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SECTION A (40 Marks)
Attempt all questions from this Section.
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1.
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a)
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Find the value of and ‘y’ if:
$2left[begin{array}{cc}x & 7 \ 9 & y-5end{array}right]+left[begin{array}{rr}6 & -7 \ 4 & 5end{array}right]=left[begin{array}{cc}10 & 7 \ 22 & 15end{array}right]$
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3
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b)
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Sonia had a recurring deposit account in a bank and deposited Rs. 600 per month for $2^{1} / 2$ years. If the rate of interest was $10 %$ p.a., find the maturity value of this account.
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3
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c)
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Cards bearing numbers $2,4,6,8,10,12,14,16,18$ and 20 are kept in a bag. A card is drawn at random from the bag. Find the probability of getting a card which is:
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4
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i.
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a prime number.
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ii.
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a number divisible by 4.
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iii.
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a number that is a multiple of 6.
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iv.
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an odd number.
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2.
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a)
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The circumference of the base of a cylindrical vessel is $132 mathrm{~cm}$ and its height is 25 $mathrm{cm}$. Find the
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3
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i.
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radius of the cylinder
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ii.
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volume of cylinder (use $pi=frac{22}{7}$ )
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b)
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If $(mathrm{k}-3),(2 mathrm{k}+1)$ and $(4 mathrm{k}+3)$ are three consecutive terms of an A.P., find the value of $mathrm{k}$.
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3
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c)
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PQRS is a cyclic quadrilateral. Given $angle mathrm{QPS}=73^{circ}, angle mathrm{PQS}=55^{circ}$ and $angle mathrm{PSR}=82^{circ}$,
calculate:

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4
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i.
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$angle QRS$
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ii.
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$angle RQS$
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iii.
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$angle PRQ$
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3.
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a)
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If $(x+2)$ and $(x+3)$ are factors of $x^{3}+a x+b$, find the values of ‘a’ and ‘b’.
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3
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b)
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Prove that $sqrt{sec ^{2} theta+operatorname{cosec}^{2} theta}=tan theta+cot theta$
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3
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c)
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Using a graph paper draw a histogram for the given distribution showing the number of runs scored by 50 batsmen. Estimate the mode of the data:
Runs scored
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3000-
4000
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4000-
5000
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5000-
6000
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6000-
7000
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7000-
8000
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8000-
9000
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9000-
10000
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No. of Batsman
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4
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18
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9
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6
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7
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2
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4
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4
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4.
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a)
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Solve the following inequation, write down the solution set and represent it on the real number line:
$-2+10 x leq 13 x+10<24+10 x, x in Z$
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3
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b)
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If the straight lines $3 x-5 y=7$ and $4 x+a y+9=0$ are perpendicular to one another, find the value of a.
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3
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c)
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Solve $x^{2}+7 x=7$ and give your answer correct to two decimal places.
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4
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SECTION B (40 Marks)
Attempt any four questions from this Section
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5.
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a)
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The $4^{text {th }}$ term of a G.P. is 16 and the $7^{text {th }}$ term is $128 .$ Find the first term and common ratio of the series.
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3
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b)
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A man invests Rs. 22,500 in Rs. 50 shares available at $10 %$ discount. If the dividend paid by the company is $12 %$, calculate:
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3
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i.
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The number of shares purchased
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ii.
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The annual dividend received.
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iii.
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The rate of return he gets on his investment. Give your answer correct to the nearest whole number.
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c)
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Use graph paper for this question (Take $2 mathrm{~cm}=1$ unit along both $x$ and y axis). $A B C D$ is a quadrilateral whose vertices are $mathrm{A}(2,2), mathrm{B}(2,-2), mathrm{C}(0,-1)$ and $mathrm{D}(0,1)$.
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4
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i.
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Reflect quadrilateral $mathrm{ABCD}$ on the $mathrm{y}$ -axis and name it as A’B’CD.
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ii.
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Write down the coordinates of $mathrm{A}^{prime}$ and $mathrm{B}^{prime}$.
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iii.
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Name two points which are invariant under the above reflection.
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iv.
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Name the polygon $mathrm{A}^{prime} mathrm{B}^{prime} mathrm{CD}$.
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6.
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a)
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Using properties of proportion, solve for $x$. Given that $x$ is positive:
$frac{2 x+sqrt{4 x^{2}-1}}{2 x-sqrt{4 x^{2}-1}}=4$
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3
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b)
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if $A=left[begin{array}{ll}2 & 3 \ 5 & 7end{array}right], B=left[begin{array}{rr}0 & 4 \ -1 & 7end{array}right]$ and $c=left[begin{array}{rr}1 & 0 \ -1 & 4end{array}right]$, find $A C+B^{2}-10 C .$
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3
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c)
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Prove that $(1+cot theta-operatorname{cosec} theta)(1+tan theta+sec theta)=2$
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4
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7.
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a)
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Find the value of $mathrm{k}$ for which the following equation has equal roots. $x^{2}+4 k x+left(k^{2}-k+2right)=0$
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3
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b)
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On a map drawn to a scale of $1: 50,000$, a rectangular plot of land ABCD has the following dimensions. $mathrm{AB}=6 mathrm{~cm} ; mathrm{BC}=8 mathrm{~cm}$ and all angles are right angles. Find: $[3]$
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3
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i.
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the actual length of the diagonal distance AC of the plot in $mathrm{km}$.
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ii.
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the actual area of the plot in sq. $mathrm{km}$.
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c)
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$mathrm{A}(2,5), mathrm{B}(-1,2)$ and $mathrm{C}(5,8)$ are the vertices of a triangle $mathrm{ABC},{ }^{prime} mathrm{M}^{prime}$ is a point on $mathrm{AB}$
such that $mathrm{AM}: mathrm{MB}=1: 2$. Find the co-ordinates of ‘ $mathrm{M}$ ‘. Hence find the equation of the line passing through the points $mathrm{C}$ and $mathrm{M}$.
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4
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8.
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a)
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Rs. 7500 were divided equally among a certain number of children. Had there been 20 less children, each would have received Rs. 100 more. Find the original number of children.
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3
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b)
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If the mean of the following distribution is 24 , find the value of ‘ $a$ ‘
Marks
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0-10
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10-20
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20-30
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30-40
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40-50
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No. of students
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7
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a
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8
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10
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5
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3
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c)
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Using ruler and compass only, construct a $triangle mathrm{ABC}$ such that $mathrm{BC}=5 mathrm{~cm}$ and $mathrm{AB}=6.5$ $mathrm{cm}$ and $angle mathrm{ABC}=120^{circ}$
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4
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i.
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Construct a circum-circle of $Delta mathrm{ABC}$
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ii.
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Construct a cyclic quadrilateral $A B C D$, such that $D$ is equidistant from $A B$ and $mathrm{BC}$.
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9.
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a)
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Priyanka has a recurring deposit account of Rs. 1000 per month at $10 %$ per annum. If she gets Rs. 5550 as interest at the time of maturity, find the total time for which the account was held.
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3
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b)
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In $Delta mathrm{PQR}, mathrm{MN}$ is parrallel to $mathrm{QR}$ and $frac{mathrm{PM}}{mathrm{MQ}}=frac{2}{3}$
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3
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i.
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Find $frac{mathrm{MN}}{mathrm{QR}}$
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ii.
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Prove that $Delta 0 M N$ and $Delta O R Q$ are similar.
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iii.
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Find, Area of $Delta 0 mathrm{MN}:$ Area of $Delta mathrm{ORQ}$

