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AB is diameter and AC is a chord of a circle with centre O such that angle BAC=30º. The tangent to the circle at C intersects AB produced in D. Show that BC = BD.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Join \[OC\] \[\angle BCD\text{ }=\angle BAC\text{ }=\] \[{{30}^{o}}~\] [Angles in the alternate segment] It’s seen that, arc \[BC\]subtends \[\angle DOC\]at the center of the circle And \[\angle...

If PQ is a tangent to the circle at R; calculate: i) ∠PRS ii) ∠ROT When O is the centre of the circle and ∠TRQ = 30degree

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: (i) As \[PQ\] is the tangent and \[OR\]is the radius. So, \[OR\bot PQ\] \[\angle ORT\text{ }=\text{ }{{90}^{o}}\] \[\angle TRQ\text{ }=\text{ }{{90}^{o}}-\text{ }{{30}^{o}}~=\text{...

In the following figure, PQ is the tangent to the circle at A, DB is a diameter and O is the centre of the circle. If ∠ADB = 30degree and ∠CBD = 60degree; calculate: ∠CDB

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: As \[BD\] is the diameter, we have \[\angle BCD\text{ }=\text{ }{{90}^{o}}\] [Angle in a semi-circle] Now in \[\vartriangle BCD\] \[\angle CDB\text{ }+\angle CBD\text{ }+\angle BCD\text{...

In the following figure, PQ is the tangent to the circle at A, DB is a diameter and O is the centre of the circle. If ∠ADB = 30degree and ∠CBD = 60degree; calculate: i) ∠QAB ii) ∠PAD

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: (i) Given, \[PAQ\]is a tangent and \[AB\]is the chord \[\angle QAB~=\angle ADB\text{ }=\text{ }{{30}^{o}}~\] [Angles in the alternate segment] (ii) \[OA\text{ }=\text{ }OD\][radii of the...

In the given figure, 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.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: (i) \[PA\text{ }=\text{ }AB\text{ }+\text{ }BP\text{ }=\text{ }\left( AB\text{ }+\text{ }4 \right)\text{ }cm\] \[PC\text{ }=\text{ }PD\text{ }+\text{ }CD\] \[=\text{ }5\text{ }+\text{...

In the given figure, tangent PT = 12.5 cm and PA = 10 cm; find AB.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

Solution: As \[PAB\]is the secant and \[PT\]is the tangent, we have \[P{{T}^{2}}~=\text{ }PA\text{ }x\text{ }PB\] \[{{12.5}^{2}}~=\text{ }10\text{ }x\text{ }PB\] \[PB\text{ }=\text{ }\left(...

(i) In the given figure, 3 x CP = PD = 9 cm and AP = 4.5 cm. Find BP. (ii) In the given figure, 5 x PA = 3 x AB = 30 cm and PC = 4cm. Find CD.

Class 10, Exercise 18B, ICSE, Maths, Selina, Tangents and Intersecting Chords

(i) (ii) Solution: (i) As the two chords \[AB\text{ }and\text{ }CD\] intersect each other at \[P\] We have \[AP\text{ }x\text{ }PB\text{ }=\text{ }CP\text{ }x\text{ }PD\] \[4.5\text{ }x\text{...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

The ratio between the altitudes of two similar triangles is same as the ratio between their sides. So, The ratio between the areas of two similar triangles is same as the square of the ratio between...

The ratio between the altitudes of two similar triangles is 3: 5; write the ratio between their: (i) medians. (ii) perimeters.

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

The ratio between the altitudes of two similar triangles is same as the ratio between their sides. So, (i) The ratio between the medians of two similar triangles is same as the ratio between their...

Two similar triangles are equal in area. Prove that the triangles are congruent.

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Let’s consider two similar triangles as \[\blacktriangle ABC\text{ }\sim\text{ }\blacktriangle PQR\] So, \[Ar\left( \blacktriangle ABC \right)/\text{ }Ar\left( \blacktriangle PQR \right)\] \[=\text{...

In the following figure, AD and CE are medians of ∆ABC. DF is drawn parallel to CE. Prove that: (i) EF = FB, (ii) AG: GD = 2: 1

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) In \[\vartriangle BFD\text{ }and\text{ }\vartriangle BEC,\] \[\angle BFD\text{ }=\angle BEC\] [Corresponding angles] \[\angle FBD\text{ }=\angle EBC\] [Common] Hence, \[\vartriangle...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: In \[\vartriangle ABC,\text{ }as\text{ }PR\text{ }||\text{ }BC\text{ }by\text{ }BPT\]we have \[AP/PB\text{ }=\text{ }AR/RC\] And, in \[\vartriangle PAR\text{ }and\text{ }\vartriangle...

In ΔABC, ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm. Find the area of ΔACD: area of ΔABC

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: As, \[\vartriangle ACD\text{ }\sim\text{ }\vartriangle BCA\] We have, \[Ar\left( \vartriangle ACD \right)/\text{ }Ar\left( \vartriangle BCA \right)\] \[=\text{ }A{{D}^{2}}/\text{...

In ΔABC, ∠ABC = ∠DAC, AB = 8 cm, AC = 4 cm and AD = 5 cm. (i) Prove that ΔACD is similar to ΔBCA. (ii) Find BC and CD

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) In \[\vartriangle ACD\text{ }and\text{ }\vartriangle BCA\] \[\angle DAC\text{ }=\angle ABC\] [Given] \[\angle ACD\text{ }=\angle BCA\][Common angles] Hence, \[\vartriangle ACD\text{...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) In \[\vartriangle PCD\text{ }and\text{ }\vartriangle PEF\] \[\angle CPD\text{ }=\angle EPF\][Vertically opposite angles] \[\angle DCE\text{ }=\angle FEP\][As DC || EF, alternate...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: Given, \[DE\text{ }||\text{ }AC\] So, \[BE/EC\text{ }=\text{ }BD/DA\text{ }\left[ By\text{ }BPT \right]\] And, \[DC\text{ }||\text{ }AP\] So, \[BC/CP\text{ }=\text{ }BD/DA\text{ }\left[...

In the following diagram, lines l, m and n are parallel to each other. Two transversals p and q intersect the parallel lines at points A, B, C and P, Q, R as shown. Prove that: AB/BC = PQ/QR

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: Let join \[AR\]such that it intersects \[BQ\]at point \[X.\] In \[\vartriangle ACR,\text{ }BX\text{ }||\text{ }CR\] By\[BPT\], we have \[AB/BC\text{ }=\text{ }AX/XR\text{ }\ldots \text{...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[\angle AXY\text{ }=~\angle AYX\] So, \[AX\text{ }=\text{ }AY\][Sides opposite to equal angles are equal.] Also, from BPT we have \[BX/AX\text{ }=\text{...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[\vartriangle ABC\text{ }\sim\text{ }\vartriangle PQR\] And, \[AD\text{ }and\text{ }PM\]are the angle bisectors. So, \[\angle BAD\text{ }=\angle QPM\]...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[\vartriangle ABC\text{ }\sim\text{ }\vartriangle PQR\] So, \[\angle ABC\text{ }=\angle PQR\] i.e. \[\angle ABD\text{ }=\angle PQM\] Also, \[\angle...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[\vartriangle ABC\text{ }\sim\text{ }\vartriangle PQR\] \[AD\text{ }and\text{ }PM\]are the medians, so \[BD\text{ }=\text{ }DC\text{ }and\text{ }QM\text{...

In the following figure, AB, CD and EF are perpendicular to the straight line BDF. If AB = x and; CD = z unit and EF = y unit, prove that: 1/x + 1/y = 1/z

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: In \[\Delta \text{ }FDC\text{ }and\text{ }\Delta \text{ }FBA,\] \[\angle FDC\text{ }=\angle FBA\text{ }\left[ As\text{ }DC\text{ }||\text{ }AB \right]\] \[\angle DFC\text{ }=\angle BFA\]...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: We already have, \[\vartriangle AEB\text{ }\sim\text{ }\vartriangle FEC\] So, \[AE/FE\text{ }=\text{ }BE/CE\] \[=\text{ }AB/FC\] \[AE/FE\text{ }=\text{ }9/13.5\] Or, \[\left( AF\text{...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) In \[\Delta \text{ }AEB\text{ }and\text{ }\Delta \text{ }FEC,\] \[\angle AEB\text{ }=\angle FEC\] [Vertically opposite angles] \[\angle BAE\text{ }=\angle CFE\] [Since, AB||DC] Hence,...

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

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[XY\text{ }||\text{ }BC\] So, In \[\Delta \text{ }AXY\text{ }and\text{ }\Delta \text{ }ABC\] \[\angle AXY\text{ }=\angle ABC\] [Corresponding angles]...

In the following figure, XY is parallel to BC, AX = 9 cm, XB = 4.5 cm and BC = 18 cm. Find: (i) AY/YC (ii) YC/AC

Class 10, Exercise 15E, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[XY\text{ }||\text{ }BC\] So, In \[\Delta \text{ }AXY\text{ }and\text{ }\Delta \text{ }ABC\] \[\angle AXY\text{ }=\angle ABC\][Corresponding angles]...

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find: (i) OA, if OA’ = 6 cm (ii) OC’, if OC = 21 cm Also, state the value of: (a) OB’/OB (b) C’A’/CA

Class 10, Exercise 15D, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

(i)\[OA\text{ }=\text{ }6\text{ }cm\] So, \[OA\text{ }\left( 3 \right)\text{ }=\text{ }OA\] \[OA\text{ }\left( 3 \right)\text{ }=\text{ }6\] Or, \[OA\text{ }=\text{ }2\text{ }cm\] (ii) \[OC\text{...

A triangle ABC is enlarged, about the point 0 as centre of enlargement, and the scale factor is 3. Find: (i) A’B’, if AB = 4 cm. (ii) BC, if B’C’ = 15 cm.

Class 10, Exercise 15D, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

According to the given question, \[\Delta \text{ }ABC\]is enlarged and the scale factor \[m\text{ }=\text{ }3\]to the \[\Delta \text{ }ABC\] (i) \[AB\text{ }=\text{ }4\text{ }cm\] So, \[AB\left( 3...

A triangle LMN has been reduced by scale factor 0.8 to the triangle L’ M’ N’. Calculate: (i) the length of M’ N’, if MN = 8 cm. (ii) the length of LM, if L’ M’ = 5.4 cm.

Class 10, Exercise 15D, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

According to the given question, \[\Delta \text{ }LMN\] has been reduced by a scale factor \[m\text{ }=\text{ }0.8\text{ }to\text{ }\Delta \text{ }LMN\] (i) \[MN\text{ }=\text{ }8\text{ }cm\] So,...

A triangle ABC has been enlarged by scale factor m = 2.5 to the triangle A’ B’ C’ Calculate: (i) the length of AB, if A’ B’ = 6 cm. (ii) the length of C’ A’ if CA = 4 cm.

Class 10, Exercise 15D, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Given that, \[\Delta \text{ }ABC\]has been enlarged by scale factor \[m\text{ }of\text{ }2.5\text{ }to\text{ }\Delta \text{ }ABC\] (i) \[AB\text{ }=\text{ }6\text{ }cm\] So, \[AB\left( 2.5...

In the given triangle PQR, LM is parallel to QR and PM: MR = 3: 4. Calculate the value of ratio: Area of Δ LQM/ Area of Δ LQN

Class 10, Exercise 15C, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: Because, \[\Delta \text{ }LQM\text{ }and\text{ }\Delta \text{ }LQN\] have common vertex at \[L\]and their bases \[QM\text{ }and\text{ }QN\] are along the same straight line. \[Area\text{...

In the given triangle PQR, LM is parallel to QR and PM: MR = 3: 4. Calculate the value of ratio: (i) PL/PQ and then LM/QR (ii) Area of Δ LMN/ Area of Δ MNR

Class 10, Exercise 15C, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) In \[\Delta \text{ }PLM\text{ }and\text{ }\Delta \text{ }PQR\] As LM || QR, corresponding angles are equal. \[\angle PLM\text{ }=\angle PQR\] \[\angle PML\text{ }=\angle PRQ\] So,...

