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ICSE Notes 2017 : Mathematics (Infant Jesus Anglo - Indian Higher Secondary School (IJHSS), Kollam)

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Maths 1. Measures of Central Tendency 1. The marks of 20 students in a test were as follows: 5, 6, 8, 9, 10, 11, 11, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 18, 19, 20. Calculate the mean, median and mode. 2. Given below are the weekly wages of 200 workers in a factory: Weekly wages in rupees Number of workers 80 100 20 100 120 30 120 140 20 140 160 40 160 180 90 Calculate the mean weekly wages of the workers. 3. If the mean of 5, 4, 6, 3, 2x + 4, 3x 6, 5, 6, and 7 is 10, find the value of x. 4. The marks obtained by a set of students in an examination are given below: Marks 5 10 15 20 25 30 No. of students 6 4 6 12 x 4 Given that the mean mark of the set is 18, calculate the value of x. 5. If the median of 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 is 24, find the value of x. 6. Calculate the mean, the median and the mode of the following numbers: 3, 1, 5, 6, 3, 4, 5, 3, 7, 2. 7. If the mean of the following distribution is 7.5, find the missing frequency f: Variable 5 6 7 8 9 10 11 12 Frequency 20 17 f 10 8 6 7 6 8. If the median of 11, 12, 14, 18, x + 2, x + 4, 30, 32, 35, 41 is 24, find the value of x. 9. The median of the following observation 11, 12, 14, 18, (x + 4), 30, 32, 35, 41 arranged in ascending order is 24. Find x. 10. For the following set of numbers find the median: 10, 75, 3, 81, 17, 27, 4, 48, 12, 47, 9, 15. 11. Find the mean of the following distribution: Class Interval 0 10 10 20 20 30 30 40 40 50 Frequency 10 6 8 12 5 Q.11. Find the mean of the following frequency distribution: Class Interval Frequency 0 50 4 50 100 8 100 150 16 150 200 13 200 250 6 250 300 3 Q.12. The following table gives the weekly wages of workers in a factory: Weekly wages in Rs No. of workers 50 55 5 55 60 20 60 65 10 65 70 10 70 75 9 75 80 6 80 85 12 85 90 8 Calculate: (i) The mean. (ii) The modal class. (iii) The number of workers getting weekly wages, below Rs80/(iv) The number of workers getting Rs65/- or more, but less than Rs85/- as weekly wages. 2. Trigonometrical Identities Q.1. Prove the following identities: secA 1 1 cosA (i) secA+1 = 1+cosA 1 1 2 s in + = (ii) sin +cos sin cos 1 2cos2 (iii) cot2A cos2A = cot2A . cos2A sec +tan 1+sin (iv) 1 sin = (v) 1 sin 1+sin sec tan sin A 1+cosA (vi) 1+cosA + sin A = 2cosecA cosA sin A (vii) 1 tanA + 1 cotA cos A+sin A 1 cos 1+sin sin (viii) 1 cosA sin A (ix) 1+cosA =1+cosA (x) sin .tan 1 cos = 1+sec (xi) (1 + tanA)2 + (1 tanA)2 = 2 sec2A Q.2.Evaluate, without the use of trigonometric tables: sin 80 sin 59 (i) cos10 + sec31 3sin 72 sec32 (ii) cos18 cosec58 (iii) 3cos80 .cosec10 +2cos59 .cosec31 cos75 sin 12 cos18 (iv) sin15 + cos78 sin72 2tan53 cot80 (v) cot37 tan10 Q.3. Without using mathematical tables. find the value of x: x= cos60 .cos30 +sin60 .sin30 cos . Q.4. Without using table, find the value of : 14sin30 +6cos60 5tan45 . Q.5. If tan x+tan y. sin x= 3 5 y= and cos 12 13 evaluate : (a) 2 sec x . (b) 2sin A 1=0. Q.6. If Show that: sin3A=3sinA 4sin3A. . 3. Height and Distance Q.1. From the top of a hill, the angle of depression of two consecutive kilometre stones, due east are found to be 30o and 45o respectively. Find the distance of the two stones from the foot of the hill. [Use 3 = 1.732] Q.2. A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60o. When he moves 50 m away from the bank, he finds the angle of elevation to be 30o. Calculate : (i) the width of the river and (ii) the height of the tree. Q.3. In the figure (not drawn to scale) TF is a tower. The elevation of T, from A is x where tan x = o o 2 5 and AF = 200 m. The elevation of T from B, where AB = 80 m is yo. Calculate : (i) the height of the tower TF. (ii) the angle y correct to nearest degree. Q.4. The angle of elevation of the top of a tower from two points P and Q at a distance of a and b respectively, from the base and in the same straight line with it are complementary. Prove that height of the tower is ab . Q.5. From a window A, 10 m above ground the angle of elevation of the 5 top C to a tower is xo , where tan xo = 2 and the angle of depression of the foot D of the tower is y , here tan y = o height CD of the tower in metres. o 1 4 . Calculate the Q.6. The shadow of a vertical tower on a level ground increases by 10 m when the altitude of the sun changes from 450 to 300. Find the height of the tower, correct to two decimal places. Q.7. From the top of a cliff 92 m high, the angle of depression of a buoy is 200. Calculate to the nearest metre, the distance of the buoy from the foot of the cliff. Q.8. Two poles standing on the same side of a tower in a straight line with it, measure the angles of elevation of the top of the tower at 250 and 500respectively. If the height of the tower is 70 m, find the distance between the two people. Q.9. Vertical tower is 20 m high. A man standing at some distance from the tower knows that the cosine of the angle of elevation of the top of the tower is 0.53. How far is he standing from the foot of the tower? Q.10. With reference to the figure given a man stands on the ground at point A, which is on the same horizontal plane as B, the foot of a vertical pole BC. The height of the pole is 10 m. The man s eye is 2 m above the ground. He observes the angle of elevation at C, the top of the pole as 2 0 0 x , where tan x = 5 . Calculate : (i) The distance AB in metre. (ii) The angle of elevation of the top of the pole when he is standing 15 m from the pole. Give your answer to the nearest degree. Q.11. The figure drawn alongside is not to the scale. AB is a tower, and two objects C and D are located on the ground on the same side of AB. When observed from the top A of the tower, their angles of depression are 450 and 600. Find the distance between the two objects if the height of the tower is 300m. Give your answer to the nearest metre. 4. Construction Question 1. Use ruler and compass only in this question. (i) Draw a circle, with centre O and radius 4 cm. (ii) Mark a point P such that OP = 7 cm. Construct the two tangents to the circle from P. Measure and record the length of one of the tangents. Question 2. Construct an angle PQR = 450. Mark a point S on QR such that QS = 4.5 cm. Construct a circle to touch PQ at Q and also to pass through S. Question 3. Construct a ABC, in which AC = 5 cm, NC = 7 cm and AB = 6 cm. (i) Mark D, the midpoint of AB. (ii) Construct the circle which touches BC at C and passes through D. Question 4. Using ruler and compass only, construct a ABC such that AB = 5 cm, ABC = 750 and the radius of the circum-circle of triangle ABC=3.5 cm. On the same figure, construct a circle touching AB at its middle point, and also touching the side AC. Question 5. Only ruler and compass may be used in this question, (i) Construct ABC, such that AB = AC = 7 cm, BC = 5 cm. (ii) Draw AX, the perpendicular bisector of side BC. (iii) Draw a circle with centre A and radius 3 cm cutting AX at Y . (iv) Construct another circle to touch the circle with centre A externally at Y and passing through B and C. Question 6. Ruler and compass 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 ABC, in which AB = 9 cm, BC = 10 cm and ABC = 450. (ii) Draw a circle, with centre A and radius 2.5 cm. Let it meet AB at D. (iii) Construct a circle to touch the circle with centre A externally at D and also to touch the line BC. Question 7. Using ruler and compass only, construct (i) A triangle ABC in which AB = 9 cm, BC = 10 cm and ABC = 450. (ii) construct a circle of radius 2 cm to touch the arms of ACB in (i) above. Question 8. Use a ruler and a pair of compasses to construct a ABC in which BC = 4.2 cm, ABC = 600 and AB = 5 cm. Construct a circle of radius 2 cm to touch both the arms of ABC of ABC. Question 9. Using a ruler, construct a triangle ABC with BC = 6.4 cm, CA = 5.8 cm and ABC = 600. Draw its in-circle. Measure and record the radius of the in-circle. Question 10. Using ruler and compass only: (i) Construct a triangle ABC with BC = 6 cm, ABC = 1200 and AB = 3.5 cm. (ii) In the above figure, draw a circle with BC as diameter. Find a point P on the circumference of the circle which is equidistant from AB and BC. (iii) Measure BCP. [Ans. = BCP = 30 ] Q.4. Construct a triangle ABC , in which AB = AC = 3 cm and BC = 2 cm. Using a ruler and compasses only, draw the reflection A B C of ABC, in BC. Draw the lines of symmetry of the figure ABA C. Q.5. Construct an isosceles triangle ABC such that AB = 6 cm, BC = AC = 4 cm. Bisect LC internally and mark a point P on this bisector such that CP = 5 cm. Find the points Q and R which are at 5 cm from P and also from the line AB. Q.6. Construct a triangle ABC with AB = 7 cm, BC = 8 cm and LABC = 600. Locate by construction the point P such that : (i) P is equidistant from B and C. (ii) P is equidistant from AB and BC. (iii) Measure and record the length of PB. [Ans. PB = 4.5 cm.] 5. Locus Q.1. Construct triangle BCP, where CB = 5 cm, BP = 4 cm, PBC = 450. Complete the rectangle ABCD such that (i) P is equidistant from AB and BC; and (ii) P is equidistant from C and D. Measure and write down the length