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If two squares are chosen at random on a chess board, the probability that they have a side in common is:

If P is a prime number, then $n^p$ - n is divisible by p when n is a

The point dividing the line joining the points (1, 2, 3)and 3,-5,-6) in the ratio 3:- is

Let $\alpha ,\beta$ and $\gamma$ be angles in the first quadrant. If $\tan (\alpha +\beta )=\frac{15}{8}$ and $\text{cosec}\gamma =\frac{17}{8}$, then which of the following is/are correct?

In the expansion of $(5^{\frac{1}{2}}+7^{\frac{1}{8}}{)}^{1024}$, the number of integral terms is

Find the sum of $\sqrt{1+\frac{1}{1^2}+\frac{1}{2^2}}+\sqrt{1+\frac{1}{2^2}+\frac{1}{3^2}}+.....+\sqrt{1+\frac{1}{{2007}^2}+\frac{1}{{2008}^2}}$.

$\int ta{n}^{-1}\sqrt{\frac{1-x}{1+x}}dx.$

A pair of tangents $OA,OB$ $(O$ is origin$)$ is drawn to a circle whose centre is $C$ and radius is $3$ units. If the combined equation of $OA$ and $OB$ is $2x^2-3xy+y^2=0,$ then the area (in sq. units) of the quadrilateral $OACB$ is equal to

${lim}_{x\to 0}\frac{5^x-1}{\sqrt{4+x}-2}$

A manufacturer has $600$ liters of an $12\%$ acid solution. The number of liters of a $30\%$ acid solution to be added to it so that acid content in the resulting mixture will be more than $15\%$ but less $18\%$ is in the interval

If the integral of the function $e^{3x}$is g(x) and $g\left(0\right)=\frac{1}{3}$, then find the value of 3 $\frac{1}{e}g\left(\frac{1}{3}\right)$

In In $\Delta ABC,$ if $8R^2=a^2+b^2+c^2,$ then the triangle is

The combined equation of two sides of a triangle is $x^2-3y^2-2xy+8y-4=0.$ The third side, which is variable always passes through the point $(-5,-1)$. If the range of values of the slope of the third line such that the origin is an interior point of the triangle is $(a,b)$, then the value of $(a+\frac{1}{b})$ is

The distance between two parallel lines is unity. A point P lies between the lines at a distance a from one of them. The length of a side of an equilateral $\Delta$PQR, vertex Q of which lies on one of the parallel lines and vertex R lies on the other line, is

1. 506

The lines $a{x}^2+2hxy+b{y}^2=0$ are equally inclined to the lines

$a{x}^2+2hxy+b{y}^2+\lambda (x^2+y^2)=0$ for

Let F be a real valued function of real and positive argument such that $F\left(x\right)+3xF\left(\frac{1}{x}\right)=2(x+1)\forall x>0$, then the value of F(10099) is

The value of $\underset{1}{\overset{3}{\int }}{e}^{\left\{x\right\}}dx$ is equal to

(where $\{.\}$ denotes fractional part function)

If line $x+y=3$ is a tangent to the ellipse with foci at $(4,3)$ and $(6,k)$ at point $(1,2),$ then the value of $k$ is

A class contains three girls and four boys. Every Saturday, a group of $5$ students go on a picnic (a different group of students is sent every week). During the picnic, each girl in the group is given a doll by the accompanying teacher. If all possible groups of five have gone for picnic once, the total number of dolls that the girls have got is

Select the exponent(s) which is(are) equal to $(a^m{)}^n.$

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f\left(x\right)=9x^4+12x^3-36x^2+25,x\in R,$ then:

Equation of smallest circle passing through points of intersection of line $x+y=1$ & circle $x^2+y^2=9$ is

Three coins are tossed. Describe
(i) two events A and B which are mutually exclusive.

(ii) three events A, B and C which are mutually exclusive and exhaustive.

(iii) two events A and B which am not mutually exclusive.

(iv) two events A and B which are mutually exclusive but not exhaustive.

If f( n + 1) = f (n) + n for all n ≥ 0 or f (0) = 1 then f (200) equals

Is it is possible to cover RD Sharma of class 11th in 5 month?

The above graph of a function is:

A coin whose faces are marked $3,5$ is tossed $4$ times. The probability that the sum of the numbers thrown is greater than $15$

The angle of intersection of curves. y = [|sin x| + |cos x|] and $x^2+y^2=5$ where [.] denotes greatest integral function is

In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ wins the game if he throws a total of $6$ before $B$ throws a total of $7$ and $B$ wins the game if he throws a total of $7$ before $A$ throws a total of six. The game stops as soon as either of the players wins. The probability of $A$ winning the game is

The value of $\Delta =\left|\begin{array}{ccc}0& \sin \alpha & -\cos \alpha \\ -\sin \alpha & 0& \sin \beta \\ \cos \alpha & -\sin \beta & 0\end{array}\right|$ is

In a class of $100$ students there are $70$ boys whose average marks in a subject is $75$. If the average mark of the complete class is $72$, then the average marks of the girls is