessel, \(\\ \frac { 2 }{ 5 } \) of the water flows out. Find the number of lead shots dropped into the vessel. (2003)
Solution:

Question 26.
The surface area of a solid metallic sphere is 616 cm². It is melted and recast into smaller spheres of diameter 3.5 cm. How many such spheres can be obtained? (2007)
Solution:

Question 27.
The surface area of a solid metallic sphere is 1256 cm². It is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate
(i) the radius of the solid sphere.
(ii) the number of cones recasts. (Use π = 3.14).
Solution:

Question 28.
Water is flowing at the rate of 15 km/h through a pipe of diameter 14 cm into a cuboid pond which is 50 m long and 44 m wide. In what time will the level of water in the pond rise by 21 cm?
Solution:

Question 29.
A cylindrical can whose base is horizontal and of radius 3.5 cm contains sufficient water so that when a sphere is placed in the can, the water just covers the sphere. Given that the sphere just fits into the can, calculate:
(i) the total surface area of the can in contact with water when the sphere is in it.
(ii) the depth of the water in the can before the sphere was put into the can. Given your answer as proper fractions.
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 17 Mensuration Ex 17.3

Question 1.
Find the surface area of a sphere of radius:
(i) 14 cm
(ii) 10.5 cm
Solution:

Question 2.
Find the volume of a sphere of radius:
(i) 0.63 m
(ii) 11.2 cm
Solution:

Question 3.
Find the surface area of a sphere of diameter:
(i) 21 cm
(ii) 3.5 cm
Solution:

Question 4.
A shot-put is a metallic sphere of radius 4.9 cm. If the density of the metal is 7.8 g per cm³, find the mass of the shot-put.
Solution:

Question 5.
Find the diameter of a sphere whose surface area is 154 cm².
Solution:

Question 6.
Find:
(i) the curved surface area.
(ii) the total surface area of a hemisphere of radius 21 cm.
Solution:

Question 7.
A hemispherical brass bowl has inner- diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm².
Solution:

Question 8.
The radius of a spherical balloon increases from 7 cm to 14 cm as air is jumped into it. Find the ratio of the surface areas of the balloon in two cases.
Solution:

Question 9.
A sphere and a cube have the same surface. Show that the ratio of the volume of the sphere to that of the cube is √6 : √π
Solution:

Question 10.
(a) If the ratio of the radii of two sphere is 3 : 7, find:
(i) the ratio of their volumes.
(ii) the ratio of their surface areas.
(b) If the ratio of the volumes of the two-sphere is 125 : 64, find the ratio of their surface areas.
Solution:

Question 11.
A cube of side 4 cm contains a sphere touching its sides. Find the volume of the gap in between.
Solution:

Question 12.
Find the volume of a sphere whose surface area is 154 cm².
Solution:

Question 13.
If the volume of a sphere is \(179 \frac { 2 }{ 3 } \) cm³, find its radius and the surface area.
Solution:

Question 14.
A hemispherical bowl has a radius of 3.5 cm. What would be the volume of water it would contain?
Solution:

Question 15.
The water for a factory is stored in a hemispherical tank whose internal diameter is 14 m. The tank contains 50 kilolitres of water. Water is pumped into the tank to fill to its capacity. Find the volume of water pumped into the tank.
Solution:

Question 16.
The surface area of a solid sphere is 1256 cm². It is cut into two hemispheres. Find the total surface area and the volume of a hemisphere. Take π = 3.14.
Solution:

Question 17.
Write whether the following statements are true or false. Justify your answer:
(i) The volume of a sphere is equal to two-third of the volume of a cylinder whose height and diameter are equal to the diameter of the sphere.
(ii) The volume of the largest right circular cone that can be fitted in a cube whose edge is 2r equals the volume of a hemisphere of radius r.
(iii) A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is 1 : 2 : 3.
Solution:

ML Aggarwal Class 10 Solutions for ICSE Maths