CBSE 2025 · Region 1 · Set 2 · Q35 · 5 marks
A two-digit number is such that the product of its digits is 12. When $\displaystyle 36$ is added to this number, the digits interchange their places. Find the number.A student scored a total of $\displaystyle 32$ marks in class tests in Mathematics and Science. Had he scored $\displaystyle 2$ marks less in Science and $\displaystyle 4$ marks more in Mathematics, the product of his marks would have been 253. Find his marks in the two subjects.
A two-digit number is such that the product of its digits is 12. When $\displaystyle 36$ is added to this number, the digits interchange their places. Find the number.
A student scored a total of $\displaystyle 32$ marks in class tests in Mathematics and Science. Had he scored $\displaystyle 2$ marks less in Science and $\displaystyle 4$ marks more in Mathematics, the product of his marks would have been 253. Find his marks in the two subjects.
Marking-scheme solution
Let unit digit be y and ten's digit \(\displaystyle =x\)
hence, the two digit number \(\displaystyle =10 \mathrm{x}+\mathrm{y}\),
ATQ
\[x y=12 \quad \ldots \ldots \text { (i) }
\]
\(\displaystyle 10 x+y+36=10 y+x\)
\[x-y+4=0 \quad \ldots \ldots \text { (ii) }
\]
From (i) and (ii)
\[\begin{aligned}
& x^{2}+4 x-12=0 \\
& (x+6)(x-2)=0
\end{aligned}
\]
Hence, \(\displaystyle x=2\)
\[\text { and } y=6
\]
∴ Number = $\displaystyle 26$
Let marks scored in Mathematics be \(\displaystyle x\) and marks scored in Science be \(\displaystyle y\)
ATQ,
\[x+y=32 \quad \ldots \ldots \text { (i) }
\]
\[\text { and }(x+4)(y-2)=253 \quad \ldots \ldots \text { (ii) }
\]
from (i) and (ii)
\[x^{2}-26 x+133=0
\]
\[(x-19)(x-7)=0
\]
\(\displaystyle x=19,7\)
\[\left.\begin{array}{l}
x=19 \Rightarrow y=13 \\
x=7 \Rightarrow y=25
\end{array}\right\}
\]
Hence, marks in Mathematics and Science are $\displaystyle 19$, $\displaystyle 13$ or $\displaystyle 7$, $\displaystyle 25$
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