CTET Paper II (Mathematics and Science) 3 Jan 2022 Solved Paper

Concept:A proper fraction has a numerator smaller than the denominator. Factorisation of a quadratic polynomial into two linear factors lets us identify the constants.Explanation:Factorise $10x^{2}+x-3$ by splitting the middle term. Find two numbers whose product is $10 \times (-3) = -30$ and sum is $1$. The numbers are $6$ and $-5$.Rewrite: $10x^{2} + 6x - 5x - 3$Group: $2x(5x+3) - 1(5x+3) = (5x+3)(2x-1)$Compare with $(ax+b)(cx+d)$. Since $a > c$, we take $a=5$, $b=3$, $c=2$, $d=-1$.Now compute $\frac{b+d}{a+c} = \frac{3 + (-1)}{5 + 2} = \frac{2}{7}$.Since numerator $2$ is less than denominator $7$, the result is a proper fraction.Answer:B. a proper fraction