Karnataka SSLC 10th Maths Question Paper 2023

The number of zeroes of a polynomial $p(x)$ is given by the number of points where the graph intersects the x-axis. Observing the provided graph, the curve crosses the x-axis at exactly 2 distinct points. Therefore, the number of zeroes is 2.

The sum of the probability of an event $E$ and its complement $\bar{E}$ is 1, i.e., $P(E) + P(\bar{E}) = 1$. Given $P(E) = 0.75$, then $P(\bar{E}) = 1 - 0.75 = 0.25$.

The total surface area of a right circular cylinder is the sum of the curved surface area ($2\pi rh$) and the areas of the two circular bases ($2 \times \pi r^2$). Factoring this expression gives $2\pi r^2 + 2\pi rh = 2\pi r(r + h)$.

Euclid's division lemma states $a = bq + r$, where $0 \le r < b$. Comparing $19 = 6 \times 3 + 1$ with $a = bq + r$, we identify $a=19$, $b=6$, $q=3$, and $r=1$.

A tangent at any point of a circle is perpendicular to the radius at the point of contact. Thus, $\angle OAP = 90^\circ$. In $\triangle OAP$, the sum of angles is $180^\circ$. So, $\angle OPA = 180^\circ - (90^\circ + 45^\circ) = 180^\circ - 135^\circ = 45^\circ$.