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Let $n_1$ and $n_2$ be the number of red and black balls, respectively, in box $I$. Let $n_3$ and $n_4$ be the number of red and black balls, respectively, in box $II$.



One of the two boxes, box $I$ and box $II$, was selected at random and a ball was drawn randomly out of this box. The ball was found to be red. If the probability that this red ball was drawn from box $II$ is $\frac{1}{3}$, then the correct option(s) with the possible values of $n_1,n_2,n_3$ and $n_4$ is(are)

Let $x$ be a real number satisfying both the equations ${\tan }^{-1}(x-1)+{\tan }^{-1}x+{\tan }^{-1}(x+1)={\tan }^{-1}3x$ and $x^3+b{x}^2+cx+d=0$. Then which of the following is/are correct:

The points $z_1,z_2,z_3,z_4$ in the complex plane are the vertices of a parallelogram taken in order, if and only if

The value of ${\sum }_{n=0}^{1947}\frac{1}{2^n+\sqrt{2^{1947}}}$ is equal to

If $Q$ be a point on the parabola $y^2=8x$ and $P(-2,0)$ be a point in the $xy$ plane. If the locus of the mid point of $PQ$ is a parabola, then its focus is

The number of ways of selecting $5$ cards from a pack of $52$, where only one even numbered card is selected is

If $T_n=si{n}^n\theta +co{s}^n\theta$, prove that

$\left(i\right)\frac{T_3-T_5}{T_1}=\frac{T_5-T_7}{T_3}$

$\left(ii\right)2T_6-3T_4+1=0$

$\left(iii\right)6T_{10}-15T_8+10T_6-1=0$

$\int \frac{(cos5x+cos4x)}{1-2cos3x}dx=$

The function $f:[0,3]\to [1,29]$, defined by $f\left(x\right)=2x^3-15x^2+36x+1$, is

Prove the following by using the principle of mathematical induction for all n ∈ N: 41ⁿ – 14ⁿ is a multiple of 27.

Find the mean , variance and standard deviation for the following data :

(i) 2,4,5,6,8,17

(ii) 6,7,10,12,13,4,8,12

(iii) 227,235,255,269,292,299,312,321,333,348

(iv) 15,22,27,11,9,21,14,9

Which of the following are equal matrixes.

$A=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]B=\left[\begin{array}{ccc}1& 2& 4\\ 3& 5& 6\\ 7& 8& 9\end{array}\right]C=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]D=\left[\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 7& 8& 9\end{array}\right]E=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]$

LMVT says that if y = f(x) be a given function which is ;

a.Continuous in [a,b]

b. Differentiable in (a,b)

Then, f'(c) = $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ for some c $ϵ$ (a,b)

Find the value of c for the function f(x) =$-x^2$+4x-5 and the interval [-1,1]

The value of the integral $\underset{0}{\overset{1}{\int }}x{\cot }^{-1}(1-x^2+x^4)dx$ is :

If the sum of the slopes of the normal from a point $P$ to the rectangular hyperbola $xy=c^2$ is equal to $\lambda (\lambda \in R^{+})$, then locus of $P$ is

If the lengths of sides of $△ABC$ are $5,7,8$ units then $A{G}^2+B{G}^2+C{G}^2=$

(where $G$ is the centroid)

Events A
and B are such that
.
State whether A and B are independent?

The equation of a plane containing the line of intersection of the planes $2x-y-4=0$ and $y+2z-4=0$ and passing through the point $(1,1,0)$ is:

If f(x) = x + tan x and f is inverse of g, then g’(x) is equal to

Let $△PQR$ be a triangle. Let $\overrightarrow{a}=\overrightarrow{QR},\overrightarrow{b}=\overrightarrow{RP}$ and $\overrightarrow{c}=\overrightarrow{PQ}.$ If $|\overrightarrow{a}|=12,|\overrightarrow{b}|=4\sqrt{3}$ and $\overrightarrow{b}.\overrightarrow{c}=24,$ then which of the following is (are) true?

${lim}_{x\to 0}\frac{x^{\frac{2}{3}-9}}{x-27}$

The length of the latus rectum of the parabola $25\left[\right(x-2{)}^2+(y-4{)}^2]=(4x-3y+12{)}^2$ is

If $\text{cosec}\theta =\frac{25}{7},$ then which of the following can be correct?

If $f\left(x\right)$ is an even function and satisfies the relation $x^{2}f\left(x\right)-2f\left(\frac{1}{x}\right)=g\left(x\right),$ where $g\left(x\right)$ is an odd function, then $f\left(5\right)$ equals

Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?

(i) f is a relation from A to B

(ii) f is a function from A to B

Justify your answer in each case.

Let $D$ be the middle point of the side $BC$ of a triangle $ABC$. If the triangle $ADC$ is equilateral, then $a^2:b^2:c^2$ is equal to

Let $f:[\frac{1}{2},1]\to R$ be a positive, non-constant and differentiable function such that $f^{\prime }\left(x\right)<2f\left(x\right)$ and $f\left(\frac{1}{2}\right)=1$. Then, the value of $\underset{\frac{1}{2}}{\overset{1}{\int }}f\left(x\right)dx$ lies in the interval

Find
the inverse of each of the matrices, if it exists.