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For an A.P, first term a = 5 and Common difference d = 3. Find the sum of first 8 terms.

The point (2,0,3) lies in which octant?

Identify the three phases of the Law of Variable Proportions from the following and also give reason behind each phase :

$\begin{array}{cccccc}\text{Units of Variable Input}& 1& 2& 3& 4& 5\\ \text{Total Physcial Product (units)}& 10& 22& 30& 35& 30\end{array}$

Let p,q,r be three statements. Then $\sim (p\vee (q\wedge r\left)\right)$ is equal to

A square is inscribed in the circle $x^2+y^2-2x+4y-93=0$ with its sides parallel to the coordinate axes. The

coordinates of its vertices are

For positive integers n₁,n₂ the value of the expression

is a real number if and only if ($1+i{)}^{n_1}$+($1+i^3{)}^{n_1}$+($1+i^5{)}^{n_2}$($1+i^7{)}^{n_2}$ where i=$\sqrt[2]{2-1}$ is a real number if and only if

Find the value of x, if the ratio of 10th term to 11th term of the expansion $(2-3x^3{)}^{20}$ is 45 : 22.

Or

Find the value of a, so that the term independent of x in ${(\sqrt{x}+\frac{a}{x^2})}^{10}$ is 405.

Show that the point (x, y) given by $x=\frac{2at}{1+t^2}$ and $y=\frac{a(1-t^2)}{1+t^2}$ lies on a circle for all real values of that such that $-1\le t\le 1$, where a is any given real numbers.

Or

Find the equations of the altitudes of the triangle whose vertices are A (7, - 1), B(- 2, 8) and C (1, 2).

A and B are two sets given in such a way that $(A\times B)$ contains 6 elements. If three elements of $(A\times B)$ be (1, 3), (2, 5) and (3, 3), find its remaining elements.

Evaluate $\underset{x\to 0}{lim}\frac{sinax}{bx}$

Number of middle terms in the expansion of $(a+b{)}^{20}$ is:

How many 3-digit even numbers can be made using the digits 1,2,3,4,6,7 if no digit is repeated?

${}^{10}{C}_1$ + ${}^{10}{C}_3$ + ${}^{10}{C}_5$ + ${}^{10}{C}_7$ + ${}^{10}{C}_9$ =

If r > 0, -$\pi$ ≤ $\theta$ ≤ $\pi$ and (r, $\theta$) satisfy r sin$\theta$ = 3 and r = 4(1 + sin$\theta$) then the number of possible solutions of the pair ( r, $\theta$) is

Find the mean deviation about the median for the given data.

13, 17, 16, 14, 11, 13, 10, 16, 11, 18, 12, 17

If $cos3x+cos2x=sin\left(\frac{3x}{2}\right)+sin\left(\frac{x}{2}\right),0\le x\le 2\pi ,thenx=$

If a, 1, c are in A.P and a, 2, c are in G.P, then for a, b, c to be H.P the value of b = __

If $z_1,z_2,z_3$ be three complex numbers which are in H.P. And the points $A\left(z_1\right),B\left(z_2\right),C\left(z_3\right)$ are non-collinear, and O is origin, then:

If $\frac{3+5+7+........+nterms}{5+8+11+........+10terms}=7,$ the value of n is

If $(x+2),3,5$ are the lengths of sides of a triangle, then $x$ lies in

The mean deviation of the variates 40, 62, 68, 76, 54 from their arithmetic mean is

The equation of S.H.M. is $y=asin(2\pi nt+\alpha )$, then its phase at time t is

[DPMT 2001]

If 3 cosx+ 2cos3x = cosy, 3sinx+2sin3x=siny, then the value of $4(cos2x{)}^2$ is

If p and q are the length of perpendiculars from the origin to the lines $xcos\theta -ysin\theta =kcos2\theta$ and $xsec\theta +ycosec\theta =k$ respectively, prove that $p^2+4q^2=k^2$

If $S_k=\frac{1+2+3+4+...+k}{k}$, find the value of $S_1^2+S_2^2+S_3^2+...+S_n^2.$

Also, determine $\sum \left(\frac{S_n}{n}\right).$

How many values of $\theta ϵ[0,\frac{\pi }{2}]$, satisfy the relation cos $\theta +cos3\theta +cos5\theta +cos7\theta =0$ ?

If ${\log }_{12}27=a$, then $\frac{3-a}{3+a}=$

If ${\log }_{0.04}(x-1)\ge {\log }_{0.2}(x-1),$ then

A golf ball has a mass of 40g and a speed of 45 m$s^{-1}$. If the speed can be measured with an accuracy of 2%, calculate the uncertainty in the position.

Which of the following points lie on the x-y plane?

For the expression f(x) = a $x^2$ + bx + c, (a > 0), the condition for both real roots of f(x) to be greater than (or) lesser than a real value M is,

If x is real and satisfies x + 2 > $\sqrt{x+4}$, then

Evaluate the following limit:
$\underset{x\to 4}{\text{lim}}\frac{4x+3}{x-2}$

If the roots of the equation $x^2+ax+b=0$ are c & d, then one of the roots of equation $x^2$ + (2c + a)x + $c^2$ + ac + b = 0 is

If S is the focus and PQ is a focal chord of the parabola $y^2=4ax$ then SP the semilatusrectum, and SQ are in

If a>0 and the equation $a{x}^2+bx+c=0$ has two real roots $\alpha$ and $\beta$ such that $\left|\alpha \right|\le 1,\left|\beta \right|\le 1,$ then

The value of $\frac{C_1}{2}$ + $\frac{C_3}{4}$ + $\frac{C_5}{6}$ + ...... is equal to

A = { 1,2,3,4,5} and B = {a,b}. The number of relations from A to B is ___ .

The distance between the parallel lines $8x+6y+5=0$ and $4x+3y-25=0$ is

(i) Evaluate $\underset{x\to 1}{lim}\frac{(2x-3)(\sqrt{x}-1)}{2x^2+x-3}$ (ii) Differentiate $\frac{x+sinx}{x+cosx}$ with respect to x.

In how many ways can the letters of the word PERMUTATIONS be arranged if the

(i) words start with P and end with S?

(ii) vowels are all together?

(iii) there are always 4 letters between P and S?

The ratio in which line segment joining the points $(-3,10)$ and $(6,-8)$ is divided by $(-1,6)$ is

If |x| < 1, then the sum of the series 1 + 2x + $3x^2$ + $4x^3$ + ....... $\infty$ will be

Minimum value of the expression $f\left(x\right)=x^2-2x+6$ is __.