Explore topic-wise InterviewSolutions in Current Affairs.

If $\alpha ,\beta \in R$ are the roots of the equation $(a^2+b^2)x^2+2x(ac+bd)+c^2+d^2=0$, then $\frac{\alpha }{\beta }$ is equal to

Value of $\lambda$ for which the function $f\left(x\right)=2x^3-3(\lambda +2)x^2+12\lambda x$ has one local maxima and one local minima in $R$, can not be

If $|5x+6|+4<1$, then $x\in$

Arti, Bharti and Seema are partners sharing profits in the proportion of 3 :2 :1 and their Balance Sheet as on March 31, 2003 stood as follows.

$\text{Balance Sheet}$
$\text{as on March 31,2003}$
$\begin{array}{cccc}\text{Capital and Liabilities}& \text{Amt. (Rs)}& \text{Assets}& \text{Amt. (Rs)}\\ \text{Bills Payable}& 12,000& \text{Building}& 21,000\\ \text{Creditors}& 14,000& \text{Cash in hand}& 12,000\\ \text{Ganeral Reserve}& 12,000& \text{Bank}& 13,700\\ \text{Capital}& & \text{Debtors}& 12,000\\ \text{Arti}20,000& & \text{Bills Receivable}& 4,300\\ \text{Bharti}12,000& & \text{Stock}& 1,750\\ \text{Seema}\underset{–}{8,000}& 40,000& \text{Investment}& 13,250\\ & \overline{\underset{–}{\underset{–}{78,000}}}& & \overline{\underset{–}{\underset{–}{78,000}}}\end{array}$

Bharti died on June 12, 2003 and according to the deed of the said partnership, her executors are entitled to be paid as under

(a) The capital to her credit at the time of her death and interest there on 10% per annum.

(b) Her proportionate share of reserve fund.

(c) Her share of profits for the intervening period will be based on the sales during that period, which were calculated as Rs. 1,00,000. The rate of profit during past three years had been 10% on sales.

(d) Goodwill according to her share of profit to be calculated by taking twice the amount of the avera ge profit of the last three years less 20%. The prof its of the previous years were

2001 - Rs. 8,200

2002 - Rs. 9,000

2003 - Rs. 9,800

The investments were sold for Rs.16,200 and her executors were paid out. Pass the necessary journal entries and write the account of the executors of Bharti.

Two systems of rectangular axes have the same origin. If a plane cuts them at distance a, b, c and a', b', c' from the origin, then

If $y=e^{ax}$. cos bx, then prove that $\frac{d^{2}y}{d{x}^2}-2a\frac{dy}{dx}+(a^2+b^2)y=0$

If $\frac{(1-3x{)}^{\frac{1}{2}}+(1-x{)}^{\frac{5}{3}}}{2}$ is approximately equal to a + bx for small values of x, then (a, b) =

Algebraic sum of the intercepts made by the plane x + 3y – 4z + 6 = 0 on the axes is :

Let f(x) = $a{x}^2+bx+c,a>0$ such that $f(-1-x)=f(-1+x)\forall x\epsilon R.$ Also given that f(x) = 0 has no real roots and b > O
Let $\alpha =4a-2b+c,\beta =9a+3b+c,\gamma =9a-3a+c.$ Then which of the following is correct?

The solution of differential equation is

[MP PET 1994]

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.
​​​​​​$\begin{array}{ccc}Column1& Column2& Column3\\ \left(I\right)x^2+y^2=a^2& \left(i\right)my=m^{2}x+a& \left(P\right)(\frac{a}{m^2},\frac{2a}{m})\\ \left(II\right)x^2+a^2{y}^2=a^2& \left(ii\right)y=mx+a\sqrt{m^2+1}& \left(Q\right)(\frac{-ma}{\sqrt{m^2+1}},\frac{a}{\sqrt{m^2+1}})\\ \left(III\right)y^2=4ax& \left(iii\right)y=mx+\sqrt{a^2{m}^2-1}& \left(R\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2+1}},\frac{1}{\sqrt{a^2{m}^2+1}})\\ \left(IV\right)x^2-a^2{y}^2=a^2& \left(iv\right)y=mx+\sqrt{a^2{m}^2+1}& \left(S\right)(\frac{-a^{2}m}{\sqrt{a^2{m}^2-1}},\frac{-1}{\sqrt{a^2{m}^2-1}})\end{array}$

The tangent to a suitable conic (Column 1) at $(\sqrt{3},\frac{1}{2})$ is found to be $\sqrt{3}x+2y=4$, then which of the following options is the only CORRECT combination?