of AB. Q.2. (i) Construct a ABC, in which BC = 6 cm, AB = 9 cm and ABC = 600. (ii) Construct the locus of all points inside ABC, which are equidistant from B and C. (iii) Construct the locus of the vertices of the triangle with BC as base, which are equal in area to triangle ABC. (iv) Mark the point Q in your construction, which would make QBC equal in area to ABC, and isosceles. (v) Measure and record the length of CQ. Q.3. Use graph paper for this question. Take2 cm = 1 unit on both axis. (i) Plot the points A(1,1), B(5,3) and C(2,7). (ii) Construct the locus of points equidistant from A and B. (iii) Construct the locus of points equidistant from AB and AC. (iv) Locate the point P such that PA = PB and P is equidistant from AB and AC. (v) measure and record the length PA in cm. Q.4. If the bisector of angles A and B of a quadrilateral ABCD intersect each other at the point P, prove that P is equidistant from AD and BC. Q.5. Using only a ruler and compasses, construct ABC = 1200, where AB = BC = 5 cm. (a) Mark two points D and E which satisfy the condition that they are equidistant from both BA and BC. (b) In the above figure, join AE and EC. Describe the figures : (i) ABCD (ii) ABD (iii) ABE. Q.6. State the locus of a point in a rhombus ABCD, which is equidistant : (i) From AB and AD. (ii) From the vertices A and C. 6. Circle Question 1. In the figure given below, there are two concentric circles and AD is a chord of larger circle. Prove that AB = CD. Question 2. In the figure given below, AOE is a diameter of a circle, write down the measure of sum of angles ABC and CDE. Give reasons of your answer. Question 3. In the figure given below, AC is a diameter of a circle with centre O. Chord BD is perpendicular to AC. Write down the angles p, q, r in terms of x . Question 4. In the given circle below, find the value of x. Question 5. A circle with centre O, diameter AB and a chord AD is drawn. Another circle is drawn with AO as diameter to cut AD at C. Prove that BD = 2OC. Question 6. In the figure given below, PT touches a circle with centre O at R. Diameter SQ when produced meets PT at P. If SPR = x0 and QRP = y0, show that x0 + 2y0 = 900. Question 7. In the figure given below, O is the centre of the circle and AOC = 1600. Prove that 3 y 2 x = 1400. Question 8. A, B and C are three points on a circle. The tangent at C meets BA produced at T. Given that ATC = 360 and that the ACT = 480. Calculate the angle subtended by AB at the centre of the circle. Question 9. In the given figure below, find TP if AT = 16 cm and AB = 12 cm. Q.10. In the figure, PM is a tangent to the circle and PA = AM. Prove that : (i) PMB is isosceles. (ii) PA PB = MB2. Q.11. In the given figure, O is the centre of the circle and PBA = 45 . Calculate the value of PQB. Q.12. In the given figure, if ACE = 430 and CAF = 620 , find the value of a, b and c. Q.13. In the given figure below, BAD = 650, ABD = 700 and BDC = 450. Find : (i) BCD. (ii) ADB. Hence show that AC is a diameter. . Q.14. In the given figure, AB is a diameter. The tangent at C meets AB produced at Q. If CAB = 340, find : (i) CBA. (ii) CQA. Q.15. PQR is a right angled triangle with PQ = 3 cm and QR = 4 cm. A circle which touches all the sides of the triangle is inscribed in the triangle. Calculate the radius of the circle. Q.16. In the figure, AB is a diameter and AC is a chord of a circle such that BAC = 300. The tangent at C intersects AB produced at D. Prove that BC = BD. Q.17. P and Q are centres of circles of radius 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of a circle of radius x cms, which touches the above circles externally. Given that PRQ = 900, write an equation in x and solve it. Q.18. In the given figure, AB is the diameter of a circle with centre O. BCD is 1200. Find: (i) DBA and (ii) BAD. Q.19. In the figure, chord AB and CD when extended meet at X. Given AB = 4 cm, BX = 6 cm, XD = 5 cm, calculate the length of CD. Q.20. In the figure, AB is a common tangent to two circles intersecting at C and D. Write down the measure of ( ACB + ADB). Justify your answer. Q.21. In the figure, AB and CD are the lines 2x y + 6 = 0 and x 2y = 4 respectively. (i) Write down the co-ordinates of A, B, C and D. (ii) Prove that triangles OAB and ODC are similar. (iii) Is figure ABCD cyclic? Give reasons for your answer. Q.22. In the figure given alongside, AD is a diameter of the circle. If LBCD = 130 , calculate : (i) DAB, (ii) ADB. Q.23. In the diagram given alongside, AC is the diameter of the circle with centre O. CD and BE are parallel. AOB = 800 and ACE = 100. Calculate : (i) BEC. (ii) BCD. (iii) CED. 7. Mensuration Q.1. In an equilateral ABC of side 14 cm, side BC is the diameter of a semi-circle as shown in the figure below. Find the area of the shaded 22 region. [Take = = 1.732] 7 and 3 Q.2. In the given figure, the area enclosed between the concentric circles is 770 cm2. If the radius of the outer circle is 21 cm, calculate the radius 22 7 of the inner circle.( = ) Q.3. In the given figure, the shape of the top of a table in a restaurant is that of a segment of a circle with centre O and BOD = 900 . BO = OD = 60 cm. Find : (i) The area of the top of the table. (ii) The perimeter of the table. [Take = 3.14] Q.4. A sheet is 11 cm long and 2 cm wide. Circular pieces of 0.5 cm in diameter are cut from it to prepare discs. Calculate the number of discs that can be prepared. Q.5. In the given figure, AC and BD are two perpendicular diameters of a circle ABCD. Given that the area of the shaded portion is 308 cm2, calculate : (i) the length of AC and 22 (ii) the circumference of the circle. [Take = 7 ] Q.6. PQRS is a diameter of a circle of radius 6 cm. The length PQ, QR and RS are equal. Semicircles are drawn on PQ and QS as diameter as shown in the figure below. Find the (i) area and (ii) of the shaded region. Q.7. In the given figure below, AB is the diameter of the circle with centre O and OA = 7 cm. Find the area of the shaded region. Q.8. Calculate the area of the shaded portion. The quadrants shown in 22 the figure are each of radius 7 cm. Take = 7 . Q.9. The figure given below shows a running track surrounding a grassed enclosure PQRSTU. The enclosure consists of a rectangle PQST with a semi-circular region at each end. PQ = 200 m; PT = 70 m. (i) Calculate the area of the grassed enclosure in m2. (ii) Given that the track is of constant width 7 m. Calculate the outer perimeter ABCDEF of the track. (Use = 22 7 ) Q.10. A solid consisting of a right circular cone, standing on a hemisphere, is placed upright, in a right circular cylinder, full of water, and touches the bottom. Find the volume of water left in the cylinder, having been given that the radius of the cylinder is 3 cm and its height is 6 cm; the radius of the hemisphere is 2 cm and the height of the one is 22 4 cm. Give your answer to the nearest cubic cm. (Take = 7 ) Q.11. With reference to the given figure, a metal container in the form of a cylinder is surmounted by a hemisphere of the same radius. The internal height of the cylinder is 7 m and internal radius is 3.5 m. Calculate : (i) the total area of the internal surface excluding the base. 22 (ii) the internal volume of the container in m3. [Take = 7 ] Q.12. The surface area of a solid metallic sphere is 1,256 cm2. It is melted and recast into solid right circular cones of radius 2.5 cm and height 8 cm. Calculate : (i) The radius of the solid sphere. (ii) The number of cones recast. [Take = 3.14] Q.13. An exhibition tent is in the form of a cylinder surmounted by a cone. The height of the tent above the ground is 85 m and the height of the cylindrical part is 50 m. If the diameter of the base is 168 m, find the quantity of canvas required to make the tent. Allow 20% extra for folds and for stitching. Give your answer to the nearest m2. Q.15. A vessel in the form of an inverted cone is filled with water to the brim. Its height is 20 cm and diameter is 16.8 cm. Two equal solid cones are dropped into it so that they are fully submerged. As a result, one third of the water in the original cone overflows. What is the volume of each of the solid cones submerged? Q.16. The surface area of a solid metallic sphere is 616 cm2. It is melted and recast into smaller spheres of diameter 3.5 cm. How many such spheres can be obtained? Q.17. A metallic sphere of radius 10.5 cm is melted and then recast into small cones, each of radius 3.5 cm and height 3 cm. Find the number of cones thus obtained. Q.18. A rectangular playground has two semicircles added to it s outside its smaller sides as diameters. If the sides of the rectangle are 120 m and 21 m, find the area of the playground. [ = 22/7]. Q.19. The figure given below, OACB is a quadrant of a circle. The radius OA = 3.5 cms. OD = 2 cm. Calculate the area of the shaded portion. Q.20. A girl fills a cylindrical bucket 32 cm in height and 18 cm in radius with sand. She empties the bucket on the ground and makes a conical heap of the sand. If the height of the conical heap is 24 cm, find : (i) Its radius and (ii) Its slant height. (Leave your answer in square root form). Q.21. A hollow sphere of internal and external diameters 4 cm and 8 cm respectively, is melted into a cone of base diameter 8 cm. Find the height of