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The number of solutions of the equation $x+2\tan x=\frac{\pi }{2}$ in the interval $[0,2\pi ]$ is :

When a system of linear equations has no solution, what does it mean?

Consider the logarithmic inequality $1+{\log }_5(x^2+1)\ge {\log }_5(a{x}^2+4x+a)$ for all real values of $x.$ The number of integers which $a$ cannot take, is

The line $\frac{x}{a}+\frac{y}{b}=1$ cuts the axis at $A$ and $B$,another line perpendicular to $AB$ cuts the axes at$P,Q$respectively.Locus of points of intersection of $AQ$ and $BP$ is

Find the value of $\tan \frac{1}{2}[{\sin }^{-1}\frac{2x}{1+x^2}+{\cos }^{-1}\frac{1-y^2}{1+y^2}],|x|<1,y<0$ and $xy<1.$

If divisor $=x+3,$ quotient $=x^3-3x^2+11x-16$, and the remainder $=0$, then the dividend is ____.

If $\frac{2-|x|}{3+|x|}\ge 5$, then $x\in$

The value of $(\sqrt{3}+\tan 1^{\circ })(\sqrt{3}+\tan 2^{\circ })(\sqrt{3}+\tan 3^{\circ })\cdots (\sqrt{3}+\tan {28}^{\circ })(\sqrt{3}+\tan {29}^{\circ })$ is

Is this chapter short??!!!

This chapter is not in syllabus of this semester in my school, therefore this chapter is hard for me now. And in study plan of this month, I have to finish it and give all tests in one day!!!

How is it possible

The value of $\underset{n\to \infty }{lim}\frac{\left[\right(n+1\left)\right(n+2)...(n+n){]}^{\frac{1}{n}}}{n}$ is:

At a telephone enquiry system the number of phone calls received regarding relevant inquiry follows probability distribution with an average of 5 phone calls during 10 minute time interval. The probability that there is at the most one phone call during a 10 minute period is___

In an entrance test that is graded on the basis of two examinations, the

probability of a randomly chosen student passing the first examination is

0.8 and the probability of passing the second examination is 0.7. The

probability of passing at least one of them is 0.95/ What is the probability

of passing both?

Let E and F be two independent events. The probability that exactly one of them occurs is $\frac{11}{25}$ and the probability that none of them occurs is $\frac{2}{25}$ . Then

The value of $\sqrt{-3}\cdot \sqrt{-75}$ is

Question 4 (ix)

Which of the following are APs? If they form an A.P. Find the common difference d and write three more terms.

(ix) 1, 3, 9, 27, …

The angle between a pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9si{n}^2\alpha +13co{s}^2\alpha =0is2\alpha .$ The equation of the locus of the point P is

The value of integral $\underset{-4}{\overset{4}{\int }}\frac{x^{2021}}{1+x^{2022}}dx,$ is

Prove that the area of the parallelogram formed by the lines $3x-4y+a=0$, $3x-4y+3a=0$, $4x-3y-a=0$ and $4x-3y-2a=0$ is $\frac{2a^2}{7}$ sq. units.

Let $S=\{1,2,3,4,5,6\}$. Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $g\left(3\right)=2g\left(1\right)$ is

Tangents are drawn to $x^2+y^2=1$ from any arbitrary point $P$ on the line $2x+y-4=0$. The corresponding chord of contact passes through a fixed point whose coordinates are

Let $f\left(x\right)$ be a continuous and differentiable function on the interval $[2,10]$. It is known that $f\left(2\right)=8$ and the derivative on the given interval satisfies the condition $f^{\prime }\left(x\right)\le 4$ for all $x\in (2,10)$. Then the maximum possible value of the function at $x=10$

A.

B.

C.

D.

Let $S$ be a circle whose centre is $C(2,3)$ and touching the $y$-axis. Tangents $OA$ and $OB$ are drawn from the origin $O$ to the circle which touches the circle at $A$ and $B.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The finite slope of the tangent OA is}& \text{(P)}& \frac{12}{13}\\ \text{(B)}& \text{If}\alpha \text{is the acute angle between the}& \text{(Q)}& \frac{54}{13}\\ & \text{tangents, then the value of}\sin \alpha \text{is}& & \\ \text{(C)}& \text{If}P(x_1,y_1)\text{is any point on the circle,}& \text{(R)}& \frac{5}{12}\\ & \text{then the product of the minimum and the}& & \\ & \text{maximum values of}OP\text{, is}& & \\ \text{(D)}& \text{A line is drawn from}S(4,5)\text{, intersecting}& \text{(S)}& 9\\ & \text{the circle at}M\text{and}N\text{. Then the value of}& & \\ & SM\cdot SN\text{is}& & \\ & & \text{(T)}& 4\end{array}$

Which of the following is the only CORRECT combination?

Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.