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$P\left(n\right):1.2+2.3+3.4+...+n(n+1)=\frac{(n+1)(n+2)}{3}$

The statement P(n) is

The probability that a bulb produced by a factory will fuse after $150$ days of use is $\mathrm{0.05.}$ Then the probability that out of $5$ such bulbs at least one will fuse after $150$ days of use is:

The value of $tan(ta{n}^{-1}\frac{1}{2}-ta{n}^{-1}\frac{1}{3})$=

If $\cos A=n\cos B$ and $\sin A=m\sin B$, then $(m^2-n^2){\sin }^{2}B=$

Match the following
Given $sinA=\frac{2}{3}andsinB=\frac{1}{4}$

$\begin{array}{cc}\left(1\right)sin(A+B)& \left(p\right)\frac{2\sqrt{15}-\sqrt{5}}{2+5\sqrt{3}}\\ \left(2\right)Cos(A-B)& \left(q\right)\frac{55}{144}\\ \left(3\right)Tan(A-B)& \left(r\right)\frac{2\sqrt{15}+\sqrt{5}}{12}\\ \left(4\right)Sin(A+B)sin(A-b)& \left(s\right)\frac{5\sqrt{3}+2}{12}\end{array}$

$1.3+{2.3}^2+{3.3}^3+\cdots +n.3^n=\frac{(2n-1)3^{n+1}+3}{4}$

The value of the integral $\int \frac{x^4+1}{x^6+1}dx$ is

$(C$ is a constant of integration$)$

Vijay wrote 4 different letters to send to 4 different addresses. For each letter, he prepared one envelope with its correct address. If the 4 letters are to be put into the 4 envelopes at random, in how many ways can we put the letters so that only two of the letters goes to the right envelopes?

If $f(\alpha ,\beta )=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1\\ \sin \alpha & \cos \alpha & 1\\ \cos (\alpha +\beta )& -\sin (\alpha +\beta )& 1\end{array}\right|,$ then

Let $f,g:R\to R$ be defined, respectively by f(x) = x + 1, g(x) = 2x - 3. Find f + g, f - g and $\frac{f}{g}.$

The range of $\frac{3x^2+9x+17}{3x^2+9x+7}$ is

The sum of (n+1) terms of $\frac{1}{1}+\frac{1}{1+2}+\frac{1}{1+2+3}+..........$ is

What can be inferred if the Range of one data set is higher than other derived from the same experiment?

The value of $\alpha$ so that the geometric mean of $x$ and $y$, where $x\ne y$ is $\frac{x^{\alpha +2}+y^{\alpha +2}}{x^{\alpha +1}+y^{\alpha +1}}$, is

Let $\omega \ne 1\text{and}{\omega }^{13}=1.\text{If}a=\omega +{\omega }^3+{\omega }^4+{\omega }^{-4}+{\omega }^{-3}+{\omega }^{-1}\text{and}b={\omega }^2+{\omega }^5+{\omega }^6+{\omega }^{-6}+{\omega }^{-5}+{\omega }^{-2},$ then the quadratic equation, whose roots are a and b is

The scalars l and m such that $l\overrightarrow{a}+m\overrightarrow{b}=\overrightarrow{c}$ where $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are given vectors, are equal to

Locus of image of the point $P(h,k)$ with respect to the line mirror which passes through the origin is

${lim}_{x\to 8}\frac{\sqrt{1+\sqrt{1+x}}-2}{x-8}=$

Let $f\left(x\right)$ be a continuous and not a constant function of all $x$ in its domain, such that

$\left(f\right(x){)}^2=\underset{0}{\overset{x}{\int }}f\left(t\right)\frac{4}{\sin 2t-4{\sin }^{2}t+4}dt$ and $f\left(0\right)=0,$ then

The transformed equation of $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ when the axes are rotated through an angle of ${90}^{\circ }$ is

