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The minimum possible value of $|x-1|+|x-2|+\cdots +|x-100|$ is

The equation of the normals to the circle $x^2+y^2-8x-2y+12=0$ at the points whose ordinate is $-1$ is/are

Which of the following equations represents a circle with centre $(g,f)$ and radius $\sqrt{g^2+f^2+c}$ ?

If z is a complex number $\overline{z^{-1}}\left(\overline{z}\right)$ = , then

Which of the following statements is/are correct for the above given figure?

1.For all the function A, B and C, domain is R and Range is $(0,\infty )$.

2.Function B is a graph for $F\left(x\right)=a^x,a>1,xϵR$.

3.Function A is a graph for $F\left(x\right)=a^{x}0<a<1,xϵR$

4.Function C is a graph for $F\left(x\right)=a^x,a<0,xϵR$

5.If the graph of B is $5^x$, then the graph of A is $y=8^x$ and the graph of C is

$y=2^x$ (you have to choose graph from $2^x,8^{x}and{5}^x)$

6.If the graph of B is $5^x$, then the graph of A is $y=2^x$ and the graph of C is

$y=8^x$ (you have to choose graph from $2^x,8^{x}and{5}^x)$

The solution of the differential equation $\frac{dy}{dx}=\frac{x+y}{x}$ satisfying the condition $y\left(1\right)=1$ is

If $2$ and $3$ are the lengths of the segments of any focal chord of a parabola $y^2=4ax$ made by the axis of the parabola, then the length of the latus rectum is

The median AD of the $\Delta ABC$ is bisected at E, BE meets AC in F, then AF: AC is equal to

The points A(4,5,1),B(0,-1,-1),C(3,9,4) and D(-4,4,4) are

[Kurukshetra CEE 2002]

If the ordinates of the points $P$ and $Q$ on the parabola $y^2=12x$ are in the ratio $1:2,$ then the locus of the point of intersection of normals to the parabola at $P$ and $Q$ is

consider the function

$f\left(x\right)=\{\begin{array}{ccc}a+bx,x<1& & \\ 4,x=1& & \\ b-ax,x>1& & \end{array}$

If ${lim}_{x\to 1}f\left(x\right)=f\left(1\right)$, then the values of a and b are

Number of solutions of the equation $(2\text{cosec}x-1{)}^{\frac{1}{3}}+(\text{cosec}x-1{)}^{\frac{1}{3}}=1$ in $(-k\pi ,k\pi )$ is $16$, then the possible value of '$k$' is

If the equations $4x^2-x-1=0$ and $3x^2+(\lambda +\mu )x+\lambda -\mu =0$ have a common root, then the rational values of $\lambda$ and $\mu$ are

Two matrices $A=[a_{ij}{]}_{p\times q},B=[b_{ij}{]}_{m\times n}$ are equal if.

Value of x at which the function f(x) = xlnx, x > 0 attains its extrema, is



x के कौनसे मान के लिए फलन f(x) = xlnx, x > 0 का चरम मान विद्यमान है

If $x=sec\theta -cos\theta ,y=se{c}^{10}\theta -co{s}^{10}\theta$ and $(x^2+4){\left(\frac{dy}{dx}\right)}^2=k(y^2+4)$, then k is equal to

The total number of ways of selecting two numbers from the set $\{1,2,3,4,\dots ,3n\},$ so that their sum is divisible by $3,$ is equal to

Centre of the ellipse $4(x-2y+1{)}^2+9(2x+y+2{)}^2$ = 5 is

If $\overrightarrow{a}=x\hat{i}+(x-1)\hat{j}+\hat{k}$ and $\overrightarrow{b}=(x+1)\hat{i}+\hat{j}+a\hat{k}$ always make an acute angle with each other for every value of $xϵR$, then

Consider the letters of the word $MATHEMATICS$. Possible number of words taking all letters at a time such that at least one repeating letter is at odd position in each word is

If $n={}^m{C}_2$, then the value of ${}^n{C}_2$ is

Which among the following is the correct graphical representation of $y=-x^2+4x+1$ ?

A quadratic polynomial $p\left(x\right)$ has $1+\sqrt{5}$ and $1-\sqrt{5}$ as its zeros and $p\left(1\right)=2$. Then the value of $p\left(0\right)$ is

An equation of a tangent drawn to the curve $y=x^2-3x+2$ from the point $(1,-1)$ is

The point of concurrence of the lines $17x+2y-28=0,x+5y+13=0$ and $7x+2y-8=0$ is

If $y\cos \theta -1=\sin \theta$, then $\sin \theta$ is

If $\sum _{j=0}^{10}{}^{30+j}{C}_{10+j}={}^m{C}_{20}-{}^p{C}_{21}$, then

The polar coordinates of the point whose Cartesian coordinates are $(-1,-\sqrt{3})$, are

The equation(s) of an standard ellipse which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$, is/are

Total number of $4$ letter words that can be formed using the letters of the word ${}^{\prime }FLOWE{R}^{\prime }$, such that the word starts with $F$ and ends with $R$ is

You can remove a removable discontinuity by

If ${(1+x-2x^2)}^6=1+a_{1}x+a_2{x}^2+a_3{x}^3+\cdots +a_{12}{x}^{12},$ then the value of

$a_2+a_4+a_6+\cdots +a_{12}$ will be

If $f_n\left(\theta \right)=\frac{\cos \frac{\theta }{2}+\cos 2\theta +\cos \frac{7\theta }{2}+\cdots +\cos \frac{(3n-2)\theta }{2}}{\sin \frac{\theta }{2}+\sin 2\theta +\sin \frac{7\theta }{2}+\cdots +\sin \frac{(3n-2)\theta }{2}},$ then which among the following is (are) CORRECT?

The Greatest co-efficient in the expansion of $(1+x{)}^{2n+2}$ is

Let $f\left(x\right)$ be a differentiable function defined on $[0,2]$ such that $f^{\prime }\left(x\right)=f^{\prime }(2-x)$ for all $x\in (0,2),$ $f\left(0\right)=1$ and $f\left(2\right)=e^2.$ Then the value of $\underset{0}{\overset{2}{\int }}f\left(x\right)dx$ is

In a group of $50$ people, $35$ speak Hindi and $25$ speak both English and Hindi. Then how many people speak only English ? (Assuming each person speaks at least one language)

The focal chord to $y^2=16x$ is tangent to $(x-6{)}^2+y^2=2$, then the possible values of the slope of this chord are

If the image of the point $P(1,–2,3)$ in the plane, $2x+3y–4z+22=0$ measured parallel to the line,$\frac{x}{1}=\frac{y}{4}=\frac{z}{5}\text{is}Q,\text{then}PQ$ is equal to:

The equation of the ellipse whose vertices are ( ±5,0) and foci are ( ± 4 , 0 ) is

Given that $A=\{1,2,3\}$, $B=\{3,4\}$ and $C=\{4,5,6\}$, then $n(A\cup (B\cap C\left)\right)=$

If $0<A+B<\frac{\pi }{2}$ and $\tan A,\tan B$ are the roots of the equation $3x^2-12x-6=0,$ then the numerical value of $\sin (A+B)\cos (A+B)-\sec (A+B)\text{cosec}(A+B)$ is

If $A=\{5,7,9,11\},B=\{9,10\}$ and $aRb$ means $a<b$ where $a\in A,b\in B,$ then which of the following are true?