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

Class 10, Exercise 15C, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

It’s given that, \[Ar\left( \Delta \text{ }APQ \right)\]\[=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\text{ }Ar\left( \Delta \text{ }ABC \right)\] \[Ar\left( \Delta \text{ }APQ \right)/\text{...

In the given figure, AX: XB = 3: 5. Find: (i) the length of BC, if the length of XY is 18 cm. (ii) the ratio between the areas of trapezium XBCY and triangle ABC.

Class 10, Exercise 15C, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[AX/XB\text{ }=\text{ }3/5\] \[\Rightarrow AX/AB\text{ }=\text{ }3/8\text{ }\ldots .\text{ }\left( 1 \right)\] (i) In \[\Delta \text{ }AXY\text{...

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

Class 10, Exercise 15C, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Suppose, \[\vartriangle ABC\text{ }\sim\text{ }\vartriangle DEF\] So, \[AB/DE\text{ }=\text{ }BC/EF\] \[=\text{ }AC/DF\] Or, \[=\text{ }\left( AB+BC+AC \right)/\left( DE+EF+DF \right)\] \[=\text{...

A line PQ is drawn parallel to the base BC of Δ ABC which meets sides AB and AC at points P and Q respectively. If AP = 1/3 PB; find the value of: (i) Area of Δ ABC/ Area of Δ APQ (ii) Area of Δ APQ/ Area of Trapezium PBCQ

Class 10, Exercise 15C, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

According to the given question, \[AP\text{ }=\text{ }\left( 1/3 \right)\text{ }PB\] So, \[AP/PB\text{ }=\text{ }1/3\] In \[\vartriangle \text{ }APQ\text{ }and\text{ }\vartriangle ABC\] As\[PQ\text{...

1. (i) The ratio between the corresponding sides of two similar triangles is 2: 5. Find the ratio between the areas of these triangles. (ii) Areas of two similar triangles are 98 sq. cm and 128 sq. cm. Find the ratio between the lengths of their corresponding sides.

Class 10, Exercise 15C, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

As per the given question, The ratio of the areas of two similar triangle are equal to the ratio of squares of their corresponding sides. Thus, (i) The ration is, (ii) The ratio is,

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

Class 10, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: According to the given question, \[\Delta \text{ }ABC\text{ }\sim\text{ }\Delta \text{ }ADE\] So, we have \[AE/AC\text{ }=\text{ }DE/BC\] \[4/11\text{ }=\text{ }6.6/BC\] Or, \[BC=\left(...

In Δ ABC, D and E are the points on sides AB and AC respectively. Find whether DE ‖ BC, if (i) AB = 9cm, AD = 4cm, AE = 6cm and EC = 7.5cm. (ii) AB = 6.3 cm, EC = 11.0 cm, AD =0.8 cm and EA = 1.6 cm.

Class 10, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

(i) In \[\vartriangle \text{ }ADE\text{ }and\text{ }\vartriangle \text{ }ABC\] \[AE/EC\text{ }=\text{ }6/7.5\text{ }=\text{ }4/5\] \[AD/BD\text{ }=\text{ }4/5\] \[\left[ BD\text{ }=\text{ }AB\text{...

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

Class 10, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

In \[\vartriangle \text{ }APQ\text{ }and\vartriangle \text{ }ABC\] \[\angle APQ\text{ }=\angle ABC\] [As PQ || BC, corresponding angles are equal.] \[\angle PAQ\text{ }=\angle BAC\] [Common angle]...

In the given figure, PQ ‖ AB; CQ = 4.8 cm QB = 3.6 cm and AB = 6.3 cm. If AP = x, then the value of AC in terms of x.

Class 10, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: As, \[\vartriangle CPQ\text{ }\sim\text{ }\vartriangle CAB\text{ }by\text{ }AA\] criterion for similarity We have, \[CP/AC\text{ }=\text{ }CQ/CB\] \[CP/AC\text{ }=\text{ }4.8/8.4\text{...

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

Class 10, Exercise 1, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) In \[\vartriangle CPQ\text{ }and\text{ }\vartriangle CAB\] \[\angle PCQ\text{ }=\angle APQ\] [As PQ || AB, corresponding angles are equal.] \[\angle C\text{ }=\angle C\] [Common angle]...

In the following figure, point D divides AB in the ratio 3: 5. If BC = 4.8 cm, find the length of DE.

Class 10, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: Because, \[\vartriangle ADE\text{ }\sim\text{ }\vartriangle ABC\text{ }by\text{ }AA\] criterion for similarity So, we have \[AD/AB\text{ }=\text{ }DE/BC\] \[3/8\text{ }=\text{ }DE/4.8\]...

In the following figure, point D divides AB in the ratio 3: 5. Find: (i) AE/AC Also if, (ii) DE = 2.4 cm, find the length of BC.

Class 10, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) In \[\vartriangle ABC,\text{ }as\text{ }DE\text{ }||\text{ }BC\] Using BPT, \[AD/DB\text{ }=\text{ }AE/\text{ }EC\] So, \[AD/AB\text{ }=\text{ }AE/AC\] Now, \[AD/AB\text{ }=\text{...

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

Class 10, Exercise 15B, ICSE, Maths, Selina, Similarity (With Applications to Maps and Models)

Solution: (i) According to the given question, \[AD/DB\text{ }=\text{ }3/5\] And \[DE\text{ }||\text{ }BC\] Using Basic Proportionality theorem, \[AD/DB\text{ }=\text{ }AE/EC\] \[AE/EC\text{...

Describe: (i) The locus of the mid-points of all chords parallel to a given chord of a circle. (ii) The locus of points within a circle that are equidistant from the end points of a given chord.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

i) The locus of the mid-points of the chords which are parallel to a given chords is the diameter perpendicular to the given chords. ii) The locus of the points within a circle which are equidistant...

Describe: (i)The locus of the centre of a given circle which rolls around the outside of a second circle and is always touching it. (ii) The locus of the centres of all circles that are tangent to both the arms of a given angle.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

i) The locus is the circumference of the circle concentric with the second circle whose radius is equal to the sum of the radii of the given two circles. ii) The locus of the centre of all circles...

Describe: (i) The locus of points at distances less than or equal to 2.5 cm from a given point. (ii) The locus of points at distances greater than or equal to 35 mm from a given point.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

i) The locus of points is the space inside and the circumference of the circle with a radius of \[2.5\text{ }cm\]and the centre as the given fixed point. ii) The locus is the space outside and...

Describe: i) The locus of points at distances less than 3 cm from a given point. ii) The locus of points at distances greater than 4 cm from a given point.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

i) The locus of the points will be the space inside of the circle whose radius is \[3\text{ }cm\]and centre as the given point. ii) The locus of the points will be the space outside of the circle...

The speed of sound is 332 meters per second. A gun is fired. Describe the locus of all the people on the Earth’s surface, who hear the sound exactly one second later.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of all the people on Earth’s surface is the circumference of a circle whose radius is \[332\text{ }m\]and centre is the point where the gun is fired.

Describe the locus for the locus of a point in rhombus ABCD, so that it is equidistant from i) AB and BC; ii) B and D.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

(i) The locus of the point in a rhombus \[ABCD\]which is equidistant from \[AB\text{ }and\text{ }BC\] will be the diagonal \[BD\]of the rhombus. (ii) The locus of the point in a rhombus...

Describe the locus for the locus of a point P, so that: AB^2 = AP^2 + BP^2, where A and B are two fixed points.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of the point \[P\]is the circumference of a circle with \[AB\] as diameter and satisfies the condition \[A{{B}^{2}}~=\text{ }A{{P}^{2~}}+\text{ }B{{P}^{2}}\]

Describe the locus for the locus of a point in space which is always at a distance of 4 cm from a fixed point.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of a point in space is the surface of the sphere whose centre is the fixed point and radius equal to \[4\text{ }cm.\]

Describe the locus for the locus of vertices of all isosceles triangles having a common base.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of vertices of all isosceles triangles having a common base will be the perpendicular bisector of the common base of the triangles.

Describe the locus for the locus of the centres of all circles passing through two fixed points.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of the centres of all the circles passing through two fixed points will be the perpendicular bisector of the line segment joining the two given fixed points.

The locus of a points inside a circle and equidistant from two fixed points on the circumference of the circle.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of the points inside the circle which are equidistant from the fixed points on the circumference of a circle will be a diameter which is the perpendicular bisector of the line joining the...

Describe the locus for The locus of the door-handle, as the door opens.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of the door handle will be the circumference of a circle with centre at the axis of rotation of the door and the radius equal to distance between the door handle and the axis of rotation...

The locus of a runner, running around a circular track and always keeping a distance of 1.5 m from the inner edge.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of the runner, running around a circular track and always keeping a distance of \[1.5\text{ }m\] from the inner edge will be the circumference of a circle where the radius is equal to the...

The locus of a stone dropped from the top of a tower.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

Solution: As per the given question, Locus of stone is dropped from the top of tower will be vertical line through the point from which the stone is dropped.

The locus of the moving end of the minute hand of a clock.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of the moving end of the minute hand of the clock will be a circle whose radius will be the length of the minute hand.

The locus of the centre of a wheel of a bicycle going straight along a level road.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of the centre of wheel, is going straight along the level road will be a straight line parallel to the road at a distance equal to the radius of the wheel.

The locus of a points at a distance of 2 cm from a fixed line.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of points which are at a distance of \[2\text{ }cm\]from a fixed line \[AB\]are a pair of straight lines \[l\text{ }and\text{ }m\]which are parallel to the given line at a distance of...

The locus of a point at a distance of 3 cm from a fixed point.

Class 10, Exercise 16B, ICSE, Loci (Locus and its Constructions), Maths, Selina

The locus of a point is at a distance of \[3\text{ }cm\]from a fixed point is circumference of a circle whose radius is \[3\text{ }cm\]and the fixed point is the centre of the circle.

In the given figure, AB is a side of a regular six-sided polygon and AC is a side of a regular eight-sided polygon inscribed in the circle with centre O. calculate the sizes of: ∠ABC.

Circles, Class 10, Exercise 17B, ICSE, Maths, Selina

Solution: According to the given question, \[AC\] is the side of a regular octagon, \[\angle AOC\text{ }=\text{ }{{360}^{o}}/\text{ }8\text{ }=\text{ }{{45}^{o}}\] Hence, \[arc\text{ }AC\]subtends...

In the given figure, AB is a side of a regular six-sided polygon and AC is a side of a regular eight-sided polygon inscribed in the circle with centre O. calculate the sizes of: (i) ∠AOB, (ii) ∠ACB,

Circles, Class 10, Exercise 17B, ICSE, Maths, Selina

Solution: (i) \[Arc\text{ }AB\] subtends\[\angle AOB\]at the centre and \[\angle ACB\]at the remaining part of the circle. \[\angle ACB\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_2}\angle AOB\]...

The given figure show a circle with centre O. Also, PQ = QR = RS and ∠PTS = 75°. Calculate: ∠PQR.

Circles, Class 10, Exercise 17B, ICSE, Maths, Selina

Join \[OP,\text{ }OQ,\text{ }OR\text{ }and\text{ }OS\] Given, \[PQ\text{ }=\text{ }QR\text{ }=\text{ }RS\] So, \[\angle POQ\text{ }=\angle QOR\text{ }=\angle ROS\] [Equal chords subtends equal...

The given figure show a circle with centre O. Also, PQ = QR = RS and ∠PTS = 75°. Calculate: (i) ∠POS, (ii) ∠QOR,

Circles, Class 10, Exercise 17B, ICSE, Maths, Selina

Solution: Join \[OP,\text{ }OQ,\text{ }OR\text{ }and\text{ }OS\] Given, \[PQ\text{ }=\text{ }QR\text{ }=\text{ }RS\] So, \[\angle POQ\text{ }=\angle QOR\text{ }=\angle ROS\] [Equal chords subtends...

