RATIO & PROPORTION
RAW- 1
RATIO
1) A metre scale is cut in two pieces in the ratio 3:2. Find the length of each piece.
2) Ram left Rs 33000 to his three sons to be divided in the ratio 3:4:5. Find the share of each.
3) Two numbers are in the ratio 13:9. Their difference is 56. Find the numbers.
4) Two numbers are in the ratio 7:9. The first is added to twice the second, the result is 100. Find the numbers.
5) A bank has rupee coins and fifty paise coins in the ratio 2:3. The total value of the coins is Rs 24.50. find the number of each type of coin.
6) Ages of Arun and Beena are in the ratio 4:5. Fourteen years ago the ratio of their ages was 2:3. How old are Arun and Beena at present?
7) The ratio of two numbers is 4:3. If 2 is added to the first and 6 is substracted from the second, the ratio becomes 5:3. Find both numbers.
8) On adding 1 to each of two numbers, the ratio of the resulting numbers is 2:5. If 1 is substracted from each, the resulting ratio is 1:3. Find the numbers.
9) The speed of two trains are in the ratio 4:7. They leave in opposite direction from a place at the same time. At the end of 7 hours, the total distance travelled by them is 154 km. Find the speed of each train.
10) There are 25 consecutive positive integers. The ratio of the first to the last integers is 3:7. Find the first integer.
PROPORTION
RAW -1
1) Given a,b,c,d are in continued proportion, show that:
a) a : b - c= cd - d²
b) 5a+ 6d : 87a - 7d = 5a³+ 6b³: 8a³- 7b³.
c) (a+ b + c)²/(a²+ b²+ c²)= (a+ b + c)/(a - b + c).
2) If (8b- 7a)/(8d - 7c)= (8b+ 7a)/(8d +7c), show a: b= c: d.
3) Given a: b= c: d , show that
a) 3a - 5b : 3c - 5d = 3a + 5b : 3c + 5d.
b) (a + c) : (b + d)= √(a²+ c²): √(b²+ d²).
4) Find x, given that the work by (x -3) men in (2x +1) days and the work done by (2x +1) men in (x +4) days is in the ratio 3:10.
5) A vessel contains water and milk in the ratio 1:4. Two litres of the mixture is removed and two litres of water are poured in the vessel. If the ratio of water to milk now is 13:12, find
a) The total amount of the mixture in the vessel
b) the amount of milk originally the vessel.
TEST PAPER -3
1)
2) Draw a neat diagram, showing the lines of symmetry and name each figure in the following cases:
a) a quadrilateral with two diagonals as lines of symmetry.
b) a quadrilateral which has just one line of symmetry.
c) a triangle with only one line of symmetry.
d) ABCD is a rhombus. Prove that AC is a line of symmetry of the Rhombus.
3) In the adjoining figure,
RT is the tangent at S, Prove that :
a) angle PSR= angle OQS.
b) the triangle PSR and SQT are similar.
c) PR. QT = RS. ST
d) If PS= 9cm and PQ= 15cm, Write down the value of (PR. RS)/(QT. ST).
4) A(0,4), B(3,0) are the two vertices of ∆ AOB.
a) Write down the coordinates of A', the reflection of A in the x-axis, of B' the reflection ofB in the y-axis.
b) Assign special name to the figure ABA'B'.
c) If C is the midpoint of AB, write down the coordinates of C', the reflection of C in the origin.
d) assign special name to the quadrilateral ABA'C'.
5) A(2,3), B(4,5) and C(7,2) are the vertices of ∆ ABC.
a) Write down the coordinates of A', B' and C', if ∆ A'B'C' is the image of ∆ ABC when reflected in the origin.
b) Write down the coordinates of A"B" and C", if ∆ A"B"C" is the image of ∆ A'B'C' when reflected in the y-axis.
c) Assign special name to the quadrilateral BCC"B".
d) Hence find its area.
6) From the adjoining ∆ ABC,
prove that ∆ ABC and ∆ ABD are similar.
Hence prove that AB²= BC. BD.
If AB= 6cm, BD= 4cm and AC= 8cm, calculatethe AD.
7) A man invests Rs 40000 in shares. He invests Rs 800 in 7%(Rs 100) shares at Rs 80, Rs 11400 in 8%(Rs 100) shares at Rs 70 and the remainder in 9%(Rs 100) shares. If the total yield from his investment is 10.25% at what price did he buy the 9% shares ? Also find the yield from the investment in 9% shares.
