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Find the equation of circle which touches $2x-y+3=0$ and pass through the points of intersection of the line $x+2y-1=0$ and the circle $x^2+y^2-2x+1=0$

In throwing a pair of dice, find the probability of getting an odd number on the first die and a total of 7 on both the sides.

The incentre of the triangle formed by (0, 0), (5,12), (16, 12) is

Find the number of dissimilar terms in the expansion of $(a^3+b^3+3ab(a+b){)}^{50}$

Find the value of ${\sum }_{i=0}^{\infty }$ $\frac{1}{5}\frac{1}{3^i}$.

Distance between the directrices of the ellipse $9x^2+5y^2-30y$ = 0 is

What are the numbers of ways in which 5 similar balls be put in 4 distinct boxes.

The minimum value of 3cosx+4sinx+5 is

[MNR 1991]

The general solution to $si{n}^{10}x+co{s}^{10}x=\frac{29}{16}co{s}^{4}2x$ is

The region of argand plane defined by |z+1|+|z-1|$\le$4 is

If $lo{g}_{7}2=m,thenlo{g}_{49}28$ is equal to

The sum of the series 1 + $\frac{1.3}{6}$ + $\frac{\mathrm{1.3.5}}{6.8}$ + ............∞ is

If $(p+q{)}^{\text{th}}$ term and $(p-q{)}^{\text{th}}$ term of a G.P. be $m$ and $n,$ then the $p^{\text{th}}$ term will be ___.

If the equation $x^2+\lambda x+\mu =0$ has equal roots and one root of the equation $x^2+\lambda x-12=0$ is 2, then (λ, μ) =

Cos 4x cos 8x - cos 5x cos 9x = 0 if

Find the principal solution of $3co{s}^2\theta -cos2\theta =1$

The coordinates of the point P which divides the line segment joining A(1,-2,3) and B(3,4,-5) internally in the ratio 2:3 is

$\begin{array}{cc}\text{Column 1}& \text{Column 2}\\ \text{1.sin}{18}^{\circ }& \text{p.}\frac{-\sqrt{5}-1}{4}\\ \text{2.cos}{18}^{\circ }& \text{q.}\frac{-1+\sqrt{5}}{4}\\ \text{3.cos}{9}^{\circ }\text{- sin}{9}^{\circ }& \text{r.}\frac{\sqrt{10+2\sqrt{5}}}{4}\\ & \text{s.}\frac{10+2\sqrt{5}}{4}\\ & \text{t.}\frac{\sqrt{5-\sqrt{5}}}{2}\\ & \text{u.}\frac{\sqrt{5-2\sqrt{5}}}{2}\end{array}$

If ${\log }_x\left({\log }_4\right({\log }_x(5x^2+4x^3)\left)\right)=0$, then the value of $x$ is

Let $a_i,i=1,2,\dots ,n$ be an A.P. If $a_7=9,$ then the value of common difference of the A.P. such that $a_1\cdot a_2\cdot a_7$ is minimum, is

Among the given functions, the one which is non-differentiable is .

Given below are the diameters of circles (in mm) drawn in a design.

$\begin{array}{cccccc}\text{Diameter}& 33-36& 37-40& 41-41& 45-48& 49-52\\ \text{Number of circles}& 15& 17& 21& 22& 25\end{array}$

Calculate the mean diameter of the circles, variance and standard deviation.

In a $\Delta ABC,$ prove that $(b^2-c^2)cota+(c^2-a^2)cotB+(a^2-b^2)cotC=0.$

If a relation $R$ is defined on the set of integers as follows: $(a,b)\in R\iff a^2+b^2=25$. Then domain of $R$ is

Explain errors of principle and give two examples with a measure to rectify them.

In drilling world's deepest hole it was found that the temperature T in degree celcius, x km below the earth's surface was given by T = 30 + 2.5 (x -3), 3 $\le x\le 15$. It was assumed that this drilling resulted

$5\%$ increase in carbon dioxide level at a temperature above

${205}^{\circ }$ and below ${155}^{\circ }C$, so drillers tried to control the depth between these two limits. At what depth will the temperature be between ${155}^{\circ }$ and ${205}^{\circ }$ What values are depicted here by the drillers for selecting this temperature range ?

Let $f\left(x\right)=[x{]}^2+\sqrt{\left\{x\right\}}$, where [x] is greatest integer function and {x} is the fractional part function, then

the function f(x) is discontinuous at.

The set of values of $x$ satisfying $1\le |x-1|\le 3$ is

Find the common interval between set A = ($-\infty$,-2) U (3,$\infty$) and set B = ($-\infty$, 7)

The number $N=6{\log }_{10}2+{\log }_{10}31$, lies between two successive integers, whose sum is equal to

Solution of $|x+2|+|2x+6|+|3x-3|=12$ is

Total number of complex numbers 'z', satisfying Re($z^2$) = 0, |z| = $\sqrt{3}$, is equal to :

If a, b, c, d are in H.P,, then ab + bc + cd is equal to

The coefficient of $x^n$ in the expansion of $(1+2x+3x^2+...{)}^{1/2}$ is equal to:

Let A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}. Find A - B and B - A.

Solve the following inequalities graphically. $2x+y\ge 4,x+y\le 3$ and $2x-3y\le 6$

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{ax+x cos x}}{\text{b sin x}}$

$1+\frac{1}{(1+2)}+\frac{1}{(1+2+3)}+\cdots +\frac{1}{(1+2+3+\cdots +n)}=\frac{2n}{(n+1)}$

Find the ${20}^{th}$ term in the binomial expansion of $(1+x{)}^{20}$ when $x=-5$

The value of ${\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+$..............

$+{\sin }^2{85}^{\circ }+{\sin }^2{90}^{\circ }$ is equal to

If ${}^8{C}_r$ = ${}^8{C}_{r+2}$, then the value of ${}^r{C}_2$ is

The co-ordinates of the extremities of the latus rectum of the parabola $5y^2=4x$ are

If A+B+C =$\pi$, prove that cos A + cos B - cos C = $\left(4\text{cos}\frac{A}{2}\text{cos}\frac{B}{2}\text{sin}\frac{C}{2}\right)-1$

The average marks of boys in class is 52 and that of girls is 42. The average marks of boys and girls combined is 50. The percentage of boys in the class is

If $\tan \theta =\frac{5}{12}$ and $\theta$ is not in the fourth quardrant then $\frac{\tan ({90}^{\circ }+\theta )-\sin ({180}^{\circ }-\theta )}{\sin ({270}^{\circ }-\theta )+cosec({360}^{\circ }-\theta )}=$

The sum of first n terms of the given series $1^2+{2.2}^2+3^2+{2.4}^2+5^2+{2.6}^2+.........is\frac{n(n+1{)}^2}{2}$, When n is even. When n is odd, the sum will be

If f is an even function defined on the interval (-5,5), then the real values of x, satisfying the equation $f\left(x\right)=f\left(\frac{x+1}{x+2}\right)$ are ___.