If two sides of a cycli-quadrilateral are parallel; prove that: (i) its other two sides are equal. (ii) its diagonals are equal.

Circles, Class 10, Exercise 17B, ICSE, Maths, Selina

Let ABCD is a cyclic quadrilateral in which\[AB\text{ }||\text{ }DC\] \[AC\text{ }and\text{ }BD\]are its diagonals. Required to prove: \[\left( i \right)\text{ }AD\text{ }=\text{ }BC\] \[\left( ii...

In the following figure, AD is the diameter of the circle with centre O. Chords AB, BC and CD are equal. If ∠DEF = 110o, calculate: (i) ∠AFE, (ii) ∠FAB.

Circles, Class 10, Exercise 17B, ICSE, Maths, Selina

Solution: Join \[AE,\text{ }OB\text{ }and\text{ }OC\] (i) As \[AOD\]is the diameter \[\angle AED\text{ }=\text{ }{{90}^{o}}~\] [Angle in a semi-circle is a right angle] But, given \[\angle DEF\text{...

In a cyclic-trapezium, the non-parallel sides are equal and the diagonals are also equal. Prove it.

Circles, Class 10, Exercise 1, Exercise 17B, ICSE, Maths, Selina

Solution: Let \[ABCD\]be the cyclic trapezium in which \[AB\text{ }||\text{ }DC,\text{ }AC\text{ }and\text{ }BD\]are the diagonals. Required to prove: \[\left( i \right)\text{ }AD\text{ }=\text{...

One card is drawn from a well shuffled deck of \[\mathbf{52}\] cards. Find the probability of getting: \[\left( \mathbf{v} \right)\] a diamond or a spade

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: \[\left( v \right)\] Number of favorable outcomes for a diamond or a spade \[=\text{ }13\text{ }+\text{ }13\text{ }=\text{ }26\] So, number of favorable outcomes \[=\text{ }26\] Hence,...

One card is drawn from a well shuffled deck of \[\mathbf{52}\] cards. Find the probability of getting: \[\left( \mathbf{iii} \right)\] the jack or the queen of the hearts \[\left( \mathbf{iv} \right)\] a diamond

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] Favorable outcomes for jack or queen of hearts \[=\text{ }1\text{ }jack\text{ }+\text{ }1\text{ }queen\] So, the number of favorable outcomes \[=\text{ }2\] Hence,...

One card is drawn from a well shuffled deck of \[\mathbf{52}\] cards. Find the probability of getting: \[\left( \mathbf{i} \right)\] a queen of red color \[\left( \mathbf{ii} \right)\] a black face card

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, Total possible outcomes \[=\text{ }52\] \[\left( i \right)\]Number queens of red color \[=\text{ }2\] Number of favorable outcomes\[~=\text{ }2\] Hence, P(queen of red color)...

A game consists of spinning arrow which comes to rest pointing at one of the numbers \[\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5},\text{ }\mathbf{6},\text{ }\mathbf{7},\text{ }\mathbf{8},\text{ }\mathbf{9},\text{ }\mathbf{10},\text{ }\mathbf{11},\text{ }\mathbf{12};\] as shown below. If the outcomes are equally likely, find the probability that the pointer will point at: \[\left( \mathbf{v} \right)\]a number less than or equal to \[\mathbf{9}\] \[\left( \mathbf{vi} \right)\] a number between \[\mathbf{3}\]and \[\mathbf{11}\]

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: \[\left( v \right)\] Favorable outcomes for a number less than or equal to \[9\text{ }are\text{ }1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6,\text{ }7,\text{ }8,\text{ }9\] So,...

A game consists of spinning arrow which comes to rest pointing at one of the numbers \[\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5},\text{ }\mathbf{6},\text{ }\mathbf{7},\text{ }\mathbf{8},\text{ }\mathbf{9},\text{ }\mathbf{10},\text{ }\mathbf{11},\text{ }\mathbf{12};\] as shown below. If the outcomes are equally likely, find the probability that the pointer will point at: \[~~~\left( \mathbf{iii} \right)\]a prime number \[\left( \mathbf{iv} \right)\] a number greater than \[\mathbf{8}\]

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: \[\left( iii \right)\]Favorable outcomes for a prime number are \[2,\text{ }3,\text{ }5,\text{ }7,\text{ }11\] So, number of favorable outcomes\[~=\text{ }5\] Hence, P(the pointer will be...

A game consists of spinning arrow which comes to rest pointing at one of the numbers \[\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5},\text{ }\mathbf{6},\text{ }\mathbf{7},\text{ }\mathbf{8},\text{ }\mathbf{9},\text{ }\mathbf{10},\text{ }\mathbf{11},\text{ }\mathbf{12};\] as shown below. If the outcomes are equally likely, find the probability that the pointer will point at: \[\left( \mathbf{i} \right)\text{ }\mathbf{6}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\] \[\left( \mathbf{ii} \right)\] an even number

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, Total number of possible outcomes \[=\text{ }12\] \[\left( i \right)\] Number of favorable outcomes for \[6\text{ }=\text{ }1\]6 Hence, \[P\left( the\text{ }pointer\text{...

A bag contains twenty Rs \[\mathbf{5}\] coins, fifty Rs \[\mathbf{2}\]coins and thirty Re \[\mathbf{1}\] coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: \[\left( \mathbf{iii} \right)\]will neither be a Rs \[\mathbf{5}\] coin nor be a Re \[\mathbf{1}\] coin?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] Number of favourable outcomes for neither Re \[1\]nor Rs \[5\]coins \[=\] Number of favourable outcomes for Rs\[~2\] coins \[=\text{ }50\text{ }=\text{ }n\left( E...

A bag contains twenty Rs \[\mathbf{5}\] coins, fifty Rs \[\mathbf{2}\]coins and thirty Re \[\mathbf{1}\] coins. If it is equally likely that one of the coins will fall down when the bag is turned upside down, what is the probability that the coin: \[\left( \mathbf{i} \right)\] will be a Re \[\mathbf{1}\] coin? \[\left( \mathbf{ii} \right)\] will not be a Rs \[\mathbf{2}\]coin?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, Total number of coins \[=\text{ }20\text{ }+\text{ }50\text{ }+\text{ }30\text{ }=\text{ }100\] So, the total possible outcomes \[=\text{ }100\text{ }=\text{ }n\left( S \right)\]...

Solution: \[\left( iii \right)\] Number of favorable outcomes for white or green ball \[=\text{ }16\text{ }+\text{ }8\text{ }=\text{ }24\text{ }=\text{ }n\left( E \right)\] Hence, probability for...

A bag contains\[\mathbf{10}\] red balls, \[\mathbf{16}\] white balls and \[\mathbf{8}\] green balls. A ball is drawn out of the bag at random. What is the probability that the ball drawn will be: \[\left( \mathbf{i} \right)\] not red? \[\left( \mathbf{ii} \right)\] neither red nor green?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: Total number of possible outcomes \[=\text{ }10\text{ }+\text{ }16\text{ }+\text{ }8\text{ }=\text{ }34\] balls So, \[n\left( S \right)\text{ }=\text{ }34\] \[\left( i \right)\] Favorable...

The probability that two boys do not have the same birthday is \[\mathbf{0}.\mathbf{897}\]. What is the probability that the two boys have the same birthday?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We know that, P(do not have the same birthday) \[+\]P(have same birthday) \[=\text{ }1\] \[0.897\text{ }+\] P(have same birthday) \[=\text{ }1\] Thus, P(have same birthday) \[=\text{...

A bag contains a certain number of red balls. A ball is drawn. Find the probability that the ball drawn is: \[\left( \mathbf{i} \right)\] black \[\left( \mathbf{ii} \right)\] red

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, Total possible outcomes = number of red balls. \[\left( i \right)\] Number of favourable outcomes for black balls \[=\text{ }0\] Hence\[,\text{ }P\left( black\text{ }ball...

If \[\mathbf{P}\left( \mathbf{E} \right)\text{ }=\text{ }\mathbf{0}.\mathbf{59}\]; find \[\mathbf{P}\left( \mathbf{not}\text{ }\mathbf{E} \right)\]

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We know that, \[P\left( E \right)\text{ }+\text{ }P\left( not\text{ }E \right)\text{ }=\text{ }1\] So, \[0.59\text{ }+\text{ }P\left( not\text{ }E \right)\text{ }=\text{ }1\] Hence,...

Which of the following cannot be the probability of an event? \[\left( \mathbf{iii} \right)\text{ }\mathbf{37}%\] \[\left( \mathbf{iv} \right)\text{ }-\mathbf{2}.\mathbf{4}\]

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution \[\left( iii \right)\text{ }As\text{ }0\text{ }\le \text{ }37\text{ }%\text{ }=\text{ }\left( 37/100 \right)\text{ }\le \text{ }1\] Thus, \[37\text{ }%\] can be a probability of an event....

Which of the following cannot be the probability of an event? \[\left( \mathbf{i} \right)~\mathbf{3}/\mathbf{7}\] \[\left( \mathbf{ii} \right)\text{ }\mathbf{0}.\mathbf{82}\]

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We know that probability of an event E is \[0\text{ }\le \text{ }P\left( E \right)\text{ }\le \text{ }1\] \[\left( i \right)\text{ }As\text{ }0\text{ }\le \text{ }3/7\text{ }\le \text{...

Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is: \[\left( \mathbf{iii} \right)\] less than or equal to \[\mathbf{12}\]

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] All the outcomes are favourable to the event \[E\text{ }=\]‘sum of two numbers \[\le ~12\] Thus, \[P\left( E \right)\text{ }=\text{ }n\left( E \right)/\text{...

Two dice are thrown at the same time. Find the probability that the sum of the two numbers appearing on the top of the dice is: \[\left( \mathbf{i} \right)\text{ }\mathbf{8}\] \[\left( \mathbf{ii} \right)\text{ }\mathbf{13}\]

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, the number of possible outcomes \[=\text{ }6~\times \text{ }6\text{ }=\text{ }36\] \[\left( i \right)\] The outcomes favourable to the event ‘the sum of the two numbers is...

In a bundle of \[\mathbf{50}\] shirts, \[\mathbf{44}\] are good, \[\mathbf{4}\] have minor defects and \[\mathbf{2}\]have major defects. What is the probability that: \[\left( \mathbf{i} \right)\] it is acceptable to a trader who accepts only a good shirt? \[\left( \mathbf{ii} \right)\] it is acceptable to a trader who rejects only a shirt with major defects?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, Total number of shirts \[=\text{ }50\] Total number of elementary events \[=\text{ }50\text{ }=\text{ }n\left( S \right)\] \[\left( i \right)\] As, trader accepts only good shirts...

In a musical chairs game, a person has been advised to stop playing the music at any time within \[\mathbf{40}\]seconds after its start. What is the probability that the music will stop within the first \[\mathbf{15}\] seconds?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: Total result \[=\text{ }0\text{ }sec\text{ }to\text{ }40\text{ }sec\] Total possible outcomes \[=\text{ }40\] So\[,\text{ }n\left( S \right)\text{ }=\text{ }40\] Favourable results...

All the three face cards of spades are removed from a well shuffled pack of \[\mathbf{52}\]cards. A card is then drawn at random from the remaining pack. Find the probability of getting: \[\left( \mathbf{iii} \right)\] a black card

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] Number of black cards left \[=\text{ }23\text{ }cards\text{ }\left( 13\text{ }club\text{ }+\text{ }10\text{ }spade \right)\] Event of drawing a black card \[=\text{...

All the three face cards of spades are removed from a well shuffled pack of \[\mathbf{52}\]cards. A card is then drawn at random from the remaining pack. Find the probability of getting: \[\left( \mathbf{i} \right)\] a black face card \[\left( \mathbf{ii} \right)\]a queen

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, Total number of cards \[=\text{ }52\] If \[3\] face cards of spades are removed Then, the remaining cards \[=\text{ }52\text{ }\text{ }3\text{ }=\text{ }49\text{ }=\] number of...

