Let $f:\mathrm{R}\to \left(0,1\right)$ , be a continuous function then, which of the following function(s), has(have) the values zero at some point in the interval, (0,1)?

Let \(\mathbf{a}=-\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) be three given vectors. If \(\mathbf{r}\) is a vector such that \(\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}\) and \(\mathbf{r} \cdot \mathbf{a}=0\), then the value of \(\mathbf{r} \cdot \mathbf{b}\) is

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle \(R\) whose sides are parallel to the coordinate axes. Another ellipse \(E_{2}\) passing through the point \((0,4)\) circumscribes the rectangle \(R\). The eccentricity of the ellipse \(E_{2}\) is

Let \(\mathrm{S}\) be the area of the region enclosed by \(y=e^{-x^{2}}\), \(y=0, x=0\) and \(x=1\); then

If \(f(x)\) is twice differentiable function such that \(f(a)=0, f(b)=2, f(c)=1, f(d)=2\), \(f(e)=0\), where \(a < b < c < d < e\), then the minimum number of zeros of \(g(x)=\left\{f^{\prime}(x)\right\}^2+f^{\prime \prime}(x) \cdot f(x)\) in the interval \([a, e]\) is

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $\mathrm{V}\mathrm{ }\mathrm{m}{\mathrm{m}}^3$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2mm and is of radius equal to the outer radius of the container.
If the volume of the material used to make the container is minimum when the inner radius of the container is 10mm, then the value of $\frac{\mathrm{V}}{250\mathrm{\pi }}\mathrm{ }$ is

Paragraph: Let $S$ be the circle in the $xy$ – plane defined by the equation $x^2 + y^2 = 4.$

Question : Let $E_1 E_2$ and $F_1 F_2$ be the chords of $S$ passing through the point $P_0 \left(\mathrm{1,1}\right)$ and parallel to the $x$ – axis and the $y$ – axis, respectively. Let $G_1 G_2$ be the chord of $S$ passing through $P_0$ and having slope $-1.$ Let the tangents to $S$ at $E_1$ and $E_2$ meet at $E_3$, the tangents to $S$ at $F_1$ and $F_2$ meet at $F_3,$ and the tangents to $S$ at $G_1$ and $G_2$ meet at $G_3$. Then, the points $E_3, F_3,$ and $G_3$ lie on the curve

A vernier calipers has \(1 \mathrm{~mm}\) marks on the main scale. It has 20 equal divisions on the vernier scale which match with 16 main scale divisions. For this vernier calipers, the least count is

Consider the motion of a positive point charge in a region where there are simultaneous uniform electric and magnetic fields \(\vec{E}=E_{0} \hat{j}\) and \(\vec{B}=B_{0} \hat{j}\). At time \(t=0\), this charge has velocity \(\vec{v}\) in the in the \(x\)-y plane, making an angle \(\theta\) with the \(x\)-axis. Which of the following option(s) is (are) correct for time \(t>0\) ?

Let $S=\left(0,1\right)\cup \left(1, 2\right)\cup \left(3, 4\right)$ and $T=\left\{0, 1, 2, 3\right\}$. Then which of the following statements is(are) true?

Paragraph :
A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the piston is pushed at a speed of \(5 \mathrm{~mms}^{-1}\), the air comes out of the nozzle with a speed of

Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$ . If the complex number $z=x+iy$ satisfies $Im\left(\frac{az+b}{z+1}\right)=y,$ then which of the following is (are) possible value (s) of $x?$

An accident in a nuclear laboratory resulted in deposition of a certain amount of radioactive material of half-life 18 days inside the laboratory. Tests revealed that the radiation was 64 times more than the permissible level required for safe operation of the laboratory. What is the minimum number of days after which the laboratory can be considered safe for use?