8) The annual salaries of a group of employees are given in the following table :
Salaries number of persons
45 3
50 5
55 8
60 7
65 9
70 4
75 7
Calculate the mean salary. Also calculate the median salary.
9) solve the following inequation and represent the solution set on a number line
x -3< 2x - 2 ≤ 9 - x, x belongs to N.
10) In cyclic quadrilateral ABCD,
AB|| DC, the bisectors of angle A meets CD at E and the circle at F. Prove that
a) EF= CF.
b) ∆ BCF ≡ ∆ DEF.
11) From a solid cylinder of height 12cm and base radius 5cm, a conical cavity of the same height and base is hollowed out. Find
a) the volume, of the remaining solid.
b) the surface, of the remaining solid.
12) A well is to be dug with 6m inside diameter and 20m in depth . Find the volume of the earth to be excavated. The earth taken out is spread all around to a width of 3 m to form an embankment . Find the height of the embankment .
13) Find the value of :
(Sin⁴30+ 2 sin²30 cos²30+ cos⁴30)(Sin²90+ cos²90+ tan²45)².
14) For the following distribution, calculate the mean:
Class frequency
10-16 2
16-22 20
22-28 10
28-34 6
34-40 12
Draw a histogram for the above data and estimate the mode.
15) A(2,5), B(4,1) and C(2,3) are the vertices of the ∆ ABC. Calculate:
a) the equation of the median AD.
b) the equation of the attitude AM.
c) the equation of AC.
d) the coordinates of E, if E is the fourth vertex of the parallelogram ABEC.
16) used ruler and compass only
a) construct a circle on AB = 8cm as diameter
b) to construct another circle of radius 3cm to touch the circle in (a) above externally and the diameter AB produced.
17) If A= 1 2 0 & B= 1 3 -1 & C= 2 & D= x
2 -1 3 2 -3 4 0 y
-1
With the relation (4A - 2B)C= D , then the value of x and y.
18) If A= 2 -1 & B= 2
4 3 -3
Find a matrix X such that AX= B.
4) Prove
a) sin²x/(1- cosx)= 1+ cosx.
b) cos²x/(1+ sinx)= 1- sinx
c) tanx + cotx = 1/sinx cosx = secx cosecx
d) (secx + tanx)(sec x - tan x)= sin²x + cos²x.
e) cosec²x sinx cosx = cotx.
f) (1+ tan²x)/(1+ cot²x)= tan²x.
g) (tan²x - sec²x)/(cot²x - cosec²x)= 1.
h) √(cosecx + cotx) √(cosecx - cotx)=1.
i) (1- cotx)/(1+ cotx) = (sinx - cosx)/(sinx + cosx).
j) √{(1- sinx)/(1+ sinx)}= secx - tanx.
TEST PAPER -2
1) From the adjoining histogram,
estimate the mode. Also construct the corresponding frequency distribution. Hence find the mean.
2) Find the values of x which satisfy the inequation:
-3+ x ≤ x/2 - 1/2 ≤ 5/6 + x; x belongs to N
graph solution set on the number line.
3) In the adjoining figure,
PQ is a tangent at Y, angle APQ=10°, angle BAY= 30° and XY is a diameter of the circle, Calculate the angles ABX, AXB, BYQ.
4) A hemispherical bowl of internal radius 18cm contains a liquid. This liquid is to be filled into cylindrical shaped small bottles of diameter 6cm and height 12cm. How many bottles can be filled to empty the bowl?
5) In a cricket match Sanjay took 3 wickets less than twice the number of wickets taken by Anshu. If the product of the number of wickets taken by them is 20, find the number of wickets taken by each.
6) Evaluate: {(1+ sin30)/cos30 + cos30/(1+ sin30)}²{sin²60/(1- cos²60 tan²60)}.
7) From the following frequency distribution, draw an accurate ogive:
Marks no. of students
1-10 10
11-20 40
21-30 80
31-40 140
41-50 170
51-60 130
61-70 100
71-80 40
81-90 20
From the ogive find
a) What percent of the candidate pass the examination, if the pass marks is 40 ?
b) What should the pass mark be, if it is decided to 80% of the candidates to pass ?
If scholarship are awarded to the top 15% of the students , what should be the lowest marks to gain a scholarship ?
