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Sin(pie-theta)=opposite/hypotenuse which is equal to opposite only . How ?

If R is the radius of the octahedral voids and 'r' is the radius of the atom in close packing, then $\frac{r}{R}$ is equal to

Let $A,B$ & $C$ be $3$ arbitrary events defined on a sample space $S$ and if, $P\left(A\right)+P\left(B\right)+P\left(C\right)=\frac{1}{2},P(A\cap B)+P(B\cap C)+P(C\cap A)=\frac{1}{4}$ & $P(A\cap B\cap C)=\frac{1}{6}$, then the probability that exactly one of the three events occurs is

At a given instant there are $25\%$ undecayed radioactive nuclei in a sample. After $10\text{sec}$ the number of undecayed nuclei reduces to $6.25\%$, the mean life of the nuclei is

The coefficients $a,b$ and $c$ of the quadratic equation, $a{x}^2+bx+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots is :

Evaluate the following limit:
$\underset{x\to -1}{\text{lim}}\frac{x^{10}+x^5+1}{x-1}$

Find the odd and even extensions of f(x) = $x^4-x^3+x^2$ (x > 0)

[ Assume the domain and range is x < 0 of the following function ]

If,
then find the value of x.

Let $f\left(x\right)=x^2,x\in R.$ For any $A\subseteq R$, define $g\left(A\right)=\{x\in R:f(x)\in A\}$. If $S=[0,4]$, then which one of the following statements is not true ?

If $O$ represents origin then, $3\overrightarrow{OD}+\overrightarrow{DA}+\overrightarrow{DB}+\overrightarrow{DC}$ is equal to

The derivative of f(x) defined by $f\left(x\right)={\tan }^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right),-\pi <x<\pi$
is

If $A=(6,-7),B=(-6,5)$ and $P,Q$ are two points on $AB$ such that $AP=PQ=QB$, then which of the following is/are correct?

Tangents are drawn from the origin to the curve y = sin x, then their point of contact lie on the curve

1. 25

A helicopter is flying along the curve given by $y-x^{\frac{3}{2}}=7,(x\ge 0)$. A soldier positioned at the point $(\frac{1}{2},7)$ wants to shoot down the helicopter when it is nearest to him. Then this nearest distance is :

If $\alpha$ is a complex constant such that $\alpha$$z^2$ + z + $\overline{\alpha }$ = 0 has a real root.Find the value of real root.

Let a and b be two positive real numbers. Then the value of
${\int }_a^b\frac{e^{\frac{x}{a}}-e^{\frac{b}{x}}}{x}dx$ is

1. 400

Let X={1,2,3} and Y ={4,5}. Find whether the following subsets of $X\times Y$ are functions from X to Y or not.
(i)g={(1,4),(2,4),(3,4)}

$\underset{n\to \infty }{lim}\frac{1+\frac{1}{2}+\frac{1}{2^2}+....+\frac{1}{2^n}}{1+\frac{1}{3}+\frac{1}{3^2}+....+\frac{1}{3^n}}\text{is equal to}$

Let $f$ is a continuous function defined as $f:R\to Z,f\left(1\right)=2$ and a matrix $A=[a_{ij}{]}_{2\times 2}$ is defined as $a_{ij}=f\left(2i\right)+f\left(2j\right).$ Then the matrix $A$ is

The value of $\underset{0}{\overset{200}{\int }}\left[{\cot }^{-1}x\right]dx$ is

If $lo{g}_{10}5+lo{g}_{10}(5x+1)=lo{g}_{10}(x+5)+1,then(x^2-9)$ is equal to:

Solve the following system of inequalities graphically: x + 2y ≤ 10, x + y ≥ 1, x – y ≤ 0, x ≥ 0, y ≥ 0

For$f\left(x\right)=a{x}^2+bx+c,a\ne 0,$ the conditions for which the graph is a downward opening parabola and f(x)=0 have a unique root with multiplicity $2$ is (Here D=discriminant)

If $f\left(x\right)=\frac{x-1}{x+1}$, then show that
(i) $f\left(\frac{1}{x}\right)=-f\left(x\right)$
(ii) $f(-\frac{1}{x})=\frac{-1}{f\left(x\right)}$

If $(1,1)$ and $(-3,5)$ are vertices of a diagonal of a square, then the equations of its sides through $(1,1)$ are