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The equation of the parabola whose focus is $(4,-3)$ and vertex is $(4,-1)$

If both the roots of $k(6x^2+3)+rx+2x^2-1=0$ and $2(6k+2)x^2+px+2(3k-1)=0$ are common, then 2r-p is equal to

Let $A,B$ and $C$ represents the angles of $△ABC$. If $\tan \frac{A}{2},\tan \frac{B}{2},\tan \frac{C}{2}$ are in H.P., then the minimum value of $\cot \frac{B}{2}$ is

Find the sum to indicated number of terms in each of the geometric progressions in $x^3,x^5,x^7,.....nterms(ifx\ne \pm 1).$

Give the formula to calculate rank correlation coefficient with
(i) non-repeated ranks and
(ii) repeated ranks

How many words (with or without meaning) can be formed from the letters of the word, 'DAUGHTER', so that

(i) all vowels occur together?

(ii) all vowels do not occur together?

If $\sum _{k=1}^{n}f\left(k\right)=\frac{n}{2(n+2)}$, then the value of $\sum _{k=1}^{10}\frac{1}{f\left(k\right)}$ is equal to

Which of the following points lie in the VIII th octant?

Suppose f(x) = $(x+1{)}^2$ for $\ge$ -1. If g(x) is the function whose graph is reflection of the graph of f(x) with respect to line y = x, then g(x) equals

If the origin is the centroid of the triangle with vertices P(2a, 2, 6), Q(-4, 3b, -10) and R(8, 14, 2c), find the values of a,b and c. Also, determine the value of $a^2+b^2-c^2$.

If a, b are non-zero real numbers of opposite signs, then which of the following is/are true?

The equation of a circle which passes through (1,2) and (2,1) and whose radius is 1 units is

If the ratio of the coefficient of third and fourth term in the expansion of $(x-\frac{1}{2x}{)}^n$ is 1:2, then the value of n will be

The points whose position vectors are $60i+3j,40i-8j$ and $ai-52j$ collinear, if

Let $a,r,s$ and $t$ be non-zero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)$ and $S(a{s}^2,2as)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is point $(2a,0)$.

If $st=1$, then the tangent at $P$ and the normal at $S$ to the parabola meet at a point whose ordinate is

The principal value of $si{n}^{-1}\left[sin\left(\frac{2\pi }{3}\right)\right]$

[IIT 1986]

If 4a+5b+6c=0 then the set of lines ax+by+c=0 are concurrent at the point

If $\int {\sin }^{-1}x{\cos }^{-1}xdx=f^{-1}\left(x\right)[Ax-x{f}^{-1}(x)-2\sqrt{1-x^2}]+\frac{\pi }{2}\sqrt{1-x^2}+2x+c,$ then

Find the sum of the series
${\sum }_n^{r=0}(-1{)}^r{}^n{C}_r[\frac{1}{2^r}+\frac{3^r}{2^{2r}}+\frac{7^r}{2^{3r}}+\frac{{15}^r}{2^{4r}}\cdots uptomterms]$

The solution of $5^{{\log }_{a}x}+5x^{{\log }_{a}5}=3(a>0,a\ne 1)$ is

If z = x + iy, $z^{\frac{1}{3}}$ = a - ib and $\frac{x}{a}$ - $\frac{y}{b}$ = $\lambda$$(a^2-b^2)$, then $\lambda$ is equal to

1. 1

The solution of $y_2-7y_1+12y=0$ is

$f\left(x\right)=\left[x\right]$ is discontinuous at $x=1$ because

If ${\cos }^{3}x\sin 2x=a_1\sin x+a_2\sin 2x+\cdots +a_n\sin nx$ $\forall x\in R$, where $a_n\ne 0$, then which of the following is/are correct ?

A tangent to the parabola $y^2=4ax$ meets x axis at a point T. This tangent meets the tangent at the vertex at point P. If rectangle TAPQ is completed then the locus of Q is.

Find the set of values of x for which the expansion of ${(9+5x)}^{\frac{-1}{2}}$ is valid in ascending powers of x.

The equation of the line passing through

(1, 2) and making an angle of 30° in clockwise direction with the positive direction of y-axis is .

If $A_1,A_2,A_3$ be the area of the incircle and exercise, then $\frac{1}{{\sqrt{A}}_1}+\frac{1}{{\sqrt{A}}_2}+\frac{1}{{\sqrt{A}}_3}$ is equal to

In a class of $400$ students, $150$ are interested in doing a statistics project, $300$ are interested in doing a machine learning project and $25$ are not interested in doing either of the projects. Then the number of students who are interested in doing both the projects, is

Number of solution(s) of equation $|x-1|-|x-2|=10$ is

The sum of the first 20 term of the sqeuence 0.7, 0.77, 0.777,...., is

(IIT JEE Main 2013)

If $\sin (A+B)=1,\cos (A-B)=1,0^{\circ }\le A+B\le {90}^{\circ }$, Find $A$ and $B$.

If odd natural numbers are arranged in groups as $\left(1\right),(3,5),(7,9,11),...$ Then the sum of the numbers in the ${10}^{\text{th}}$ group is

$\begin{array}{ccc}\text{Function}& \text{Domain}& \text{Range}\\ \text{(i) sinx}& \text{(P) R}& \text{(L) [-1, 1]}\\ \text{(ii) cos x}& \text{(Q) R-n}\pi & \text{(M) R}\\ \text{(iii) tan x}& \text{(R) R- {(2n+1)}}& \text{(N)}(-\infty ,\text{-1]}\cup \text{[1,}\infty )\\ \text{(iv) cosec x}& & \\ \text{(v) sec x}& & \\ \text{(vi) cot x}& & \end{array}$

How many of the following are matched correct?

(i) -P-L, (ii) -P-L, (iii)-R-M, (iv) -Q-N, (v) -R-N, (vi) -Q-M

Evaluate:

$\cos \left(\frac{\pi }{2}-{\sin }^{-1}\left(-\frac{1}{2}\right)\right\}.$

If the truth value of the statement $p\to (\sim q\vee r)$ is false $\left(F\right),$ then the truth values of the statements $p,q,r$ are respectively :

Solve the inequality $\frac{3-x}{x+2}$ > 1

Equation of the straight line passing through point of intersection of the line $\frac{x}{a}+\frac{y}{b}=1$ and $\frac{x}{b}+\frac{y}{a}=1$ and is making an angle $\frac{\pi }{4}$ with the line $2x-2y+3=0$ is

The relation f is defined by

$f\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 2\\ 3x,& 3\le x\le 10\end{array}$

The relation g is defined by

$g\left(x\right)=\{\begin{array}{cc}{x}^2,& 0\le x\le 3\\ 3x,& 2\le x\le 10\end{array}$

Show that f is a function and g is not a function.

Solution of the differential equation $2ysinx\frac{dy}{dx}=2sinxcosx-y^{2}cosx$ satisfying $y\left(\frac{\pi }{2}\right)=1)$is given by

If $y^x-x^y=1$ then $\frac{dy}{dx}$ at $x=1$ is

The letters of the word SACHIN are permuted and are arranged in an alphabetical order as in an English dictionary. Then, the rank of the word SACHIN is _____

Solve: $\sqrt{x-2}$ ≥ -1

The product of two complex numbers $1+i$ and $2-5i$ is