8)
9) A company with 15000 shares of nominal value of Rs 100, declares annual dividend of 10% to the shareholders.
a) find the total dividend paid by the company.
b) Mukesh had bought 250 shares of the company at Rs 125 per share . Calculate the dividend he receives and the percentage return on his investment.
10) A(7,6) and B(-5,-6) are the opposite vertices of a rhombus . Find the equations of its diagonals.
11) Using ruler and compass only , draw a circle of radius 3cm, Extend AB, a diameter of this circle to C, so that BC= 3cm.
Construct a circle to touch AB at C and to touch the circle externally .
12) If the matrices
A= 4 1 3 & B= 3 2 4 & C= 1 & D= x
0 -1 -3 -6 1 -3 3 y
-2
With the relation (3A - 2B)C= D then find x and y.
13) A model of a gas cylinder is made to a scale of 1:100. The gas cylinder consists of a cylindrical part and two hemispherical parts, as shown in the adjoining model.
a) The length of the model is 5 cm. Calculate the length of the cylinder in m.
b) The area of the gas cylinder is 10π m². Calculate the area of the model.
c) Calculate the volume of the gas cylinder in π litres.
TEST PAPER - 1
1) The shadow of a flag post 25m high is 25√3m. Find the angle of elevation of the Sun.
2) A conical tent has a circular base area 0.375 hectares. if its height is 20m, finds its capacity.
3)
4) OX and OY are the co-ordinate axes. AB = 6cm.
The point A slides along OX and point B slides along OY. Find the locus of the mid point of AB.
5) in the given figure
AB || CD and O is the centre of the circle. If angle BED= 35°, find angle ACD.
6) a) The line x - y = 3 divides the join of (3,4) and (8,3) in the ratio m: n. Find the ratio.
7) If A=2 0 & B= 14 0
-3 4. 45 44
find the value of scalar factors x and y, such that xA²+ yA= B.
8) 4x³- 12x²+ ax + b has x -3 is a factor but when it is divided by x+ 2 the reminder is -755. Find a and b.
9) A's income is Rs 140 more than B's and C's income is Rs 80 more than D's. If the ratio of A's and C's income is 2:3 and the ratio of B's and D's income is 1:2, find the income of each.
10) Three numbers are in continued proportion. Their sum is 38 and the sum of their squares is 532. Find the numbers.
11) Mrs. Mehta plans to invest Rs 8456 in shares. She partly invests in 17% shares at Rs 140 and the remaining amount in 9% share at Rs 112. Her income from the second investment is Rs 58 more than the first invesr. How much did she invest in shares at Rs 112?
Section II
12)
13) Mrs. Bhagat deposits Rs 1500 every month for 36 months in a bank and receives Rs 65655 at the end of 36 months. Find the rate of simple interest paid by the bank on the recurring deposit.
14) Solve the equation and represent it on the number line
x/2 + 3 ≤ x/3 + 4 < 4x -7, x belongs to R.
15) From the following table, find the frequency distribution and calculate the mean marks:
Marks no of students
less than 8 4
Less than 16 10
less than 24 22
Less than 32 41
less than 40 50
16) Find the values of x and y if the matrices
A= x+ y y & B= 2 & C= 3
2x x- y -1 2 with the relation AB = C.
17) Prove: sin⁶x + cos⁶x = 1 - 3 sin²x + 3 sin⁴x.
18) Two spheres of the same metal weight 1kgf and 7kgf. The radius of the smaller sphere is 2.5cm. The spheres are melted to form a single big sphere . Find the diameter of bigg sphere.
19) MT and NT are tangents to two circles . Prove that M,B,N and T are concyclic points.
(Use alternate segment property and prove that angle MBN + Angle T = 180°)
20) ∆ ABC and ∆ PQR are similar and their areas are 1089cm² and 2304 cm² respectively. If AB= 22cm, find PQ.
21) If A(3,2), B(-2,4) and C(3,-2) are the vertices of ∆ ABC, find the equation of the line perpendicular to AB and passing through the mid-point of BC.
22) The difference between the reciprocals of two consecutive multiples of 3 is 1/468. Find the numbers.
CYLINDER
1) The diameter of the base of a right circular cylinder is 10cm and its height is 21 cm. Find the cost of painting the curved surface at the rate of Rs1.50 per cm².
2) The diameter of the roller, 1m 40cm long, is 80 cm. if it takes 600 complete revolutions to level a playground, find the cost of levelling the ground at 75 paise per square metre.
3) A cylinder is 12cm high and the circumference of its base is 44cm. Find the curved surface area and total surface area.
