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Mathematics · 2024 · 5 marks
CBSE 2024 · Region 4 · Set 2 · Q35
The angles of depression of the top and the bottom of a $\displaystyle 50$ m high building from the top of a tower are $\displaystyle 45$° and $\displaystyle 60$°, respectively. Find the height of the tower. (Use $\displaystyle \sqrt{3}=1 \cdot 73$ )
Marking-scheme solution
Let AB be the tower of height H m and CD be the building.
In \(\displaystyle \triangle \mathrm{ABD}\),
\[\begin{aligned}
& \tan 60^{\circ}=\sqrt{3}=\frac{\mathrm{H}}{\mathrm{x}} \\
& \Rightarrow \mathrm{H}=\sqrt{3} \mathrm{X}
\end{aligned}
\]
In \(\displaystyle \triangle \mathrm{AEC}\)
\[\tan 45^{\circ}=1=\frac{\mathrm{H}-50}{\mathrm{x}}
\]
Solving equations (i) & (ii)
\[H=25(3+\sqrt{3})=25(3+1.73)=118.25
\]
The height of tower is $\displaystyle 118.25$ m
(i)
Volume of cuboidal carton \(\displaystyle =30 \times 32 \times 15\)
\[=14400 \mathrm{~cm}^{3}
\]
(ii) (a) Total surface area of milk packet \(\displaystyle =2(15 \times 8+8 \times 5+5 \times 15)\)
\[=470 \mathrm{~cm}^{2}
\]
OR
(ii) (b) Number of milk packets in carton \(\displaystyle =\frac{30 \times 32 \times 15}{15 \times 8 \times 5}\)
\[=24
\]
(iii) Capacity of the cup \(\displaystyle =\frac{22}{7} \times 5 \times 5 \times 7\)
\[=550 \mathrm{~cm}^{3} \text { or } 550 \mathrm{ml}
\]
Based on the above, answer the following questions :
(i) Find the mid-point of the segment joining F and G.
(ii) (a) What is the distance between the points A and C ? OR
(b) Find the coordinates of the point which divides the line segment joining the points A and B in the ratio \(\displaystyle 1: 3\) internally.
(iii) What are the coordinates of the point D ?
(i) Mid point of FG is \(\displaystyle \left(\frac{-3+1}{2}, \frac{0+4}{2}\right)=(-1,2)\)
(ii) (a)
\[\begin{aligned}
\mathrm{AC} & =\sqrt{(-1-3)^{2}+(-2-4)^{2}} \\
& =\sqrt{52} \text { or } 2 \sqrt{13}
\end{aligned}
\]
OR
(ii) (b) The coordinates of required point are \(\displaystyle \left(\frac{1 \times 3+3 \times 3}{1+3}, \frac{1 \times 2+3 \times 4}{1+3}\right)\)
\[\text { i.e. }\left(3, \frac{7}{2}\right)
\]
(iii) D$\displaystyle (-2, -5)$
(i) Which number is on first spot?
(ii) (a) Which spot is numbered as $\displaystyle 112$ ? OR
(b) What is the sum of all the numbers on the first $\displaystyle 10$ spots ?
(iii) Which number is on the \(\displaystyle (\mathrm{n}-2)^{\text {th }}\) spot?
(i)
Number on the first spot \(\displaystyle =20+4 \times 1=24\)
(ii)
$\displaystyle 20$ + 4n = $\displaystyle 112$
\[\text { ⇒ n = } 23
\]
\(\displaystyle \mathrm{d}=4\)
\[\begin{aligned}
S_{10} & =\frac{10}{2}[2 \times 24+9 \times 4] \\
& =420
\end{aligned}
\]
(iii)
Number on the \(\displaystyle (\mathrm{n}-2)^{\mathrm{th}}\) spot \(\displaystyle =20+4(\mathrm{n}-2)\)
\[\text { = } 12 \text { + 4n }
\]
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CBSE Class 10 Mathematics past-paper question from the 2024board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.