A box contains \[\mathbf{7}\] red balls, \[\mathbf{8}\]green balls and \[\mathbf{5}\] white balls. A ball is drawn at random from the box. Find the probability that the ball is: \[\left( \mathbf{i} \right)\] white \[\left( \mathbf{ii} \right)\] neither red nor white

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We have, Total number of balls in the box \[=\text{ }7\text{ }+\text{ }8\text{ }+\text{ }5\text{ }=\text{ }20\] balls Total possible outcomes \[=\text{ }20\text{ }=\text{ }n\left( S...

A man tosses two different coins (one of \[\mathbf{Rs}\text{ }\mathbf{2}\] and another of \[\mathbf{Rs}\text{ }\mathbf{5}\]) simultaneously. What is the probability that he gets: \[\left( \mathbf{i} \right)\] at least one head? \[\left( \mathbf{ii} \right)\] at most one head?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: We know that, When two coins are tossed simultaneously, the possible outcomes are \[\left\{ \left( H,\text{ }H \right),\text{ }\left( H,\text{ }T \right),\text{ }\left( T,\text{ }H...

A and B are friends. Ignoring the leap year, find the probability that both friends will have: \[\left( \mathbf{i} \right)\] different birthdays? \[\left( \mathbf{ii} \right)\] the same birthday?

CBSE, Class 10, Exercise 13.3, Selina, Statistics and Probability

Solution: Out of the two friends, A’s birthday can be any day of the year. Now, B’s birthday can also be any day of \[365\] days in the year. We assume that these \[365\] outcomes are equally...

In a match between A and B: \[\left( \mathbf{i} \right)\] the probability of winning of A is \[\mathbf{0}.\mathbf{83}\]. What is the probability of winning of B? \[\left( \mathbf{ii} \right)\] the probability of losing the match is \[\mathbf{0}.\mathbf{49}\] for B. What is the probability of winning of A?

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( i \right)\]We know that, The probability of winning of A \[+\]Probability of losing of A \[=\text{ }1\] And, Probability of losing of A \[=\] Probability of winning of B...

From a well shuffled deck of \[\mathbf{52}\] cards, one card is drawn. Find the probability that the card drawn is: \[\left( \mathbf{v} \right)\] a card with number less than \[\mathbf{8}\] \[\left( \mathbf{vi} \right)\] a card with number between \[\mathbf{2}\] and \[\mathbf{9}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( v \right)\] Numbers less than \[8\text{ }=\text{ }\left\{ \text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6,\text{ }7 \right\}\]\[\] Event of drawing a card with number less than...

From a well shuffled deck of \[\mathbf{52}\] cards, one card is drawn. Find the probability that the card drawn is: \[\left( \mathbf{iii} \right)\]a queen of black card \[\left( \mathbf{iv} \right)\]a card with number \[\mathbf{5}\text{ }\mathbf{or}\text{ }\mathbf{6}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] Event of drawing a queen of black colour \[=\text{ }\left\{ Q\left( spade \right),\text{ }Q\left( club \right) \right\}\text{ }=\text{ }E\] So,\[~n\left( E...

From a well shuffled deck of \[\mathbf{52}\] cards, one card is drawn. Find the probability that the card drawn is: \[\left( \mathbf{i} \right)\] a face card \[\left( \mathbf{ii} \right)\] not a face card

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: We have, the total number of possible outcomes \[=\text{ }52\] So, \[n\left( S \right)\text{ }=\text{ }52\] \[\left( i \right)~\]No. of face cards in a deck of \[52\]cards \[=\text{...

A dice is thrown once. What is the probability of getting a number: \[\left( \mathbf{i} \right)\]greater than \[\mathbf{2}\]? \[\left( \mathbf{ii} \right)\] less than or equal to \[\mathbf{2}\]?

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: The number of possible outcomes when dice is thrown \[=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}\] So\[,\text{ }n\left( S \right)\text{ }=\text{ }6\]...

A bag contains \[\mathbf{3}\] red balls, \[\mathbf{4}\] blue balls and \[\mathbf{1}\] yellow ball, all the balls being identical in shape and size. If a ball is taken out of the bag without looking into it; find the probability that the ball is: \[\left( \mathbf{iii} \right)\]not yellow \[\left( \mathbf{iv} \right)\] neither yellow nor red

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] Probability of not drawing a yellow ball \[=\text{ }1\text{ }\] Probability of drawing a yellow ball Thus, probability of not drawing a yellow ball \[=\text{...

A bag contains \[\mathbf{3}\] red balls, \[\mathbf{4}\] blue balls and \[\mathbf{1}\] yellow ball, all the balls being identical in shape and size. If a ball is taken out of the bag without looking into it; find the probability that the ball is: \[\left( \mathbf{i} \right)\] yellow \[\left( \mathbf{ii} \right)\] red

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: The total number of balls in the bag \[=\text{ }3\text{ }+\text{ }4\text{ }+\text{ }1\text{ }=\text{ }8\] balls So, the number of possible outcomes \[=\text{ }8\text{ }=\text{ }n\left( S...

The monthly income of a group of 320 { employees in a company is given below: $$\begin{tabular}{|l|l|} \hline Monthly Income (thousands) & No. of employees \\ \hline $6-7$ & 20 \\ \hline $7-8$ & 45 \\ \hline $8-9$ & 65 \\ \hline $9-10$ & 95 \\ \hline $10-11$ & 60 \\ \hline $11-12$ & 30 \\ \hline $12-13$ & 5 \\ \hline \end{tabular}$$ Draw an ogive of the given distribution on a graph paper taking $2 \mathrm{~cm}=\mathrm{Rs} 1000$ on one axis and $2 \mathrm{~cm}=$ 50 employees on the other axis. From the graph determine:
(i) if salary of a senior employee is above Rs 11,500 , find the number of senior employees in the company.
(ii) the upper quartile.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|} \hline Monthly Income (thousands) & No. of employees (f) & Cumulative frequency \\ \hline $6-7$ & 20 & 20 \\ \hline $7-8$ & 45 & 65 \\ \hline $8-9$ & 65 & 130 \\...

The monthly income of a group of 320 { employees in a company is given below: $$\begin{tabular}{|l|l|} \hline Monthly Income (thousands) & No. of employees \\ \hline $6-7$ & 20 \\ \hline $7-8$ & 45 \\ \hline $8-9$ & 65 \\ \hline $9-10$ & 95 \\ \hline $10-11$ & 60 \\ \hline $11-12$ & 30 \\ \hline $12-13$ & 5 \\ \hline \end{tabular}$$ Draw an ogive of the given distribution on a graph paper taking $2 \mathrm{~cm}=\mathrm{Rs} 1000$ on one axis and $2 \mathrm{~cm}=$ 50 employees on the other axis. From the graph determine:
(i) the median wage.
(ii) number of employees whose income is below Rs 8500 .

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|} \hline Monthly Income (thousands) & No. of employees (f) & Cumulative frequency \\ \hline $6-7$ & 20 & 20 \\ \hline $7-8$ & 45 & 65 \\ \hline $8-9$ & 65 & 130 \\...

The mean of the following distribution is 52 and the frequency of class interval $30-40$ is ‘f’. Find $f$. $$\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Class Interval & $10-20$ & $20-30$ & $30-40$ & $40-50$ & $50-60$ & $60-70$ & $70-80$ \\ \hline Frequency & 5 & 3 & $f$ & 7 & 2 & 6 & 13 \\ \hline \end{tabular}$$

Class 11, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|l|} \hline C.I. & Frequency(f) & Mid value (x) & fx \\ \hline $10-20$ & 5 & 15 & 75 \\ \hline $20-30$ & 3 & 25 & 75 \\ \hline $30-40$ & $f$ & 35 & $35 f$ \\ \hline...

The distribution, given below, shows the marks obtained by 25 students in an aptitude test. Find the mean, median and mode of the distribution. $$\begin{tabular}{|l|l|l|l|l|l|l|} \hline Marks obtained & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline No. of students & 3 & 9 & 6 & 4 & 2 & 1 \\ \hline \end{tabular}$$

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|l|} \hline Marks obtained(x) & No. of students (f) & c.f. & fx \\ \hline 5 & 3 & 3 & 15 \\ \hline 6 & 9 & 12 & 54 \\ \hline 7 & 6 & 18 & 42 \\ \hline 8 & 4 & 22 &...

Using a graph paper, draw an ogive for the following distribution which shows a record of the width in kilograms of 200 students. $$\begin{tabular}{|l|l|} \hline Weight & Frequency \\ \hline $40-45$ & 5 \\ \hline $45-50$ & 17 \\ \hline $50-55$ & 22 \\ \hline $55-60$ & 45 \\ \hline $60-65$ & 51 \\ \hline $65-70$ & 31 \\ \hline $70-75$ & 20 \\ \hline $75-80$ & 9 \\ \hline \end{tabular}$$ Use your ogive to estimate the following:
(i) The number of students who are (a) underweight (b) overweight, if $55.70 \mathrm{~kg}$ is considered as standard weight.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) (a) underweight students when $55.70 kg$ is standard $= 46$ (approximate) from graph (b) overweight students when $55.70 kg$ is standard $= 200 – 55.70 = 154$ (approximate) from...

Using a graph paper, draw an ogive for the following distribution which shows a record of the width in kilograms of 200 students. $$\begin{tabular}{|l|l|} \hline Weight & Frequency \\ \hline $40-45$ & 5 \\ \hline $45-50$ & 17 \\ \hline $50-55$ & 22 \\ \hline $55-60$ & 45 \\ \hline $60-65$ & 51 \\ \hline $65-70$ & 31 \\ \hline $70-75$ & 20 \\ \hline $75-80$ & 9 \\ \hline \end{tabular}$$ Use your ogive to estimate the following:
(i) The percentage of students weighing $55 \mathrm{~kg}$ or more
(ii) The weight above which the heaviest $30 \%$ of the student fall

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) No. of students weighing more than 55 kg $= 200 – 44 = 156$ Therefore, the percentage of students weighing 55 kg or more $= (\frac{156}{200}) \times 100 =$ 78 % (ii) 30% of students...

The marks obtained by 120 students in a mathematics test is given below: $$\begin{tabular}{|l|l|} \hline Marks & No. of students \\ \hline $0-10$ & 5 \\ \hline $10-20$ & 9 \\ \hline 20-30 & 16 \\ \hline 30-40 & 22 \\ \hline 40-50 & 26 \\ \hline 50-60 & 18 \\ \hline 60-70 & 11 \\ \hline 70-80 & 6 \\ \hline 80-90 & 4 \\ \hline $90-100$ & 3 \\ \hline \end{tabular}$$ Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for your ogive. Use your ogive to estimate:

(i) the number of students who did not pass in the test if the pass percentage was 40 .
(ii) the lower quartile

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) No. of students who obtained less than 40% marks in the test $= 52$ (from the graph; $x = 40$, $y = 52$) (ii) Lower quartile $= Q_1 = 120 \times {(\frac{1}{4})} = 30^{th}$ term $=...

The marks obtained by 120 students in a mathematics test is given below: $$\begin{tabular}{|l|l|} \hline Marks & No. of students \\ \hline $0-10$ & 5 \\ \hline $10-20$ & 9 \\ \hline 20-30 & 16 \\ \hline 30-40 & 22 \\ \hline 40-50 & 26 \\ \hline 50-60 & 18 \\ \hline 60-70 & 11 \\ \hline 70-80 & 6 \\ \hline 80-90 & 4 \\ \hline $90-100$ & 3 \\ \hline \end{tabular}$$ Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for your ogive. Use your ogive to estimate:
(i) the median
(ii) the number of students who obtained more than $75 \%$ in test.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) Median $=\frac{ (120 + 1)}{2} = 60.5^{th}$ term Draw a parallel line to $x-axis$ through mark $60.5$ which meets the curve at A. From A draw a perpendicular to $x-axis$ meeting it at...

