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Let $A(3,0,-1),B(2,10,6)$ and $C(1,2,1)$ be the vertices of a triangle and $M$ be the mid point of $AC$. If $G$ divides $BM$ in the ratio $2:1,$ then $\cos \left(\angle GOA\right)$ ($O$ being the origin) is equal to :

If A(1, 2, 3), B(-1, -1, -1) be the points, then the distance AB is

[MP PET 2001; Pb. CET 2001]

The value of the integral $\underset{-2}{\overset{2}{\int }}\frac{{\sin }^{2}x}{\left[\frac{x}{\pi }\right]+\frac{1}{2}}dx$

(where $\left[x\right]$ denotes the greatest integer less than or equal to $x$ ) is:

If $n=2$ then the value of $\underset{x\to 0}{lim}\frac{e^x-1-x-\frac{x^2}{2!}-....-\frac{x^n}{n!}}{x^{n+1}}$ is

What is the expansion of (x + a) (x + b)

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 5y² – 9x² = 36

A bats man scores runs in 10 innings as 38,70,48, 34,42,55,63,46,54 and 44. The mean deviation about mean is

By reversing the order of integration. ${\int }_0^2{\int }_{x^2}^{2x}f(x,y)dydx$ may be resperesented as

If | $\frac{z_1-2z_2}{2-z_1\overline{z_2}}$| = 1, |$z_2$| $\ne$ 1, then |$z_1$| =

Let $O(0,0)$ and $A(0,1)$ be two fixed points. Then the locus of a point $P$ such that the perimeter of $△AOP$ is $4$, is:

Find the equations of all lines having slope 0 which are tangent to the curve $y=\frac{1}{x^2-2x+3}$

Find the volume of water which can be filled in a bucket of height 8 cm and radii of ends of the bucket are 9 cm and 3 cm.

Equation of the plane which passes through the point of intersection of lines $\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}$ and $\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ and at the greatest distance from the point (0, 0, 0) is

The value of the equals

Find the total number of selections of A + B + C + D objects where A are alike (of one kind), B are alike (of second kind), C are alike (of third kind) and D are alike (of fourth kind) if he has to select at least one of each kind of object.

On the basis of the information given below calculate the amount of Stationery to be debited to the 'Income and Expenditure Account' of Good Health Sports Club for the year ended 31st March 2017 :

$\begin{array}{ccc}& \text{Apirl 1, 2016}& \text{March 31, 2017}\\ \text{Stock of Stationery}& 8,000& 6,000\\ \text{Creditors for Stationery}& 9,000& 11,000\end{array}$

Stationery purchased during the year ended 31-3-2017 was Rs 47,000.

100 surnames were randomly picked up from a local telephone directory and the frequency distribution of the number of letters in the English alphabet in the surnames was obtained as follows:

$\begin{array}{ccccccc}Numberofletters& 1-4& 4-7& 7-10& 10-13& 13-16& 16-19\\ Numberofsurnames& 6& 30& 40& 16& 4& 4\end{array}$

Determine the median number of letters in the surnames. Find the mean number of letters in the surnames. Also, find the modal size of the surnames.

How to find log easily sir plaese help me with the log.

Find the equation of the tangent to the
curve

which is parallel to the line 4x − 2y + 5 = 0.

Consider the following code segment:

x = u - t;

y = x * v;

x = y + w;

y = t - z;

y = x * y;

The minimum number of total variables required to convert the above code segment to static single assignment form is

The complete solution set of the inequality $\frac{1}{1+\ln x}+\frac{1}{1-\ln x}>2$ is

If $A$ has coordinates $(-1,5)$ and $\overrightarrow{a}$ is a position vector whose tip is $(1,-3)$. Then the coordinates of the point B such that $\overrightarrow{AB}=\overrightarrow{a}$ is

If $\frac{1}{{\alpha }_k+i}({\alpha }_k\in R)$ are $8$ vertices of a regular octagon for $k=1,2,3,\dots ,8$, where $i=\sqrt{-1}$, then the area of the octagon is

Let $PQ$ be a chord of the parabola $y^2=8x$. A circle drawn with $PQ$ as diameter passes through the vertex $V$ of the parabola. If area of $\Delta PVQ=80$ square unit, then the coordinates of $P$ are

P, Q and R sharing profits and losses in the ratio of 3 : 2 : 1, decide to share profits and losses equally with effect from 1st April, 2017. Following is an extract of their Balance Sheet as at 31st March, 2017:

$\begin{array}{cccc}\text{Liabilities}& \text{Rs}& \text{Assets}& \text{Rs}\\ \text{Investment Fluctuation Reserve}& 30,000& \text{Investments (At Cost)}& 5,00,000\end{array}$

Show the accounting treatment under the following alternative cases:

Case (i) If there is no other information.
Case (ii) If the market value of Investments is Rs 5,00,000.
Case (iii) If the market value of Investments is Rs 4,88,000.
Case (iv) If the market value of Investments is Rs 4,46,000.
Case (v) If the market value of Investments is Rs 5,06,000.

Find the equation of the circle orthogonal to the circles $x^2+y^2+3x-5y+6=0and4x^2+4y2-28x+29=0$ and whose center lies on the line 3x + 4y + 1 = 0.

The solution of the differential equation is

[AISSE 1989, 90]

The equation of the focal chord of the parabola $y^2=8x$ whose sum of ordinates of the endpoints of the chord is $8$, is

Draw the graph of {sin x}

The value of $\int \frac{x+\sin x}{1+\cos x}dx$ is

$($where $C$ is integration constant$)$

Let $n\left(U\right)=700,n\left(A\right)=200,n\left(B\right)=300$ and $n(A\cap B)=100$, then $n(A^c\cap B^c)$ equals to

If the roots of the quadratic equation $x^2+6x+b=0$ are real and distinct and they differ by at most $4,$ then the range of $b$ is

The value of $2cos\theta -cos3\theta -cos5\theta -16co{s}^3\theta si{n}^2$ is