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Let the harmonic mean and the geometric mean of two positive numbers be in the ratio $4:5.$ Then the two numbers are in the ratio

A continuous function y = f(x) is defined on [–7, 5]. A(–7, –4), B(–2, 6), C(0, 0), D(1, 6), E(5, –6) are consecutive points on the graph of ‘f’ and AB, BC, CD, DE are line segments. The number of real roots of the equation f[f(x)] = 6 is ___

A line meets the co-ordinate axes in A & B, a circle is circumscribed about the triangle OAB. If $d_{1}and{d}_2$ are the distances of the tangent to the circle at the origin O from the points A and B respectively the diameter of the circle is:

The Value of x, for which the 6-th term in the expansion of

${\left\{2^{lo{g}_2\sqrt{(9^{y-1}+7)}+\frac{1}{2^{\left[\right(1/5\left)lo{g}_2\right(3^{x-1}+1\left)\right]}}}\right\}}^7$is 84, is equal to:

If $f\left(x\right)$ is twice differentiable function in $[c_1-1,c_2+1]$ and $f^{\prime }\left(c_1\right)=f^{\prime }\left(c_2\right)=0,f^{\prime \prime }\left(c_1\right)\cdot f^{\prime \prime }\left(c_2\right)<0,f\left(c_1\right)=9,f\left(c_2\right)=0$. Let $k$ and $m$ be the minimum number of the roots of $f\left(x\right)=0$ and $f^{\prime }\left(x\right)=0$ respectively,in $[c_1-1,c_2+1]$

$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \text{(I) If}{f}^{\prime \prime }\left(c_1\right)-f^{\prime \prime }\left(c_2\right)>0,\text{then k =}& \text{(P) 1}\\ \text{(II) If}{f}^{\prime \prime }\left(c_1\right)-f^{\prime \prime }\left(c_2\right)<0,\text{then k =}& \text{(Q) 2}\\ \text{(III) If}{f}^{\prime \prime }\left(c_1\right)-f^{\prime \prime }\left(c_2\right)>0,\text{then m =}& \text{(R) 3}\\ \text{(IV) If}{f}^{\prime \prime }\left(c_1\right)-f^{\prime \prime }\left(c_2\right)<0,\text{then m =}& \text{(S) 4}\end{array}$

Which of the following is only CORRECT combination?

The domain of the function defined by $f\left(x\right)=si{n}^{-1}\sqrt{x-1}$ is

(a) $[1,2]$ (b) $[-1,1]$ (c) $[0,1]$ (d) None of these

The particular solution of the differential equation $\frac{dy}{dx}=e^{4x-2y-2}$, given $y\left(1\right)=1,$ is

Let $A=\left(\begin{array}{ccc}0& 2q& r\\ p& q& -r\\ p& -q& r\end{array}\right)$. If $A{A}^T=I_3$, then $|p|$ is:

A determinant is chosen at random from the set of all determinant of order 2 with elements 0 or 1 only. Find the probability that the determinant chosen is nonzero.

f(ln6) =

If $3\sin \theta +5\cos \theta =5$, the value of $5\sin \theta -3\cos \theta$ is

Let $d\in R$, and
$A=\left[\begin{array}{ccc}-2& 4+d& \left(\sin \theta \right)-2\\ 1& \left(\sin \theta \right)+2& d\\ 5& \left(2\sin \theta \right)-d& (-\sin \theta )+2+2d\end{array}\right]$,
$\theta \in [0,2\pi ].$ If the minimum value of $det\left(A\right)$ is $8$, then a value of $d$ is:

If points (a, 0), (0, b) and (x, y) are collinear, then will hold true.

The value(s) of $\underset{x\to \infty }{lim}(\sin x{)}^x$ can be

How many 6-digit telephone numbers can be constructed with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each number starts with 35 and no digit appears more than once?

If the sum of two of the roots of $x^3+p{x}^2+qx+r=0$ is zero, then pq =

If $f$ is an invertible function given by $f\left(x\right)=\ln \left(x\right)+4x-8,$ then the value of $f^{-1}(-4)$ is

Area of the region bounded by the curve
y² = 4x, y-axis and the line y
= 3 is

A. 2

B.

C.

D.

If $I=\sum _{k=1}^{98}\underset{k}{\overset{k+1}{\int }}\frac{k+1}{x(x+1)}dx,$ then

Let $PQRS$ is a parallelogram where $P=(2,2),Q=(6,-1),\text{and}R=(7,3)$. Then equation of the line through $S$ and perpendicular to $QR$ is

The sum of $20$ terms of the progression $\frac{1}{4},-\frac{1}{2},1,-2,4,\dots \dots$ is

The number of positive integral solutions of abc=30

Show that the given differential equation is homogeneous and then solve it.

$\{xcos\left(\frac{y}{x}\right)+ysin\left(\frac{y}{x}\right)\}ydx=\{ysin\left(\frac{y}{x}\right)-xcos\left(\frac{y}{x}\right)\}xdy$

For the equation ${\cos }^{-1}x+{\cos }^{-1}2x+2\pi =0$, the number of real solution(s) is

If the $n^{th}$ term of a sequence is given by $t_n=7n-9$, then the sum of first $100$ terms is

$\text{If}{\left(\frac{1-i}{1+i}\right)}^{100}=a+ib,\text{find (a, b)}.$