Algebra
Quadratic EquationsSequences and SeriesPermutations and CombinationsProbabilitySets and RelationsBinomial TheoremVector AlgebraThree Dimensional GeometryMatrices and DeterminantsStatisticsLinear ProgrammingComplex NumbersTrigonometry
Trigonometric EquationsProperties of TrianglesCalculus
FunctionsLimits, Continuity and DifferentiabilityApplication of DerivativesDefinite IntegrationArea Under The CurvesDifferential EquationsCoordinate Geometry
Straight Lines and Pair of Straight LinesCircleFunctions
Practice QuestionsMCQ (Single Correct Answer)
1
Let $f: Q \rightarrow Q$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbf{Q}$, and $f(1)=$ 10. Which one of the following statements is Correct?
IAT (IISER) 2024
2
Let $[x]$ denote the greatest integer less than or equal to $x$. The number of positive integer solutions of the equation
$$ \left[\frac{x}{19}\right]=\left[\frac{x}{20}\right] $$
are:
IAT (IISER) 2020
3
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions such that
$$ g \circ f(x)= \begin{cases}x^2, & \text { if } x \geq 0 \\ e^x-1, & \text { if } x<0\end{cases} $$
Then which of the following statements is correct?
IAT (IISER) 2020
4
For what value of $t$ does $f(x)=x^3+t x^2+(2-t) x-3$ have only real roots?
IAT (IISER) 2020