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Find the Derivative of f(x) = $x^2$ .at x = 0

Find the mean, variance and standard deviation using short-cut
method

If a convex polygon has $35$ diagonals, then the number of triangles formed by joining the vertices of the polygon is :

The relation between time taken in min (x) and distance covered in km (y) is given below. Which of the following holds true for the condition that represents the distance covered is 12 km throughout?

Consider the first-order logic sentence $F:\forall x\left(\exists yR(x,y)\right)$. Assuming non-empty logical domains, which of the sentences below are impied by F ?

I $\exists y\left(\exists xR(x,y)\right)$

II $\exists y\left(\forall xR(x,y)\right)$

III $\forall y\left(\exists xR(x,y)\right)$

IV $\rightharpoondown \exists x(\forall y\rightharpoondown R(x,y))$

Examine
the consistency of the system of equations.

5x
− y + 4z = 5

2x
+ 3y + 5z = 2

5x
− 2y + 6z = −1

Let $PQ$ be a focal chord of $y^2=4ax$. The tangents to parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a(a>0)$. If the chord $PQ$ subtends an angle $\theta$ at the vertex of prabola then $\tan \theta =$

If (3,4) and (6,8) are the foci of an ellipse passing through the Origin, then the eccentricity of the ellipse is

If $a_n={\sum }_{k=0}^n\frac{(lo{g}_{e}10{)}^n}{k!(n-k)!}$ for $n\ge 0$ then $a_0+a_1+a_2+a_3+\dots \dots$ upto $\infty$ is equal to $t^s$
(where ‘t’ is a least possible two digit number and ‘s’ an even number). Which of the following is correct?

Complete set of values of '$m$' for which function $f\left(x\right)=e^{\cos x}+2m\cos x+1$ is increasing $\forall xϵ(0,\frac{\pi }{2})$ is

The number of words that can be formed using all the letters of the word "$KANPUR$" when the vowels are in even places is

$\text{What does the equation}(a-b)(x^2+y^2)-2abx=0becomeiftheoriginisshiftedtothepoint(\frac{ab}{a-b},0)withoutrotation?$

The point of inflection for the function $f\left(x\right)=\frac{\ln x}{x}$ is:

The parametric equation of the circle whose center is $(3,-5)$ and touches the $x-$ axis is

$\underset{x\to 0}{Lt}\underset{0}{\overset{X}{\int }}\frac{si{n}^{2}tcostdt}{x^3}=$

If $x\in [-3,2]$, then $2x+7$ lies in

The value of following integral $\underset{\pi /4}{\overset{3\pi /4}{\int }}\ln \left(\tan (x-\frac{\pi }{4})\right)dx$ is equal to

Let $f\left(x\right)=\left|\begin{array}{ccc}{\omega }^3& {\omega }^4& {\omega }^5\\ \sin (m-1)x& \sin mx& \sin (m+1)x\\ \cos (m-1)x& \cos mx& \cos (m+1)x\end{array}\right|$,



where $m\in N$ and $\omega$ is the cube root of unity.

If $\underset{0}{\overset{\pi /2}{\int }}f\left(x\right)dx=a\omega +b{\omega }^2,$ then $(a,b)=$

A certain bookcase has 2 shelves of books. On the upper shelf, the book with the greatest number of pages has 400 pages. On the lower shelf, the book with the least number of pages has 475 pages. What is the median number of pages for all of the books on the 2 shelves?

(1) There are 25 books on the upper shelf. (2) There are 24 books on the lower shelf.

If $(p\wedge \sim r)\Rightarrow (q\wedge r)$ is false and q and r are both false, then p is ___.

If $x\ge 1then2ta{n}^{-1}x+si{n}^{-1}\left(\frac{2x}{1+x^2}\right)$=

t-(2t+5)-(1-2t)=(3+4t)-2(t-4)

With checking

If $f\left(x\right)=\frac{\ln (x^2+e^x)}{\ln (x^4+e^{2x})},$ then $\underset{x\to \infty }{\text{lim}}f\left(x\right)$ is equal to

If $\text{A}$ and $\text{B}$ are two equal sets, then which of the following is not true?

If $f\left(x\right)=\{\begin{array}{cc}\frac{\sin (p+1)x+\sin x}{x},& x<0\\ q,& x=0\\ \frac{\sqrt{x+x^2}-\sqrt{x}}{x^{3/2}},& x>0\end{array}$ is continuous at $x=0$, then the ordered pair $(p,q)$ is equal to: