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 LinesCircleDefinite Integration
Practice QuestionsMCQ (Single Correct Answer)
1
Let $I=\int_{e^{-\pi / 2}}^{e^{\pi / 2}}\left(\sin ^2(\log (x))+\sin \left(\log \left(x^2\right)\right)\right) d x$. What is the value of $I$ ?
IAT (IISER) 2024
2
Let $f: \mathbf{R} \rightarrow(0, \infty)$ be a continuous decreasing function. Suppose $f(0), \dot{f}(1), \ldots, f(10)$ are in a geometric progression with common ratio $\frac{1}{5}$. In which of the following intervals does the value of $\int_0^{10} f(x) d x$ lie?
IAT (IISER) 2023
3
Let $f:(-1,2) \rightarrow \mathbf{R}$ be a differentiable function such that $f^{\prime}(x)=\frac{2}{x^2-5}$ and $f(0)=0$. Then in which of the following intervals does $f(1)$ lie?
IAT (IISER) 2023
4
For a natural number $n$, let $C_n$ be the curve in the $X Y$-plane given by $y=x^n$, where $0 \leq$ $x \leq 1$. Let $A_n$ denote the area of the region bounded between $C_n$ and $C_n+1$. Then the largest value of $A_n$ is
IAT (IISER) 2022
5
Let $f$ be a continuous function on $[0,1]$ and $F$ be its antiderivative. If $F(0)=1$ and $\int_0^1 f(x) d x=1$, then $F(1)$ is
IAT (IISER) 2022
6
The value of the integral
$$ \int_1^{100} \frac{[x]}{x} d x $$
where $[x]$ is the greatest integer less than or equal to $x$ for any real number $x$, is
IAT (IISER) 2022
7
If $p(t)=\frac{t(t-1) \cdots(t-2019)}{2019!}$, then the value of
$$ \int_0^1\left(\frac{1}{t+1}+\frac{1}{t+2}+\cdots+\frac{1}{t+2020}\right) p(-t-1) d t $$
is:
IAT (IISER) 2020
8
$F(x)=\int_0^{e^x}\left(t^3+2 t^2-t-2\right) d t$, then for how many real numbers $x$ does $F^{\prime}(x)=0$ ?
IAT (IISER) 2020