Let $\alpha, \beta, \gamma, \delta$ be real numbers such that $\alpha^2+\beta^2+\gamma^2 \neq 0$ and $\alpha+\gamma=1$. Suppose the point $(3,2,-1)$ is...

Let 𝒇:ℝ→ℝ and 𝒈:ℝ→ℝ be functions satisfying 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦)+𝑓(𝑥)𝑓(𝑦) and 𝑓(𝑥)=𝑥𝑔(𝑥) for all 𝑥,𝑦∈ℝ. If $\lim _{x \rightarrow 0} g(x)=1$, then...

Let b be a nonzero real number. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is a differentiable function such that $(0)=1$. If...

The value of the limit
$...

Let the functions : (−1,1)→ℝ and 𝑔 : (−1,1)→(−1,1) be defined by ƒ(𝑥) = |2𝑥−1|+|2𝑥+1| and 𝑔(𝑥) = 𝑥−[𝑥], where...

The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a...

Let O be the centre of the circle $\mathrm{x}^2+\mathrm{y}^2=\mathrm{r}^2$, where $\mathrm{r}>\frac{\sqrt{5}}{2}$. Suppose $P Q$ is a chord of this circle...

The probability that a missile hits a target successfully is 0.75. In order to destroy the target completely, at least...

For a complex number z, let $\operatorname{Re}(z)$ denote the real part of $z$. Let $S$ be the set of all...

Let e denote the base of the natural logarithm. The value of the real number a for which the right...

For a polynomial $\mathrm{g}(\mathrm{x})$ with real coefficient, let $\mathrm{m}_{\mathrm{g}}$ denote the number of distinct real roots of $\mathrm{g}(\mathrm{x})$. Suppose S...

Let $\mathrm{x} \in \mathbb{R}$ and let $...

A container with 1 kg of water in it is kept in sunlight, which causes the water to get warmer...

In a triangle PQR , let $\overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{QR}}, \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{RP}}$ and $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{PQ}}$. If $|\overrightarrow{\mathrm{a}}|=3,|\overrightarrow{\mathrm{~b}}|=4$ and $\frac{\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{b}})}{\overrightarrow{\mathrm{c}} \cdot(\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}})}=\frac{|\overrightarrow{\mathrm{a}}|}{|\overrightarrow{\mathrm{a}}|+|\overrightarrow{\mathrm{b}}|}$, then the value of...

Let f : [0, 2] -> R be the function defined by
$...

The inductors of two $L R$ circuits are placed next to each other, as shown in the figure. The values...

A cubical solid aluminium (bulk modulus $=-V \frac{\mathrm{dP}}{\mathrm{dV}}=70 \mathrm{GPa}$ ) block has an edge length of 1 m on the...

Let $a_1, a_2, a_3, \ldots$. be a sequence of positive integers in arithmetic progression with common difference 2 . Also,...