3 Marks Question

Question 13 Marks

A school has invited $42$ Mathematics teachers, $56$ Physics teachers and $70$ Chemistry teachers to attend a Science workshop. Find the minimum number of tables required, if the same number of teachers are to sit at a table and each table is occupied by teachers of the same subject.

Answer

$\operatorname{HCF}(42,56,70)=14$
$ \text { Minimum number of tables required } =\frac{42}{14}+\frac{56}{14}+\frac{70}{14}$
$ =12$

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Question 23 Marks

Find the values of $x$ and $y$ from the following pair of linear equations:
$62 x+43 y=167$
$43 x+62 y=148$

Answer

$62 x+43 y=167 \ldots \text { (i) }$
$43 x+62 y=148 \ldots \text { (ii) }$
Adding $(i)$ and $(ii)$ and simplifying, we get $x + y =3 \ldots (iii)$
Subtracting $(ii)$ from $(i)$ and simplifying, we get $x-y=1 \ldots (iv)$
Solving $(iii)$ and $(iv)$ to get $x=2$ and $y=1$

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Question 33 Marks

The sum of the digits of a $2 -$digit number is $12$ . Seven times the number is equal to four times the number obtained by reversing the order of the digits. Find the number.

Answer

Let the unit's place digit be $x$ and ten's place digit be $y$
$\therefore \text { Number }=10 y+x$
According to question,
$x+y=12 \ldots \text { (i) }$
$\text { and } 7(10 y+x)=4(10 x+y)$
$x-2 y=0 \ldots \text { (ii) }$
Solving $(i)$ and $(ii)$, we get
$x=8 \text { and } y=4$
Hence the reauired number is $48$

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Question 43 Marks

Find a quadratic polynomial whose sum of the zeroes is 8 and difference of the zeroes is 2 .

Answer

Let the zeroes be $\alpha$ and $\beta$
$
\therefore \alpha+\beta=8 \text { and } \alpha-\beta=2
$
Solving above two equations, we get $\alpha=5$ and $\beta=3$
So, the quadratic polynomial is $x^2-8 x+15$

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Question 53 Marks

In the given figure, two concentric circles have radii $3 \ cm$ and $5 \ cm$ . Two tangents $TR$ and $TP$ are drawn to the circles from an external point $T$ such that $TR$ touches the inner circle at $R$ and $TP$ touches the outer circle at $P$. If $T R=4 \sqrt{10} \ cm$, then find the length of $T P$.

Answer

Join $\text{OR, OP}$ and $OT$
In $\triangle ORT,$
$OT^2=OR^2+TR^2=3^2+(4 \sqrt{10})^2=169$
$\therefore OT=13 \ cm$
In $\triangle OPT$,
$TP^2=OT^2-TP^2=13^2-5^2=144$
$\therefore TP=12 \ cm$

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Question 63 Marks

If a hexagon $\text{PQRSTU}$ circumscribes a circle, prove that,
$\text{PQ+RS+TU=QR+ST+UP}$

Answer

In the given figure,$PA=PF \ldots(1)$
$AQ=BQ \ldots \text { (2) }$
$RC=RB \ldots(3)$
$CS=DS \ldots \text { (4) }$
$ET=TD \ldots(5)$
$UE=UF \ldots(6)$
Adding $\text{(1), (2), (3), (4), (5)}$ and $\text{(6),}$
$\text{PA+AQ+RC+CS+ET+UE=PF+BQ+BR+DS+TD+UF}$
$\Rightarrow \text{PQ+RS+TU=UP+ST+QR}$

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Question 73 Marks

Prove that:
$\left(\frac{1+\tan ^2 A}{1+\cot ^2 A}\right)=\frac{(1-\tan A)^2}{(1-\cot A)^2}$

Answer

$ \text { LHS } =\frac{1+\tan ^2 A}{1+\frac{1}{\tan ^2 A}}$
$ =\tan ^2 A$
$ \text { RHS } =\frac{(1-\tan A)^2}{\left(1-\frac{1}{\tan A}\right)^2}$
$ =\tan ^2 A$
$\therefore \text { LHS }=\text { RHS }$

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Question 83 Marks

The government rescued $100$ people after a train accident. Their ages were recorded in the following table. Find their mean age.

Age $($in years$)$	Number of people rescued
$10-20$	$9$
$20-30$	$14$
$30-40$	$15$
$40-50$	$21$
$50-60$	$23$
$60-70$	$12$
$70-80$	$6$

Answer

Age (in years)	Number of people rescued $(f_i)$	$X_i$	$U_i$	$x_iu_i$
$10-20$	$9$	$15$	$-3$	$-27$
$20-30$	$14$	$25$	$-2$	$-28$
$30-40$	$15$	$35$	$-1$	$-15$
$40-50$	$21$	$45$	$0$	$0$
$50-60$	$23$	$55$	$1$	$23$
$60-70$	$12$	$65$	$2$	$24$
$70-80$	$6$	$75$	$3$	$18$
$Total$	$100$			$-5$

$\text { Mean Age } =45+\frac{(-5)}{100} \times 10$
$ =44.5$
Hence$,$ mean age is $44.5$ years

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Question 93 Marks

A school has invited $42$ Mathematics teachers, $56$ Physics teachers and $70$ Chemistry teachers to attend a Science workshop. Find the minimum number of tables required, if the same number of teachers are to sit at a table and each table is occupied by teachers of the same subject.

Answer

$\operatorname{HCF}(42,56,70)=14$
$ \text { Minimum number of tables required } =\frac{42}{14}+\frac{56}{14}+\frac{70}{14}$
$ =12$

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Maths 3 Marks Question

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