Torque and magnetization measurements in magnetic fields $H$ up to 14 T were performed on CeCoIn$_5$ single crystals. The amplitude of the paramagnetic torque shows an $H^{2.3}$ dependence in the mixed state and an $H^{2}$ dependence in the normal state. In addition, the mixed-state magnetizations for both $Hparallel c$ and $Hparallel ab$ axes show anomalous behavior after the subtraction of the corresponding paramagnetic contributions as linear extrapolations of the normal-state magnetization. These experimental results point towards a nonlinear paramagnetic magnetization in the mixed state of CeCoIn$_5$, which is a result of the fact that both orbital and Pauli limiting effects dominate in the mixed state.
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