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Let $f\left(x\right)=\left(\right[a{]}^2-5\left[a\right]+4)x^3+(6\{a{\}}^2-5\{a\}+1)x-\left(\tan x\right)\times \text{sgn}x$ be an even function for all $x\in R,$ then the sum of all possible values of $a$ is (where $[.],\{.\},\text{sgn}x$ represents greatest integer function , fractional part function and signum function respectively)

A team of $12$ railway station masters is to be divided into two groups of $6$ each, one for day duty and the other for night duty. The number of ways in which this can be done if two specified persons $A,B$ should not included in the same group is

The values of x and y for which the numbers 3+i$x^2$y and $x^2$ +y+4i are conjugate complex can be

The area bounded by $y=x^2+2$ and $y=2|x|-\cos \pi x$ is

General solutions of $x$ for which $2\sin x+1=0$ is :

Let $y=y\left(x\right)$ be the solution of the differential equation $\cos x(3\sin x+\cos x+3)dy=(1+y\sin x(3\sin x+\cos x+3))dx,0\le x\le \frac{\pi }{2},y\left(0\right)=0.$ Then, $y\left(\frac{\pi }{3}\right)$ is equal to:

The equation of the circle passing through the points (0, 0), (0, b) and (a, b) is

The number of quadratic equation(s), with real roots which remain unchanged even after squaring its roots, is

Area bounded by the inverse function of $y=x^3+1$ between the ordinates $x=-7$ and $x=2$ and $x-$axis is

Differentiate the
function with respect to x.

Three ladies have brought one child each for admission to a school. The principal wants to interview the six persons one by one subject to the condition that no mother is interviewed before her child. The number of ways in which interviews can be arranged, is

If,
then show that

Find the equation of a straight line:

(i) with slope 2 and y-intercept 3; (ii) with slope -1/3 and y-intercept -4.

(iii) with slope -2 and intersecting the x-axis at a distance of 3 units to the left of origin.

Let ${}^{\prime }{f}^{\prime }$ be a real valued function such that $0\le f\left(x\right)\le \frac{1}{2}$ and for some fixed $a>0,f(x+a)=\frac{1}{2}-\sqrt{f\left(x\right)-\left(f\right(x){)}^2},\forall x\in R$, then the period of the function $f\left(x\right)$ is

If h(x) = min {x,x²} for x ϵ R. Find LHD and RHD at x = 1.

The approximate value of square root of 25.2 is

If $y={\cos }^2({45}^{\circ }+x)+(\sin x-\cos x{)}^2$, where $x\in (0,\frac{\pi }{2}]$, then the possible value(s) of $y$ is/are

If in a $\Delta ABC,A\equiv (1,10),$ circumcentre $\equiv (-\frac{1}{3},\frac{2}{3})$ and orthocentre $\equiv (\frac{11}{3},\frac{4}{3}),$ then the equation of side $BC$ is

For a positive integer n, find the value of $(1-i{)}^n{(1-\frac{1}{i})}^n.$

The number of telephone calls received at an exchange in 245 successive one-minute intervals are shown in the following frequency distribution :

$\begin{array}{ccccccccc}\text{Number of calls}& 0& 1& 2& 3& 4& 5& 6& 7\\ \text{Frequency}& 14& 21& 25& 43& 51& 40& 39& 12\end{array}$

Compute the mean deviation about median.

The value of $\int x^{\sin x}(\cos x\ln x+\frac{1}{x}\sin x)dx$ is

(where $C$ is constant of integration)

The value of the integral $\underset{-1}{\overset{1}{\int }}\log (x+\sqrt{x^2+1})dx$ is

The number $x$ is $111$ when written in base $b,$ but it is $212$ when written in base $b-2.$ What is $x$ in base $10$?
(correct answer + 3, wrong answer 0)