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If $f$ is a continuous function on $[0,2],$ differentiable in $(0,2)$ such that $f\left(2\right)=0,$ then which of the following options is(are) correct for atleast one value of $c\in (0,2)$

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is :

The first and the last terms of an A.P. are a and l respectively. Show that the sum of nth term from the beginning and nth term from the end is a + l.

If the end points of one axis of an ellipse are $(-12,4)$ and $(14,4)$ and eccentricity $\frac{12}{13}$, then the equation(s) of the ellipse is/are

Let the primitive of $\sin \left(\ln x\right),$ $x>0$ be of the form $f\left(x\right)\left(\sin g\right(x)-\cos h(x\left)\right)+C,$ where $C$ is constant of integration. If $a=\underset{x\to 2}{lim}f\left(x\right),$ $b=g\left(e^5\right)+g\left(e^3\right)-6$ and $g\left(1\right)=h\left(1\right)=0,$ then point $(a,b)$ lies on the curve(s)

If an unbiased die is thrown 5 times and every throw resulted in a 6, what is the probability of getting a 6 on the sixth throw?

What should be added to x² + 2x + 0.5 to make it a perfect square?

If a, b, c are in G.P., prove that:

(i) $a(b^2+c^2)=c(a^2+b^2)$

(ii) $a^2{b}^2{c}^2(\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3})=a^3+b^3+c^3$

(iii) $\frac{(a+b+c{)}^2}{a^2+b^2+c^2}=\frac{a+b+c}{a-b+c}$

(iv) $\frac{1}{a^2-b^2}+\frac{1}{b^2}=\frac{1}{b^2-c^2}$

(v) $(a+2b+2c)(a-2b+2c)=a^2+4c^2$.

(v) $(a+2b+2c)(a-2b+2c)=a^2+4c^2$.

Find the equation of an ellipse, the distance between the foci is 8 units and the distance between the directions is 18 units.

The number of terms in the expansion of $(x+y+z+u{)}^5$ is:

Given $A,B$ and $C$ are events such that $P\left(A\right)=P\left(B\right)=P\left(C\right)=\frac{1}{5},P(A\cap B)=P(B\cap C)=0$ and $P(A\cap C)=\frac{1}{10}.$ Then the probability that atleast one of the events $A,B$ or $C$ occurs is

A bag contains 15 red, 18 white, 12 blue and 16 black marbles. If a marble is drawn at random from the bag, then the probability of getting a black marble is

The equation of the plane containing the lines $2x-5y+z=3,$ $x+y+4z=5$ and parallel to the plane $x+3y+6z=1,$ is:

Draw the lines x = -3, x = 2, y = -2, y = 3 and write the coordinates of the vertices of the square so formed.

${lim}_{n\to \infty }\frac{(n+2)!+(n+1)!}{(n+2)!-(n+1)!}$ is

Let $f\left(x\right)=\{\begin{array}{c}\sqrt{\left\{x\right\}},x\notin Z\\ 1,x\in Z\end{array}$ and $g\left(x\right)=\{x{\}}^2,$ the area bounded by $f\left(x\right)$ and $g\left(x\right)$ for $x\in [0,n];(n\in Z)$ is denoted by $S\left(n\right)$, then which of the following is correct?

The plane passing through the point $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ also passes through the point :

In how many ways 3 friends Ram, Rajat and Rupesh having 6 one rupee coins, 7 one rupee coins, 8 one rupee coins respectively donate 10 rupee coin collectively?

In how many ways a commitee of six members be formed from 7 men and 5 women if that commitee contains at least 2 women?

If the sum of the roots of the equation $\lambda x^2+2x+3\lambda =0$ be equal to their product, then λ =

Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): (ax + b) (cx + d)²

The area of the region $(x,y)$ : $xy\le 8,1\le y\le x^2$ is

The ratio of the sums of m and n terms of an A.P is m² /n² show that the ratio of mth and nth term is (2m-1)/(2n-1)

Two different digits are chosen at random from the set $\{1,2,3,4,5,6,7,8\}.$ Then the probability that both the digits are less than $3,$ is

Solve $xdx+ydy=\frac{xdy-ydx}{x^2+y^2}$

If the equation $a{x}^n+3x^5+b{x}^2+c=0,$ $n\in Z^{+}$ has infinite number of real solutions, then $a+b+c+n$ is equal to

Prove that following identities:

$sin5\theta =5sin\theta -20si{n}^3\theta +16si{n}^5\theta$

The middle term in the expansion of ${(x^2+\frac{1}{x})}^n$ is $924x^6$. If $n$ is even, then $n$ is equal to

The equation of the plane passing through the points (3,2,2) and (1,0,-1) and parallel to the line $\frac{x-1}{2}=\frac{y-1}{-2}=\frac{z-2}{3}$, is

The coefficient of $x^{11}$ in the expansion of $(1+2x+2x^2{)}^6$ is

Find the equation for the ellipse that satisfies the given conditions,
major axis on the x - axis and passes through the point (4,3) and (6,2).

The value of $\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}$ - 1 =