4) A rectangular sheet of 88cm x 45cm is the rolled along its length to form a cylinder. Find the curved surface area of the cylinder.
5) The area of the curved surface of a right circular cylinder with radius of the base as 10cm is 880cm². Determine its height.
6) The total surface area of a right circular cylinder is 165π cm². if the radius of its base is 5 cm, find its height.
7) It times costs Rs 2200 to paint the inner curved surface of a cylindrical vessel 10m deep. if the cost of painting is 20 per m², find radius of the base.
8) Find the ratio of the curved surface area of a cylinder to its total surface area, given that its height and radius are 13.5cm and 7.5cm respectively.
9) The cost of polishing the total surface area of a closed cylindrical tank at the rate of 20 paise per dm² is Rs 154. if the height is one and a half times the radius of the base, determine the radius and height of the closed cylindrical tank.
10) A metal pipe is 77cm long.
The inner diameter of the cross section is 4cm, the outer diameter being 4.4 cm, Find its
a) inner curved surface area
b) outer curved surface area of
c) total surface area.
REFLECTION
1) A Triangle ABC is such that the coordinates A, B and C are (2,0),(1,1) and (0,2) respectively. Write down the coordinates of the triangle obtained by reflecting ∆ ABC in the line y=0. Also reflect (2,0) in the line x=0.
2) Draw the unit square, whose vertices are (2,2),( 4,2),(4,4) and (2,4). Reflect the square in the y-axis and then reflect the image in the origin. What single transformation would give the same final result ?
3) A man leaving point A must take water from a river and deliver it to a man at point B. Use reflection to find the shortest path.
PAPER - 6
1) Ashok invested Rs 12500 in shares of a company paying 8% per annum. If he bought Rs 20 shares for Rs 25, find his annual dividend.
2) The mid-point of AB is P(-2,4). The coordinates of the point A and B are (a,0) and (0,b) respectively. Find a and b.
3) If A= 4 3 & B= x & C= 6
-5 0 -2 y with the relation AB= C then 5 find the value of x and y.
4) Calculate the median and mode of the following set of numbers:
9, 0, 2, 8, 5, 3, 5, 4 ,1, 5, 2, 7.
5) Solve the following inequation and represent the solution set on the number line.
30 - 4(2x - 1)> - 8. x belongs to positive integers.
6) Solve : y - √(3y -6)= 2.
7) ay triangle whose area is 12cm², is transferred under enlightenment about a point in space. If the area of the image is 108cm², find the dilation factor of the enlargement.
8) Point P(a,b) is reflected in x-axis to (5,-2).
a) Write down the values of a and b.
b) P" is the image of P when reflected in the y-axis. Write down the coordinates of P''.
c) Name the single transformation that maps P to P".
9) If A= 1 2 & B= 2 1 & C= 1 3
-2 3 3 2 3 1
Find C(B - A).
10) In the figure, AB || CD and O is the centre of the circle.
If angle ADC=24°, find angle AEB.
11) If a,b,c are in continued proportion, show that:
(a²+ b²)/b(a+ c) = b(a+ c)/(b²+ c²).
12) Mr. Gupta invested Rs 8000 in 8%(Rs 100) shares, selling at Rs80. After a he sold these shares at Rs 75 each and invested the proceed in Rs 100 shares selling at Rs 90 with a dividend of 12%. Calculate
a) his income from the first investment.
b) his income from the second investment.
c) the increased percentage return on his original investment.
13) If -5 is a root of the quadratic equation x²+ kx - 130= 0, find k. Hence, find the other root.
14) An open cylindrical vessel of internal diameter 49cm and height 64 cm stands on a horizontal platform. Inside this is placed a solid metallic right circular cone whose base has a diameter of 21/2cm and whose height is 12cm. Calculate the volume of water required to fill the tamk. Take π to be 22/7.
15) The perimeter of a rectangular plot is 180m and its area is 1800m². If the length is x m, Express the breadth in terms of x. Hence , form an equation in x. Solve the equation and find the length and the breadth of the rectangle.
16) Prove : (1+ tan²x)/(1+ cot²x)= sin²x/cos²x.
17) The I.Q of 50 pupils was recorded as follows :
I. Q scores no of pupils
80-90. 6
90- 100 9
100-110 16
110-120 13
120-130 4
130-140 2
Draw a histogram for the above data and estimate the mode.