If a, b, c $ϵ$ R and a $\ne$ 0, c > 0, the graph of $f\left(x\right)$ = $a{x}^2+bx+c$ for which $f\left(x\right)=0$ has only imaginary roots, will look like

The pressure applied from all directions on a cube is P. How much its temperature should be raised to maintain the original volume ? The volume elasticity of the cube is $\beta$ and the coefficient of volume expansion is $\alpha$

Let $f\left(x\right)$ be positive, continuous, and differentiable on the interval $(a,b)$ and $\underset{x\to a^{+}}{lim}f\left(x\right)=1,\underset{x\to b^{-}}{lim}f\left(x\right)=3^{1/4}.$ If $f^{\prime }\left(x\right)\ge f^3\left(x\right)+\frac{1}{f\left(x\right)}$, then the greatest value of $b-a$ is

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Area of a triangle with adjacent sides determined by vectors}\overrightarrow{a}\text{and}\overrightarrow{b}\text{is}1\text{. Then the area of}& \text{(P)}& 6\\ & \text{the triangle with adjacent sides determined by}(3\overrightarrow{a}+4\overrightarrow{b})\text{and}(\overrightarrow{a}-3\overrightarrow{b})\text{is}& & \\ \text{(B)}& \text{Volume of parallelopiped determined by vectors}\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\text{is}\frac{1}{4}\text{. Then the volume of the}& \text{(Q)}& 9\\ & \text{parallelopiped determined by vectors}3(\overrightarrow{a}+\overrightarrow{b}),(\overrightarrow{b}+\overrightarrow{c}),4(\overrightarrow{c}+\overrightarrow{a})\text{is}& & \\ \text{(C)}& \text{Area of a parallelogram with adjacent sides determined by vectors}\overrightarrow{a}\text{and}\overrightarrow{b}\text{is}8\text{. Then the}& \text{(R)}& 13\\ & \text{area of the parallelogram with adjacent sides determined by vectors}(2\overrightarrow{a}-\overrightarrow{b})\text{and}\overrightarrow{b}\text{is}& & \\ \text{(D)}& \text{Volume of tetrahedron determined by vectors}\overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{is}\frac{1}{2}\text{. Then the volume of the}& \text{(S)}& 16\\ & \text{tetrahedron determined by vectors}2(\overrightarrow{a}\times \overrightarrow{b})\text{,}3(\overrightarrow{b}\times \overrightarrow{c})\text{and}(\overrightarrow{c}\times \overrightarrow{a})\text{is}& & \end{array}$



Which of the following is the only CORRECT combination?

If the coefficient of $x^7$ in $(a{x}^2+\frac{1}{bx}{)}^{11}$ is equal to

the coefficient of $x^{-7}$ in $(ax-\frac{1}{b{x}^2}{)}^{11}$, then ab =

Number of positive integers which have characteristic $2$ when base is $10$, is

A spherical iron ball of $10\text{cm}$ radius is coated with a layer of ice of uniform thickness that melts at the rate of $50{\text{cm}}^3/\text{min}$. When the thickness of ice is $5\text{cm}$, then the rate (in $\text{cm/min}$) at which the thickness of ice decreases, is :

Find the cente and radius for the following circle.
$2x^2+2y^2-x=0$

Which of the following are equivalent statements to the implication $p\to q$.

Find the number of terms in the expansion of $(x_1+x_2+x_3....x_k{)}^n$

The point on the parabola $y^2=36x$ whose ordinate is three times its abscissa is

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are non coplanar non zero vectors such that $\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a},\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$ and $\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b},$ then which of the following is not correct?

Find the $4^{th}$ term in the expansion of $(x-2y{)}^{12}$.

The ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of ${(2^{\frac{1}{3}}+\frac{1}{2(2{)}^{\frac{1}{3}}})}^{10}$ is:

The vertices of a hyperbola are (2, 0), (–2, 0) and the foci are (3, 0), (–3, 0). The equation of the hyperbola is

The number of possible outcomes in a throw of $5$ ordinary dice in which at least one of the dice shows an odd number is