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Algebra (March 1999)
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SELECTION MENU
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ALGEBRA
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Time: 2 ½ Hours
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MARCH – 1999
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Marks: 75
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| Q.1. |
Attempt any five of the following: |
10 |
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i) |
If
 ,
find the value of  |
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ii) |
4x
+ 3y = 18 ----- (I)
3x + 2y ----- (II)
Write new equations by multiplying equation (I) by 3 and
equation (II) by 4. |
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iii) |
Complete the following table assuming m a n: |
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m |
5 |
6.5 |
--- |
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n |
20 |
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28 |
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iv) |
Solve the quadratic equation by factorisation:p2 - p - 2 = 0 |
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v) |
Fill
in the blanks and rewrite:
If sin q = 1, then q
= -------°. If cos
q = 1, then q = -------° |
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vi) |
In
D PQR, Ð
PQR = 90, PR = 7Ö2, PQ = 7, QR
= 7.
Write the values of :
1) sin R;
2) cot P.
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vii) |
Rewrite the following expression arranging the
terms in descending powers of b:
a2b - a2c - ab2 + ac2
+ b2c - bc2. |
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| Q.2. |
Attempt any five of the following: |
15 |
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i) |
If
 ,
show that each ratio =  ,
(x + y + z ¹ 0) |
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ii) |
Draw
a histogram from the data given below: (Scale : 1 cm = 5 units) |
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Class
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200-300
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300-400
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400-500
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500-600
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Frequency
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21
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53
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28
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12
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iii) |
Write the equation x + 5/x = 9 in the form of ax2
+ bx + c = 0 and then find the values of a, b, c. Also find
the value of 4b2 + ac. |
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iv) |
In
a parallelogram ABCD, the measure of Ð A is thrice the measure of ÐB. Find the measures of Ð A and Ð
B. |
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v) |
Fill
in the blanks in the following: 
\7a = --------
\ 7a - 3b = ----------
\  |
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vi) |
If  ,
find x. |
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vii) |
Study the adjoining graph and answer the questions
given below:
1) Write the co-ordinates of the point in which line
AB intersects
X-axis.
2) To which axis is line CD parallel?
3) In which quadrant does point C lie? |
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| Q.3. |
a) |
Solve any two of the following: |
6 |
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i) |
There is direct variation between the cost of books
(P) and the copies (N) of the book. If 10 copies of the book
cost Rs. 25, find the cost of 12 copies of the same book. |
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ii) |
Solve using the formula:3m2 + 7m + 3 = 0 |
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iii) |
If
2 cosec2 A = sec 60 + cosec2 45, find
the measure of Ð A. |
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b) |
Solve any one of the following: |
4 |
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i) |
The
length of a rectangle is less than twice its breadth by 9.
The perimeter of the rectangle is 54 cm. Find the area of
the rectangle. |
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ii) |
Three numbers are in continued proportion. The
middle number is 20 and sum of the other two numbers is 58.
Find the numbers. |
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| Q.4. |
a) |
Solve any two of the following: |
6 |
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i) |
Factorise using the formula 2x2 - 3x
- 9. |
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ii) |
2
pens and 3 pencils cost Rs. 26 while 3 pens and 2 pencils
cost Rs. 34. Find the cost of a pen and a pencil. |
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iii) |
The
following data show the average daily consumption of electricity
units of some offices in a city. Prepare a frequency distribution
table and answer the questions: |
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30 |
20
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40 |
20 |
10 |
50 |
40 |
50 |
20 |
10 |
30 |
20 |
30 |
40 |
20 |
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1) How many of the offices consume average
20 units of electricity daily? |
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2) Write the range of the data. |
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3) Write the number of offices consuming
maximum average. |
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b) |
Attempt any one of the following: |
4 |
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i) |
Plot
the following points, on a graph paper:
(Scale: 1 cm = 1 on both the axes)
A
(-1, 6), B (1, 4), C (0, -1), D (4, 3).
Draw line AB and CD. Write the coordinates of the point of
intersection. |
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ii) |
Draw
a frequency polygon showing the following information: (Scale:
1 cm = 5 doctors) |
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| Age (years) |
25-30
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30-35
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35-40
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40-45
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45-50
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| No. of Doctors |
26
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58
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52
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36
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20
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| Q.5. |
Attempt any three of the following: |
15 |
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i) |
If
sin A =  ,
then prove that tan A.
Cosec A =  |
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ii) |
Draw
graphs of the following equations on the same co-ordinate
system:
y
= - x - 4; and 4)
y = - x + 5.
Write the co-ordinates of the point of intersection
of the diagonals of the quadrilateral formed by these lines. |
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iii) |
The
following frequency polygon is based on the scores obtained
by students in a test of 100 marks. Observe it carefully and
answer the questions given below it:
1) Prepare a frequency distribution table of the data.
2) If the minimum marks for passing were 60, find how
many students passed and how many failed in the examination. |
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iv) |
Solve the equation: 3y4 - 13y2
+ 12 = 0. |
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| Q.6. |
Attempt any three of the following: |
15 |
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i) |
If
 and  , find the value of m and n. |
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ii) |
The
area of a regular polygon of given number of sides is proportional
to the area of its circumcircle and also proportional to the
square of its side. When the side is 3 then area of the polygon
is  and
when its area is  then the area of its circumcircle is  .
What is the area of the circumcircle when its side is 5? |
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iii) |
If
(x - 2) = (y - 3) = (z - 4), find the numerical value of
x (y2 - z2) + y (z2 - x2)
+ z (x2 - y2). |
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iv) |
The
average of a, b, c is 5 more than the average of b, c, d.