the cone. Q.22. On a map drawn to a scale of 1 : 25000, a rectangular plot of land, ABCD, has the following measurements, AB = 12 cm and BC = 16 cm. Angles A, B, C and D are 90 each. Calculate : (i) The diagonal distance of the plot in km, (ii) The area of the plot in sq. km. 8. Symmetry Question 1. Using ruler and compass only, construct a rectangle ABCD with AB = 5 cm and AD = 3 cm and construct its lines of symmetry. How many lines of symmetry are there ? Question 2. Using ruler and compass only, construct a regular hexagon of side 2.5 cm. Draw all its lines of symmetry. Question 3. Construct a ABC in which AB = AC = 3 cm and BC = 2 cm. Using a ruler and compass only, draw the reflection A BC of ABC, in BC. Draw the lines of symmetry of the figure ABA C. Question 4. Use graph paper for this question. Plot the points A(8,2) and B(6,4). These two points are the vertices of a figure which is symmetrical about x = 6 and y = 2. Complete the figure on the graph. Write down the geometrical name of the figure. Q.5. Use a ruler and compass only in this question. (i) Construct the quadrilateral ABCD in which AB = 5 cm, BC = 7 cm and LABC = 120 , given that AC is its only line of symmetry. (ii) Write down the geometrical name of the quadrilateral. (iii) Measure and record the length of BD in cm. Q.6. Parts of a geometrical figure is given in each of the diagrams below. Complete the figures so that the line m , in each case, is the line of symmetry of the completed figure. Recognizable free hand sketches would be awarded full marks. 9. Equation of a Straight Line Question 1. Find the equation of a line passing through (2, 3) and inclined at an angle of 1350 with positive direction of x-axis. Question 2. Find the equation of the line parallel to 3x + 2y = 8 and passing through the point (0, 1). Question 3. The line 4x 3y + 12 = 0 meets the x-axis at A. Write down the co-ordinates of A. Determine the equation of the line passing through A and perpendicular to 4x 3y + 12 = 0. 3 Question 4. Write down the equation of the line whose gradient is 2 and which passes through P, where P divides the line segment joining A( 2,6) and B(3, 4) in the ratio 2 : 3. Question 5. ABCD is a square. The co-ordinates of A and C are (3,6) and ( 1,2) respectively. Write down the equation of BD. Question 6. If 3y 2x 4 = 0 and 4y ax 2 = 0 are perpendicular to each other, find the value of a. 10. Distance and Section Formulae Question 1. The midpoint of the line segment joining (2a, 4) and ( 2, 2b) is (1, 2a + 1). Find the value of a and b. Question 2. If the points (2,1) and (1, 2) are equidistant from the point (x,y), show that x + 3y = 0. Question 3. For what value of a , the point (a,1), (1, 1) and (11,4) are collinear. Question 4. Find the value of k if the triangle formed by A(8, 10), B(7, 3) and C(0,k) is right angled at B. Question 5.The three vertices of a parallelogram are (3,4), (3,8) and (9,8). Find the fourth vertex. Question 6.( 2, 2), (x,8), (6,y) are three cyclic points whose centre is (2,5). Find the values of x and y. Question 7. In what ratio does the y-axis divide the line AB, where A( 4, 1) and B(17.10)? Question 8. Two vertices of a triangle are (3, 5) and ( 7,4). If its centroid is (2, 1), find the third vertex. 11. Matrices Question 1. Find x and y if: Question 2. Evaluate without using tables : Question 3. Find the matrix X such that A + X = 2 B + C. 12. Factor Theorem Question 1. Show that (x 1) is a factor of x3 7x2 + 14x 8. Hence, factorize the above polynomial completely. Question 2. Find the remainder when f(x) = x3 6x2 + 9x + 7 is divided by g(x) = x 1. Question 3. For what value of a , the polynomial g(x) = x a is a factor of f(x) = x3 ax2 +x + 2. Question 4. Find the value of p and q, if (x + 3) and (x 4) are the factors of x3 px2 qx + 24. Question 5. Find the value of q if the polynomial f(x) = x3 + 2x2 13x + q is divisible by g(x) = x 2. Hence find all the factors. Q.6. Find the value of a, if x a is a factor of x3 a2x + x + 2. Q.7. Find the value of p and q, if (x + 3) and (x 4) are the factors of x3 px2 qx + 24. Q.8. Find the value of the constants a and b, if (x 2) and (x + 3) are both factors of the expression x3 + ax2 + bx 12. Q.9. Find the value of q if the polynomial f(x) = x3 + 2x2 13x + q is divisible by g(x) = x 2. Hence find all the factors. Q.10. (x 2) is a factor of the expression x3 + ax2 + bx + 6. When this expression is divided by (x 3), it leaves the remainder 3. Find the value of a and b. Q.11. Use the factor theorem to factorize completely. x3 + x2 4x 4. Q.12. Using Factor theorem, show that (x 3) is a factor of x3 7x2 + 15x 9. Hence, factorise the given expression completely. Q. 4. Use factor theorem to factorize completely 13. Ratio and Proportion Question 1. If A:B = 4:5, B:C = 6:7and C:D = 14:15, find A:D. Question 2. If x : y = 9:10, find the value of (5x + 3y) : (5x 3y). Question 3. What must be subtracted from each term of 5 :7, so that it is equal to 3 : 4 ? Question 4. What must be added to each of 7, 16, 43 and 79 so that they become proportion ? Find the number, which are in proportion. Question 5. In a regiment, the ratio of number of officers to the number of soldiers was 3 : 31 before a battle. In the battle 6 officers and 22 soldiers were killed. The ratio between the numbers of officers and the number of soldiers now is 1 : 13. Find the number of officers and soldiers in the regiment before the battle. Question 6.Find the mean proportion of : 25, 64 Question 7. If x : y : : y :z , show that x :z = x2 : y2. Question 8. If (4a + 9b) : (4c + 9d) = (4a 9b) : (4c 9d) , show that a : b = c : d. a+3b+ a 3b 2 Question 9. If x = a+3b+ a 3b . Prove that [ ] 3bx 2ax+3b=0. Q.3. If a : b = 5 : 3, find (5a + 8b) : (6a 7b). 3x+5y 7 = Q.4. If 3x 5y 3 Find x : y. Q.5. Two numbers are in the ratio of 3 : 5, if 8 is added to each number, the ratio becomes 2 : 3. Find the numbers. Q.8. What number must be added to each of the numbers 6, 15, 20 and 43 to make them proportional? Q.9. What number should be subtracted from each of the following numbers 23, 30, 57 and 78, so that the remainders are in proportion? 3a+4b 3a 4b a c Q.14. If 3c+4d = 3c 4d Prove that b = d . Q.15. Given a c = b d 3a 5b 3c 5d prove that 3a+5b = 3c+5d . Q.17. The work done by (x 3) men in (2x + 1) days and the work done by (2x + 1) men in (x + 4) days are in the ratio of 3 : 10. Find the value of x. 14. Quadratic Equation Question 1. Solve the following quadratic equation for x and give your answer correct to two decimal places : x2 3x 9 = 0 7x+1 3x+1 Question 2. Solve 7x+5 =5x+1 . Question 3. Solve the quadratic equation : 21x2 8x 4 = 0 2 Question 4. Solve the equation : 3x 2=2x 1. Question 5. Solve the quadratic equation : Q.1. Solve the equation: 2x 1 x 7 3x2+10x 8 3 . . Write your answer correct to two decimal places. Q.3. Solve the following equation and give your answer up to two decimal places : x2 5x 10 = 0. Q.4. Solve the equation 3x2 x 7 = 0 and give your answer correct to two decimal places. Q.9. Solve using quadratic formula : x2 4x + 1 = 0. Q.10. Solve for x and give your answer correct to 2 decimal places. x2 10x + 6 = 0. Q. 64. If the roots of the quadratic equation x + px + q = 0 are tan30 and tan15 ,respectively then the value of p and q if 2 + q p is 3 Q. 66. For what value of k, the quadratic equation 16x2 9kx + 1 = 0 has real and equal roots Q. 67. Determine the values of p for which the quadratic equation: 2x2 + px + 8 = 0 has real roots 4 5 3 Q. 77. Rewrite as a quadratic equation in x and then solve for x: x 3=2x+3 2 ; where x 0. 15. Problems on Quadratic Equations. Question 6. A train covers a distance of 600 km at x km/hr. Had the speed been (x + 20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. Write down an equation in x and solve it to evaluate x. Question 7. In an auditorium, seats were arranged in rows and columns. The number of rows was equal to the number of seats in each row. When the number of rows was doubled and the number of seats in each row was reduced by 10, the total number of seats increased by 300. Find: (i) The number of rows in the original arrangement. (ii) The number of seats in the auditorium after re-arrangement. Question 8. A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, The digits interchange their places. Find the number. Question 9. Five years ago, a woman s age was the square of her son s age. Ten years later her age will be twice that of her son s age. Find : (i) The age of the son five years ago. (ii) The present age of the woman. Q.5. A shopkeeper buys a certain number of books for Rs720. If the cost per book was Rs5 less the number of books that could be brought for Rs720 would be 2 more. Taking the original cost of each book to be Rs. x, write an equation in x and solve it. Q.6. By increasing the speed of a car by 10 km/hr, the time of journey for a distance of 72 km is reduced by 36 minutes. Find the original speed of the car. Q.7. An aeroplane traveled a distance of 400 km at an average speed of x km/hr. On the return journey, the speed was increased by 40 km/hr. Write down an expression for the time taken for: (i) the onward journey. (ii) The return journey. If the return journey took 30 minutes less than the onward journey, write down an equation in x