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Find the value of k , if the matrix $\left[\begin{array}{cc}2& 3\\ 4& 5\end{array}\right]=\left[\begin{array}{cc}x& 3\\ 2x& 5\end{array}\right]$

Let $y=f\left(x\right)$ is a positive function which satisfies equation $\sqrt{y^2+2x}+\sqrt{y^2-2x}=2x^2,\text{then}\frac{dy}{dx}$ is

The points A(4,5,1),B(0,-1,-1),C(3,9,4) and D(-4,4,4) are
[Kurukshetra CEE 2002]

If a normal to the hyperbola $xy=c^2$ at $(c{t}_1,\frac{c}{t_1})$ meets the curve again at $(c{t}_2,\frac{c}{t_2})$ then:

An equilateral triangle has two vertices (-2,0) and (2,0) and its third vertex lies below the x-axis, the equation of the circumcircle of the triangle is

Two aeroplanes 1 and 2 bomb a target in succession.The probability of 1 and 2 scoring a hit correctly are 0.3 and 0.2 respectively. The second plane will bomb only when 1st one misses the target.The probability that the target is hit by 2nd plane is?

Write the integral of $\frac{1}{x\sqrt{x^2-1}}$ with respect to $x,x>1$ .

Is there any trick to find the products of two or three matrices of order 3×3 quickly?

If $\left(4^3\right)\left(4^6\right)\left(4^{12}\right).......\left(4^{3x}\right)=(0.0625{)}^{-54}$, the value of x is

Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector $3\hat{i}+5\hat{j}-6\hat{k}.$

The equation $16x^2+y^2+8xy-74x-78y+212=0$ represents

If $tanA=\frac{1}{7}andtanB=\frac{1}{3}$, show that cos 2A = sin 4B.

If in a $\Delta ABC,A\equiv (1,10),$ circumcentre $\equiv (-\frac{1}{3},\frac{2}{3})$ and orthocentre $\equiv (-\frac{11}{3},\frac{4}{3}),$ then the equation of side $BC$ is

Which of the following functions is differentiable at $x=0$?

If the number of ways of selecting $3$ numbers out of $1,2,3,\dots ,2n+1$ such that they form an increasing arithmetic progression is $441,$ then the sum of the divisors of $n$ is equal to

The set of all real values of ${}^{\prime }{a}^{\prime }$ so that the range of function $y=\frac{x^2+a}{x+1},x\in R-\{-1\}$ is $R$, is

If ${}^n{C}_r+{}^n{C}_{r+1}={}^{n+1}{C}_x,$ then x =

Among the following, how many are incorrect with respect to enthalpy of formation?

$\Delta H_f^0\left(\text{C, graphite}\right)=0\Delta H_f^0\left(B{r}_2\text{liquid}\right)=0\Delta H_f^0\left(\text{S, rhombic}\right)\ne 0\Delta H_f^0\left(\text{P, white}\right)=0\Delta H_f^0\left(\text{C, diamond}\right)\ne 0\Delta H_f^0\left(\text{S, monoclinic}\right)=0$
$\Delta H_f^0\left(\text{P, black}\right)=0$___

A point from a vector starts is called ...... and where it ends is called its ......

The value of cosx in the second quadrant

If $f\left(x\right)=3x^2-5x-1$ and $(f\circ g)\left(x\right)=3x^2+7x+1,$ then which of the following option is $\text{INCORRECT}?$

Find inverse by row transformations
A=[1 2 5
2 3 1
-1 1 1]

The equation of the parabola whose vertex and focus lie on the $x-$axis at distances $a$ and $a_1$ $(0<a<a_1)$ from the origin respectively, is

Given $L_1$ = x-2y+11 = 0 and $L_2$ = 3x+6y+5 = 0

The largest term in the expansion of $(3+2x{)}^{50}$ where $x=\frac{1}{5}$ is

If pth, qth and rth terms of an A.P. are in G.P., then the common ratio of this G.P. is

For the differential equation in given question find the general solution.​​​​

$\frac{dy}{dx}=(1+x^2)(1+y^2)$

${lim}_{x\to 0}\frac{sin{x}^0}{x}$ is equal to

For the differential equation in given question find the general solution.​​​​

$x^5\frac{dy}{dx}=-y^5$

Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect at point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at $(-\frac{1}{3\sqrt{2}},0)$ and $(0,\beta )$, then $\beta$ is equal to:

If the graph $y=g\left(x\right)$ has a minimum point at $(1,2),$ then minimum point of the graph $y=g(x-3)-4$ is

If the algebraic sum of the perpendiculars drawn from the points $(2,0),(0,2),(1,1)$ to a variable line is zero, then the line will always pass through a fixed point whose co-ordinates are

If $2x+y=p$ is a chord to the parabola $y^2=16x$ whose midpoint is $(h,k)$, then which of the following is/are true?

ABCD is a square whose side is a ; taking AB and AD as axes, prove that die equation of the circle circumscribing the square is $x^2+y^2-a(x+y)=0$.

The number of distinct normals that can be drawn from $(-2,1)$ to the parabola $y^2-4x-2y-3=0$, is

Find the values of $\theta$ and p, if the equation x $cos\theta +ysin\theta =p$ is the normal form of the line $\sqrt{3}x+y+2=0.$

A five digit number divisible by 30 is to be formed using the digits 0,1,2,3,4,5 without repetition of the digits. The number of ways it can be done is :

$(c^2-a^2+b^2)tanA=(a^2-b^2+c^2)tanB=(b^2-c^2+a^2)tanC$

Represent to solution set of each of the following in equations graphically in two dimensional plane :
$x\le 8-4y$

In a survey of 100 persons it was found that 28 read magazine A, 30 read magazine B, 42 read magazine C, 8 read magazines A and B, 10 read magazines A and C, 5 read magazines B and C and 3 read all the three magazines. Find :

(i) How many read none of three magazines ?

(ii) How many read magazine C only ?

$\frac{cos2A}{a^2}-\frac{cos2B}{b^2}=\frac{1}{a^2}-\frac{1}{b^2}$

If $f\left(x\right)=4x-x^2,xϵR$, then write the value of f(a + 1) - (a - 1).