18) in the given figure, find 
a) the co-ordinates of the points A, B and C.
b) the slope of BC .
c) the equation of the line AP (|| BC).
d) the coordinate of the point X and Y where line AP meets the x-axis and y-axis respectively.
e) the ratio in which point A devidas the line segment XY.
19) Factorise , by factor theorem, the expression 2x³+ 13x²+ 17x -12.
20) in the given figure,
if angle BAD= 65°, angle ABD=70° and angle BDC=45°, calculate
a) angle BCD, ADB
b) Show that AC is a diameter.
21) The angle of elevation of a cloud from a point 50m above a lake is 30° and the measure of the angle of its depression of its reflection in the lake is 60°. Find the height of the cloud.
22) A solid cylinder of radius 14cm and height 21cm is melted down and recast into spheres of radius 3.5cm each. Calculate the number of spheres that can be made. (π= 22/7).
PAPER- 5
1) Solve: 21x²- 8x -4=0.
2) Find the coordinates of the image of (5,-4) after reflection in
a) x= 0
b) y= 2.
3) List the solution set of the following and inequation and graph the solution set:
(1/2) + 8x > 5x - 3/2, x belongs to Z.
4) Calculate the ratio in which the line joining A(6,5) and B(4,3) is divided by the line y= 2.
5) In the figure,
BC is parallel to DW.
Area of triangle ABC= 25 cm², area of trapezium BCED= 24 cm² and DE = 21cm.
Calculate the length of BC.
6) Calculate the mean, median and mode of the following numbers :
13,11,15,13,14,15,13,17,12,16.
7) Given A= 1 1
8 3 evaluate A² - 3A.
8) In the figure,
I is the incentre of the circle. AI produced to meet the circle in D. Calculate
a) angle DCB b) angle IBC c) angle BID d) angle BIC
Given angle BAC= 50° and angle ABC= 64°.
9) Show that: √{(1- cosA)/(1+ cosA)}= sinA/(1+ cosA)
10) In the figure,
AB it is a common tangent to two circles intersection at C and D. Write down the measure of (angle ACB + angle ADB).
11) The surface area of a solid metallic sphere is 1256 cm½. It is melted and recast into right circular cones of radius 2.5 cm and height 8cm. Calculate
a) the radius of the solid sphere.
b) the number of cones recast (π= 3.14).
12) A dividend of 9% was declared on Rs 100 shares selling at a certain price. If the rate of return is 15/2%, calculate
a) the market value of the share.
b) the amount to be invested to obtain an annual dividend of Rs 630.
13) in the figure AB and CD are the lines 2x - y +6=0 and x - 2y = 4 respectively.
a) write down the coordinates of A, B, C and D.
b) prove that the triangles OAB and ODC are similar .
c) Is figure ABCD cyclic ?
14) The hotel bill for a number 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 number of people staying overnight.
15) ABCD is a rhombus. The coordinates of A and C are (5,8) and (-1,2) respectively. Write down the equation of BD.
16) The following cable shows the distribution of the heights of a group of factory workers:
Ht(cm) no of workers
140-145 6
145-159 12
150-155 18
155-160 20
160-165 13
165-170 8
170-175 6
a) Determine the cumulative frequencies
b) draw the cumulative frequency curve on a graph
c) From your graph, write down the median height in cm.
PAPER -4
1) A Colour TV is marked for sale for Rs16500 which includes GST at 10%. Calculate the tax in rupees.
2) Find the remainder when 2x³- 3x²+ 7x -8 is divided by x -2.
3) Given a/b = c/d, prove that: (3a - 5b)/(3a + 5b)= (3c - 5d)/(3c + 5d).
4) Two numbers are the ratio of 7: 11. If 15 is added to each number, the ratio becomes 5 : 7. Find the numbers .
5) Find the value of x, which satisfies the inequation:
-2≤ 1/2 - 2x/3 ≤ 11/6, x belongs to N.
Graph the solution on the number line.
6) Priti deposited Rs 1500 per month in a bank for 8 months under the recurring deposit scheme. What will be the maturity value of her deposits , if the rate of interest is 12% per annum and interest is calculated at the end of every month.
7) solve for x and give your answer correct 2 decimal places:
3x²- 5x = 1.
8) The catalogue price of washing machine is Rs 16000. The shopkeeper gives a discount of 5% on the listed price. He gives a further off season discount of 12% on the balance. But GST at 5% is charged on the remaining amount. Find :
a) The GST paid by the customer.
b) The final price he has to pay for the washing machine.