and find its value. Q.7. The hotel bill for a number of people for overnight stay is Rs4, 800. If there were 4 more, the bill each person had to pay would have reduced by Rs200. Find the number of people staying overnight. Q.8. A trader buys x articles for a total cost of Rs600. (i) Write down the cost of one article in terms of x. If the cost per article were Rs5 more, the number of articles that can be bought for Rs600 would be four less. (ii) Write down the equation in x for the above situation and solve it to find x. Q.9. The distance travel by a car between two towns A and B is 216 km and by rail it is 208 km. A car travels at a speed of x km/hr and the train travels at a speed which is 16 km/hr faster than the car. Calculate : (i) The time taken by the car to reach town B from A in terms of x. (ii) The time taken by the train to reach town B from A in terms of x. (iii) If the train takes 2 hours less than the car to reach town B, obtain the equation in x and solve it. (iv) Hence find the speed of the train. Q. 1. Find the length of a diagonal of a square whose sides are 10 m long. Give an exact answer and an approximation to two decimal places. Q. 2. The hypotenuse of a right triangle has a length of 13 cm. The sum of the lengths of the two legs is 17 cm. Find the lengths of the legs. Q. 3.The area of a rectangle is 180 square meters. If the rectangle's length is increased by 3 meters and the width is decreased by 2 meter, its area does not change. Find the perimeter of the original rectangle. Q. 4.If from the square of a number, five times the number is subtracted we get 50, what is the number? Q. 5.The sum of 2 digits of a number is 9. The sum of the squares of the digits is 5/7 of the number itself. Find the number Q. 6.Find two Consecutive odd integers whose product is 1599 Q. 7.A rectangular playing ground is twice as long as it is wide, If both dimensions are increased by 4 m, the area is increased by 88 sq m. Find the dimension of the original ground Q. 8.Find two consecutive odd positive integers, sum of whose squares is 290. Q. 9. A rectangular garden is to be designed whose breadth is 3 m less than its length. Its area is to be 4 sq.m more than the area of a garden that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular garden and of altitude 12 m. find its length and breadth. Q. 10. The sum of a number and its square is 90. Find the number. Q. 11. A two-digit number is such that the product of digits is 8. When 18 is added to it, they interchange their places. Determine the number. Q. 12. The age of the father is square of the age of his daughter. 15 years later, the father is double as old as daughter. Find their present ages. Q. 13. The sum of a number and its square is 90. Find the number. Q. 14. Sum of the squares of 3 consecutive numbers is 194. Find them. Q. 15. The product of two consecutive odd numbers is 323. Find them. Q. 16. The perimeter of a rectangle is 36 cm and its area is 80 sq.cm. Find its dimensions. Q. 17. The area of a rectangular land is 240 m2. If 8 m is decreased from its length it will become a square. Determine the length and breadth of the land. Q. 18. The sides of a right angle triangle are (x 1.8) cm, (x + 1.8) cm and (x + 1) cm. Find the area of the triangle. Q. 19. The sum of ages of a father and the son is 45 years. Five years hence the product of their ages will be 600. Find their present ages. Q. 20. The product of age of a man, 6 years before, and 10 years later is 960. Find his present age. Q. 21. Five times a certain number is equal to three less than twice the square of the number. Find the number. Q. 22. Find two consecutive positive even integers whose product is 224. Q. 23. The length of a veranda is 3 m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter. Find the length and breadth of the veranda. Q. 24. In a class of 60 students, each boy contributed rupees equal to the number of girls and each girl contributed rupees equal to the number of boys. If the total contribution then collected is Rs.1600. How many boys are there in the class? Q. 25. In a flowerbed, there are 23 rose plants in the first row, 19 in the second, 15 in the third, and so on. There are 7 rose plants in the last row. How many rows are in the flowerbed? Q. 26. If twice the father s age is added to the son s age, the sum is 77. Five years ago, the father was 15 times the age of his son, then. Find their present ages. Q. 27. The hypotenuse of a right triangle is 20m. If the difference between the length of the other side is 4m. Find the sides. Q. 28. A teacher on attempting to arrange the students for mass drill in the form of a solid square found that 24 students were left over. When he increased the size of the square by one student he found he was short of 25 students. Find the number of students Q. 29. A pole has to be erected at a point on the boundary of a circular park of diameter 13m in such a way that the differences of its distances from two diametrically opposite fixed gates A & B on the boundary in 7m. Is it possible to do so? If answer is yes at what distances from the two gates should the pole be erected. Q. 30. X and Y are centres of circles of radius 9cm and 2cm and XY = 17cm. Z is the centre of a circle of radius 4 cm, which touches the above circles externally. Given that XZY=90o, write an equation in x and solve it Q. 31. A ROSE nursery bed is set up measuring 20 feet by 30 feet. A walkway around the bed is made that reduces the area by 264 square feet, how wide is the walkway Q. 32. In Goa sea beach an old woman had a vast quantity of eggs to sell, when asked how many she said if you divide the number of eggs by 2 there will be one egg left. If you divide the number of eggs by 3 there will be one egg left. If you divide the number of eggs by 4 there will be one egg left. If you divide the number of eggs by 5 there will be one egg left. Finally if you divide the number of eggs by 7 there will be NO EGGS left. How many eggs did the old lady have Q. 33. A speedboat takes 1 hour longer to go 24miles up a river than to return. If the boat cruises at 10 miles per hour in still water, what is the rate of the current Q. 34. A cloth shop owner earns 3000 rupees a week on the sale of one type of shirt. If he reduces the price by Rs 10. per shirt, he can generate more business and sell 10 more shirts per week while still generating the same Rs 3000.00. At what price did he sell each shirt originally? Q. 35. Some students of Holy Cross School went for the picnic total cost for the food was Rs 720. 4 students among the group failed to go, so the cost for the food raised Rs. 15 for each .Find the total no. of students went for the picnic? Q. 36. A new computer can process a company's monthly payroll in 1 hour less time than the old computer. To really save time, the manager uses both computers to finish the payroll in 3hrs. How long would it take the new computer to do the payroll itself? Q. 37. The profit on a watch is given by P =x 13x 80, where x is the number of watches sold per day. How many watches were sold on a day when there was 50 rupees loss? Q. 38. The Hooghly River flows at a rate of 3mph. A patrol boat travels 60 miles upriver, and returns in a total time of 9 hours. What is the speed of the boat in still water? Q. 39. Neel drives his car for 20 km at a certain speed. At a distance, he increases his speed by 4km per hour and drives for an additional 22 km. If the total trip taken is 1 hour, what is his original speed? Q. 40. Two trains leave Howrah Railway Station. The first train travels due west towards Shibpur and the second train due North towards Srirampur. The first train travels 10 km per hour faster than the second train. If after 2 hours they are 100 km apart, find the average speed of each train. Q. 41. A train covers a distance of 600 km with speed x. Had the speed been (x + 20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. Write down an equation in x and Solve it to evaluate x. Q. 42. The distance travel by car between two towns A and B is 216 km and by rail 208 km. A car travels at a speed of x km/hr and the train travels in a speed which is 16 km/hr faster than the car. Find time taken by car to reach from A to B, time taken by rail from A to B in terms of x, if the train takes 2 hours less than the car - obtain the equation in x and solve it , hence find the speed of the train . Q. 43. The hotel bill for a no of people for overnight stay is Rs 4800. If there were 4 more the bill each person had to pay would have reduced by Rs 200,.find the no of people Q. 44. An airplane traveled a distance of 400 kms with speed x km/hr while the speed on the return journey was increased by 40 km/hr. Write down the expression for i. forward journey ii. Return journey if the R journey took half an hour less than the onward journey write the equation of x and find its value. Q. 45. In an auditorium seats are arranged in rows and columns. The number of rows is equal to the no of seats in each row. When the no of rows is doubled and no of seats in each row is reduced by 10the total no of seats increased by 300. Find the no of rows in original arrangements and number of seats after rearrangements. 