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In a triangle $ABC,\cos A\cos B+\sin A\sin B\sin C=1,$ then $a:b:c$ is:

If a>0, b>0, c>0 are in G.P., then $lo{g}_{a}x,lo{g}_{b}x,lo{g}_{c}x$ are in

Let $p,q$ and $r$ be the roots of the equation $y^3-3y^2+6y+1=0$. If the vertices of a triangle are $(pq,\frac{1}{pq}),$ $(qr,\frac{1}{qr})$ and $(rp,\frac{1}{rp}),$ then the coordinates of its centroid are

The eccentricity of the conjugate hyperbola of $\frac{x^2}{16}-\frac{y^2}{9}=1$ is .

The value of $\int ({\sin }^{4}x+{\cos }^{4}x)dx$ is

(where $C$ is constant of integration)

Let $F\left(\alpha \right)=\left[\begin{array}{ccc}cos\alpha & -sin\alpha & 0\\ sin\alpha & cos\alpha & 0\\ 0& 0& 1\end{array}\right]$, where $\alpha ϵR.$
Then $\left[F\right(\alpha ){]}^{-1}$ is equal to

If $f"\left(x\right)>0\forall xϵR$ then for any two real numbers $x_{1}and{x}_2,(x_1\ne x_2)$

Statements:

All puppets are dolls.

All dolls are toys.

Conclusions:

Some toys are puppets.

All toys are puppets.

Area of the triangle formed by the tangents drawn from $(2,2\sqrt{3})$ to the circle $x^2+y^2=4$ and the chord of contact joining the tangency points is

If the term independent of $x$ in the expansion of ${(\frac{3}{2}{x}^2-\frac{1}{3x})}^9$ is $k$, then $18k$ is equal to:

Let $S$ be the area of the region enclosed by $y=e^{-x^2},y=0,x=0$ and $x=1$. Then

The value of $p$ such that the length of the sub-tangent and sub-normal is equal for the curve $y=e^{px}+px$ at the point $(0,1)$ is

Area under the curve $y=x^2-4x$ within the x-axis and the line x=2, is

Let $A$ and $B$ be independent events such that $P\left(A\right)=p,P\left(B\right)=2p.$ The largest value of $p$ for which $P\text{(exactly one of}A,B\text{occurs)}=\frac{5}{9},$ is

1. 291

For any sets $A\&B,$ select the correct statements.

If a + ib is a root of the equation. $x^2$ + x + 1 = 0, where a, b ∈ R and a ≠ 0, b ≠ 0, find the value of a.

Find the range of values of l for which the variable line 3x+4y-l=0 lies between the circles x²+y²-2x-2y+1=0 and x²+y²-18x-2y+78=0 without intercepting a chord on either circle.

The critical point(s) of the function $f\left(x\right)=(x-2{)}^{2/3}(2x+1)$ is(are)

Write the number of ways in which 5 red and 4 white balls can be drawn from a bag containing 10 red and 8 white balls.

For the quadratic equation $a{x}^2+bx+c=0.$ Match the following graphs with the conditions given.

The value of ${lim}_{n\to \infty }{\left\{\frac{(n^3+1)(n^3+2^3)(n^3+3^3)........(n^3+n^3)}{n^3}\right\}}^{\frac{1}{n}}$ is equal to $\alpha .e^{\beta \int \frac{x^3}{1+x^3}dx}$, where $\alpha ϵN,\beta ϵR$,then find $\alpha -\beta$.___

Let $f\left(x\right)$ be a real valued function. Then the domain of $f\left(x\right)=\sqrt{x-\sqrt{1-x^2}}$ is

Which of the following functions would have only odd powers of x in its Taylor series expansion about the point x=0?