How to solve Monty hall problem by using conditional probability?

The eccentricity of ellipse $36x^2+4y^2=144$ is

The radius of the circle which touches the line $x+y=0$ at $M(-1,1)$ and cuts the circle $x^2+y^2+6x-4y+18=0$ orthogonally, is

The radius of a right circular cylinder increases at the rate of $0.1$ cm/min, and the height decreases at the rate of $0.2$ cm/min. The rate of change of the volume of the cylinder, in ${\text{cm}}^3/\text{min},$ when the radius is $2$ cm and the height is $3$ cm, is

Area bounded between two latus-rectum of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1;a>b$ is
(where $e$ is eccentricity of the ellipse)

If $x^2+y^2=1$, then $\frac{d^{2}y}{d{x}^2}$ is

The principal value of $co{s}^{-1}(-\frac{1}{\sqrt{2}})$ is

The equation of the hyperbola whose centre is(6,2) one focus is (4,2) and of eccenticity 2 is

If cos 2x + 2 cos x = 1 then, $(2-co{s}^{2}x)si{n}^{2}x$ is equal to

Find the coordinatesof the foce, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse.
$\frac{x^2}{16}+\frac{y^2}{9}=1$

Find the equations of the medians of a triangle, the coordinates of whose vertices are (-1, 6), (-3, -9) and (5, -8).

The critical angle of a medium is ${\text{sin}}^{-1}\left(\frac{3}{5}\right)$. The polarizing angle of the medium will be

The solution of the differential equation $\left[\right(\cos x)dx–dy](1+x^2)+e^x\left[\right(1+x^2){\tan }^{–1}x+1]dx=0$ is
(where $C$ is arbitrary constant)

The equation of the line belonging to the family of lines $(x+y)+\lambda (2x-y+1)=0$ and farthest from point $(1,-3)$ is

If x=ct and $y=\frac{c}{t}$, find $\frac{dy}{dx}$ at $t=2.$

If the parabola $y=-x^2-2x+k$ touches the parabola $y=-\frac{1}{2}{x}^2-4x+3$, then the value of $k$ is

Equation of the plane which contains the line $L:\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{2}$ and its image about the plane $x+y+z=3$

Find the equation of the locus of a point such that the sum of its distance from (0,2) and (0,-2) is 6

If $\log (x+z)+\log (x-2y+z)=2\log (x-z)$, then

If radii of director circles of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are $2r$ and $r$ respectively and $e_e$ and $e_h$ be the eccentricities of the ellipse and the hyperbola respectively then

For the relation $R_1$ defined on R by the rule $(a,b)ϵR_1\iff 1+ab>0.$

Prove that : $(a,b)ϵR_1$ and $(b,c)ϵR_1$

$\Rightarrow (a,c)ϵR_1$ is not true for all $a,bcϵR$

The least positive integer k for which the value $k\times n^2(n^2-1^2)(n^2-2^2)....(n^2-(n-1{)}^2)$ turns into a factorial of some positive integer is

The common roots of the equations
$z^3+(1+i)z^2+(1+i)z+i=0,$ (where $i=\sqrt{-1}$) and $z^{1993}+z^{1994}+1=0$ are

Find the area bounded by the curve $x^2=4y$ and the line x = 4y - 2.

Using properties of determinants, prove that: $\left|\begin{array}{ccc}x+y& x& x\\ 5x+4y& 4x& 2x\\ 10x+8y& 8x& 3x\end{array}\right|=x^3$

Which of the following is the principal value branch of $cose{c}^{-1}x?$

(a) $(\frac{-\pi }{2},\frac{\pi }{2})$ (b) $[0,\pi ]-\left\{\frac{\pi }{2}\right\}$ (c) $[\frac{\pi }{2},\frac{\pi }{2}]$ (d) $[\frac{-\pi }{2},\frac{\pi }{2}]-\left[0\right]$

The first term of a G.P. is $1$. If the sum of the third and fifth terms of the G.P. is $90$, then the common ratio of the G.P. is

If $\theta =ta{n}^{-1}\frac{d}{1+a_1{a}_2}+ta{n}^{-1}\frac{d}{1+a_2{a}_3}+\cdots +ta{n}^{-1}\frac{d}{1+a_{n-1}{a}_n}$, where $a_1,a_2,a_3,\cdots a_n$ are in A.P. with common difference d, then $tan\theta =$

Integrate the following functions.
$\int \frac{6x+7}{\sqrt{(x-5)(x-4)}}dx$

Find the equation of the tangent and normal to the given curve at the given points

$x=cost,y=sint=\frac{\pi }{4}$