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Find the sum of square of the roots $(\sum x_1^2)$ of biquadratic equation $x^4+2g{x}^3+d{x}^2+2f{c}^{2}x+c^4=0$ if $x_1.x_2.x_3.x_4$ are the roots of the equation.

If $\sin x=\frac{12}{13}$ and $x\in [0,\pi ]$, then the value(s) of $\sec x+\tan x$ is/are

Find the equation of straight line passing through (-2, -7) and having an intercept of length 3 between the straight lines 4x + 3y = 12 and 4x + 3y = 3.

If the normals at two points $P$ and $Q$ of a parabola $y^2=4ax$ intersect at $R$ on the curve, then the product of ordinates of $P$ and $Q$ is

If $\alpha ,\beta$ are the roots of the equation $4^x-3\left(2^{x+3}\right)+128=0$, then ${\alpha }^2+{\beta }^2=$

The number of triangles whose vertices are at the vertices of an octagon but none of whose sides happen to come from the octagon, is

$\sqrt{2+\sqrt{2+2cos4\theta }}$ =

For the equation of ellipse, $\frac{x^2}{4}+\frac{y^2}{9}=1$ what does S correspond to

Range of $f\left(x\right)=\frac{tan\left(\pi [x^2-x]\right)}{1+sin\left(cosx\right)}$ is where $\left[x\right]$ denotes the greatest integer function

If $A=\left[\begin{array}{c}12\\ -13\end{array}\right]B=\left[\begin{array}{c}40\\ 15\end{array}\right]C=\left[\begin{array}{c}20\\ 1-2\end{array}\right]$ a = 4, and b = - 2, then show that

A (BC) = (AB) C

Find the second order derivative of the given functions.

log (log x).

If $a,b,c$ are three complex numbers such that $a^2+b^2+c^2=0$ and
$\left|\begin{array}{ccc}(b^2+c^2)& ab& ac\\ ab& (c^2+a^2)& bc\\ ac& bc& (a^2+b^2)\end{array}\right|=K{a}^2{b}^2{c}^2$ then $K$ is -

$\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$ is equal to

A function f(x) is defined on [a,b]. What should be the condition for the function to be a strictly decreasing function at x = b ?

If $cos\left(2si{n}^{-1}x\right)=\frac{1}{9}$, then x=

The probability that a ${}^{\prime }P$ and $C^{\prime }$ question wil be asked in $IITJEE$ is $\frac{2}{5}$ and probability question is asked is $\frac{4}{7}$. If the probability of getting at least one of them is $\frac{2}{3}$ what is the probability that questions from both the topics are asked.

Find the equation for the ellipse that satisfies the given conditions,
Length of minor axis 16, foci $(0,\pm 6)$

Find the equation of the plane through the points (2,1,-1), (-1,3,4) and perpendicular to the plane x-2y+4z=10.

Find $\frac{dy}{dx}$in the following questions:

2x+3y=sin x.

The inegral $\int \frac{3x^{13}+2x^{11}}{(2x^4+3x^2+1{)}^4}dx$ is equal to :
(where $C$ is a constant of integration)

If vertices of any quadrilateral are (0, -1), (2,1), (0, 3) and (- 2,1), then it is a

Solve the equation, if $\left|\begin{array}{ccc}x+a& x& x\\ x& x+a& x\\ x& x& x+a\end{array}\right|=0,a\ne 0$,

The value of $sin\left[2ta{n}^{-1}\right(0.75\left)\right]$ is

(a) 0.75 (b) 1.5 (c) 0.96 (d) sin 1.5

Let $\ast$ be a binary operation on the set Q of rational number as follows:
$\left(i\right)a\ast b=a^2+b^2$
Find which of the binary operation are commutative and which are associative?

On her vacation, Veena visits four cities A, B, C and D in a random order, what is the probability that she visits

(i) A before B ?

(ii) A before B and B before C?

(iii) A first and B last?

The probabilities of three mutually exclusive events are 2/3, 1/4 and 1/6. The statement is [MNR 1987; UPSEAT

The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1.$

The compound A is

A ladder20 ft long has one end on the ground and the other end in contact with a vertical wall. The lower end slips along the ground. If the lower end of the ladder is 16 ft away from the wall, upper end is moving $\lambda$ times as fast as the lower end, then $\lambda$ is

The sum of the real values of $x$ for which the middle term in the binomial expansion of ${(\frac{x^3}{3}+\frac{3}{x})}^8$ equals $5670$ is :

If $\frac{\pi }{2}<\theta <\pi$, then write the value of $\sqrt{2+\sqrt{2+2cos2\theta }}$ in the simplest form.

Let A be the set of first natural numbers and let R be a relation on A defined as follows :

$(x,y)ϵR\iff x\le y$

Express R and $R^{-1}$ as sets of ordered pairs. Determine also

(i) The domain of $R^{-1}$

(ii) The range of R.

Find the values of K for which the line $(K-3)x-(4-K^2)y+K^2-7K+6=0$ is

(a) Parallel to the x-axis

(b) Parallel to the y-axis,

(c) Passing through the origin.

If the linear function is of the form $y=mx+c$, where $m<0$ and $c>0$, then the graph will be

Which of the following can be insert between $3$ and $19$ as arithmetic means?

If tangent at $P$ and at vertex of the parabola $y^2=4ax$ meets at $Q$, then the value of $\angle SQP$, where $S$ is the focus, is

Show that the straight lines given by. $(2+k)x+(1+k)y=5+7k$ for different values of k pass through a fixed point. Also find that point.

1. undefined
2. undefined
3. undefined
4. undefined

Locus of a point through which three normals of parabola $y^2=4ax$ are passing, two of which are making angles $\alpha$ and $\beta$ with positive $x-$ axis and $\tan \alpha \cdot \tan \beta =2$ is

One root of the following given equation $2x^5-14x^4+31x^3-64x^2+19x+130=0$ is