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$f\left(x\right)=\frac{{27}^x-9^x-3^x+1}{\sqrt{2}-\sqrt{1+cosx}}$ is continuous at x = 0 and f(0)= $k\sqrt{2}(log3{)}^2$ then k =

Question 4

If $\frac{1}{2}$ is a root of equation $x^2+kx-\frac{5}{4}=0$, then the value of k is

$\left(A\right)2$

$\left(B\right)–2$

$\left(C\right)\frac{1}{4}$

$\left(D\right)\frac{1}{2}$

The differential equation of the family of curves for which the length of the normal is equal to a constant k, is given by

[Pb. CET 2004]

Find the area between the curves y
= x and y = x²

Evaluate $\int \frac{dx}{(5x-2)(2x+7)}$

(where $C$ is constant of integration)

Find the value of x, y and z from the following equations:
$\left(i\right)\left[\begin{array}{cc}4& 3\\ x& 6\end{array}\right]$

$\left(ii\right)\left[\begin{array}{cc}x+2& 2\\ 5+z& xy\end{array}\right]=\left[\begin{array}{cc}6& 2\\ 5& 8\end{array}\right]$

$\left(iii\right)\left[\begin{array}{c}x+y+z\\ x+z\\ y+z\end{array}\right]=\left[\begin{array}{c}9\\ 5\\ 7\end{array}\right]$

$\frac{x}{3}>\frac{x}{2}+1$

Evaluate P
(A ∪ B), if 2P (A) = P (B)
=and
P(A|B) =

Venn diagram representation of A - B for 2 sets A & B is

Find the
value of p so that the three lines 3x + y –
2 = 0, px + 2y – 3 = 0 and 2x – y
– 3 = 0 may intersect at one point.

Vector A:- Magnitude 4cm

Direction ${30}^{\circ }$North of East

Vector B:- Magnitude 8 cm

Direction ${60}^{\circ }$North of West

Find $\overrightarrow{C}=\overrightarrow{A}+\overrightarrow{B}$

Find
the value of a,
b, c,
and d from
the equation:

For a real number x, let [x] denote the largest integer less than or equal to x, and let {x} = x - [x]. The number of solutions x to be equation [x]{x} = 5 with is 0 $\le$ x $\le$ 2015 is

In a triangle with one angle $\frac{2\pi }{3},$ the length of the sides form an A.P. If the length of the greatest side is $7cm.$ The radius of the circumcircle of triangle is

Let $a_1,a_2,a_3,\dots$ be a G.P. such that $a_1<0,a_1+a_2=4$ and $a_3+a_4=16$. If $\sum _{i=1}^9{a}_i=4\lambda$, then $\lambda$ is equal to :

If $\sin 2A+\sin 2B=\frac{1}{2}$ and $\cos 2A+\cos 2B=\frac{3}{2}$, then the value of $|\cos (A-B)|$ is

If derivative of ${\tan }^{-1}\left(\frac{4\sqrt{x}\cdot x^2}{4-x^5}\right)$ is $g\left(x\right)$ for some $x\in R,$ then $g\left(1\right)$ equal to

If ${}^{18}{C}_{2r}={}^{18}{C}_{r+3}$, then the value of $r$ is/are

If A is a square matrix of order 2 such that $A^2=0$, then

Prove that:-

Cos [tan^{​​​​-1}{sin(cot^{​​​​-1}​​x)}] = √1+x^{​​​​​​2}/√2+x^{​​​​​​2}

1. 0.3

Let $\ast$ be the binary operation on N given by $a\ast b=LCM$ of a and b.
(i) Is $\ast$ associative?

The area of the region bounded by the lines $x=1,x=2,$ and the curves $x(y-e^x)=\sin x$ and $2xy=2\sin x+x^3$ is

Prove the following question.${\int }_{-1}^1{x}^{17}co{s}^{4}xdx=0.$

If $(2\le r\le n),$ then ${}^n{C}_r+2{\cdot }^n{C}_{r+1}{+}^n{C}_{r+2}$ is equal to

Let $P$ and $Q$ be two distinct points on a circle which has centre at $C(2,3)$ and which passes through origin $O.$ If $OC$ is perpendicular to both the line segments $CP$ and $CQ$, then the set $\{P,Q\}$ is equal to

In a party, there are 10 married couples. Each person shake hands with every persion other than her or his spouce. The total numberof hand shakes exchanged in that party is ______?

The limit $\underset{x\to \infty }{lim}{x}^2\underset{0}{\overset{x}{\int }}{e}^{t^3-x^3}dt$ equals

If $f\left(x\right)=\underset{x-1}{\overset{x+1}{\int }}{e}^{-(t-1{)}^2}dt$, then the maximum value of $f\left(x\right)$ will occur at $x$ equal to
(correct answer + 2, wrong answer - 0.50)

The number of solutions of the equation :

$3{\cos }^{2}x{\sin }^{2}x-{\sin }^{4}x-{\cos }^{2}x=0$ in the interval $[0,2\pi ]$ is: