ICSE

Two straight lines PQ and PK cross each other at P at an angle of 750. S is a stone on the road PQ, 800 m from P towards Q. By drawing a figure to scale 1 cm = 100 m, locate the position of a flagstaff X, which is equidistant from P and S, and is also equidistant from the road.

We know that \[\begin{array}{*{35}{l}} 1\text{ }cm\text{ }=\text{ }100\text{ }cm  \\ 800\text{ }m\text{ }=\text{ }8\text{ }cm  \\ \end{array}\] Following are the steps of Construction: (i) Construct...

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Draw a line segment AB of length 12 cm. Mark M, the mid-point of AB. Draw and describe the locus of a point which is (i) at a distance of 3 cm from AB. (ii) at a distance of 5 cm from the point M. Mark the points P, Q, R, S which satisfy both the above conditions. What kind of quadrilateral is PQRS? Compute the area of the quadrilateral PQRS.

Following are the steps of Construction: (i) Construct a line AB = 12 cm. (ii) Take M as the midpoint of line AB. (iii) Construct straight lines CD and EF which is parallel to AB at 3 cm distance....

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Ruler and compasses only may be used in this question. All construction lines and arcs must be clearly shown and be of sufficient length and clarity to permit assessment. (i) Construct a triangle ABC, in which BC = 6 cm, AB = 9 cm and ∠ABC = 600. (ii) Construct the locus of all points, inside Δ ABC, which are equidistant from B and C.(iii) Construct the locus of the vertices of the triangle with BC as base, which are equal in area to Δ ABC. (iv) Mark the point Q, in your construction, which would make Δ QBC equal in area to Δ ABC and isosceles. (v) Measure and record the length of CQ.

Following are the steps of Construction: (i) Construct \[AB\text{ }=\text{ }9\text{ }cm\] (ii) At the point B construct an angle of 600 and cut off \[BC\text{ }=\text{ }6\text{ }cm.\] (iii) Now join...

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Use ruler and compasses only for this question. Draw a circle of radius 4 cm and mark two chords AB and AC of the circle of length 6 cm and 5 cm respectively. (i) Construct the locus of points, inside the circle, that are equidistant from A and C. Prove your construction. (ii) Construct the locus of points, inside the circle, that are equidistant from AB and AC.

Following are the steps of Construction: (i) Taking O as centre and 4 cm radius construct a circle. (ii) Mark a point A on this circle. (iii) Taking A and centre and 6 cm radius construct an arc...

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While boarding an aeroplane, a passengers got hurt. The pilot showing promptness and concern, made arrangements to hospitalize the injured and so the plane started late by 30 minutes. To reach the destination, 1500 \mathrm{~km} away, in time, the pilot increased the speed by 100 \mathrm{~km} / hour. Find the original speed of the plane. Do you appreciate the values shown by the pilot, namely promptness in providing help to the injured and his efforts to reach in time?

Let the original speed of the plane be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Actual speed of the plane $=(x+100) \mathrm{km} / \mathrm{h}$ Distance of the journey $=1500 \mathrm{~km}$ Time...

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Without using set-squares or protractor construct: (i) Triangle ABC, in which AB = 5.5 cm, BC = 3.2 cm and CA = 4.8 cm. (ii) Draw the locus of a point which moves so that it is always 2.5 cm from B. (iii) Draw the locus of a point which moves so that it is equidistant from the sides BC and CA. (iv) Mark the point of intersection of the loci with the letter P and measure PC.

Steps of Construction: (i) Construct \[BC\text{ }=\text{ }3.2\text{ }cm\] long. (ii) Taking B as centre and 5.5 cm radius and C as centre and 4.8 cm radius construct arcs intersecting each other at...

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Points A, B and C represent position of three towers such that AB = 60 mm, BC = 73 mm and CA = 52 mm. Taking a scale of 10 m to 1 cm, make an accurate drawing of Δ ABC. Find by drawing, the location of a point which is equidistant from A, B and C and its actual distance from any of the towers.

According to ques, \[AB\text{ }=\text{ }60\text{ }mm\text{ }=\text{ }6\text{ }cm\] \[BC\text{ }=\text{ }73\text{ }mm\text{ }=\text{ }7.3\text{ }cm\] \[CA\text{ }=\text{ }52\text{ }mm\text{ }=\text{...

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In the diagram, A, B and C are fixed collinear points; D is a fixed point outside the line. Locate:(iii) the points R on AB such that DR = 4 cm. How many such points are possible? (iv) the points S such that CS = DS and S is 4 cm away from the line CD. How many such points are possible?