3 Q. 46. Sum of 2 numbers =15, and the sum of their reciprocals = 10 Find them. Q. 47. A Cyclist traveled 80 kilometres from Kharagpur to Jamshedpur at a certain speed. Had he gone 4 km/h faster, the adventure would have taken 1 hour less. Find the speed of cyclist Q. 55. A 2 digit no is 7 times sum of its digits, and also is equal to 12 times less than 3 times the product of its digits. find the number Q. 56. 300 apples are distributed equally among a certain number of students. H ad thee been 10 more students each would have received one apple less. Find the number of students Q. 57. if twice the area of a smaller square is subtracted from the area of a larger square the result 14 cm2 if the twice area of larger square is added to three times the area of the smaller square the result is 203 cm2 find the sides of the squares. Q. 58. 300 apples are distributed among a certain no of students. Had there been 10 more students each would have received one apple less. Find the number of students Q. 59. A person on tour has 360 for his expenses. If he extends his tour for 4 days he has to cut down his daily exp by 3 rupees. Find the original duration of his tour Q. 60. A plane left 30 minutes later than schedule time and in order to reach its destination 1500 km away in time it has to increase its speed by 250 km/h from its usual speed. Find its usual speed. Q. 61. The side of a square exceeds the sides of another square by 4 cm and the sum of areas of two squares is 400 Esq. find the dimensions of the square 40 Q. 62.Two pipes running together can fill a cistern in 13 minutes. if one pipe takes 3 minutes more than the other to fill it find the time in which each pipe would fill the cistern Q. 63. THE angry Arjun carried some arrows for fighting with Vishma. With half the arrows he cut down the arrows thrown by Vishma on him and with six arrows he killed the rath driver of Vishma. With one arrow each he knocked down respectively the rath, flag and the bow of Vishma. Finally with one more than four times the square root of arrows he laid Vishma unconscious on an arrow bed. Find the total no of arrows Arjun had Q. 65. A trader bought a number of articles for Rs.1200. Ten were damaged and he sold each of the rest at Rs.2 more than what he paid for it thus getting a profit of Rs.60 on the whole transaction. Find the number of articles bought. Q. 68. A plane left 30 minutes later than its scheduled time. In order to reach its destination 1500 km away on time, it has to increase its speed by 250 km/hr than its usual speed. Find its usual speed Q. 69. The sum of areas of two squares is 468m2. If the difference of their perimeters is 24cm, find the sides of the two squares Q. 70. A dealer sells a toy for Rs.24 and gains as much percent as the cost price of the toy. Find the cost price of the toy. Q. 71. A fox and an eagle lived at the top of a cliff of height 6m, whose base was at a distance of 10m from a point A on the ground. The fox descends the cliff and went straight to the point A. The eagle flew vertically up to a height x meters and then flew in a straight line to a point A, the distance traveled by each being the same. Find the value of x. Q. 72. A lotus is 2m above the water in a pond. Due to wind the lotus slides on the side and only the stem completely submerges in the water at a distance of 10m from the original position. Find the depth of water in the pond. Q. 73. A train travels a distance of 240 km at constant speed, if the speed of the train is increased by 4 km/hr, the journey would have been 2 hours less. Find the original speed of the train. Q. 74. In a CINEMA hall, the number of rows was equal to the number of seats in each row. If the number of rows is doubled and the number of seats in each row is reduced by 5, then the total number of seats is increased by 375. How many rows were there? Q. 75. The sum of the digits of a two-digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18. Find the number. Q. 76. The speed of a boat in still water is 11 km/hr. It can go 12 km upstream and return downstream to the original point in 2 hours 45 minutes. Find the speed of the stream Q. 78. A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/hour more, it would have taken 30 minutes less for the journey. Find the original speed of the train Q. 79. A train travels a distance of 300 km at a uniform speed. If the speed of the train is increased by 5km an hour, the journey would have taken two hours less. Find the original speed of the train Q. 80. A passenger train takes two hours less for a journey of 300 kms. Its speed is increased by 5 kms / hr from usual speed. What is its usual speed? Q. 81. If one of the quadratic equation x2 + px +q = 0 and x2 +qx + p = 0 are common then prove that p q and p + q + 1 = 0. Q. 82. Divide 29 into two parts so that the sum of the squares of the parts is 425 Q. 83. Find two consecutive numbers, whose square have sum 85. Q. 84. The sum of the squares of two positive integers is 208. If the square of the larger number is 18 times the smaller, find the numbers. Q. 85. A passenger train takes 2 hours less for a journey of 300 km if its speed is increased by 5 km/hr from its usual speed. Find the usual speed of the train Q. 87. A motorboat whose speed is 18km/hr in still water takes 1 hour more to go 24km upstream than to return downstream to the same spot. Find the speed of the stream Q. 88. The sum of the squares of two consecutive odd numbers is 394. Find the integers Q. 89. A fast train takes 3 hours less than a slow train for a journey of 600 km. If the speed of the slow train is 10 km/hour less than that of the fast train, find the speeds of the two trains Q. 90. A train covers a distance of 600 km at x km/hr. Had speed been (x + 20) km/hr, the time taken to cover the distance would have been reduced by 5 hours. Write down an equation in x and solve it. Q. 91. The speed of a boat in still water is 15 km/hr; it can go 30 km upstream and return downstream to original point in 4hrs30 minutes. Find speed of the stream Q. 92. Two trains leave a railway stn at the same time. The first one went westwards and second one due north. First one travels 5km/h faster than the second. After two hours they are 50 km apart. Find their speed Q. 93. A shopkeeper sold an article for Rs 56 and gained as much percent as the cost price of the article find the cost price Q. 94. Sum of the squares of three consecutive positive numbers is 50 find them Q. 95. A factory kept increasing its output by the same percentage every year. find the percentage increase if is known that the output is doubled in the last 2 yrs. Q. 96. A takes 10 days less than the time taken by B to finish a piece of work. If both together cam finish it in 12 days find the time taken by B to finish the work alone Q. 97. One fourth of a herd of buffaloes was seen in the forest. Twice the square root of the herd had gone to mountains, and the remaining 15 were seen on the bank of the river. find the total no of buffaloes Q. 98. The Jones family had a square patch of lawn in their backyard. Its original area was x . They increased its length by 3 and its width by 5. Write a polynomial to describe the new area Q. 99. If 2 3x 2x+7=0 , then find the value of 1 3 2 ( ) x . Q. 100. A rectangle has the following sides. One side must have an odd integer length. Find the perimeter. 16. Shares and Dividends Question 1. Ajay owns 560 shares of a company. The face value of each share is Rs. 25. The company declares a dividend of 9%. Calculate : (i) The dividend that Ajay will get. (ii) The rate of interest on his investment if Ajay had paid Rs. 30 for each share. Question 2. Aman invests Rs.29929 in shares of par value Rs.26 at 10% premium. The dividend is 15% per annum. Calculate : (i) the number of shares (ii) the dividend received by him annually (iii) the rate of interest he gets on his money. Question 3. Which is better investment ? 7% Rs.100 shares at Rs.120 or 8% Rs.10 shares at Rs.13.50 ? Question 4. A company with 10000 shares of Rs.100 each, declares an annual dividend of 5%. (i) What is the total amount of dividend paid by the company ? (ii) What would be the annual income of a man, who has 72 shares, in the company ? (iii) If he received only 4% on his investment, find the price he paid for each share. Question 5. A man invested Rs.45000 in 15% Rs.100 shares quoted at Rs.125. When the market value of these shares rose to Rs.140, he sold some shares, just enough to raise Rs.8400. Calculate : (i) the number of shares he still holds. (ii) the dividend due to him on these shares. Q.2. A man invests Rs.20020 in shares of par value Rs.26 at 10% premium. The dividend is 15% per annum. Calculate : (i) the number of shares (ii) the dividend received by him annually (iii) the rate of interest he gets on his money. Q.6. A man wants to buy 62 shares available at Rs132(par value of Rs100). (i) How much should he invest? (ii)If the dividend is 7.5%, what will be his annual income? (iii) If he wants to increase his annual income by Rs150, how many extra shares should he buy? Q.7. A dividend of 9% was declared on Rs100 shares selling at a certain price. If the rate of return is 7 %, calculate : (i) the market value of the share. (ii) the amount to be invested to obtain an annual dividend of Rs630. Q.8. A man invest Rs8800 on buying shares of face value of rupees hundred each at a premium of 10%. If he earns Rs1200 at the end of year as dividend, find : (i) the number of shares he has in the company. (ii) the dividend percentage per share. Q.9. Mr. Ram Gopal invested Rs8,000 in 7% Rs100 shares at Rs80. After a year he sold these shares at Rs75 each and invested the proceeds (including his dividend) in 18%, Rs25 shares at Rs41. Find : (i) His dividend for the first year. (ii) His annual income in the second year. (iii) The percentage increase in his return on his original investment. Q.9. Mr. Tiwari invested Rs29,040 in 15% Rs100 shares quoted at a premium of 20%. Calculate : (i) The number of shares bought by Mr. Tiwari. (ii) Mr. Tiwari s income from the investment. (iii) The percentage return on his investment. Q.10. A man invest Rs1,680 in buying shares of nominal value Rs24 and selling at 12% premium. The dividend on the shares is 15% per annum. (i) Calculate the number of shares he buys; (ii) Calculate the dividend he receives annually. 