NIMCET 2027 2022 Previous Year Questions (PYQs) – Page 1 of 12
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Inverse of the function $f(x)=\frac{10^x-10^{-x}}{10^{x}+10^{-x}}$ is
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Solution
Let \( y = \dfrac{10^x - 10^{-x}}{10^x + 10^{-x}} \).
Multiply numerator and denominator by \(10^x\): \( y = \dfrac{10^{2x} - 1}{10^{2x} + 1} \).
Put \( t = 10^{2x} \), then \( y = \dfrac{t-1}{t+1} \). Solving, \( t = \dfrac{1+y}{1-y} \).
Hence, \( 10^{2x} = \dfrac{1+y}{1-y} \). Taking log base 10: \( 2x = \log_{10}\!\Big(\dfrac{1+y}{1-y}\Big) \).
✅ Final Answer
The inverse function is:
\[
f^{-1}(y) = \tfrac{1}{2}\,\log_{10}\!\left(\dfrac{1+y}{1-y}\right), \quad |y|<1
\]
Let $\vec{a}, \vec{b}, \vec{c} $ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$ , $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then c is
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Solution
$\vec{a}=a\hat{i}+a\hat{j}+c\hat{k}\, ,\, \vec{b}=\hat{i}+\hat{k}\, \&\, \vec{c}=c\hat{i}+c\hat{j}+b\hat{k}$ are coplanar.
$\Rightarrow\begin{vmatrix}{a} & {a} & {c} \\ {1} & {0} & {1} \\ {c} & {c} & {b}\end{vmatrix}=0$
$\Rightarrow-ac-ab+ac+{c}^2=0$
$\Rightarrow{c}^2=ab$
The correct expression for $cos^{-1} (-x)$ is
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Solution
You should learn it as an important formula.
Suppose that the temperature at a point (x,y), on a metal plate is $T(x,y)=4x^2-4xy+y^2$, An ant, walking on the plate, traverses a circle of radius 5 centered at the origin. What is the highest temperature encountered by the ant?
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Solution
The 10th and 50th percentiles of the observation 32, 49, 23, 29, 118 respectively are
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Solution
Angles of elevation of the top of a tower from
three points (collinear) A, B and C on a road
leading to the foot of the tower are 30°, 45°
and 60° respectively. The ratio of AB and BC is
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Solution
If the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{81}=\frac{1}{25}$ are coincide, then the value of $b^2$
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Solution
A particle is at rest at the origin. It moves
along the x −axis with an acceleration $x-x^2$
, where x is the distance of the
particle at time t. The particle next comes to
rest after it has covered a distance
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Solution
If $a{\lt}b$ then $\int ^b_a\Bigg{(}|x-a|+|x-b|\Bigg{)}dx$ is equal to
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Solution
The domain of the function $f(x)=\frac{{\cos }^{-1}x}{[x]}$ is
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Solution
NIMCET 2027
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NIMCET 2027
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and More.
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