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c)
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The following figure represents a solid consisting of a right circular cylinder with a hemisphere at one end and a cone at the other. Their common radius is $7 mathrm{~cm}$. The height of the cylinder and cone are each of $4 mathrm{~cm}$. Find the volume of the solid.

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4
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10.
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a)
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Use Remainder theorem to factorize the following polynomial:
$2 x^{3}+3 x^{2}-9 x-10$
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3
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b)
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In the figure given below ‘ 0 ‘ is the centre of the circle. If $Q R=0 P$ and $angle O R P=20^{circ}$. Find the value of ‘x’ giving reasons.

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3
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c)
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The angle of elevation from a point $mathrm{P}$ of the top of a tower $mathrm{QR}, 50 mathrm{~m}$ high is $60^{circ}$ and that of the tower PT from a point $Q$ is $30^{circ} .$ Find the height of the tower $P T$, correct to the nearest metre.

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4
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11.
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a)
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The $4^{text {th }}$ term of an A.P. is 22 and $15^{text {th }}$ term is 66 . Find the first terns and the common difference. Hence find the sum of the series to 8 terms.
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4
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b)
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Use Graph paper for this question.
A survey regarding height (in $mathrm{cm}$ ) of 60 boys belonging to Class 10 of a school was conducted. The following data was recorded:
Height in cm.
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135-140
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140-145
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145-150
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150-155
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155-160
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160-165
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165-170
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No. of boys
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4
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8
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20
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14
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7
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6
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1
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Taking $2 mathrm{~cm}=$ height of $10 mathrm{~cm}$ along one axis and $2 mathrm{~cm}=10$ boys along the other axis draw an ogive of the above distribution. Use the graph to estimate the following:
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6
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i.
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the median
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ii.
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lower Quartile
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iii.
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if above $158 mathrm{~cm}$ is considered as the tall boys of the class. Find the number of boys in the class who are tall.
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