Question
Factorise : $2 x^2+5 x-3$
✓
Answer
$(x+3)(2 x-1)$
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In the given figure, the area of $\| gm ABCD$ is $90 cm^2$.
State giving reasons :
(i) $\operatorname{ar}(\|$ gm ABEF)
(ii) $\operatorname{ar}(\triangle ABD )$
(iii) ar ( $\triangle BEF )$.
→State giving reasons :
(i) $\operatorname{ar}(\|$ gm ABEF)
(ii) $\operatorname{ar}(\triangle ABD )$
(iii) ar ( $\triangle BEF )$.
If $2^x=3^y=12^z$, show that $: \frac{1}{z}=\frac{1}{y}+\frac{2}{x}$.
[ Hint. Let $2^x=3^y=12^z=k$. Then, $2=k^{\frac{1}{x}}, 3=k^{\frac{1}{y}}$ and $12=k^{\frac{1}{z}}$.
Now, $\left.12=2^2 \times 3 \Rightarrow k^{\frac{1}{2}}=\left(k^{\frac{1}{x}}\right)^2 \times\left(k^{\frac{1}{y}}\right) \Rightarrow k^{\frac{1}{2}}=k^{\left(\frac{2}{x}+\frac{1}{y}\right)}\right]$
→[ Hint. Let $2^x=3^y=12^z=k$. Then, $2=k^{\frac{1}{x}}, 3=k^{\frac{1}{y}}$ and $12=k^{\frac{1}{z}}$.
Now, $\left.12=2^2 \times 3 \Rightarrow k^{\frac{1}{2}}=\left(k^{\frac{1}{x}}\right)^2 \times\left(k^{\frac{1}{y}}\right) \Rightarrow k^{\frac{1}{2}}=k^{\left(\frac{2}{x}+\frac{1}{y}\right)}\right]$
Evaluate the following without using log table:
$\frac{\log 9-\log 3}{\log 27}$
→$\frac{\log 9-\log 3}{\log 27}$
The population of a town is increasing at the rate of 10% per annum. If its present population is 36300, find:
(i) its population after 2 years,
(ii) its population 2 years ago.
(i) its population after 2 years,
(ii) its population 2 years ago.
The area of a rectangular is $640\ m^2.$ Taking its length as $x\ cm;$ find in terms of $x,$ the width of the rectangle. If the perimeter of the rectangle is $104 \ m$; find its dimensions.
→The ratio between an exterior angle and an interior angle of a regular polygon is $2 : 3.$ Find the number of sides in the polygon.
→Factorise the following:$p^4 + 23p^2q^2 + 90q^4$
→In the adjoining figure, ABCD is a parallelogram and O is any point on its diagonal AC.
Show that : ar $(\triangle AOB )=\operatorname{ar}(\triangle AOD )$.
→Show that : ar $(\triangle AOB )=\operatorname{ar}(\triangle AOD )$.
A 90% acid solution is mixed with a 97% acid solution to obtain 21 litres of a 95% solution. Find the quantity of each of the solutions to get the resultant mixture.
Find the perimeter and area of quadrilateral ABCD in which
$\begin{array}{l}AB=9 cm, AD=12 cm, BD=15 cm \\
CD=17 cm \text { and } \angle CBD=90\end{array}$.
→$\begin{array}{l}AB=9 cm, AD=12 cm, BD=15 cm \\
CD=17 cm \text { and } \angle CBD=90\end{array}$.