If two coins are tossed once, what is the probability of getting: \[\left( \mathbf{iii} \right)~\]both heads or both tails

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] E = event of getting both heads or both tails \[=\text{ }\left\{ HH,\text{ }TT \right\}\] \[n\left( E \right)\text{ }=\text{ }2\] Hence, probability of getting both...

If two coins are tossed once, what is the probability of getting: (i) both heads. (ii) at least one head.

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: We know that, when two coins are tossed together possible number of outcomes = {HH, TH, HT, TT} So, \[n\left( S \right)\text{ }=\text{ }4\] \[\left( i \right)\]E = event of getting both...

The marks of 20 students in a test were as follows: 2, 6, 8, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 18, 19 and 20. Calculate:
(i) the mode

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the terms in ascending order: $2,6,8,9,10,11,11,12,13,13,14,14,15,15,15,16,16,18,19,20$ No. of terms $=20$ $\sum x=2+6+8+9+11+11+12+13+13+14+14+15+15+15+15+16+16+18+19+20=257$ (i)...

A pair of dice is thrown. Find the probability of getting a sum of \[\mathbf{10}\] or more, if \[\mathbf{5}\] appears on the first die

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: In throwing a dice, total possible outcomes \[=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}\] So\[,\text{ }n\left( S \right)\text{ }=\text{ }6\] For two...

The marks of 20 students in a test were as follows: 2, 6, 8, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 18, 19 and 20. Calculate:
(i) the mean
(ii) the median

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the terms in ascending order: $2,6,8,9,10,11,11,12,13,13,14,14,15,15,15,16,16,18,19,20$ No. of terms $=20$ $\sum x=2+6+8+9+11+11+12+13+13+14+14+15+15+15+15+16+16+18+19+20=257$ (i)...

The income of the parents of 100 students in a class in a certain university are tabulated below. $$\begin{tabular}{|l|l|l|l|l|l|} \hline Income (in thousand Rs) & $0-8$ & $8-16$ & $16-24$ & $24-32$ & $32-40$ \\ \hline No. of students & 8 & 35 & 35 & 14 & 8 \\ \hline \end{tabular}$$
(i) Calculate the Arithmetic mean.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) Calculating the Arithmetic Mean Mean $=\sum \mathrm{f} x / \sum \mathrm{f}=1832 / 100=18.32$

A book contains \[\mathbf{85}\] pages. A page is chosen at random. What is the probability that the sum of the digits on the page is \[\mathbf{8}\]?

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: We know that, Number of pages in the book \[=\text{ }85\] Number of possible outcomes \[=\text{ }n\left( S \right)\text{ }=\text{ }85\] Out of \[85\]pages, pages that sum up to \[8\text{...

The income of the parents of 100 students in a class in a certain university are tabulated below. $$\begin{tabular}{|l|l|l|l|l|l|} \hline Income (in thousand Rs) & $0-8$ & $8-16$ & $16-24$ & $24-32$ & $32-40$ \\ \hline No. of students & 8 & 35 & 35 & 14 & 8 \\ \hline \end{tabular}$$
(i) Draw a cumulative frequency curve to estimate the median income.
(ii) If 15% of the students are given freeships on the basis of the basis of the income of their parents, find the annual income of parents, below which the freeships will be awarded.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) Cumulative Frequency Curve Plot the points $(8,8),(16,43),(24,78),(32,92)$ and $(40,100)$ to obtain the curve as follows: Here, $\mathrm{N}=100$ $\mathrm{N} / 2=50$ At $y=50$, affix...

A die is thrown once. Find the probability of getting a number: \[(\mathbf{iii})\] less than \[\mathbf{8}\] \[\left( \mathbf{iv} \right)\] greater than \[\mathbf{6}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] On a dice, numbers less than \[8\text{ }=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}\] So\[,\text{ }n\left( E \right)\text{ }=\text{...

A die is thrown once. Find the probability of getting a number: \[\left( \mathbf{i} \right)\] less than \[\mathbf{3}\] \[\left( \mathbf{ii} \right)\] greater than or equal to \[\mathbf{4}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: We know that, In throwing a dice, total possible outcomes \[=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}\] So\[,\text{ }n\left( S \right)\text{ }=\text{...

In a malaria epidemic, the number of cases diagnosed were as follows: $$\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|} \hline Date (July) & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline Number & 5 & 12 & 20 & 27 & 46 & 30 & 31 & 18 & 11 & 5 & 0 & 1 \\ \hline \end{tabular}$$ On what days do the mode and upper and lower quartiles occur?

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) Mode $=5^{\text {th }}$ July as it has maximum frequencies. (ii) Total no. of terms $=206$ Upper quartile $=206 \mathrm{x}(3 / 4)=154.5^{\text {th }}=7^{\text {th }}$ July Lower...

The mean of 1, 7, 5, 3, 4 and 4 is m. The numbers 3, 2, 4, 2, 3, 3 and p have mean m – 1 and median q. Find p and q.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Mean of $1, 7, 5, 3, 4$ and $4 = {\frac{(1 + 7 + 5 + 3 + 4 + 4)} {6}} = {\frac{24}{6}} = 4$ Therefore, $m = 4$ It is given that The mean of $3, 2, 4, 2, 3, 3$ and $p = m -1 = 4 – 1 = 3$...

From identical cards, numbered one card is drawn at random. Find the probability that the number on the card drawn is a multiple of: \[\left( \mathbf{iii} \right)\text{ }\mathbf{3}\text{ }\mathbf{and}\text{ }\mathbf{5}\] \[\left( \mathbf{iv} \right)\text{ }\mathbf{3}\text{ }\mathbf{or}\text{ }\mathbf{5}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] From numbers \[1\text{ }to\text{ }25\], there is only one number which is multiple of \[3\text{ }and\text{ }5\text{ }i.e.~\left\{ 15 \right\}\] So, favorable number...

Draw ogive for the data given below and from the graph determine:
(i) the median marks.
(ii) the number of students who obtained more than 75% marks. $$\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|} \hline Marks & $10-$ & 20 & $30-$ & $40-$ & $50-$ & $60-$ & $70-$ & $80-$ & $90-$ \\ \hline & 19 & $-29$ & 39 & 49 & 59 & 69 & 79 & 89 & 99 \\ \hline No. of students & 14 & 16 & 22 & 26 & 18 & 11 & 6 & 4 & 3 \\ \hline \end{tabular}$$

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|} \hline Marks & No. of students & Cumulative frequency \\ \hline $9.5-19.5$ & 14 & 14 \\ \hline $19.5-29.5$ & 16 & 30 \\ \hline $29.5-39.5$ & 22 & 52 \\ \hline...

The following distribution represents the height of 160 students of a school. $$\begin{tabular}{|l|l|} \hline Height (in cm) & No. of Students \\ \hline $140-145$ & 12 \\ \hline $145-150$ & 20 \\ \hline $150-155$ & 30 \\ \hline $155-160$ & 38 \\ \hline $160-165$ & 24 \\ \hline $165-170$ & 16 \\ \hline $170-175$ & 12 \\ \hline $175-180$ & 8 \\ \hline \end{tabular}$$ Draw an ogive for the given distribution taking $2 \mathrm{~cm}=5 \mathrm{~cm}$ of height on one axis and $2 \mathrm{~cm}=20$ students on the other axis. Using the graph, determine:
(i). The number of students whose height is above $172 \mathrm{~cm}$.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|} \hline \text { Height (in cm) } & \text { No. of Students } & \text { Cumulative frequency } \\ \hline 140-145 & 12 & 12 \\ \hline 145-150 & 20 & 32 \\ \hline...

The following distribution represents the height of 160 students of a school. $$\begin{tabular}{|l|l|} \hline Height (in cm) & No. of Students \\ \hline $140-145$ & 12 \\ \hline $145-150$ & 20 \\ \hline $150-155$ & 30 \\ \hline $155-160$ & 38 \\ \hline $160-165$ & 24 \\ \hline $165-170$ & 16 \\ \hline $170-175$ & 12 \\ \hline $175-180$ & 8 \\ \hline \end{tabular}$$ Draw an ogive for the given distribution taking $2 \mathrm{~cm}=5 \mathrm{~cm}$ of height on one axis and $2 \mathrm{~cm}=20$ students on the other axis. Using the graph, determine:
(i). The median height.
(ii). The interquartile range.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|} \hline \text { Height (in cm) } & \text { No. of Students } & \text { Cumulative frequency } \\ \hline 140-145 & 12 & 12 \\ \hline 145-150 & 20 & 32 \\ \hline...

From \[\mathbf{25}\]identical cards, \[\text{ }\mathbf{1},\text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5},\text{ }\ldots \ldots ,~\mathbf{24},\text{ }\mathbf{25}:\]numbered one card is drawn at random. Find the probability that the number on the card drawn is a multiple of\[\left( \mathbf{i} \right)\text{ }\mathbf{3}\] \[\left( \mathbf{ii} \right)\text{ }\mathbf{5}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: We know that, there are \[25\] cards from which one card is drawn. So, the total number of elementary events \[=\text{ }n\left( S \right)\text{ }=\text{ }25\] \[\left( i \right)\]From...

The following table shows the expenditure of $60$ boys on books. Find the mode of their expenditure: $$\begin{tabular}{|l|l|} \hline \text { Expenditure } & \text { No. of } \\ \text { (Rs) } & \text { students } \\ \hline \text { 20-25 } & 4 \\ \hline \text { 25-30 } & 7 \\ \hline 30-35 & 23 \\ \hline 35-40 & 18 \\ \hline 40-45 & 6 \\ \hline \text { 45-50 } & 2 \\ \hline \end{tabular}$$

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: We can clearly see that, Mode is in $30-35$ because it has the maximum frequency.

Find the mode of following data, using a histogram: $$\begin{tabular}{|l|l|l|l|l|l|} \hline Class & $0-10$ & $10-20$ & $20-30$ & $30-40$ & $40-50$ \\ \hline Frequency & 5 & 12 & 20 & 9 & 4 \\ \hline \end{tabular}$$

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: We can clearly say that, Mode is in $20-30$, because in this class there are $20$ frequencies.

The following table shows the frequency distribution of heights of 50 boys: $$\begin{tabular}{|l|l|l|l|l|l|} \hline Height (cm) & 120 & 121 & 122 & 123 & 124 \\ \hline Frequency & 5 & 8 & 18 & 10 & 9 \\ \hline \end{tabular}$$ Find the mode of heights.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: We can clearly say that, Mode is $122 \mathrm{~cm}$ because it has occurred the maximum number of times. Therefore, frequency is 18.

Find the mode of the following data:
(i) $7,9,8,7,7,6,8,10,7$ and 6
(ii) $9,11,8,11,16,9,11,5,3,11,17$ and 8

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) We can see that 7 occurs 4 times in the given data. As a result, mode $=7$ (ii) Mode $=11$ As it can be seen that 11 occurs 4 times in the given data.

$Hundred identical cards are numbered from \[\mathbf{1}\text{ }\mathbf{to}\text{ }\mathbf{100}\] The cards The cards are well shuffled and then a card is drawn. Find the probability that the number on card drawn is: \[\left( \mathbf{v} \right)\] less than \[\mathbf{48}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( v \right)\]From numbers \[1\text{ }to\text{ }100\], there are \[47\] numbers which are less than \[48\text{ }i.e.~\{1,\text{ }2,\text{ }\ldots \ldots \ldots ..,\]\[46,\text{...

