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Find the direction cosines of the vector joining the points $A(1,2,-3)andB(-1,-2,1)$, directed from A to B.

A hypothetical experiment was conducted to determine Young's Formula $Y=\frac{T^x.\tau \cos \theta }{l^3}$. If $Y$ is young's modulus ($T$ is time, $\tau$ torque, $l=$ length). Find the value of $x$.
$\left(\right[Y]=M{L}^{-1}{T}^{-2},[\tau ]=M{L}^2{T}^{-2})$

Let $y=mx+c$ is a tangent at point $P$ to the curve $y^2=2x,$ meets the curve again at $Q.$ Then which of the following condition holds true

The half life of a ratio active source is T. If radio activities of the samples are $R_1-R_2$ at time

$T_{1}and{T}_2$ Respectively, then the number of atoms disintegrated in time $(T_2-T_1)$ is proportional to

Which of the following is a set?

The equation(s) of normal(s) to the curve $3x^2-y^2=8$ which is (are) parallel to the line x + 3y = 4 is (are)

If $\varphi \left(x\right)={\int }_0^{x^2}\sqrt{t}dt.$ then $\frac{d\varphi }{dx}$ is:

Refer to question 15. Determine the maximum distance that the man can travel.

If Q = 10^-4 C; then the 100 V equipotential surface is drawn at a distance of ?

Let $A$ and $B$ be two points with position vector $2\hat{i}+\hat{j}-3\hat{k}$ and $\beta \hat{i}+{\alpha }^2\hat{j}$ respectively such that $|\overrightarrow{AB}|=3$ then $\underset{x\to \beta }{lim}(x-{\alpha }^2{)}^{\frac{1}{x-(\alpha +1)}}$ may be equal to:

Fill in the blanks:

$1{\mathrm{m}}^3={\mathrm{\_\_\_\_\_\_\_\_cm}}^3$

If a cube of volume V cubic units is completely opened up, then what is the area of the surface (in sq units) it will cover on a floor?

Find the sum to n
terms of each of the series in Exercises 1 to 7.

3 ×
1² + 5 × 2²
+ 7 × 3² + …

The value of $sin\left[2co{s}^{-1}\frac{\sqrt{5}}{3}\right]$ is

In the triangle ABC, which statement is not true?

Number of real solutions of the equation $|x-3{|}^{3x^2-10x+3}=1$ is

The mean deviation for n observations $x_1,x_2,....x_n$ from their mean $\overline{X}$ is given by

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}.$ Let $Q$ be the mirror image of the point $(2,3,–1)$ with respect to $L.$ Let a plane $P$ be such that it passes through $Q,$ and the line $L$ is perpendicular to $P.$ Then which of the following points is on the plane $P$:

The equation of the plane through intersection of planes $x+2y+3z=4$ and $2x+y-z=-5,$ and perpendicular to the plane $5x+3y+6z+8=0$ is

The total number of matrices
$A=\left[\begin{array}{ccc}0& 2y& 1\\ 2x& y& -1\\ 2x& -y& 1\end{array}\right],(x,y\in R,x\ne y)$

for which $A^{T}A=3I_3$ is :

The number of terms in the expansion of $(x+y+z{)}^{10}$ is

What fraction of the figure is colored green?

If $f\left(x\right)$ and $g_a\left(x\right)$ are real valued function defined as $g_a\left(x\right)=\frac{a^x}{a^x+\sqrt{a}},$ where $a>0$ and $f\left(x\right)=g_9\left(x\right)$, then the value of $\left[\sum _{r=1}^{1995}f\left(\frac{r}{1996}\right)\right]$ is

(where [.] denotes greatest integer function)

If $f\left(x\right)=$$\{\begin{array}{c}\frac{1}{|x|};|x|\ge 1\\ a{x}^2+b;|x|<1\end{array}$ is a differentiable at every point of the domain, then the values of $a$ and $b$ are respectively:

Sketch the graph of the following functions:
y= $si{n}^2$ x,y = sin x

In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?

Few dice are taken and each die is painted with three different colours such that the opposite faces have the same colour. If three such identical dice are thrown simultaneously, then what is the probability of getting different colours on all the three?

The largest number common to both the sequences $1,11,21,31,\cdots \text{upto}100\text{terms}$ and $31,36,41,46,\cdots \text{upto}100\text{terms}$ is

The line $2x+y=1$ is tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If this line pasess through the point of intersection of the nearest directrix and the X-axis, then the eccentricity of the hyperbola is ___