9) If 3 tan²A - 1=0, then show that cos3A= 4 cos³A - 3 cosA.
10) A plot of land has an area of 400000 m². it is represented on the map by an area of 40 cm². Find:
a) the scale factor of the map.
b) what distance on the map would a distance of 2.4km.
11) Use graph paper for this question.
The point A(4,7) was reflected in the origin to get the image A'.
a) write down the coordinate of A'.
b) If M is the foot of the perpendicular from A to the x-axis. find the coordinates of M.
c) If N is the foot of the perpendicular from A' to the x-axis, find the coordinates of N.
d) name the figure AMA'N.
e) find the area of the figure AMA'N.
12) Prove: sinx(1+ tan x)+ cos x(1+ cot x)= cosecx + secx.
13) A(14,7), B(6,-3) and C(8,1) are the vertices of a triangle ABC . P is the midpoint of AB, and Q is the midpoint of AC. Write down the coordinates of P and Q. Show that BC= 2PQ.
14) A, B and T are 3 points on a circle.
The tangent at T meets BA produced at P. Given that angle ATB= 32 and that the angle APT= 78, calculate the angle subtended by BT at the centre of the circle.
15) If A= 4 3 & B= x & C= 6
-5 0 -2 y with the relation AB= C. Find x and y.
16) A ma invests Rs7500 on buying shares of face value of Rs 100 each at a premium of 50% in a company. If he earns Rs 550 at the end of the year as dividend, find
a) the number of shares he has in the company.
b) what is dividend percentage per share ?
17) write down the equation of the line whose gradient is 4/3 and which passes through P, where P divides the line segment joining A(-2,-3) and B(5,4), in the ratio 2:5.
18) A vertical Tower is 40m high . A man standing at some distance from the tower knows that cosines of the angle of elevation of the top of the Tower is 30°. How far is he standing from the foot of the tower?
19) An exhibition tent is in the form of cylinder surmaunted by a cone. The height of the tent above the ground is 67m and the height of the cylindrical part is 40m. If the diameter of the base is 144m, find the quantity of canvas required to make the tent. Allow 10% extra for folds and for stitching. Give your answer to the nearest m².
20) Using the data given below , construct the cumulative frequency table and draw the ogive. From the ogive determine the median.
Mark no of students
00-10 3
10-20 8
20-30 12
30-40 14
40-50 10
50-60 6
60-70 5
70-80 2
21) In the given figure,
find TP if AT= 20cm and AB= 15cm.
22) Factorise the expression with the help of the factor theorem f(x)= 6x³- 7x²- 7x + 6. Hence, find the values of x when f(x)= 0.
PAPER- 3
1) The price of a TV set inclusive GST of 9% is 40221. Find the marked price.
2) If x: y= 4:3, find (5x +8y): (6x - 7y).
3) Using the reminder theorem, find the remainder when y³- 7y¹+ 15y - 19 is divided by y- 3.
4) State and draw the locus of a point eqidistance from two parallel lines.
5) The given figure, the medians QS and RT of a ∆ PQR meet at G. prove that:
a) ∆ TGS~ ∆ RGQ
b) QG= 2 GS from (a) above.
6) Solve the following inequation and graph the solution on the number line:
2x -5≤ 5x +4 < 11, x belongs to R.
7) 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, 16,16 18, 19 20. Calculate:
a) the mean
b) the median
c) the mode
8) If the matrix
A= 1 -4 & B= -3 2 & C= 4 0
4 1 4 0 0 -3 find
a) A² b) BC c) A²+ BC .
8) The point A(3,4) is reflected to A' in the x-axis, and O' is the image of O(the origin) when reflected in the AA'. Using graph paper, give
a) the coordinates of A' and O'.
b) the lengths of the segments AA' and OO'.
c) the perimeter of the quadrilateral AOA'O'.
d) the geometrical name of the figure AOA'O'.
9) Prove the following identity:
1/(sinA + cosA) + 1/(sinA - cosA)= 2sinA/(2 sin²A -1).
10) In the given figure, AB is the diameter of a circle with centre O. Angle BCD is 130°. Find
a) angle DBA
b) angle BAD.
11) Find the equation of a line passing through the point (-4,6) and having the x-intercept of 8 units.
12) A man wants to buy 72 shares available at Rs 150 (per value of Rs 100).
a) How much should he invest ?
b) if the dividend is 7.5%, what will be his annual income ?
c) if he wants to increase his annual income by Rs 300, how many extra shares should be buy ?
