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Find sin $\frac{x}{2}$, cos $\frac{x}{2}$ and tan $\frac{x}{2}$ in each of the following:

tan x = - $\frac{4}{3}$, x in quadrant II.

Find the value of $\underset{x\to 0}{lim}\frac{(1+x{)}^5-1}{x}$

If the equation $x^2+2x+3\lambda =0and2x^2+3x+5\lambda =0$ have a non - zero common root, then $\lambda$=

If z = x + i y, then the area of the triangle whose vertices are points z, iz and z + iz is

A particle is undergoing SHM along a straight line so that its period is 12 s. The time it takes in traversing a distance equal to half its amplitude from the equilibrium position is

If the ellipse $\frac{x^2}{4}+\frac{y^2}{1}=1$ meet the ellipse $\frac{x^2}{1}+\frac{y^2}{a^2}=1$ in four distinct points and $a=b^2-10b+25$, then the value b does not satisfy

Prove that $0!=1.$

If $a{x}^2+bx+c,a,b,cϵR$, a < 0 has no real zeros and a - b+c<0, then the

value of ac

If $\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$ , then the real values of x and y are given by :

Find the equations of tangents to the ellipse
$\frac{x^2}{25}+\frac{y^2}{16}=1$
which are parallel to $3x+2y=25$

Match the columns I and II select the correct option.

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ 1.& \text{Lycopodium}& \text{p.}& \text{Alga}\\ 2.& \text{Sequoia}& \text{q.}& \text{Moss}\\ 3.& \text{Polytrichum}& \text{r.}& \text{Pteriophyte}\\ 4.& \text{Dictyota}& \text{s,}& \text{Gymnosperm}\end{array}$

For the complex number z, which of the following is true?

The solution set of the inequality $|[|x|-5]|-7<0$ is
$\left(\right[.]$ denotes the greatest integer function$)$

${\sum }_{r=0}^{7}ta{n}^2\left(\frac{\pi r}{16}\right)=$___

Three vertices of a parallelogram ABCD are $A(3,-1,2)B(1,2,-4)$ and $C(-1,1,2).$ Find the coordinates of the fourth vertex.

If a, b and c are three positive real numbers, which one of the following are true?

How many ways we can get a sum of atmost 17 by throwing six distinct dies.

The sum of two numbers is 6 times their geometric mean. Show that numbers are in the ratio $(3+2\sqrt{2}):(3-2\sqrt{2})$

The number of integral terms in the expansion of $(\sqrt{6}+\sqrt{7}{)}^{32}$ is:

The coefficient of $x^7$ in the expansion of $\frac{(1-x-x^2+x^3{)}^6}{(1-x{)}^{12}}$ is

$1^3+2^3+3^3+\cdots +n^3={\left[\frac{n(n+1)}{2}\right]}^2$

The equation $\frac{x^2}{12-k}+\frac{y^2}{8-k}=1$ represents.

If x = 1 + a + $a^2$ .......... to $\infty$ (|a|<1), y = 1 + b + $b^2$ ......... to $\infty$ (|b| < 1), then Z = 1 + ab + $a^2$ $b^2$ + $a^3$ $b^3$..... to ∞ is

Equation of the hyperbola passing through (2,1) and having the distance between the Directrices is $\frac{4}{\sqrt{3}}$ is

The new coordinates of a point (4, 5), when the origin is shifted to the point (1,-2) are

In $\Delta ABC,ccos(A-\alpha )+\alpha cos(C+\alpha )=$

How many 4-letter codes can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?

Find the sum to indicated number of terms in each of the geometric progressions in $\sqrt{7},\sqrt{21},3\sqrt{7},.....$ n terms.

Which of the following functions are identical to f(x) =
$f\left(n\right)=\{\begin{array}{cc}x,& 1\le x<2\\ x^2,& 2\le x<3\end{array}$

(1 $\le$ x < 3)

In how many ways can five keys be put in a ring

In fig. Blocks A and B move with velocities $v_1$ and $v_2$ along horizontal direction. Find the ratio of $v_1/v_2$.

Find the coordinatesof the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$\frac{x^2}{36}+\frac{y^2}{16}=1.$

Let A be a non-empty set such that A $\times$ A has 9 elements among which are found (-1,0) and (0,1), then

$\begin{array}{cc}Graphs& Functions\\ a& x^{\frac{1}{2}}\\ b& x^{\frac{1}{3}}\\ c& x^{\frac{1}{4}}\\ d& x^{\frac{1}{5}}\end{array}$

$iff:N\to Nisdefinedbyf\left(n\right)=n-(-1{)}^n,then$

Middle term in the expansion of

$(1+3x+3x^2+x^3{)}^6$ is

Find the value $si{n}^4\theta$ - $co{s}^4\theta$ + 2$co{s}^2\theta$, when $\theta =\frac{\Pi }{7}$

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

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

[Karnataka CET 1999]

All the pairs $(x,y)$ that satisfy the inequality $2^{\sqrt{{\sin }^{2}x-2\sin x+5}}\cdot \frac{1}{4^{{\sin }^{2}y}}\le 1$ also satisfy the equation :

Find the point where the graph of the function Sgn (lnx) breaks (or becomes discontinuous)
(Sgn is the Signum function)

If $a={\pi }^{3e},b=3^{\pi e}andc=e^{3\pi },$ then

The straight lines x+2y-9=0, 3x+5y-5=0 and ax+by-1=0 are concurrent if the straight line 22x-35y-1=0 passes through the point