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The sum of value(s) of $k$ for which the equation $(({\log }_{5}k{)}^2+({\log }_{5}k)-2)x^2-(2^{2k}-34\cdot 2^k+64)x+(k^2+7k-60)=0$ possesses more than two roots, is

The solution set of the inequality ${\log }_3(x+2)(x+4)+{\log }_{1/3}(x+2)<\frac{1}{2}{\log }_{\sqrt{3}}7$ is

If $z$ is a complex number of unit modulus and argument $\theta$, then $\text{arg}\left(\frac{1+z}{1+\overline{z}}\right)=$

Solve the following quadratics $8x^2-9x+3=0$

The equation of the circle in diameter form with centre $(4,–2)$ and passing through the point $(2,-2)$ is

A circle passes through the points $(2,3)$ and $(4,5)$. If its centre lies on the line, $y-4x+3=0$, then its radius is equal to :

$1+2^1+2^2+2^3.........{.2}^{1999}$ =

Let $f\left(x\right)=cosx(sinx+\sqrt{si{n}^{2}x+si{n}^2\theta }),\theta$ is a given const, then max of f(x)is

Suppose you have a box with 3 blue marbles, 2 red marbles, and 4 yellow marbles. You are going to pull out one marble, record its color, put it back in the box and draw another marble. What is the probability of pulling out a red marble followed by a blue marble?

The integral value(s) of

$x$ satisfying $\sqrt{-x^2+10x-16}<x-2$ is/are

If $\sin \theta =\left(\frac{z-1}{i}\right),$ where $z$ is non real, $\theta$ represents angle of a triangle, then

From a point $P(\lambda ,\lambda ,\lambda )$, perpendiculars PQ and PR are drawn respectively on the lines y = x, z = 1 and y = -x, z = -1. If P is such that $\angle QPR$ is a right angle, then the possible value(s) of

$\lambda$ is (are)

Consider all the permutations of the word $BENGALURU$. The number of words in which vowels occur at even places is given as $A$ and the number of words in which the letters of the word $GLUE$ appear together in that order is given as $B$. Find the value of $A-B$

If $\alpha$ and $\beta$ are the roots of a quadratic equation satisfying the conditons $\alpha \beta =4$ and $\frac{\alpha }{\alpha -1}+\frac{\beta }{\beta -1}=\frac{a^2-7}{a^2-4},\alpha ,\beta ,a\in R$. For what values of ${}^{\prime }{a}^{\prime }$ will the quadratic equation have equal roots?

The complete set of values of $x$ satisfying $5x+2<3x+8$ and $\frac{x+2}{x-1}<4,$ is

A man has $7$ letters for his $7$ friends. The letter are kept in the envelopes at random. The number of ways in which exactly $3$ letters are going to correct envelope and rest $4$ letters are going to the wrong envelopes is

The ratio of coefficient of $x^{15}$ to the term independent of $x$ in the expansion of ${(x^2+\frac{1}{2x})}^{15}$ is

The values of $m$ such that exactly one root of $x^2+2(m-3)x+9=0$ lies between $1$ and $3$, is

If $\frac{9x^2-4}{\sqrt{5x^2-1}}\le 3x+2,$ then $x\in$

If $si{n}^{-1}x=\theta +\beta$ and $si{n}^{-1}y=\theta -\beta$ then 1+xy is equal to

Equation of the hyperbola with focus (-3,4) directrix 3x-4y+5=0 and e = $\frac{5}{2}$ is

If $z$ is a complex number satisfying $z-12=i(9-2\overline{z}),$ then the value of $z+\overline{z}$ is

Let $y=a{x}^2+bx+c(a\ne 0)$ and $a,b,c\in R$. If $abc>0$, then which of the following graph(s) satisfy the given condition?