13) The following table gives the weekly wages of workers in a factory:
weekly wages (Rs). No. of workers
150-150 5
155-160 20
160-165 10
165-170 10
170- 175 9
175-180 6
180-185 12
185- 190 8 Calculate
a) the mean
b) the model class
c) the numbers workers getting weekly wages, below Rs 180.
d) the number of workers getting Rs 165 or more, but less than Rs 185 as weekly wages.
14) A hollow sphere of internal and external diameters 8 cm and 16 cm respectively , is melted into a cone of base diameter 16 cm. Find the height of the cone.
15) The shadow of a vertical tower AD on level ground is increased by 30m, when the altitude of the sun changes from 45° to 30° as shown in the given figure.
Find the height of the tower and give your answer correct to 1/10 of a metre.
16) The marks obtained by 240 students in a mathematics test is given below:
Marks No. if students
00-10 10
10-20 18
20-30 32
30-40 44
40-50 52
50-60 26
60-70 22
70-80 12
80-90 16
90-100 8
Draw an ogive for the given distribution on a graph sheet. Use a suitable scale for your ogive and using ogive, estimate:
a) the median
b) the lower quartile
c) the number of student who obtained more than 75% in the test :
d) the number students who did not passing inthe test if the pass percentage was 40.
17) P(2,4), Q(3,3) and R(7,5) are the vertices of a ∆ PQR. Find
a) the coordinates of the centroid G of ∆ PQR.
b) the equation of a line, through G and parallel to PQ.
18) An aeroplane travelled a distance of 800 km at an average speed of x kmph. On the return journey, the speed was increased by 40 kmph. Write down an expression for the time taken for:
a) the onward journey .
b) the return journey .
If the return journey took 40 minutes less then the onward journey, write down an equation in x and find its value.
Paper - 2
1) The point P(a,b) is reflected in the x-axis to obtain the point Q(3,-4). Find a and b. (1)
2) If A= a 3a & B= 2 & C= 5
b 4b 1 12 find a and b when the relation AB= C. (1)
3) The mean of the number 6, y, 7, x and 14 is 8. Express y terms of x. (1)
4) Solve using the quadratic formula, x²- 5x -2=0. Give your answer correct to 3 significant figures. (2)
5) If (8a + 5b)/(8c + 5d)= (8a - 5b)/(8c - 5d), prove that a/b = c/d. (1)
6) Find the value of k, if x - k is a factor of x³- kx²+ x + 4. (1)
7) Solve 1< 3x -3≤ 11, x ∈ R and mark it on a number line. (1)
8) Calculate the mean, median and mode of the following numbers : 12, 11, 10, 11, 12, 13, 14, 13, 15, 13. (2)
9) In the diagram,
chords AB and CD of the circle are produced to meet at O. Given that CD= 4cm, DO= 12cm and BO= 6cm, calculate AB . (2)
10) If cosA= 4/5 and cosB= 24/25; evaluate
a) cosec²A
b) cotA + cotB. (2)
11) on a map drawn to a scale 1:125000, a triangular plot of land has the following measurements :
PQ=10cm, QR= 8cm, angle PRQ= 90°. Calculate
a) the actual length of PQ in km.
b) the area of the plot in square kilometres. (2)
12) The work done by (2x -3) men in (3x +1) days and work done by (3x +1) men in (x +8) days are in the ratio of 11:15. Find the value of x. (2)
13) Find the mean of the following frequency distribution:
Class interval frequency
00-30 3
30-60 7
60-90 15
90-120 14
120-150 7
150-180 4 (3)
14) A man invests Rs 30800 in buying shares of nominal value Rs 56 at 10% premium . The dividend on the shares is 18% per annum. Calculate
a) The number of shares he buys.
b) The dividend he receives annually.
c) The rate of interest he gets on his money. (3)
15) prove that: sinx/(1- cotx) + cosx/(1- tanx)= sinx + cosx. (2)
16) A straight line passes through the points A(-2,8) and B(10,-4). It intersects the coordinate axes at points E and F. P if the midpoint of the segment EF. 