The ages of 37 students in a class are given in the following table: $$\begin{tabular}{|l|l|l|l|l|l|l|} \hline Age (in years) & 11 & 12 & 13 & 14 & 15 & 16 \\ \hline Frequency & 2 & 4 & 6 & 10 & 8 & 7 \\ \hline \end{tabular}$$ Find the median.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|} \hline Age (in years) & Frequency & Cumulative \\ \hline 11 & 2 & Frequency \\ \hline 12 & 4 & 2 \\ \hline 13 & 6 & 6 \\ \hline 14 & 10 & 12 \\ \hline 15 & 8 &...

From the following data, find:
(i) Inter-quartile range
25, 10, 40, 88, 45, 60, 77, 36, 18, 95, 56, 65, 7, 0, 38 and 83

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the data given in ascending order $0,7,10,18,25,36,38,40,45,56,60,65,77,83,88,95$ (i) Interquartile range is given by, $\mathrm{q}_{1}=16^{\text {th }} / 4 \text { term }=18 ;...

From the following data, find:
(i) Median
(ii) Upper quartile
25, 10, 40, 88, 45, 60, 77, 36, 18, 95, 56, 65, 7, 0, 38 and 83

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the data given in ascending order $0,7,10,18,25,36,38,40,45,56,60,65,77,83,88,95$ (i) Median is the mean of $8^{\text {th }}$ and $9^{\text {th }}$ term Therefore, median $=(40+45)...

Hundred identical cards are numbered from The cards The cards are well shuffled and then a card is drawn. Find the probability that the number on card drawn is: \[\left( \mathbf{iii} \right)\] between \[\mathbf{40}\] and \[\mathbf{60}\] \[\left( \mathbf{iv} \right)\] greater than \[\mathbf{85}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] From numbers \[1\text{ }to\text{ }100\], there are \[19\] numbers which are between \[40\text{ }and\text{ }60\text{ }i.e.~\{41,\text{ }42\], \[43,\text{ }44,\text{...

The marks obtained by 19 students of a class are given below: $27,36,22,31,25,26,33,24,37,32,29,28,36,35,27,26,32,35$ and 28. Find:
(i) upper quartile
(ii) interquartile range

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the given data in ascending order: $22,24,25,26,26,27,27,28,28,29,21,32,32,33,35,35,36,36,37$ (i) Upper quartile = $\begin{array}{l} q_{3}=\left[\frac{3(n+1)}{4}\right]^{\text {th...

The marks obtained by 19 students of a class are given below: $27,36,22,31,25,26,33,24,37,32,29,28,36,35,27,26,32,35$ and 28. Find:
(i) median
(ii) lower quartile

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the given data in ascending order: $22,24,25,26,26,27,27,28,28,29,21,32,32,33,35,35,36,36,37$ (i) The middle term is $10^{\text {th }}$ term that is 29 , As a result, median $=29$...

The weights (in $\mathrm{kg}$ ) of 10 students of a class are given below:
$21,28.5,20.5,24,25.5,22,27.5,28,21$ and 24 . Find the median of their weights.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the data given in descending order: $28.5,28,27.5,25.5,24,24,22,21,21,20.5$ It can be clearly seen that, The middle terms are $24$ and $24,5^{\text {th }}$ and $6^{\text {th }}$...

A student got the following marks in 9 questions of a question paper. $3,5,7,3,8,0,1,4$ and 6 Find the median of these marks.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: Arrange the data given in descending order: $8,7,6,5,4,3,3,1,0$ We can clearly see, the middle term is 4 which is the $5^{\text {th }}$ term. As a result, median $=4$

The mean of the following frequency distribution is $21 \frac{1}{7}$. Find the value of ‘f’. $$\begin{tabular}{|l|l|l|l|l|l|} \hline C. I. & $0-10$ & $10-20$ & $20-30$ & $30-40$ & $40-50$ \\ \hline freq & 8 & 22 & 31 & $\mathbf{f}$ & 2 \\ \hline \end{tabular}$$

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: It is given that $\bar{x}=21 \frac{1}{7}=\frac{148}{7}$ $$\begin{tabular}{|l|l|l|l|} \hline C. I. & frequency & Mid-value $\left(\mathrm{x}_{\mathrm{i}}\right)$ & $\mathrm{f}_{\mathrm{i}}...

Hundred identical cards are numbered from \[\mathbf{1}\text{ }\mathbf{to}\text{ }\mathbf{100}\]. The cards The cards are well shuffled and then a card is drawn. Find the probability that the number on card drawn is: \[\left( \mathbf{i} \right)\] a multiple of \[\mathbf{5}\] \[\left( \mathbf{ii} \right)\] a multiple of \[\mathbf{6}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: We kwon that, there are \[100\] cards from which one card is drawn. Total number of elementary events \[=\text{ }n\left( S \right)\text{ }=\text{ }100\] \[\left( i \right)~\] From numbers...

multiple Nine cards (identical in all respects) are numbered . A card is selected from them at random. Find the probability that the card selected will be: \[\left( \mathbf{iii} \right)\] an even number and a multiple of \[\mathbf{3}\] \[\left( \mathbf{iv} \right)\] an even number or a of \[\mathbf{3}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] From numbers \[2\text{ }to\text{ }10\], there is one number which is an even number as well as multiple of \[3\text{ }i.e.\text{ }6\] So, favorable number of events...

Nine cards (identical in all respects) are numbered \[\mathbf{2}\text{ }\mathbf{to}\text{ }\mathbf{10}\]. A card is selected from them at random. Find the probability that the card selected will be: \[\left( \mathbf{i} \right)\]an even number \[\left( \mathbf{ii} \right)\] a multiple of \[\mathbf{3}\]

CBSE, Class 10, Exercise 13.2, Selina, Statistics and Probability

Solution: We know that, there are totally \[9\] cards from which one card is drawn. Total number of elementary events \[=\text{ }n\left( S \right)\text{ }=\text{ }9\] \[\left( i \right)\] From...

Find mean by step – deviation method: $$\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline C. I. & $\mathbf{6 3 – 7 0}$ & $\mathbf{7 0 – 7 7}$ & $\mathbf{7 7 – 8 4}$ & $\mathbf{8 4 – 9 1}$ & $\mathbf{9 1 – 9 8}$ & $\mathbf{9 8 – 1 0 5}$ & $\mathbf{1 0 5 – 1 1 2}$ \\ \hline Freq & 9 & 13 & 27 & 38 & 32 & 16 & 15 \\ \hline \end{tabular}$$

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|l|l|} \hline C. I. & Frequency $\left(\mathrm{f}_{\mathrm{i}}\right)$ & Mid-value $\mathrm{x}_{\mathrm{i}}$ & \multicolumn{2}{|l|}{ $\mathrm{A}=87.50$...

In a T.T. match between Geeta and Ritu, the probability of the winning of Ritu is \[\mathbf{0}.\mathbf{73}\]. Find the probability of: \[\left( \mathbf{i} \right)\] winning of Geeta \[\left( \mathbf{ii} \right)\] not winning of Ritu

CBSE, Class 10, Exercise 13.1, Selina, Statistics and Probability

Solution: \[\left( i \right)\] Winning of Geeta is a complementary event to winning of Ritu Thus, P(winning of Ritu) \[+\]P(winning of Geeta) \[=\text{ }1\] P(winning of Geeta) \[=\text{ }1\text{...

The following table gives the weekly wages of workers in a factory. $$\begin{tabular}{|l|l|} \hline Weekly Wages (Rs) & No. of Workers \\ \hline $\mathbf{5 0 – 5 5}$ & 5 \\ \hline 55-60 & 20 \\ \hline 60-65 & 10 \\ \hline 65-70 & 10 \\ \hline 70-75 & 9 \\ \hline 75-80 & 6 \\ \hline 80-85 & 12 \\ \hline 85-90 & 8 \\ \hline \end{tabular}$$ Calculate the mean by using:
(i) Direct Method
(ii) Short – Cut Method

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (i) Direct Method $\text { Mean }=\sum f_{i} x_{i} / \sum f_{i}=5520 / 80=69$ (ii)Short – cut method Here, $A=72.5$ $\bar{x}=A+\frac{\sum f_{i} d_{i}}{\sum...

The following table gives the ages of 50 students of a class. Find the arithmetic mean of their ages. $$\begin{tabular}{|l|l|l|l|l|l|} \hline Age – Years & $16-18$ & $18-20$ & $20-22$ & $22-24$ & $24-26$ \\ \hline No. of Students & 2 & 7 & 21 & 17 & 3 \\ \hline \end{tabular}$$

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|l|l|} \hline Age in years C.I. & $\mathrm{x}_{\mathrm{i}}$ & Number of students $\left(\mathrm{f}_{\mathrm{i}}\right)$ & $\mathrm{x}_{\mathrm{i}} \mathrm{fi}$ \\...

The ages of 40 students are given in the following table:
$$\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Age( in yrs) & 12 & 13 & 14 & 15 & 16 & 17 & 18 \\ \hline Frequency & 2 & 4 & 6 & 9 & 8 & 7 & 4 \\ \hline \end{tabular}$$
Find the arithmetic mean.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: $$\begin{tabular}{|l|l|l|} \hline Age in yrs & Frequency & $\mathrm{f}_{\mathrm{i}} \mathrm{x}_{\mathrm{i}}$ \\ $\mathrm{x}_{\mathrm{i}}$ & $(\mathrm{fi})$ & \\ \hline 12 & 2 & 24 \\...

The mean of the number $6, ‘ y^{\prime}, 7, ‘ x^{\prime}$ and 14 is 8. Express ‘ $y$ ‘ in terms of ‘ $x$ ‘.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: It is given that, Number of terms $(n)=5$ and mean $=8$ Therefore, the sum of all terms $=5 \times 8=40 \ldots \ldots$ (i) But the sum of numbers $=6+y+7+x+14=27+y+x \ldots \ldots$ (ii) On...

If the mean of $6,4,7$, ‘ $a$ ‘ and 10 is 8 . Find the value of ‘ $a$ ‘

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: It is given that, No. of terms $(n)=5$ Mean $=8$ The sum of all terms $=8 \times 5=40 \ldots \ldots$ (i) But, the sum of numbers $=6+4+7+a+10=27+a \ldots \ldots$ (ii) On equating eq.(i)...

(a) Find the mean of $7,11,6,5$, and 6
(b) If each number given in (a) is diminished by 2, find the new value of mean.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (a) Mean $=\sum x / n$, here $n=5$ $=(7+11+6+5+6) / 5=35 / 5=7$ (b) If number 2 is subtracted from each number, then the mean will be changed to $7-2=5$

Find the mean of the natural numbers from 3 to 12 .

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: The nos. between 3 to 12 are $3,4,5,6,7,8,9,10,11$ and 12 Here $n=10$ Mean $=\sum \mathrm{x} / \mathrm{n}$ $=(3+4+5+6+7+8+9+10+11+12) / 10=75 / 10=7.5$

Marks obtained (in mathematics) by 9 student are given below: $60,67,52,76,50,51,74,45$ and 56
(a) find the arithmetic mean)
(b) if marks of each student be increased by $4 ;$ what will be the new value of arithmetic mean.

Class 10, ICSE, Maths, Measures of Central Tendency, Selina

Solution: (a) Mean $=\sum x / n$ Here, $\mathrm{n}=9$ Therefore, $\text { Mean }=(60+67+52+76+50+51+74+45+56) / 9=531 / 9=59$ (b) If the marks of each student is increased by 4 then the new...

\[\left( \mathbf{i} \right)\] If A and B are two complementary events then what is the relation between \[\mathbf{P}\left( \mathbf{A} \right)\] and \[\mathbf{P}\left( \mathbf{B} \right)\]? \[\left( \mathbf{ii} \right)\] If the probability of happening an event A is \[\mathbf{0}.\mathbf{46}\]. What will be the probability of not happening of the event A?

CBSE, Class 10, Exercise 13.1, Selina, Statistics and Probability

Solution: \[\left( i \right)\] Two complementary events, taken together, include all the outcomes for an experiment and the sum of the probabilities of all outcomes is \[1.\]...