Twice of a is 7 less than thrice of d. Find the average of
a and d. |
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Algebra (October 1999)
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SELECTION MENU
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ALGEBRA
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Time: 2 ½ Hours
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October - 1999
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Marks: 75
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Q.1.
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Solve any five of the following sub-questions:
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10
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i)
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Find
the fourth proportional to 15,12, 35.
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ii)
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,
if d = 5 , then f = 16. Find the value of f when d = 10.
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iii)
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3x
- 5y = -1 and x + y = 13, then find the value of x - y.
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iv)
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Write the equation in the form of ax2
+ bx + c = 0 : x +  =
7.
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v)
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Verify whether the given expression is cyclic or
not:
ab2 + bc2 + ca2
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vi)
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If
cos A =  ,
find the values of sin A and cot A.
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vii)
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Find
the value of 
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Q.2.
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Solve any five of the following sub-questions:
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15
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i)
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The
marks secured by students in a class-test of Mathematics
out of 50 are:
36,
09, 17, 35, 22, 38, 14, 27, 41, 18, 31, 24, 26, 41, 35,
06, 04,
28, 15, 30, 23, 34, 17, 44, 26, 19, 22, 29, 25, 10, 23,
26, 30, 28,
34, 29, 02, 29, 16, 21, 08, 33, 35, 28, 36, 16, 35, 44.
Prepare a grouped frequency distribution table
of this data taking
classes 1-10, 11-20, -------, 41-50.
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ii)
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If
tan A =  ,
find sin A, sec A, and (sin A - cos A).
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iii)
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The
ratio of the present ages of John and Jim is 4:3. Six years
hence it will be 5:4. Find their present ages.
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iv)
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Write the values of b2 - 4ac for the
quadratic equation:
2x2 - 5x - 1 = 0.
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v)
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Solve the following simultaneous equations:
7x
- 5y = -1
5x
- 7y = -11.
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vi)
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Complete the following table for drawing the graph
of y = 3x -2 :
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x
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0
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-1
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y
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---
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4
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(x, y)
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---
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---
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---
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vii)
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If
 ,
find the values of the following ratios:
a)
b : a b)  c)
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| Q.3. |
A)
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Solve any two of the following sub-questions:
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6
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i)
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The
cost of sugar is directly proportional to its weight. 12
kg of sugar cost Rs. 138. Find the cost of 5 kg of sugar.
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ii)
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Solve the following simultaneous equations:
7x
+ 5y = 4;
4x
+ 3y = 2.
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iii)
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If
7 cos A - 24 sin A = 0,
find the value of tan A, sec A and cosec A.
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B)
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Solve any one of the following sub-questions:
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4
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i)
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Solve the following simultaneous equations:
15x
+ 17y = 81;
17x
+ 15y = 79.
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ii)
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The
following table shows the distribution of 90 apprentice
workers in a factory according to trade:
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Trade
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Fitting
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Turning
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Welding
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Moulding
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Spray
Painting
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Number of Workers
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25
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30
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8
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15
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12
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Show
the above data by a pie-diagram.
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| Q.4. |
A)
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Solve any two of the following sub-questions:
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6
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i)
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The
members of the Maharashtra Yuvak Mandal collected the following
amounts in rupees to help the earthquake affected people:
158,
238, 453, 134, 240, 343, 495, 230, 178, 275, 245, 175, 334,
248, 305, 120, 225, 210, 437, 160, 235, 290, 200, 320, 190,
240, 420, 225, 320, 150, 201, 105, 298, 240, 330, 101, 155,
410, 451, 221
Prepare a grouped frequency distribution table
of the data taking classes 100-200, -------, 400-500. Hence
prepare a table showing the cumulative frequency less than
the upper class limit.
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ii)
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Solve the following quadratic equation by factorisation:
x2 + 10x - 21 = 0.
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iii)
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Factorise the following quadratic expression by
using the formula:
2x2 + 4x -5.
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B)
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Solve any one of the following sub-questions:
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4
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i)
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If
a, b, c, d are in proportion, then show that: 
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ii)
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Answer the following questions with the help of
the given graph:
a)
Write the equation of line AB.
b)
Name the line whose equation is y = 4.
c)
Write the co-ordinates of the point of intersection of lines
PQ and AB.
d)
Write the equation of line EF.
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Q.5.
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Solve any three of the following sub-questions:
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15
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i)
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Factorise the following cyclic expression:
x
(y + z) 2 + y (z + x) 2 + z (x + y)
2 - 4xyz.
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ii)
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If
sin (A + B) =  and
sin B =  , then
find the values of all trigonometric ratio of Ð A.
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iii)
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Draw
a line parallel to X-axis, through the point of intersection
of the lines y = x - 3 and 2x + 3y = -9. Write the equation
of the line.
(Scale: 1 cm = 1 unit on both the axes)
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iv)
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Prepare the grouped frequency distribution table
from the cumulative frequency distribution table given below.
Hence draw the frequency polygon:
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Marks
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Cumulative Frequency
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Marks
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Cumulative Frequency
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Less than 5
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0
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Less than 45
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44
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Less than 15
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3
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Less than 55
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70
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Less than 25
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5
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Less than 65
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91
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Less than 35
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15
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Less than 75
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106
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Q.6.
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Solve any three of the following sub-questions:
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15
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i)
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If
x varies directly as a4 and y varies inversely
as a3 and z varies inversely as xy, then prove
that z varies inversely as a.
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ii)
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If
yza = zxb = xyc, then
prove that 
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iii)
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The
sum of the two natural numbers is 50 and the sum of their
reciprocals is  ,
then find those natural numbers,
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iv)
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Sharad bought a table and a fan together for Rs.5,000.
After some time, he sold the table at a gain of 25% and
the fan at the gain of 20%. Thus he gained 23% on the whole,
find the cost price of the fan.
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