17. Banking Question 1. Shyam deposited Rs. 150 per month in his bank for eight months under the Recurring Deposit Scheme. Find the maturity value of his deposit, if the rate of interest is 8% per annum and the interest is calculated at the end of every month ? Question 2. Mr. Ajay Kumar has a saving account in a bank. His passbook has the following entries : Date Year Particula Withdrawal Deposits Rs. BalanceRs. P 2007 rs s Rs.P P January 01 B/F ------- ------- 1276.38 January 09 By cheque -------- 2307.25 3583.63 March 07 To self 2000.00 ------- 1583.63 March 25 By cash ------- 6200.00 7783.63 June 10 To cheque 4500.00 ------- 3283.63 July 16 By clearing ------- 2628.70 5912.33 October 20 To cheque 524.50 ------- 5387.83 October 25 To self 2700.00 ------- 2687.83 November 5 By cash ------- 1500.00 4187.83 December 3 To cheque 1000.00 --------- 3187.83 December 25 By transfer ------- 2927.50 6115.33 Calculate the interest due to him for the year 2007 at 4.5% per annum if the interest is paid once in a year at the end of December. Also, find the total amount he will receive on 11th January, 2008, if he closes his account. Question 3. The entries in a Saving Bank Passbook are as given below : Date Particular Withdr s awal ------- Deposit Balance ------- Rs.1400 0 01-01-07 B/F 01-02-07 By cash ------- Rs.1150 0 Rs.2550 0 12-02-07 To cheque Rs.50 00 ------- Rs.2050 0 05-04-07 By cash ------- Rs.3750 Rs.2425 0 15-04-07 To cheque Rs.425 0 ------- Rs.2000 0 09-05-07 By cash ------- Rs.1500 Rs.2150 0 04-06-07 By cash ------- Rs.1500 Rs.2300 0 Calculate the interest for six months (January to June) at 4% per annum on the minimum balance on or after the tenth day of each month. Question 4. Bharati has a recurring deposit account in a bank for 5 years at 9% per annum simple interest. If she gets Rs.51607.50 at the time of maturity, find the monthly instalment. Question 5. Mira Kumar has an account with a bank. The following entries are from her pass-book : Particula Withdrawal Deposits Balance rs s Rs. P. Rs. p. Rs. p. Date 08 02 08 B/F 18 02 08 To self 12 04 08 By cash 15 06 08 To self 08 07 08 By cash 8500.00 4000.00 2238.00 5000.00 6000.00 Compute the above page of her pass book and calculate the interest for the six months. February to July 2008, at 4.5% per annum. Q.4. Mr. Siva Kumar has a Saving Bank account in Punjab National Bank. His pass book has the following entries : Date Particulars Withdrawa Deposit l (in Rs) (in Rs) Balance (in Rs) April 1, 2007 B/F 3220.00 April 15 By transfer May 8 To cheque 298.00 July 15 By clearing July 29 By cash Sept. 10 To self 698.00 2010.00 5230.00 4932.00 4628.00 9560.00 5440.00 15,000.00 8020.00 Jan. 10, 2008 By cash 8000.00 16020.00 Calculate the interest due to him at the end of financial year (March 31st 2008) at the rate of 6% per annum. Q.7. A page from passbook of Mrs. Rama Bhalla is given below : Date Year 2004 Particulars Withdrawa Deposits ls Rs P Rs. p. Jan 1 B/F Jan 9 By cash 200.00 22000.00 Feb 10 To cheque 500.00 300.00 1700.00 Feb 24 By cheque July 29 To cheque 200.00 Nov 7 By cash 300.00 2100.00 Dec 8 By cash 200.00 2300.00 Balance Rs. p. 20000.00 20000.00 1800.00 Calculate the interest due to Mrs. Bhalla for the period from January 2004 to December 2004, at the rate of 5% per annum Q.8. Pratibha has a recurring deposit account in a bank for 5 years at 9% p.a. simple interest. If he gets Rs516075 at the time of maturity, find the monthly instalment. Q.9. Mr. R. K. Nair gets Rs6,455at the end of one year at the rate of 14% per annum in a recurring deposit account. Find the monthly instalment. Q.10. Amit deposited Rs150/- per month in a bank for 8 months under the Recurring Deposit Scheme. What will be the maturity value of his deposits, if the rate of interest is 8% per annum and interest is calculated at the end of every month? Q.11. Mohan deposits Rs80 per month in a cumulative deposit account for six years. Find the amount payable to him on maturity, if the rate of interest is 6% per annum. Q.12. Mr. Ashok has an account in the Central Bank of India. The following entries are from his pass book : Date Year 2004 Particulars Withdrawa Deposits ls Rs P Rs. p. Balance Rs. p. 01 01 05 B/F 1200.00 07 01 05 By cash 500.00 1700.00 17 01 05 To cheque 400.00 1300.00 10 02 05 By cash 800.00 25 02 05 To cheque 500.00 20 09 05 By cash 21 11 05 To cheque 600.00 05 12 05 By cash 1600.00 700.00 2100.00 2300.00 300.00 1700.00 2000.00 If Mr. Ashok gets Rs83.75 as interest at the end of the year where the interest is compounded annually, calculate the rate of interest paid by the bank in his Savings Bank Account on 31st December, 2005. Q.13. Given the following details, calculate the simple interest at the rate of 6% per annum up to June, 30: Date Jan, 1 Jan, 20 Debit (Rs) 5,000.00 Credit (Rs) Balance (Rs) 24,000.00 24,000.00 19,000.00 Jan, 29 10,000.00 29,000.00 March, 15 8,000.00 37,000.00 April, 3 7,653.00 44,653.000 May, 6 May, 8 3,040.00 5,087.00 41,613.00 46,700.00 18. Sales Tax and Value added Tax Question 1. Dinesh bought an article for Rs. 374, which included a discount of 15% on the marked price and a sales tax of 10% on the reduced price. Find the marked price of the article. Question 2. Ms. Chawla goes to a shop to buy a leather coat which costs Rs.735. The rate of sales tax is 5%. She tells the shopkeeper to reduce the price to such an extent that she has to pay Rs.735, inclusive of sales tax. Find the reduction needed in the price of the coat. Question 3. A shopkeeper buys an article at a rebate of 30% on the printed price. He spends Rs.40 on transportation of the article. After charging sales tax at the rate of 7% on the printed price, he sells the article for Rs.856.Find his profit percentage. Question 4. The catalogue price of a colour TV is Rs.2400. The shopkeeper gives a discount of 8% on the listed price. He gives a further off-season discount of 5% on the balance. But sales tax at the rate of 10% is charged on the remaining amount. Find : (i) the sales tax amount a customer has to pay. (ii) the final price he has to pay for the colour TV. Question 5. [VAT] A manufacturer produces a good which cost him Rs.500. He sells it to a wholesaler at a price of Rs.500 and wholesaler sells it to retailer at a price of Rs.600. The retailer sells it to the customer at a price of Rs.800. If the sales tax charged is 5%. Find the tax charged under VAT by : (i) manufacturer, (ii) wholesaler and (iii) retailer. Find the tax paid by the customer. Q.2. The price of a T.V. set inclusive of sales tax of 9% is Rs13407. Find its marked price. If the sales tax is increased to 13%, how much more does the customer pay for the T.V.? Q.4. A shopkeeper buys an article at a rebate of 30% on the printed price. He spends Rs.40 on transportation of the article. After charging sales tax at the rate of 7% on the printed price, he sells the article for Rs.856.Find his profit percentage. Q.5. The price of a washing machine, inclusive of sales tax, is Rs13530. If the sales tax is 10%, find its basic price. Q.6. A colour T.V. is marked for sale for Rs17,600 which includes sales tax at 10%. Calculate the sales tax in rupees. Q.8. The catalogue price of a computer set is Rs45,000. The shopkeeper gives a discount of 7% on the listed price. He gives a further off-season discount of 4% on the balance. However, sales tax at 8% is charged on the remaining amount. Find : (i) The amount of sales tax a customer has to pay. (ii) The final price he has to pay for the computer set. Q.8. Kiran purchases an article for Rs5,400 which includes 10% rebate on the marked price and 20% sales tax on the remaining price. Find the marked price of the article. 