Karnataka SSLC 10th Maths Question Paper 2025

By definition, the sum of the first $n$ terms is $S_n = a_1 + a_2 + ... + a_{n-1} + a_n$. Since $S_{n-1} = a_1 + a_2 + ... + a_{n-1}$, subtracting $S_{n-1}$ from $S_n$ leaves only $a_n$. Thus, $a_n = S_n - S_{n-1}$. Options A and D are algebraic rearrangements that do not hold true, and C is identical to B (though B is the standard representation).

According to the theorem on tangents to a circle, the tangent at any point of a circle is perpendicular to the radius through the point of contact. Therefore, the angle between them is always 90°.

For a system of linear equations in two variables, having no solution corresponds to the lines being parallel, meaning they never intersect at any point. Coincident lines have infinitely many solutions, and intersecting lines have a unique solution.

The volume of a cuboid is calculated as the product of its length, breadth, and height. Thus, Volume = $l \times b \times h = lbh$ cubic units. Other options correspond to perimeter or surface area formulas.

The probability $P(E)$ of an event must satisfy $0 \leq P(E) \leq 1$. Option A ($1/5 = 0.2$) is within this range. Option B ($1.5$) and C ($2.5$) are greater than 1, and option D ($-5$) is less than 0, so none of those can be probabilities.