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For given vectors, $a=2\hat{i}-\hat{j}+2\hat{k}andb=-\hat{i}+\hat{j}-\hat{k}$, find the unit vector in the direction of the vector $a+b$.

The value of the integral: $\underset{1}{\overset{3}{\int }}\left[x\right]\cos \left(\frac{\pi }{2}\right(x-\left[x\right]\left)\right)dx$

where $\left[x\right]$ denotes the largest integer not exceeding $x$ is :

Prove that:
(i) $2sin\frac{5\pi }{12}sin\frac{\pi }{12}=\frac{1}{2}$
(ii) $2cos\frac{5\pi }{12}cos\frac{\pi }{12}$
(iii) $2sin\frac{5\pi }{12}cos\frac{5\pi }{12}=\frac{\sqrt{3}+2}{2}$

The 2nd, 3rd and 4th terms in the expansion of $(x+y{)}^n$ are 240, 720 and 1080 respectively. Find x, y, n.

The equation of the curve $y=f\left(x\right)$ in first quadrant with positive tangent slope such that the sum of the lengths of the tangent and subtangent at any point on it is proportional to the product of the coordinates of the point (proportionality factor is $k$) is:

(where $c$ is arbitrary constant)

Find the value of following function;

$co{s}^{-1}\left(cos\frac{13\pi }{6}\right)$

If $1+3{\log }_{10}\sqrt{2+x}+4{\log }_{10}\sqrt{2-x}=3{\log }_{10}\sqrt{4-x^2}$, then the value of $x$ is

The sum of squares of the perpendiculars drawn from the points (0, 1) and (0, -1) to any tangent to a curve is 2. Then, the equation of the curve is

Show that the sum of (m
+ n)^th and (m – n)^th
terms of an A.P. is equal to twice the m^th
term.

A parabolic vertical curve is being designed to join a road of grade +5 % with a road of grade -3 %. The length of the vertical curve is 400 m measured along the horizontal. The vertical point of curvature VPC is located on the road of grade +5 %. The difference in height between VPC and vertical point of intersection (VPI) (in m, round of to the nearest integer) is

1. 10

The function ‘t’
which maps temperature in degree Celsius into temperature in degree
Fahrenheit is defined by.

Find (i) t
(0) (ii) t
(28) (iii) t
(–10) (iv) The value of C, when t(C)
= 212

Let $\omega$ is the root of the equation $x^2+x+1=0$ whose imiginary part is postive and $|z-\omega |=|z+\omega |$, then $\arg \left(z\right)$ is

The range of $f\left(x\right)=[|\sin x|+|\cos x|],$ where $[.]$ denotes the greatest integer function, is

The number of real solutions of the equation, $x^2-|x|-12=0$ is

If $m,n$ are the roots of the quadratic equation $x^2-3x+5=0,$ then the equation whose roots are $(m^2-3m+7)\&(n^2-3n+7)=$

Find the slope of the tangent to the
curve y = 3x⁴ − 4x at x
= 4.

Lines are drawn parallel to the line $4x-3y+2=0,$ at a distance $\frac{3}{5}$ units from the origin. Then which of the following points lies on any of these lines ?

The locus of the point of intersection of tangents to the parabola $y^2=4ax$ which includes an angle $\alpha$ is

If A and B are two sets such that $AsubsetB,$ then find :

(i) $A\cap B$ (ii) $A\cup B$

Prove that
is
the general solution of differential equation,
where c is a parameter.

Determine the maximum value of Z=3x+4y, if the feasible region (shaded) for a LPP is shown in following figure.

In acute angled triangle $ABC,r+r_1=r_2+r_3$ and $\angle B>\frac{\pi }{3}$ then

Write the value of ${lim}_{x\to -\infty }(3x+\sqrt{9x^2-x}).$

The least value of $f\left(x\right)=|x-a|+|x-b|+|x-c|+|x-d|$, where $a<b<c<d$ are fixed real numbers, is

Match the given functions in the first column with their first derivatives in the second column.

The value of $\underset{x\to 2}{lim}\frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2}$ is

The sum of $(1+x)+(1+x+x^2)+(1+x+x^2+x^3)+\dots$ upto $n$ terms is

Usng properties of determinants , prove that $\left|\begin{array}{ccc}1& 1& 1+3x\\ 1+3y& 1& 1\\ 1& 1+3z& 1\end{array}\right|=9(3xyz+xy+yz+zx)$

Integrate the following functions.
$\int \frac{1}{(1+cotx)}dx.$