Here the points A, B and C are collinear and D is any point which is outside AB.   (iii) Taking D as centre and 4 cm radius construct an arc which intersects AB at R and R’ now R and R’ are the...

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Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 1050. Hence: (i) Construct the locus of points equidistant from BA and BC. (ii) Construct the locus of points equidistant from B and C. (iii) Mark the point which satisfies the above two loci as P. Measure and write the length of PC.

Following are the Steps of Construction: Construct a triangle ABC with AB = 5.5 cm, AC = 6 cm and ∠BAC = 1050. (i) The points which are equidistant from BA and BC lies on the bisector of ∠ABC. (ii)...

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Use ruler and compasses only for this question.(i) Construct the locus of points inside the triangle which are equidistant from B and C. (ii) Mark the point P which is equidistant from AB, BC and also equidistant from B and C. Measure and record the length of PB.

Since, In Δ ABC, AB = 3.5 cm, BC = 6 cm and ∠ABC = 600 (i) Construct a perpendicular bisector of BC which intersects BY at point P. (ii) It is given that point P is equidistant from AB, BC and also...

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A straight line AB is 8 cm long. Locate by construction the locus of a point which is: (i) Equidistant from A and B. (ii) Always 4 cm from the line AB. (iii) Mark two points X and Y, which are 4 cm from AB and equidistant from A and B. Name the figure AXBY.

Following are the Steps of Construction, (i) Construct a line segment AB = 8 cm. (ii) Using compasses and ruler, construct a perpendicular bisector l of AB which intersects AB at the point O. (iii)...

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Describe completely the locus of points in each of the following cases: (v) centre of a circle of varying radius and touching two arms of ∠ADC. (vi) centre of a circle of varying radius and touching a fixed circle, centre O, at a fixed point A on it.

(v) Construct the bisector BX of ∠ABC. hence, this bisector of an angle is the locus of the centre of a circle having different radii. now, any point on BX is equidistant from BA and BC which are...

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A farmer wishes to grow a

    \[100{{m}^{2}}\]

rectangular vegetable garden. Since he has with him only

    \[30\]

m barbed wire, he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side fence. Find the dimensions of his garden.

Given, Area of rectangular garden = \[100{{m}^{2}}\] Length of barbed wire = \[30\] m Let’s assume the length of the side opposite to wall to be x And the length of other side = \[\left( 30\text{...

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A model of a ship is made to a scale of 1: 250 calculate:
(i) The length of the ship, if the length of model is 1.6 m.
(ii) The area of the deck of the ship, if the area of the deck of model is 2.4 m2.
(iii) The volume of the model, if the volume of the ship is 1 km3.

Solution:- From the question it is given that, a model of a ship is made to a scale of 1 : 250 (i) Given, the length of the model is 1.6 m Then, length of the ship = (1.6 × 250)/1 = 400 m (ii)...

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The hypotenuse of grassy land in the shape of a right triangle is

    \[1\]

metre more than twice the shortest side. If the third side is

    \[7\]

metres more than the shortest side, find the sides of the grassy land.

Let’s consider the shortest side to be ‘x’ cm Hypotenuse = \[2x\text{ }+\text{ }1\] And third side = \[x\text{ }+\text{ }7\] Now, by Pythagoras theorem we have \[\begin{array}{*{35}{l}} {{\left(...

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The perimeter of a rectangular plot is

    \[180\]

m and its area is

    \[1800\]

    \[{{m}^{2}}\]

. Take the length of the plot as x m. Use the perimeter

    \[180\]

m to write the value of the breadth in terms of x. Use the values of length, breadth and the area to write an equation in x. Solve the equation to calculate the length and breadth of the plot. (1993)

Given, The perimeter of a rectangular field = \[180\]m And area = \[1800\] \[{{m}^{2}}\] Let’s assume the length of the rectangular field as ‘x’ m We know that, Perimeter of rectangular field =...

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A rectangular garden

    \[10\]

m by

    \[16\]

m is to be surrounded by a concrete walk of uniform width. Given that the area of the walk is

    \[120\]

square meters, assuming the width of the walk to be x, form an equation in x and solve it to find the value of x. (1992)

Given: Length of garden = \[16\]cm Width = \[10\]cm Let the width of walk be ‘x’ meter Outer length = \[16+2x\] Outer width = \[10+2x\] So according to the question, \[\begin{array}{*{35}{l}} \left(...