20. Compound Interest Question 1. Ramesh invests Rs. 12800 for three years at the rate of 10% per annum compound interest. Find : (i) The sum due to Ramesh at the end of the first year. (ii) The interest he earns for the second year. (iii) The total amount due to him at the end of the third year. Question 2. A man borrows Rs.5000 at 12% compound interest per annum, interest payable after six months. He pays back Rs.1800 at the end of every six months. Calculate the third payment he has to make at the end of 18 months in order to clear the entire loan. Question 3. A man invests Rs.5000 for three years at a certain rate of interest, compounded annually. At the end of one year it amounts to Rs.5600. Calculate : (i) the rate of interest per annum. (ii) the interest occurred in the second year. (iii) the amount at the end of the third year. Question 4. The compound interest on a certain sum of money at 5% per annum for two years is Rs.246. Calculate the simple interest on the same sum for three years at 6% per annum. Question 5. On a certain sum of money, the difference between the compound interest for a year, payable half-yearly, and the simple interest for a year is Rs.180. Find the sum lent out, if the rate of interest in both the cases is 10%. Question 6. A certain sum amounts to Rs.5292 in 2 years and to Rs.5556.60 in 3 years at compound interest. Find the rate and the sum. Question 7. The cost of car, purchased 2 years ago, depreciates at the rate of 20% every year. If its present worth is Rs.315600, find : (i) its purchase price (ii) its value after 4 years. Q.1. What sum of money will amount to Rs3704.40 in 3 years at 5% compound interest. Q.2. What sum of money will amount to Rs3630 in two years at 10% per annum compound interest? Q.3. Calculate the compound interest for the second year on Rs8,000 invested for 3 years at 10% p.a. Q.4. At what rate percent p.a. compound interest would Rs80000 amount to Rs88200 in two years, interest being compounded yearly. Also find the amount after 3 years at the above rate of compound interest. Q.11. A sum of money is lent out at compound interest for two years at 20% p.a. C.I. being reckoned yearly. If the same sum of money is lent out at compound interest at the same rate percent per annum, C.I. being reckoned half-yearly. It would have fetched Rs482 more by way of interest. Calculate the sum of money lent out. Q.13. A person invests Rs10,000 for two years at a certain rate of interest compounded annually. At the end of one year this sum amounts to Rs12,000. Calculate : (i) The rate of interest per annum. (ii) The amount at the end of second year. Q.14. If the interest is compounded half yearly, calculate the amount when the Principal is Rs7,400, the rate of interest is 5% per annum and the duration is one year. Q.15. A man invests Rs46,875 at 4% per annum compound interest for 3 years. Calculate: (i)The interest for the first year. (ii) The amount standing to his credit at the end of the second year. (iii) The interest for the third year. 21. Linear Inequation in One Unknown Q.1. Solve the following inequation and graph the solution on the 8 1 10 x+ < ;x R number line. 3 33 Q.2. Find the range of values of x, which satisfy the inequation 1 3x 2 +1< ;x R 5 10 5 . Graph the solution set on the number line. Q.3. Solve the following inequation, and graph the solution on the number line: 2x 5 5x+4<1 ;x R . Q.4. Solve the following inequation and graph the solution set on the number line : 2x 3 < x + 2 3x + 5, x R. Q.5. Given that x R, solve the following inequality and graph the solution on the number line : 1 3 + 4x 23. Q.6. Given that x I, solve the inequation and graph the solution on the x x 4 + number line: 3 2 2 3. Q.7. Solve : 2 2x 3 5, x R and mark it on a number line. 1 2x 11 2 ;x N Q.8. Find the value of x which satisfies the inequation : 236 Q.9. Solve the inequation : 3 3 2x < 9, x R. Represent your solution on a number line. 15x Q.10. Solve the inequation : 12 + 6 5 + 3x, x R. Represent the solution on a number line. Q. 2. Solve the inequality 5x 2 3(3 x) where x is a set of real numbers Q. 3. Solve the inequality and represent the solution set on a number line: 7 x+3 3x 5;x Z. 22. Reflection Q.1. Use graph paper for this question. The point A (2, 3), B (4, 5) and C (7, 2) are the vertices of a ABC, (i) Write down the coordinates of A , B , C if A B C is the image of ABC, when reflected in the origin. (ii) Write down the co-ordinates of A , B , C if A B C is the image of ABC, when reflected in the x-axis. (iii) Mention the special name of the quadrilateral BCC B and find its area. Q.2. (i) Point P(a,b) is reflected in the x-axis to P (5, 2). Write down the values of a and b. (ii) If P is the image of P when reflected in the y-axis. Write down the co-ordinates of P . (iii) Name the single transformation that maps P to P . Q.3. (i) Plot A(3, 2) and B(5, 4) on graph paper . Take 2 cm = 1 unit on both axes. (ii) Reflect A and B in the x-axis to A , B . Plot these on the same graph paper. (iii) Write down : (a) the geometrical name of the figure ABB A . (b) the axis of symmetry of ABB A . (c) the measure of the angle ABB . (d) the image A of A, when A is reflected in the origin. (e) the single transformation that maps A to A . Q.4. Use graph paper for this question. The point P(2, 4)is reflected in the x = 0 to get the image P . (i) Write down the co-ordinates of P . (ii) Point P is reflected in the line y = 0, to get the image P . Write down the co-ordinates of P . (iii) Name the figure PP P . (iv) Find the area of the figure PP P . Q.5. A (3,2), B (4,0) and C ( 5,0) are the vertices of ABC. (i) Write down the co-ordinates of (a) B1and C1, the image of B and C under the reflection in y-axis. (b) B2 and C2, the image of B and C under the reflection in (0,0). (ii) Write down the co-ordinates of (a) A3 and C3, the image of A and C by reflection in x-axis. (b) What type of figure is formed by ABC and its image taken together. Q.6. The points (4,1), (4, 1), ( 4, 1) and ( 4, 1) are the vertices of a rectangle. If the rectangle is reflected in the line x = 5, find the coordinates of the reflected rectangle also find the area and perimeter of the reflected rectangle. Q.7. Use a graph paper for this question.(Take 10 small division = 1 unit on both axes). P and Q have co-ordinates (0, 5) ( 2, 4). (i) P is invariant when reflected in an axis. Name the axis. (ii) Find the image of Q on reflection in the axis found in (i). (iii) (0, k) on reflection in the origin is invariant. Write the value of k. (iv) Write the co-ordinates of the image of Q obtained by reflecting it in the origin followed by reflection in x-axis. Q.8. Use a graph paper for this question. A(1, 1), B(5, 1), C(4, 2) and D(2, 2) are the vertices of a quadrilateral. Name the quadrilateral ABCD. A, B, C and D are reflected in the origin onto A , B , C and D on the graph sheet and write their co-ordinates. Are D, A, A and D collinear? Q.9. Use a graph paper for this question. (Take 10 small divisions = 1 unit on both axes). Plot the points P(3, 2) and Q( 3, 2) from P and Q, draw perpendiculars PM and QN on the x-axis. (a) Name the image of P on reflection in the origin. (b) Assign the special name to the geometrical figure PMQN and find its area. (c) Write the co-ordinates of the point to which M is mapped on the reflection in (i) x-axis; (ii) y-axis; (iii) origin. Q.10. The point (3, 4) is reflected to P in the x-axis; and O is the image of O (the origin) when reflected in the line PP . Using graph paper, give : (i)The co-ordinates of P and O . (ii) The length of the segments PP and OO . (iii) The perimeter of the quadrilateral POP O . (iv) The geometrical name of the figure POP O . Q.11. Use graph paper for this question. The point P(5, 3) was reflected in the origin to get the image P . (a) Write down the co-ordinate of P . (b) If M is the foot of the perpendicular from P the x-axis, find the coordinates of M. (c) If N is the foot of perpendicular from P to the x-axis, find the co-ordinates of N. (d) Name the figure PMP N. (e) Find the area of the figure PMP N. Q.12. Write down the co-ordinates of the image of the point (3, 2) when : (i) reflected in the x-axis. (ii) reflected in the y-axis. (iii) reflected in the x-axis followed by reflection in the y-axis. (iv) reflected in the origin. Q.13. Use graph paper for this question. (i) Plot the point A(3, 5) and B ( 2, 4). Use 1 cm = 1 unit on both the axes. (ii) A is the image of A when reflected in the x-axis. Write down the co-ordinates of A and plot it on the graph paper. (iii) B is the image of B when reflected in the y-axis, followed by reflection in the origin. Write down the co-ordinates of B and plot it on the graph paper. (iv) Write down the geometrical name of the figure AA BB . (v) Name two invariant points under reflection in the x-axis. 23. Distance and Section Formulae Q.1. The midpoint of the line segment joining (2a, 4) and ( 2, 2b) is (1, 2a + 1). Find the value of a and b. Q.2. If the points (2,1) and (1, 2) are equidistant from the point (x,y), show that x + 3y = 0. Q.3. Find the ratio in which the point (1 , 1) divides the line joining the points (a, 1) and (11, 4). Also find the value of a. Q.4. Find the value of k if the triangle formed by A(8, 10), B(7, 3) and C(0,k) is right angled at B. Q.5. The three vertices of a parallelogram are (3,4), (3,8) and (9,8). Find the fourth vertex. Q.6. ( 2, 2), (x, 8), (6, y) are three cyclic points whose centre is (2,5). Find the values of x and y. Q.7. In what ratio does the y-axis divide the line AB, where A( 4, 1) and B(17, 10) ? Q.8. Calculate the ration in which the line joining A(6, 5) and B(4, 3) is divided by the line y = 2. Q.9. Two vertices of a triangle are (3, 5) and ( 7,4). If it s centroid is (2, 1), find the third vertex. Q.10. If