Construct a frequency distribution table for the number given below, using the class intervals 21-30, 31-40 … etc. 75, 67, 57, 50, 26, 33, 44, 58, 67, 75, 78, 43, 41, 31, 21, 32, 40, 62, 54, 69, 48, 47, 51, 38, 39, 43, 61, 63, 68, 53, 56, 49, 59, 37, 40, 68, 23, 28, 36, 47 Use the table obtained to draw:
(i) a histogram
(ii) an ogive

Class 10, Graphical Representations, ICSE, Maths, Selina

Solution: (i) (ii)

Draw an ogive for each of the following distributions:
(i) $$\begin{tabular}{|l|l|l|l|l|l|} \hline Marks Obtained & less than 10 & less & less & less & less \\ \hline No. of Students & 8 & 25 & 38 & 50 & than 50 \\ \hline \end{tabular}$$
(ii) $$\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Age in years (less than) & 10 & 20 & 30 & 40 & 50 & 60 & 70 \\ \hline Cumulative Frequency & 0 & 17 & 32 & 37 & 53 & 58 & 65 \\ \hline \end{tabular}$$

Class 10, Graphical Representations, ICSE, Maths, Selina

Solution: (i) $$\begin{tabular}{|l|l|} \hline Marks Obtained & No. of students (c.f.) \\ \hline less than 10 & 8 \\ \hline less than 20 & 25 \\ \hline less than 30 & 38 \\ \hline less than 40 & 50...

From a well shuffled deck of \[\mathbf{52}\] cards, one card is drawn. Find the probability that the card drawn will:(v) be a face card of red colour

CBSE, Class 10, Exercise 13.1, Selina, Statistics and Probability

Solution: \[\left( v \right)\]There are \[26\] red cards in a deck, and \[6\] of these cards are face cards (\[2\] kings, \[2\]queens and \[2\]jacks). The number of favourable outcomes for the event...

Draw cumulative frequency curve (ogive) for each of the following distributions:
(i) $$\begin{tabular}{|l|l|l|l|l|l|l|} \hline Class Interval & $10-15$ & $15-20$ & $20-25$ & $25-30$ & $30-45$ & $35-40$ \\ \hline Frequency & 10 & 15 & 17 & 12 & 10 & 08 \\ \hline \end{tabular}$$
(ii) $$\begin{tabular}{|l|l|l|l|l|l|} \hline Class Interval & $10-19$ & $20-29$ & $30-39$ & $40-49$ & $50-59$ \\ \hline Frequency & 23 & 16 & 15 & 20 & 12 \\ \hline \end{tabular}$$

CBSE, Class 10, Graphical Representations, ICSE, Maths, Selina

Solution: (i) $$\begin{tabular}{|l|l|} \hline Class Interval & Frequency \\ \hline 10-15 & 10 \\ \hline 15-20 & 15 \\ \hline 20-25 & 17 \\ \hline 25-30 & 12 \\ \hline $30-35$ & 10 \\ \hline $35-40$...

Draw histograms for the following frequency distributions:
(i) $$\begin{tabular}{|l|l|l|l|l|l|} \hline Class Interval & $30-39$ & $40-49$ & $50-59$ & $60-69$ & $70-79$ \\ \hline Frequency & 24 & 16 & 09 & 15 & 20 \\ \hline \end{tabular}$$
(ii) $$\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline Class Marks & 16 & 24 & 32 & 40 & 48 & 56 & 64 \\ \hline Frequency & 8 & 12 & 15 & 18 & 25 & 19 & 10 \\ \hline \end{tabular}$$

CBSE, Class 10, Graphical Representations, ICSE, Maths, Selina

Solution: (i) $$\begin{tabular}{|l|l|l|} \hline Class Interval (Inclusive form) & Class Interval (Exclusive Form) & Frequency \\ \hline $30-39$ & 29.5-39.5 & \\ \hline $40-49$ & 39.5-49.5 & 24 \\...

From a well shuffled deck of \[\mathbf{52}\] cards, one card is drawn. Find the probability that the card drawn will: \[\left( \mathbf{iii} \right)\] be a red card. \[\left( \mathbf{iv} \right)\] be a face card

CBSE, Class 10, Exercise 13.1, Selina, Statistics and Probability

Solution: \[\left( iii \right)\] Number of red cards in a deck \[=\text{ }26\] The number of favourable outcomes for the event of drawing a red card \[=\text{ }26\] Then, probability of drawing a...

Draw histograms for the following frequency distributions:
(i) $$\begin{tabular}{|l|ccccc|c|} \hline Class Interval & $0-10$ & $10-20$ & $20-30$ & $30-40$ & $40-50$ & $50-60$ \\ \hline Frequency & 12 & 20 & 26 & 18 & 10 & 6 \\ \hline \end{tabular}$$
(ii) $$\begin{tabular}{|l|l|l|l|l|l|} \hline Class Interval & $10-16$ & $16-22$ & $22-28$ & $28-34$ & $34-40$ \\ \hline Frequency & 15 & 23 & 30 & 20 & 16 \\ \hline \end{tabular}$$

Class 10, Graphical Representations, ICSE, Maths, Selina

Solution: (i) $$\begin{tabular}{|l|l|} \hline Class Interval & Frequency \\ \hline $0-10$ & 12 \\ \hline 10-20 & 20 \\ \hline 20-30 & 26 \\ \hline $30-40$ & 18 \\ \hline 40-50 & 10 \\ \hline 50-60 &...

From a well shuffled deck of 52 cards, one card is drawn. Find the probability that the card drawn will: \[\left( \mathbf{i} \right)\] be a black card. \[\left( \mathbf{ii} \right)\] not be a red card

CBSE, Class 10, Exercise 13.1, Selina, Statistics and Probability

Solution: We know that, Total number of cards \[=\text{ }52\] So, the total number of outcomes \[=\text{ }52\] There are \[13\] cards of each type. The cards of heart and diamond are red in colour....

In a single throw of a die, find the probability that the number: \[\left( \mathbf{i} \right)\]will be an even number. \[\left( \mathbf{ii} \right)\] will not be an even number.

CBSE, Class 10, Exercise 13.1, Selina, Statistics and Probability

Solution: Here, the sample space \[=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}\] \[n\left( s \right)\text{ }=\text{ }6\] \[\left( i \right)\] If \[E\text{ }=\]event...

In a single throw of a die, find the probability of getting a number: \[\left( \mathbf{iii} \right)\]not greater than \[\mathbf{4}\]

CBSE, Class 10, Selina, Statistics and Probability

\[\left( iii \right)\text{ }E\text{ }=\] event of getting a number not greater than \[4\text{ }=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4 \right\}\] So\[,\text{ }n\text{ }\left( E...

In a single throw of a die, find the probability of getting a number: \[\left( \mathbf{i} \right)\] greater than \[\mathbf{4}.\] \[\left( \mathbf{ii} \right)\] less than or equal to \[\mathbf{4}.\]

CBSE, Class 10, Exercise 13.1, Selina, Statistics and Probability

Solution: Here, the sample space \[=\text{ }\left\{ 1,\text{ }2,\text{ }3,\text{ }4,\text{ }5,\text{ }6 \right\}\] So\[,\text{ }n\text{ }\left( s \right)\text{ }=\text{ }6\] \[\left( i \right)\]If...

A bag contains \[\mathbf{3}\] white, \[\mathbf{5}\] black and \[\mathbf{2}\] red balls, all of the same shape and size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is: \[\left( \mathbf{v} \right)\]not a black ball.

CBSE, Class 10, Selina, Statistics and Probability

Solution: \[\left( v \right)\] There are \[3\text{ }+\text{ }2\text{ }=\text{ }5\] balls which are not black So, the number of favourable outcomes \[=\text{ }5\] Thus, P(getting a white ball)...

A bag contains \[\mathbf{3}\] white, \[\mathbf{5}\] black and \[\mathbf{2}\] red balls, all of the same shape and size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is: \[\left( \mathbf{iii} \right)\] a white ball. \[\left( \mathbf{iv} \right)\] not a red ball.

CBSE, Class 10, Selina, Statistics and Probability

Solution \[\left( iii \right)\]There are \[3\] white balls So, the number of favourable outcomes \[=\text{ }3\] Thus, P(getting a white ball) \[=~3/10\text{ }=\text{ }3/10\] \[\left( iv...

A bag contains \[\mathbf{3}\] white, \[\mathbf{5}\] black and \[\mathbf{2}\] red balls, all of the same shape and size. A ball is drawn from the bag without looking into it, find the probability that the ball drawn is: \[\left( \mathbf{i} \right)\]a black ball. \[\left( \mathbf{ii} \right)\] a red ball.

CBSE, Class 10, Probability, Selina

Total number of balls \[=\text{ }3\text{ }+\text{ }5\text{ }+\text{ }2\text{ }=\text{ }10\] So, the total number of possible outcomes \[=\text{ }10\] \[\left( i \right)~\] There are \[5\] black...

\[\]A coin is tossed once. Find the probability of: (i) getting a tail (ii) not getting a tail

CBSE, Class 10, Probability, Selina

Here, the sample space \[=\text{ }\left\{ H,\text{ }T \right\}\] \[i.e.\text{ }n\left( S \right)\text{ }=\text{ }2\] (i) If A = Event of getting a tail \[=\text{ }\left\{ T \right\}\] Then\[,\text{...

From a window A, 10 m above the ground the angle of elevation of the top C of a tower is xo, where tan xo = 5/2 and the angle of depression of the foot D of the tower is yo, where tan yo = 1/4. Calculate the height CD of the tower in metres.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

SOLUTION: Since, \[AB\text{ }=\text{ }DE\text{ }=\text{ }10\text{ }m\] So, in ∆ABC \[\begin{array}{*{35}{l}} DE/AE\text{ }=\text{ }tan\text{ }y\text{ }=\text{ }{\scriptscriptstyle 1\!/\!{ }_4} \\...

The angles of elevation of the top of a tower from two points on the ground at distances a and b meters from the base of the tower and in the same line are complementary. Prove that the height of the tower is √ab meter.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

SOLUTION: Let AB to be the tower of height h meters and let C and D be two points on the level ground such that BC = b meters, BD = a meters, ∠ACB = α, ∠ADB = β. Given, \[\alpha \text{ }+\text{...

With reference to the given figure, a man stands on the ground at point A, which is on the same horizontal plane as B, the foot of the vertical pole BC. The height of the pole is 10 m. The man’s eye s 2 m above the ground. He observes the angle of elevation of C, the top of the pole, as xo, where tan xo = 2/5. Calculate: (i) the distance AB in metres; (ii) angle of elevation of the top of the pole when he is standing 15 metres from the pole. Give your answer to the nearest degree.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

Let AD to be the height of the man, AD = 2 m. \[=>\text{ }CE\text{ }=\text{ }\left( 10\text{ }-\text{ }2 \right)\text{ }=\text{ }8\text{ }m\] (i) In ∆CED, \[\begin{array}{*{35}{l}} CE/DE\text{...

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height h meter. At a point on the plane, the angle of elevation of the bottom of the flagstaff is α and at the top of the flagstaff is β. Prove that the height of the tower is h tan α/ (tan β – tan α).

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

SOLUTION: Let AB be the tower of height x metre, surmounted by a vertical flagstaff AD. Let C be a point on the plane such that ∠ACB = α, ∠ACB = β and AD = h. In ∆ABC, \[\begin{array}{*{35}{l}}...

At a point on level ground, the angle of elevation of a vertical tower is found to be such that its tangent is 5/12. On walking 192 meters towards the tower, the tangent of the angle is found to be 3/4. Find the height of the tower.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

SOLUTION: Let AB to be the vertical tower and C and D be the two points such that CD = 192 m. And let ∠ACB = θ and ∠ADB = α \[\begin{array}{*{35}{l}} tan\text{ }\theta \text{ }=\text{ }5/12 \\...

