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If the circles $x^2+y^2+2ax+cy+a=0$ and $x^2+y^2-3ax+dy-1=0$ interesect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for

(2005)

If $x^2+ax+10=0$ and $x^2+bx-10=0$ have a common root, then $a^2-b^2$ is equal to

Difference between the greatest and the least values of the function $f\left(x\right)=x(\ln x-2)$ on $[1,e^2]$ is

cos
6x
= 32 cos⁶
x
– 48 cos⁴
x
+ 18 cos²
x –
1

Integrate the function.
$\int x$ log 2x dx.

If the coordinates of the four vertices of a quadrilateral are $(-2,4),(-1,2),(1,2)$ and $(2,4)$ taken in order, then the equation of line passing through the vertex $(-1,2)$ and dividing the quadrilateral in two equal areas is

If the straight lines $x=1+s,y=-3-\lambda s,z=1+\lambda s$ and $x=\frac{t}{2},y=1,z=2-t$, with parameters s and trespectively, are co-planar, then $\lambda$ equals
[AIEEE 2004]

Let $f\left(x\right)=\underset{x}{\overset{2}{\int }}\frac{dy}{\sqrt{1+y^3}}.$ Then the value of $\underset{0}{\overset{2}{\int }}xf\left(x\right)dx$ is equal to

The
maximum value of

is

(A) (B)

(C) 1 (D) 0

Let $S_1,S_2,S_3,\dots ,S_n$ be squares such that for each $n\ge 1$, the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10$ cm, then for which of the following values of $n$ is the area of $S_n$ less than $1$ sq. cm?

Let $a_1,a_2,\dots ,a_n$ be a given A.P. whose common difference is an integer and $S_n=a_1+a_2+\dots +a_n$. If $a_1=1$, $a_n=300$ and $15\le n\le 50$, then the ordered pair $(S_{n-4},a_{n-4})$ is equal to

If $f_n\left(x\right)=\frac{1}{n}({\cos }^{n}x+{\sin }^{n}x),$ for $n=1,2,3,....,$ then $f_4\left(x\right)-f_6\left(x\right)$ is equal to

The Cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$, write its vector form.

​​​a+b=π/2

b+c=a

tan a=__________

1. 2(tan b+tan c)

2. tan b+tan c

3. tan b+2.tan c

4. 2.tan b+tan c

If $\overrightarrow{c}=2\overrightarrow{a}-3\overrightarrow{b}$ where $|\overrightarrow{a}|=2$ and $|\overrightarrow{b}|=3$ and $\overrightarrow{a}\cdot \overrightarrow{b}=6$, also $\overrightarrow{c}$ is coplanar with $\overrightarrow{p}=\hat{i}+\hat{j}-2\hat{k}$ and $\overrightarrow{q}=\hat{i}-2\hat{j}+\hat{k}$ and perpendicular to $\overrightarrow{r}=\hat{i}+\hat{j}+2\hat{k}$ then $\overrightarrow{c}$ can be

1. 5

The number of pairs of consecutive even natural numbers whose sum is less than $16$ is

A man saved Rs. 66000 in 20 years. In each succeeding year after the first year he saved Rs. 200 more than what he saved in the previous year. How much did he save in the first year ?

The equation of the parabola whose focus lies at the intersection point of the lines $x+y=3$ and $x-y=1$ and directrix is $x-y+5=0$

The largest non-negative integer k such that ${24}^k$ divides 13! is.

The length of the latus-rectum of the parabola $4y^2+2x-20y+17=0$

Find $\frac{dy}{dx}$, if x and y are connected parametrically by the equations given in questions without eliminating the parameter.

$x=4t,y=\frac{4}{t}.$

Let the circles $C_1:x^2+y^2=9$ and $C_2:(x-3{)}^2+(y-4{)}^2=16,$ intersect at the points $X$ and $Y.$ Suppose that another circle $C_3:(x-h{)}^2+(y-k{)}^2=r^2$ satisfies the following conditions:



$\left(i\right)$ centre of $C_3$ is collinear with the centres of $C_1$ and $C_2.$



$\left(ii\right)$$C_1$ and $C_2$ both lie inside $C_3$, and



$\left(iii\right)$ $C_3$ touches $C_1$ at $M$ and $C_2$ at $N$



Let the line through $X$ and $Y$ intersect $C_3$ at $Z$ and $W$, and let a common tangent of $C_1$ and $C_3$ be the tangent to the parabola $x^2=8\alpha y.$

There are some expressions given in $List-I$ whose values are given in $List-II$ below:



$\begin{array}{cccc}& \text{List}I& & \text{List}II\\ \left(I\right)& 2h+k& \left(P\right)& 6\\ \left(II\right)& \frac{\text{length of}ZW}{\text{length of}XY}& \left(Q\right)& \sqrt{6}\\ \left(III\right)& \frac{\text{Area of triangle}MZN}{\text{Area of triangle}ZMW}& \left(R\right)& \frac{5}{4}\\ \left(IV\right)& \alpha & \left(S\right)& \frac{21}{5}\\ & & \left(T\right)& 2\sqrt{6}\\ & & \left(U\right)& \frac{10}{3}\end{array}$

Which of the following is the only CORRECT combination?

Solve for x. $|5x-75|>45-x$

x is one of the four harmonic means inserted between 2/3 and 2/13. The value/s of x can be:

Find out the appropriate word which fits the 7th blank.