the line joining the points A(4, 5) and B(4, 5) is divided by the AP 2 point P such that AB = 5 , find the co-ordinates of P. Q.11. The line segment joining A (2, 3) and B (6, 5) is intersected by the x-axis at the point K. Write the ordinate of the point K. Hence find the ratio in which K divides AB. Q.12. Find the co-ordinates of the centroid of a triangle whose vertices are : A ( 1, 3 ), B(1, 1) and C(5, 1). Q.13. A(1, 4), B(3, 2) and C(7, 5) are the vertices of a ABC. Find : (i) The co-ordinate of the centroid G of ABC. (ii) The equation of a line, through G and parallel to AB. Q.14. KM is a straight line of 13 units. If K has the co-ordinates (2, 5) and M has the co-ordinates (x, 7) find the possible value of x. Q.15. A (10, 5), B (6, 3) and C (2, 1) are the vertices of a triangle ABC. L is the mid-point of AB and M is the mid-point of AC. Write down the coordinates of L and M. Show that LM = 1/2 BC. Q.16. ABCD is a rhombus. The co-ordinates of A and C are (3, 6) and ( 1, 2) respectively. Write down the equation of BD. Q.17. In the figure, line APB meets the x-axis at A. y-axis at B. P is the point ( 4, 2) and AP : PB = 1 : 2. Write down the co-ordinates of A and B. Q.18. The centre of a circle of radius 13 units is the point (3, 6), P(7, 9) is a point inside the circle. APB is a chord of the circle such that AP = PB. Calculate the length of AB. Q.19. The centre O, of a circle has the co-ordinates (4, 5) and one point on the circumference is (8, 10). Find the co-ordinates of the other end of the diameter of the circle through this point. 24. Equation of a Straight Line Q.1. Find the equation of a line passing through (2, 3) and inclined at an angle of 1350 with positive direction of x-axis. Q.2. Find the equation of the line parallel to 3x + 2y = 8 and passing through the point (0,1). Q.3. Find the equation of the line passing through (0, 4) and parallel to the line 3x + 5y + 15 = 0. Q.4. The line 4x 3y + 12 = 0 meets the x-axis at A. Write down the coordinates of A. Determine the equation of the line passing through A and perpendicular to 4x 3y + 12= 0. Q.5. Write down the equation of the line whose gradient is 3/2 and which passes through P, where P divides the line segment joining A( 2,6) and B(3, 4) in the ratio 2 :3. Q.6. ABCD is a square. The co-ordinates of A and C are (3,6) and ( 1,2) respectively. Write down the equation of BD. Q.7. If 3y 2x 4 = 0 and 4y ax 2 = 0 are perpendicular to each other, find the value of a. Q.8. If the lines y = 3x + 7 and 2y + px = 3 are perpendicular to each other, find the value of p. Q.9. Find the value of k for which the lines kx 5y + 4 = 0 and 4x 2y + 5 = 0 are perpendicular to each other. Q.11. P(3, 4), Q(7, 2) and R( 2, 1) are the vertices of triangle PQR. Write down the equation of the median of the triangle, through R. Q.12. The line joining P ( 4, 5) and Q (3, 2), intersects the y-axis at R. PM and QN are perpendiculars from P and Q on the x-axis. Find: (i) The ratio PR : RQ. (ii) The co-ordinates of R. (iii) The area of the quadrilateral PMNQ. Q.13. A straight line passes through the points P ( 1, 4) and Q (5, 2). It intersects the co-ordinate axes at points A and B. M is the midpoint of the segment AB. Find (i) The equation of the line. (ii) The co-ordinates of A and B. (iii) The co-ordinates of M. Q.14. Find the equation of a line passing through the point ( 2, 3) and having x-intercept of 4 units. Q.15. (i) Find the equation of a line, which has y-intercept 4 and is parallel to the line 2x 3y = 7. (ii) Find the co-ordinates of the point where it cuts the x-axis. 25. Similarity Q.1. Area of two similar triangles ABC and PQR are 25 cm2 and 49 cm2 respectively. If QR = 9.8 cm, find BC. Q.2. Two isosceles triangles have equal vertical angles and their areas are in the ratio 4 : 9. Find the ratio of their corresponding heights. Q.3. P and Q are points on the side AB and AC respectively of a triangle ABC such that PQ is parallel to BC and divides triangle ABC into two parts, equal in area. Find PB : AB. Q.4. In the figure given below, DE is parallel to BC, AD = 4 cm, BD = 2 cm and the area of ABC = 12 cm2. Calculate : (i) Area of ADE (ii) (Area of ADE)/(Area of trapezium DBCE). Q.5. In the given figure, ABC is a triangle, DE is parallel to BC and AD/DB = 3/2. (i) Determine the ratio AD/AB, DE/BC. (ii) Prove that DEF is similar to CBF. Hence, find EF/FB. (iii) What is the ratio of the areas of DEF and BFC ? Q.6. In the figure given below, PB and QA are perpendiculars to the line segment AB. If PO = 6 cm, QO = 9 cm and the area of POB = 120 cm2, find the area of QOA. Q.7.In the given figure, AB and DE are perpendicular to BC. If AB = 9 cm, DE = 3 cm and AC = 24 cm, calculate AD. Q.8. In the given figure, the median BD and CE of a triangle ABC meet at G. Prove that : (i) EGD ~ CGB and (ii) BG = 2GD from (i) above. Q.9. In the figure alongside, BC || DE. Area of triangle ABC = 25 cm2, area of trapezium BCED = 24 cm2, DE = 14 cm. Calculate the length of BC. 26. Graphical Representation of Data Q.1. Draw a histogram to represent the following data : Pocket money in Rs. No. of students 150 200 10 200 250 5 250 300 7 300 350 4 350 400 3 Q.2. For the following frequency distribution draw a histogram. Hence calculate the mode. Class Frequency 0 5 5 10 10 15 15 20 20 25 25 30 2 7 18 10 8 5 Q.3. Draw a histogram and hence estimate the mode for the following frequency distribution : Class 0 10 10 20 20 30 30 40 40 50 50 60 Frequenc y 2 8 10 5 4 3 Q.4. The marks obtained by 200 students in an examination are given below : Marks No. of students 0 10 05 10 20 10 20 30 11 30 40 20 40 50 27 50 60 38 60 70 40 70 80 29 80 90 14 90 100 06 Using a graph paper, draw an Ogive for the above distribution. Use your Ogive to estimate : i. The median; ii. The lower quartile; iii. The number of students who obtained more than 80% marks in the examination and iv. The number of students who did not pass, if the pass percentage was 35. Q.5. The table below shows the distribution of the scores obtained by 120 shooters in a shooting competition. Using a graph sheet, draw an ogive for the distribution. Score obtained Number of shooters 0 10 5 10 20 9 20 30 16 30 40 22 40 50 26 50 60 18 60 70 11 70 80 6 80 90 4 90 100 3 Use your ogive to estimate : i. The median. ii. The inter quartile range. iii. The number of shooters who obtained more than 75% scores. iv. Q.6. The marks obtained by 120 students in a Mathematics test is given below : Marks No. of Student 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 80 90 90 100 5 9 16 22 26 18 11 6 4 3 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 lower quartile. (iii) The number of students who obtained more than 75% in the test. (iv) The number of students who did not pass in the test if the pass percentage was 40. Q.7. The daily wages of 160 workers in a building project are given below : Wages in Rs 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 No. of Workers 12 20 30 38 24 16 12 8 Using a graph paper, draw an Ogive for the above distribution. Use your Ogive to estimate : (i) The median wage of the workers. (ii) The upper quartile wage of the workers. (iii) The lower quartile wage of the workers. (iv) The percentage of workers who earn more than Rs45 a day. Q.8. Using a graph paper, draw an Ogive for the following distribution which shows a record of the weight in kilograms of 200 students. Weight Frequency 40 45 5 45 50 17 50 55 22 55 60 45 60 65 51 65 70 31 70 75 20 75 80 9 Use your Ogive to estimate : i. The percentage of students weighing 55 kg or more. ii. The weight above which the heaviest 30% of the students fall. iii. The number of students who are : (1) under-weight and (2) over-weight, if 55.70 kg is considered as standard weight. Solution : Q.9. Using the data given below construct the cumulative frequency table and draw the ogive. From the ogive determine the median. Marks 0 10 10 20 20 30 30 40 40 50 50 60 60 70 70 80 3 8 12 14 10 6 5 2 No. of Student Q.10. The following table shows the distribution of the heights of a group of factory workers : Height (cm) No. of workers 150 155 155 160 160 165 165 170 170 175 175 180 180 185 6 12 18 20 13 8 6 Determine the cumulative frequencies. Draw the cumulative frequency curve on a graph paper. Use 2 cm = 5 cm height on one axis and 2 cm = 10 workers on the other. iii. From your graph, write down the median height in cm. Q.11. Use graph paper for this question. The table given below shows the monthly wages of some factory workers; (i) Using the table, calculate the cumulative frequencies of workers. (ii) Draw the cumulative frequency curve. Use 2 cm = Rs500, starting the origin at Rs6,500 on x-axis, and 2 cm = 10 workers on yaxis. (iii) Use your graph to write down the median wage in Rs. i. ii. Wages in Rs. No. of workers 6500 7000 10 7000 7500 18 7500 8000 22 8000 8500 25 8500 9000 17 Cumulative Frequency 9000 9500 10 9500 10000 8 Q.12. The daily profits in Rupees of 100 shops in a department store are distributed as follows : Profit per shop (in Rs) No. of shops 0 100 12 100 200 18 200 300 27 300 400 20 400 500 17 500 600 6 Draw a histogram of the data given above, on graph paper, and estimate the mode. *********************

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