The radius of a circle is given as 15 cm and chord AB subtends an angle of 131o at the centre C of the circle. Using trigonometry, calculate: (i) the length of AB; (ii) the distance of AB from the centre C.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

Since, CA = CB = 15 cm and ∠ACB = 131o Constructing a perpendicular CP from centre C to the chord AB,we get CP bisects ∠ACB as well as chord AB. =>∠ACP = 65.5o In ∆ACP, \[\begin{array}{*{35}{l}}...

Calculate AB.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

SOLUTION: In ∆AMOB, \[\begin{array}{*{35}{l}} cos\text{ }{{30}^{o}}~=\text{ }AO/MO \\ \surd 3/2\text{ }=\text{ }AO/6 \\ AO\text{ }=\text{ }5.20\text{ }m \\ \end{array}\] In ∆BNO,...

Calculate BC.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

SOLUTION: In ∆ADC, \[\begin{array}{*{35}{l}} CD/AD\text{ }=\text{ }tan\text{ }{{42}^{o}} \\ CD\text{ }=\text{ }20\text{ x }0.9004\text{ }=\text{ }18.008\text{ }m \\ \end{array}\] In ∆ADB,...

In the following diagram, AB is a floor-board; PQRS is a cubical box with each edge = 1 m and ∠B = 60o. Calculate the length of the board AB.

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

SOLUTION: In ∆PSB, \[\begin{array}{*{35}{l}} PS/PB\text{ }=\text{ }sin\text{ }{{60}^{o}} \\ PB\text{ }=\text{ }2/\text{ }\surd 3\text{ }=\text{ }1.155\text{ }m \\ \end{array}\] In ∆APQ,...

Find AD in FIG-1 and FIG-2

Class 10, Exercise 22C, Heights and Distances, ICSE, Selina

(i) FIG-1 SOLUTION: In ∆AEB, \[\begin{array}{*{35}{l}} AE/BE\text{ }=\text{ }tan\text{ }{{32}^{o}} \\ AE\text{ }=\text{ }20\text{ x }0.6249\text{ }=\text{ }12.50\text{ }m \\ AD\text{ }=\text{...

The angle of elevation of the top of a tower is observed to be 60o. At a point, 30 m vertically above the first point of observation, the elevation is found to be 45o. Find: (i) the height of the tower, (ii) its horizontal distance from the points of observation.

Class 10, Exercise 22B, Heights and Distances, ICSE, Selina

Let AB to be the tower of height h meters and let the two points be C and D be such that CD = 30 m, ∠ADE = 45o and ∠ACB = 60o (i) In ∆ADE, \[\begin{array}{*{35}{l}} AE/DE\text{ }=\text{ }tan\text{...

From the figure, given below, calculate the length of CD.

Class 10, Exercise 22B, Heights and Distances, ICSE, Selina

SOLUTION: In ∆AED, \[\begin{array}{*{35}{l}} AE/\text{ }DE\text{ }=\text{ }tan\text{ }{{22}^{o}} \\ AE\text{ }=\text{ }DE\text{ }tan\text{ }{{22}^{o}}~=\text{ }15\text{ }x\text{ }0.404\text{...

Two pillars of equal heights stand on either side of a roadway, which is 150 m wide. At a point in the roadway between the pillars the elevations of the tops of the pillars are 60o and 30o; find the height of the pillars and the position of the point.

Class 10, Exercise 22B, Heights and Distances, ICSE, Selina

Let AB and CD be the two towers of height h m each and let P be a point in the roadway BD such that BD = 150 m, ∠APB = 60o and ∠CPD = 30o In ∆ABP, \[\begin{array}{*{35}{l}} AB/BP\text{ }=\text{...

From the top of a light house 100 m high, the angles of depression of two ships are observed as 48o and 36o respectively. Find the distance between the two ships (in the nearest metre) if: (i) the ships are on the same side of the light house. (ii) the ships are on the opposite sides of the light house.

Class 10, Exercise 22B, Heights and Distances, ICSE, Selina

Let AB to be the lighthouse and the two ships be C and D such that ∠ADB = 36o and ∠ACB = 48o In ∆ABC, \[\begin{array}{*{35}{l}} AB/BC\text{ }=\text{ }tan\text{ }{{48}^{o}} \\ BC\text{ }=\text{...

Find the height of a building, when it is found that on walking towards it 40 m in a horizontal line through its base the angular elevation of its top changes from 30o to 45o.

CBSE, Class 12, Exercise 22B, Heights and Distances, Selina

Let AB to be the building of height h meters. and let the two points be C and D be such that CD = 40 m, ∠ADB = 30o and ∠ACB = 45o In ∆ABC, \[\begin{array}{*{35}{l}} AB/BC\text{ }=\text{ }tan\text{...

Find the height of a tree when it is found that on walking away from it 20 m, in a horizontal line through its base, the elevation of its top changes from 60 to 30.

Class 10, Exercise 22B, Heights and Distances, ICSE, Selina

Let AB to be the height of the tree, h m. and let the two points be C and D be such that CD = 20 m, ∠ADB = 30o and ∠ACB = 60o In ∆ABC, \[\begin{array}{*{35}{l}} AB/BC\text{ }=\text{ }tan\text{...

In the figure, given below, it is given that AB is perpendicular to BD and is of length X metres. DC = 30 m, ∠ADB = 30 and ∠ACB = 45. Without using tables, find X.

Class 10, Exercise 22B, Heights and Distances, ICSE, Selina

In ∆ABC, \[\begin{array}{*{35}{l}} AB/BC\text{ }=\text{ }tan\text{ }{{45}^{o}}~=\text{ }1 \\ =>\text{ }BC\text{ }=\text{ }AB\text{ }=\text{ }X \\ \end{array}\] In ∆ABD,...

A boy, 1.6 m tall, is 20 m away from a tower and observes the angle of elevation of the top of the tower to be (i) 45, (ii) 60. Find the height of the tower in each case.

Class 10, Exercise 22A, Heights and Distances, ICSE, Selina

Let the height of the tower to be ‘h’ m. (i) Since, \[\begin{array}{*{35}{l}} \theta \text{ }=\text{ }{{45}^{o}} \\ tan\text{ }{{45}^{o}}~=\text{ }\left( h\text{ }-\text{ }1.6 \right)/\text{ }20 ...

A kite is attached to a string. Find the length of the string, when the height of the kite is 60 m and the string makes an angle 30o with the ground.

Class 10, Exercise 22A, Heights and Distances, ICSE, Selina

Let the length of the rope to be x meters. \[\begin{array}{*{35}{l}} sin\text{ }{{30}^{o}}~=\text{ }60/x \\ {\scriptscriptstyle 1\!/\!{ }_2}\text{ }=\text{ }60/x \\ x\text{ }=\text{ }120\text{ }m ...

Two persons are standing on the opposite sides of a tower. They observe the angles of elevation of the top of the tower to be 30o and 38o respectively. Find the distance between them, if the height of the tower is 50 m.

Class 10, Exercise 22A, Heights and Distances, ICSE, Selina

Let one of the persons be A , at a distance of ‘x’ meters and the second person be B at a distance of ‘y’ meters from the foot of the tower. The angle of elevation of A is 30o...

A ladder is placed along a wall such that its upper end is resting against a vertical wall. The foot of the ladder is 2.4 m from the wall and the ladder is making an angle of 68o with the ground. Find the height, up to which the ladder reaches.

Class 10, Exercise 22A, Heights and Distances, ICSE, Selina

Let the height upto which the ladder reaches as ‘h’ meters. the angle of elevation is 68o \[\begin{array}{*{35}{l}} =>\text{ }tan\text{ }{{68}^{o}}~=\text{ }h/\text{ }2.4 \\ 2.475\text{ }=\text{...

The angle of elevation of the top of a tower from a point on the ground and at a distance of 160 m from its foot, is found to be 60o. Find the height of the tower.

Class 10, Exercise 22A, Heights and Distances, ICSE, Selina

Let the height of the tower to be h meters. the angle of elevation is 60o => \[\begin{array}{*{35}{l}} tan\text{ }{{60}^{o}}~=\text{ }h/160 \\ \surd 3\text{ }=\text{ }h/160 \\ h\text{ }=\text{...

The height of a tree is √3 times the length of its shadow. Find the angle of elevation of the sun.

Class 10, Exercise 22A, Heights and Distances, ICSE, Selina

Let the length of the shadow of the tree to be x meters. Therefore, the height of the tree = √3 x meters If θ is the angle of elevation of the sun, \[\begin{array}{*{35}{l}} =>\text{ }tan\text{...

If m and n are roots of the equation: 1/x – 1/(x-2) = 3: where x ≠ 0 and x ≠ 2; find m x n.

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to ques, \[~1/x\text{ }-\text{ }1/\left( x-2 \right)\text{ }=\text{ }3\] \[\left( x\text{ }-\text{ }2\text{ }\text{ }x \right)/\text{ }\left( x\left( x\text{ }-\text{ }2 \right)...

Find the solution of the quadratic equation 2×2 – mx – 25n = 0; if m + 5 = 0 and n – 1 = 0.

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to ques, \[m\text{ }+\text{ }5\text{ }=\text{ }0\text{ }and\text{ }n\text{ }\text{ }1\text{ }=\text{ }0\] hence, \[m\text{ }=\text{ }-5\text{ }and\text{ }n\text{ }=\text{ }1\] putting...

If p – 15 = 0 and 2×2 + px + 25 = 0: find the values of x.

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to the given question, \[p\text{ }\text{ }15\text{ }=\text{ }0\] And \[2{{x}^{2}}~+\text{ }px\text{ }+\text{ }25\text{ }=\text{ }0\] Thus, \[p\text{ }=\text{ }15\] Using\[p\]in the...

Show that one root of the quadratic equation x2 + (3 – 2a)x – 6a = 0 is -3. Hence, find its other root.

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to the given equation, \[{{x}^{2}}~+\text{ }\left( 3\text{ }\text{ }2a \right)x\text{ }\text{ }6a\text{ }=\text{ }0\] By putting \[x\text{ }=\text{ }-3\]we get \[{{\left( -3...

One root of the quadratic equation 8×2 + mx + 15 = 0 is ¾. Find the value of m. Also, find the other root of the equation.

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to the given question, \[8{{x}^{2~}}+\text{ }mx\text{ }+\text{ }15\text{ }=\text{ }0\] One of the roots is\[~{\scriptscriptstyle 3\!/\!{ }_4},\]and it satisfies the given equation So,...

Solve: 2x – 3 = √(2×2 – 2x + 21)

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to the given question, \[2x\text{ }\text{ }3\text{ }=\text{ }\surd (2{{x}^{2}}~\text{ }2x\text{ }+\text{ }21)\] Squaring on both sides, we get \[{{\left( 2x\text{ }\text{ }3...

Solve: . \[\left( \mathbf{i} \right)\text{ }\mathbf{3}\surd \mathbf{2}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{5x}\text{ }\text{ }\surd \mathbf{2}\text{ }=\text{ }\mathbf{0}\]

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to the given question, \[3\surd 2{{x}^{2}}~\text{ }5x\text{ }\text{ }\surd 2\text{ }=\text{ }0\] Or, \[3\surd 2{{x}^{2}}~\text{ }6x\text{ }+\text{ }x\text{ }\text{ }\surd 2\text{ }=\text{...

solve: \[\left( \mathbf{ii} \right)\text{ }\mathbf{3}{{\mathbf{x}}^{\mathbf{2}}}~\text{ }\mathbf{2}\surd \mathbf{6x}\text{ }+\text{ }\mathbf{2}\text{ }=\text{ }\mathbf{0}\]

Class 10, Exercise 5F, ICSE, Maths, Quadratic Equations, Selina

According to the given equation, \[3{{x}^{2}}~\text{ }2\surd 6x\text{ }+\text{ }2\text{ }=\text{ }0\] Or, \[3{{x}^{2}}~\text{ }\surd 6x\text{ }\text{ }\surd 6x\text{ }+\text{ }2\text{ }=\text{ }0\]...