The point(s) on the $x-$axis which is (are) at a distance of $5$ units from the point $(6,-3),$ is (are)

If $\alpha ,\beta$ are the roots of $2x^2+3x+1=0$, then the equation whose roots are $\frac{1}{\alpha },\frac{1}{\beta }$ is

The value of $\sin ({45}^{\circ }+\theta )-\cos ({45}^{\circ }-\theta )$ is

Mr. A lives at origin on the Cartesian plane and has his office at (4, 5). His friend lives at (2, 3) on the same plane. Mr. A can go to his office travelling one block at a time either in the +y or +x direction. If all possible paths are equally likely then the probability that Mr. A passed his friends house is (shortest path for any event must be considered)

If $\theta$ is the angle between the two tangents to $y^2=12x$ drawn from the point $(1,4)$, then $\tan \theta$ is equal to

The focus of the parabola $x^2-2x=2y$ is

Solution of $|x+2|+|2x+6|+|3x-3|=12$ is

$y=4sin3x$ is a solution of the differential equation

[AI CBSE 1986]

If $\theta$ be the angle subtended at the focus by the chord which is normal at the point $(\lambda ,\lambda ),\lambda \ne 0$ to the parabola $y^2=4x,$ then the equation of the line making angle $\theta$ with positive $x-$axis and passing through $(1,2)$ is

Equation of the circle passing through the point $(1,1)$ and point of intersection of circles $x^2+y^2=6$ and $x^2+y^2-6x+8=0$ is

The unit vector in ZOX plane and making angle ${45}^{\circ }$ and ${60}^{\circ }$ respectively with $\overrightarrow{a}=2\hat{i}+2\hat{j}-\hat{k}$ and $\overrightarrow{b}=0\hat{i}+\hat{j}-\hat{k}$

Find $\int \sec \left(x\right)\tan \left(x\right)dx$

$\underset{x\to 0}{lim}$ $\frac{x(e^x-1)}{1-cosx}$ =

Which of the following is/are a function?

Calculate mean deviation about median for following readings

$\begin{array}{ccccc}Class& 20-40& 40-60& 60-80& 80-100\\ F_i& 20& 44& 30& 40\end{array}$

If ${\alpha }_0,{\alpha }_1,{\alpha }_2\dots ,{\alpha }_{n-1}$ be the n, nth roots of the unity, then the value of ${\sum }_{i=0}^{n-1}\frac{{\alpha }_i}{(3-{\alpha }_i)}$ is equal to

Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :



$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}\frac{z-(1+i)}{2+i}\text{is purely real, then}Re\left(z\right)+Im\left(z\right)\text{can be equal to}& \text{(P)}& 9\\ \text{(B)}& \text{If}\sum _{r=0}^8{}^{r+2}{C}_r={}^{11}{C}_b,\text{then a possible value of}b\text{is}& \text{(Q)}& 10\\ \text{(C)}& \text{The coefficient of}{x}^5\text{in the expansion of}{(x^2+x+2)}^5\text{is divisible by}& \text{(R)}& 3\\ \text{(D)}& \text{If the least area of triangle formed by tangent to the circle}{x}^2+y^2=1& \text{(S)}& 8\\ & \text{and}x=0\text{,}y=0\text{is}A,\text{then}A\text{is co-prime with}& & \\ & & \text{(T)}& 7\end{array}$



Which of the following is the only CORRECT combination?

The number of solutions of the equation $z^2$ + $\overline{z}$ = 0 is

If the line $y=11x+(b-4)$ passes through the origin, then the value of $b$ is

What is the condition for the line y = mx + c to be a secant of the circle $x^2+y^2=a^2$

Let $a,b,c$ be in arithmetic progression. Let the centroid of the triangle with vertices $(a,c),$ $(2,b)$ and $(a,b)$ be $(\frac{10}{3},\frac{7}{3}).$ If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+1=0,$ then the value of ${\alpha }^2+{\beta }^2-\alpha \beta$ is

The centre of circle passing through (0, 0) and (1, 0) and touching the circle $x^2+y^2=9$ is

(2002)

In a $△ABC,AB=ri+j,AC=si-j$ if the area of triangle is of unit magnitude, then