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A model of a ship is made to a scale of 1 : 200.
(i) If the length of the model is 4 m, find the length of the ship.
(ii) If the area of the deck of the ship is 160000 m², find the area of the deck of the model.
(iii) If the volume of the model is 200 liters, find the volume of the ship in m³. (100 liters = 1 m³)

Solution:- From the question it is given that, a model of a ship is made to a scale of 1 : 200 (i) Given, the length of the model is 4 m Then, length of the ship = (4 × 200)/1 = 800 m (ii) Given,...

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The model of a building is constructed with the scale factor 1 : 30. (i) If the height of the model is 80 cm, find the actual height of the building in metres. (ii) If the actual volume of a tank at the top of the building is 27 m³, find the volume of the tank on the top of the model.

Solution:- From the question it is given that, The model of a building is constructed with the scale factor 1 : 30 So, Height of the model/Height of actual building = 1/30 (i) Given, the height of...

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In a certain positive fraction, the denominator is greater than the numerator by

    \[3\]

. If 1 is subtracted from both the numerator and denominator, the fraction is decreased by

    \[1/14\]

. Find the fraction.

Let the denominator be ‘x’ So the numerator will be ‘\[8-x\]’ The obtained fraction is \[8-x/x\] So according to the question, By cross multiplying, \[\begin{array}{*{35}{l}} 35\left( 4x\text{...

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On a map drawn to a scale of 1 : 25000, a rectangular plot of land, ABCD has the following measurements AB = 12 cm and BG = 16 cm. Calculate:
(i) the distance of a diagonal of the plot in km.
(ii) the area of the plot in sq. km.

Solution:- From the question it is given that, Map drawn to a scale of 1: 25000 AB = 12 cm, BG = 16 cm Consider the ∆ABC, From the Pythagoras theorem, AC2 = AB2 + BC2 AC = √(AB2 + BC2) = √((12)2 +...

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ABC is a right angled triangle with ∠ABC = 90°. D is any point on AB and DE is perpendicular to AC. Prove that:
(i) ∆ADE ~ ∆ACB.
(ii) If AC = 13 cm, BC = 5 cm and AE = 4 cm. Find DE and AD.
(iii) Find, area of ∆ADE : area of quadrilateral BCED.

Solution:- From the question it is given that, ∠ABC = 90° AB and DE is perpendicular to AC (i) Consider the ∆ADE and ∆ACB, ∠A = ∠A … [common angle for both triangle] ∠B = ∠E … [both angles are equal...

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In the adjoining figure, ABCD is a parallelogram. P is a point on BC such that BP : PC = 1 : 2 and DP produced meets AB produced at Q. If area of ∆CPQ = 20 cm², find
(i) area of ∆BPQ.
(ii) area ∆CDP.
(iii) area of parallelogram ABCD.

Solution:- From the question it is given that, ABCD is a parallelogram. BP: PC = 1: 2 area of ∆CPQ = 20 cm² Construction: draw QN perpendicular CB and Join BN. Then, area of ∆BPQ/area of ∆CPQ =...

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In the figure (iii) given below, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O and EF || BC. If AE : EB = 2 : 3, find
(i) EF : AD
(ii) area of ∆BEF : area of ∆ABD In the figure
(iii) given below, ABCD is a parallelogram
(iv) area of ∆FEO : area of ∆OBC.

Solution:- From the question it is given that, ABCD is a parallelogram. E is a point on AB, CE intersects the diagonal BD at O. AE : EB = 2 : 3 (i) We have to find EF : AD So, AB/BE = AD/EF EF/AD =...

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In the adjoining figure, ABC is a triangle. DE is parallel to BC and AD/DB = 3/2,
(i) Determine the ratios AD/AB, DE/BC0
(ii) Prove that ∆DEF is similar to ∆CBF. Hence, find EF/FB.
(iii) What is the ratio of the areas of ∆DEF and ∆CBF?

Solution:- (i) We have to find the ratios AD/AB, DE/BC, From the question it is given that, AD/DB = 3/2 Then, DB/AD = 2/3 Now add 1 for both LHS and RHS we get, (DB/AD) + 1 = (2/3) + 1 (DB + AD)/AD...

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Given that ∆s ABC and PQR are similar. Find: (i) The ratio of the area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1 : 3. (ii) the ratio of their corresponding sides if area of ∆ABC : area of ∆PQR = 25 : 36.

Solution:- From the question it is given that, (i) The area of ∆ABC to the area of ∆PQR if their corresponding sides are in the ratio 1 : 3 Then, ∆ABC ~ ∆PQR area of ∆ABC/area of ∆PQR = BC2/QR2 So,...

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