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Using
section formula, show that the points A (2, –3, 4), B (–1,
2, 1) and
are
collinear.

The random variable $X$ has a probability distribution $P\left(X\right)$ of the following form,

$P\left(X\right)=\{\begin{array}{cc}k,& x=0\\ 2k,& x=1\\ 3k,& x=2\\ 0,& \text{otherwise}\end{array}$

then value of $k$ is:

If the roots of \(x^2+ax+b\)=0 are $se{c}^2\Pi /8$ and $cose{c}^2\Pi /8$ ,then which of the following is correct?

How can we calculate inverse tan for determining angle between vectors using Natural tangents table?

If $z$ is a complex number satisfying $z+\overline{z}=0$, then

An infinite G.P. has first term 'x' and sum '5', then x belongs to

The set of the solutions for $\frac{2x-5}{(x-1)(x-7)}\le 0$ is

Let $PQ$ be a focal chord of the parabola $y^2=4ax.$ The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,$ $a>0.$ Length of chord $PQ$ is

If $A(\alpha ,\beta )=\left[\begin{array}{ccc}\cos \alpha & \sin \alpha & 0\\ -\sin \alpha & \cos \alpha & 0\\ 0& 0& e^{\beta }\end{array}\right]$, then $\left(A\right(\alpha ,\beta ){)}^{-1}$ is equal to

Let $A=[a_{ij}{]}_{4\times 4}$ be a matrix such that $a_{ij}=\{\begin{array}{cc}2,& \text{if}i=j\\ 0,& \text{if}i\ne j\end{array}.$

Then the value of $\left\{\frac{det(adj\left(adjA\right))}{7}\right\}$ is

$\left(\right\{.\}$ represents the fractional part function $)$

Prove the following by using the principle of mathematical induction for all n ∈ N:

Let $AB$ be a sector of a circle with centre $O$ and radius $d$, $\angle AOB=\theta (<\frac{\pi }{2})$ , and $D$ be a point on $OA$ such that $BD$ is perpendicular $OA$. Let $E$ be the midpoint of $BD$ and $F$ be a point on the arc $AB$ such that $EF$ is parallel to $OA$. Then the ratio of length of the arc $AF$ to the length of the arc $AB$ is

The range of the function $f\left(x\right)=co{s}^2\frac{x}{4}+sin\frac{x}{4},xϵR$ is

State
whether each of the following statement is true or false. Justify
your answer.

(i) {2,
3, 4, 5} and {3, 6} are disjoint sets.

(ii) {a,
e,
i,
o,
u
} and {a,
b,
c,
d}
are disjoint sets.

(iii) {2,
6, 10, 14} and {3, 7, 11, 15} are disjoint sets.

(iv) {2,
6, 10} and {3, 7, 11} are disjoint sets.

If sets $A$ and $B$ are defined as

$A=\left\{\right(x,y)|y=\frac{1}{x},x\ne 0,x\in R\},$

$B=\left\{\right(x,y)|y=-x,x\in R\},$ then

A rail road curve is to be laid out on a circle. What radius should be used if the track is to change direction by ${25}^{\circ }$ in a distance of 40 metres?

The convolution of $y\left[n\right]=\frac{1}{4^n}u\left[n\right]\ast u[n+2]$ is

What is the maximum value of the function $sinx+cosx$?

Find the centre and radius of the circle x² + y² – 4x – 8y – 45 = 0

Which of the following is negative?

In the tetrahedron $ABCD,A=(1,2,-3)$ and $G(-3,4,5)$ is the centroid of the tetrahedron. If $P$ is the centroid of the $\Delta BCD$, then $AP=$

Area of the triangle formed by the lines x-y=0, x+y=0 and any tangent to the hyperbola

$x^2-y^2=a^2$ is

The number of distinct real roots of $\left|\begin{array}{ccc}\sin x& \cos x& \cos x\\ \cos x& \sin x& \cos x\\ \cos x& \cos x& \sin x\end{array}\right|=0$ in the interval $-\frac{\pi }{4}\le x\le \frac{\pi }{4}$ is

‘A’ lives at origin on the cartesian plane and his office at (4, 5). His friend lives at (2, 3) on the same space. ‘A’ can go to his office travelling one block at a time either in the +y or +x direction. If all possible paths are equally likely, then the probability that ‘A’ passed his friend’s house is

Using the property of determinants and without expanding.
$\left|\begin{array}{ccc}a-b& b-c& c-a\\ b-c& c-a& a-b\\ c-a& a-b& b-c\end{array}\right|=0$

The sum of the following series
$1+6+\frac{9(1^2+2^2+3^2)}{7}+\frac{12(1^2+2^2+3^2+4^2)}{9}+\frac{15(1^2+2^2+...+5^2)}{11}+...\text{up to}15\text{terms}$, is :

The number of solutions of the equation $min\{|x|,|x-1|,|x+1|\}=\frac{1}{2}$ is

The equation $\left|\begin{array}{ccc}2& 1& 1\\ 1& 1& -1\\ y& x^2& x\end{array}\right|$ represents a parabola passing through the points