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Let \(f:[0,1] \rightarrow \mathbb{R}\) (the set of all real numbers) be a function. Suppose the function \(f\) is twice differentiable, \(f(0)=f(1)=0\) and satisfies \(f^{\prime \prime}(x)-2 f^{\prime}(x)+f(x) \geq e^{x}, x \in[0,1] .\)


Question:

If the function \(\mathrm{e}^{-x} f(x)\) assumes its minimum in the interval \([0,1]\) at \(x=\frac{1}{4}\), which of the following is true?

Consider a triangle $PQR$ having sides of lengths $p,q$ and $r$ opposite to the angles $P,Q$ and $R$, respectively. Then which of the following statements is (are) TRUE?

Let $n_1 and n_2$ be the number of red and black balls, respectively, in box I. Let $n_{3}and n_4$ be the number of red and black balls, respectively, in box II.
A ball is drawn at random from box I and transferred to box II. If the probability of drawing a red ball from box I, after this transfer, is $\frac{1}{3}$ , then the correct options(s) with the possible values of $n_{1}and n_2$ is(are)

Let \(\mathbb{R}\) denote the set of all real numbers. Then the area of the region \(\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x>0, y>\frac{1}{x}, 5 x-4 y-1>0,4 x+4 y-17 <0\right\}\) is

Let \(S=\left\{A=\left(\begin{array}{lll}0 & 1 & c \\ 1 & a & d \\ 1 & b & e\end{array}\right): a, b, c, d, e \in\{0,1\}\right.\) and \(\left.|A| \in\{-1,1\}\right\}\), where \(|A|\) denotes the determinant of \(A\). Then the number of elements in \(S\) is

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Consider the functions defined implicitly by the equation \(y^3-3 y+x=0\) on various intervals in the real line. If \(x \in(-\infty,-2) \cup(2, \infty)\), the equation implicitly defines a unique real valued differentiable function \(y=f(x)\). If \(x \in(-2,2)\), the equation implicitly defines a unique real valued differentiable function \(y=g(x)\), satisfying \(g(0)=0\).Question:
The area of the region bounded by the curve \(y=f(x)\), the \(X\)-axis and the lines \(x=a\) and \(x=b\), where \(-\infty < a < b < -2\), is

Let \(f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]\) be the function defined by \(f(x)=\sin ^2 x\) and let \(g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)\) be the function defined by \(g(x)=\sqrt{\frac{\pi x}{2}-x^2}\).
The value of \(\frac{16}{\pi^3} \int_0^{\frac{\pi}{2}} f(x) g(x) d x\) is

A particle of mass $m$ is moving in the $xy$-plane such that its velocity at a point $(x, y)$ is given as $\overrightarrow{v}=\alpha \left(y\stackrel{\wedge }{x}+2x\stackrel{\wedge }{y}\right)$ where $\alpha$ is a non-zero constant. What is the force $\overrightarrow{F}$ acting on the particle?

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then

A charged shell of radius $\mathrm{R}$ carries a total charge $\mathrm{Q}.$ Given $\mathrm{\Phi }$ as the flux of electric field through a closed cylindrical surface of height $\mathrm{h},$ radius $\mathrm{r}$ and with its center same as that of the shell. Here, center of the cylinder is a point on the axis of the cylinder which is equidistant from its top and bottom surfaces. Which of the following option(s) is/are correct? [ ${\in }_0$ is the permittivity of free space]

To form a complete monolayer of acetic acid on \(1 \mathrm{~g}\) of charcoal, \(100 \mathrm{~mL}\) of \(0.5 \mathrm{M}\) acetic acid was used. Some of the acetic acid remained unadsorbed. To neutralize the unadsorbed acetic acid, 40 \(\mathrm{mL}\) of \(1 \mathrm{M} \mathrm{NaOH}\) solution was required. If each molecule of acetic acid occupies \(\mathbf{P} \times 10^{-23} \mathrm{~m}^2\) surface area on charcoal, the value of \(\mathbf{P}\) is _______ [Use given data: Surface area of charcoal \(=1.5 \times 10^2 \mathrm{~m}^2 \mathrm{~g}^{-1} ;\) Avogadro's number \(\left(\mathrm{N}_{\mathrm{A}}\right)=6.0 \times 10^{23}\) \(\left.\mathrm{mol}^{-1}\right]\)

On decreasing the $\mathrm{pH}$ from $7$ to $2$, the solubility of a sparingly soluble salt $\left(\mathrm{MX}\right)$ of a weak acid $\left(\mathrm{HX}\right)$ increased from ${10}^{-4} \mathrm{mol} {\mathrm{L}}^{-1}$ to ${10}^{-3} \mathrm{mol} {\mathrm{L}}^{-1}$. The ${\mathrm{pK}}_{\mathrm{a}}$ of $\mathrm{HX}$ is

Let \(\mathbb{R}\) denote the set of all real numbers. Let \(z_1=1+2 i\) and \(z_2=3 i\) be two complex numbers, where \(i=\sqrt{-1}\).
Let \(S=\{(x, y) \in \mathbb{R} \times \mathbb{R}:\left|x+i y-z_1\right|=2\)\(\left|x+i y-z_2\right|\}\).
Then which of the following statements is (are) TRUE?