Find
a) the equation of the line.
b) the coordinate of E and F.
c) the coordinates of the point P. (3)
17) 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 15, the total number of seats increased by 400. Find
a) The number of rows in the original arrangement.
b) the number of seats in the auditorium after rearrangement. (3)
18) Draw a histogram and hence estimate the mode for the following frequency distribution:
Class frequency
00-20 3
20-40 8
40-60 10
60-80 6
80-100 4
100-120 3 (3
19) A man standing on the bank of a river observes that the angle of elevation of a tree on the opposite bank is 60°. When he moves 40m away from the bank, he finds the angle of elevation to be 30°. Calculate:
a) the width of the river and
b) the height of the tree. (3)
20) Find a and b, if
a= 3 -2 & B= 2a & C= 4 & D= 2
-1 4 1 5 b with the relation AB + 4C = 3D. (2)
21) A vessel is in the form of an inverted cone. Its height is 15cm and the diameter of its top which is open, is 5cm. It is filled with water up to the rim. When lead shots, each of which is a sphere of diameter 5mm are dropped into the vessel, 1/3 of the water flows out. Find the number of lead shots dropped into the vessel. (3)
22) In the given circle
with diameter AB, find the value of x. (2)
23) Find the value of k for which the lines kx - 7y + 5=0 and 6x - 2y +9=0 are perpendicular to each other. (3)
1) Find the rate of GST levied on a car that was sold at a price 3 times its marked price. (1)
2) When 7x²- 3x + 8 is divided by (x -4), find the remainder (using remainder theorem). (1)
3) Calculate the length of the tangent drawn to a circle of diameter 8cm from a point 5cm away from the centre of the circle. (1)
4) If x²,4 and 9 are in continued proportion , find the value of x. (1)
5) If x ∈Z, find the solution set for the inequation 5< 2x -3≤ 14 and graph the solution on a number line. (1)
6) Find p and q if g(x)= x +2 is a factor of f(x)= x³- px + x + q and f(2)= 4. (2)
7) 1 -2 0
If X= -3 4 & Y= 1
a) Find the matrix Z such X + Z is a zero matrix.
b) Find the matrix M such that X + M = X.
c) Find XY. (3)
8) a) If 7 is the mean of 5, 3, 0.5, 4.5, b, 8.5, 9.5, find b
b) If each observation is decreased in value by 1 unit, what would the new mean be ? (2)
9) In the figure below,
AB is a chord of the circle with centre O and BT is tangent to the circle at B, if angle OAB= 32°, Find the value of x and y. (2)
10) Construct a regular pentagon of side 3cm. Draw the lines of symmetry. (2)
11) The volume of a cylinder 14cm long is equal to that of a cube having an edge 11cm. Calculate the radius of the cylinder. (3)
12) A piece of butter 3cm by 5cm by 12cm is placed on a hemispherical bowl of radius 3.25cm. Will the butter overflow when it melts completely. (3)
13) A company with 10000 shares of Rs 50 each declares an annual dividend of 5%.
a) What is the total amount of dividend paid by the company ?
b) What would be the annual income of a man who has 72 shares in the company?
c) if he receives only 4% on his investment, find the price he paid for each share. (3)
14)a) State the equation of the mirror line, if point A(5,0) on reflection is mapped as A'(-5, 0).
b) State the equation of the the mirror line, if point B(4,-3) on reflection is mapped as B'(4,3).
c) Point C(-3,5) on reflection in y=2 is mapped as C'. Find the coordinates of C. (3)
15) Tanya standing on a vertical cliff in a jungle observes two rest-horses in a line with her on opposite sides deep in the Jungle below. If their angles of depression are 30° and 45° and the distance between them is 200mp, find the height of the cliff. (3)
16) Find the equation of a line that passes through (1,3) and is parallel to the line y= -3x +2. (2)
17) In the given figure,
calculate
a) angle APB
b) angle AOB. (2)
18) The midpoint of the line joining A(2,p) and B(q,4) is (3,5). Find the numerical values of p and q. (2)
19) From the following table, find:
a) average wage of a worker. Give your answer, to the nearest paise
b) Modal class.
Wages in Rs No of workers
Less than 10 15
Less than 20 35
Less than 30 60
Less than 40 80
Less than 50 96
Less than 60 127
Less than 70 190
Less than 80 200 (3)
20) Examine the ogive given below
which shows the marks obtained out of 100 by a set of students in an examination and answer the following questions:
a) How many students are there in the set ?
b) How many students obtained 40% marks ?
c) How many students obtained 90% and above ?
d) What is the median marks? (4)
21) Show that: √{(1+ cosx)/(1- cosx)}= cosecx + cot x. (2)
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