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A variable straight line of slope $4$ intersects the hyperbola $xy=1$ at two points. The locus of the point which divides the line segment between these two points in the $1:2$ is

$\int e^x\left(\sin \left(x\right)+\cos \left(x\right)\right)dx$ is equal to -

Which among the following equations represents a pair of straight lines?

Three machines $E_1,E_2,E_3$ in a certain factory produced 50%, 25% and 25%, respectively, of the total daily output of electric tubes. It is known that 6% of the tubes produced on each of machines $E_1$ and $E_2$ is defective and that 5% of those produced on $E_3$ is defective. If one tube is picked up at random from a day’s production, calculate the probability that it is defective.

Solve $\sqrt{x-2}\ge -1$

One hundred identical coins, each with probability p of showing up heads are tossed once. If 0 < p < 1 and the probability of heads showing 50 coins is equal to that head showing 51 coins, then the value of p is

Evaluate the following limit:
$\underset{x\to 0}{\text{lim}}\frac{\text{cos 2x-1}}{\text{cos x-1}}$

Let $C$ be the circle with centre $(0,0)$ and radius $3$ units. Locus of the point $P$ from which the chord of contact subtends an angle of ${60}^{\circ }$ at any point on the circumference of $C$ is

In a Searle's experiment, the diameter of the wire as measured by a screw gauge with least count 0.001 cm is 0.050 cm. The length, measured by a scale of least count 0.1 cm, is 110.0 cm. When a weight of 50 N is suspended from the wire, the extension is measured to be 0.125 cm by a micrometer of least count 0.001 cm. Find the maximum error in the measurement of Young's modulus of the material of the wire from these data.

$y=\frac{W}{A}\times \frac{L}{X}$

If $lo{g}_{m^x}>lo{g}_{m^y}$ which of the following statements is / are true ?

1.$x>yifm>1$

2.$x<yifm>1$

3.$x>yif0<m<1$

4.$x<yif0<m<1$

Locus of the point P if $A{P}^2-B{P}^2=18$, where A ≡ (1, 2, –3) and B ≡ (3, –2, 1) is .

What is the period of f(x) = Sinx. Cos3x ?

Which of the following(s) is(are) correct in the interval of $x\in (0,\frac{\pi }{2})$

The value of $\underset{x\to 0}{lim}\frac{(1+x{)}^{1/x}-e+\frac{1}{2}ex}{x^2}$ is

The equation of the circle passing through (1, 1) and the points of intersection of $x^2+y^2+13x-3y=0$ and $2x^2+2y^2+4x-7y-25=0$ is

Match the domain of the following functions:

If $|x+3|$ = - x - 3 , then

Let $f$ and $g$ be continuous function on $[0,a]$ such that $f\left(x\right)=f(a-x)$ and $g\left(x\right)+g(a-x)=4,$ then $\underset{0}{\overset{a}{\int }}f\left(x\right)g\left(x\right)dx$ is equal to :

$\frac{d}{dx}\left[log|secx+tanx|\right]=$

If sin $\theta$ = $-\frac{1}{\sqrt{2}}$ and tan $\theta$ = 1, then $\theta$ lies in which quadrant.

If a rubber ball is taken at the depth of 200 m in a pool, its volume decreases by 0.1%. If the density of the water is $1\times {10}^{3}kg/m^3$ and $g=10m/s^2$, then the volume elasticity in $n/m^2$ will be

Graph of f(x) is given. Find the value of left hand limit as x approaches 3.

Solve the equation $co{s}^2\left[\frac{\pi }{4}\right(sinx+\sqrt{2}co{s}^{2}x\left)\right]-ta{n}^2(x+\frac{\pi }{4}ta{n}^{2}x)=1$

If $(1+2x+3x^2{)}^{10}=a_0+a_{1}x+a_2{x}^2+...+a_{20}{x}^{20},$ then $a_1$ equals

If $2x-y+1=0$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{16}=1$, then which of the following $CANNOT$ be sides of a right angled triangle?

The equation of a circle passing through points of intersection of the circles $x^2+y^2+13x-3y=0$ and $2x^2+2y^2+4x-7y-25=0$ and point (1, 1) is

Let $\bigcup _{i=1}^{50}{X}_i=\bigcup _{i=1}^n{Y}_i=T,$ where each $X_i$ contains $10$ elements and each $Y_i$ contains $5$ elements. If each element of the set $T$ is an element of exactly $20$ of sets $X_i$'s and exactly $6$ of sets $Y_i$'s, then $n$ is equal to

How many values of $xϵ[0,2\pi ]$ satisfies the equation sin 2x + 5 sin x + 1 + 5 cos x = 0?

The coefficient of $x^n$ in the expansion of $(1-x)e^x$ is

Find the value of the expression

$3{\sec }^{-1}\left(2\right)-6{\sin }^{-1}\left(\frac{1}{2}\right).$

If $f\left(x\right)=max\{1+\sin x,1,1-\cos x\},\forall x\in [0,2\pi ]$ and $g\left(x\right)=max\{1,x-1\}\forall x\in R$, then

Equation of tangent drawn to circle $|z|=r$ at the point $A\left(z_0\right)$ is

Find the value of ${\sec }^{2}x$ - ${\text{cosec}}^{2}x$.

The distance of the point (-4,1,-3) from the y-z plane is ___ units.

If two roots of the equation $x^3-3x+2=0$ are same, then the roots will be

What is the angle between the tangents to the curve $y=x^2-5x+6$ at the points (2, 0) and (3, 0)

Find the coefficient of x in the expansion of $(1-3x+7x^2)(1-x{)}^{16}$

A point is on the x-axis. What are its y-coordinates and z-coordinates?

If sin x + cos x = t, then sin 3x - cos 3x is equal to

The combined equation of straight lines joining the origin to the points of intersection the line y=2x+1 with the curve $x^2+y^2+2xy+4=0$ is given by

Find the point to which the origin be shifted after a translation, so that the equation $x^2+y^2-4x-8y+3=0$will have no first degree terms.

If $P_1$ and $P_2$ are the perpendiculars from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ on its asymptotes, then :

If $p\Rightarrow (q\vee r)$ is false, then the truth values of p,q,r are respectively

The general solution of the inequality $-9\le 1-2(x+3)<1$ and $-5\le \frac{x-9}{2}\le 8$ is

Suppose $f$ and $g$ are differentiable functions on $(0,\infty )$ such that $f\text{'}\left(x\right)=-\frac{g\left(x\right)}{x}$ and $g\text{'}\left(x\right)=-\frac{f\left(x\right)}{x}$, for all $x>0$. Further, $f\left(1\right)=3$ and $g\left(1\right)=-1$.



$f\left(10\right)$ is equal to

If f : D $\to$R $f\left(x\right)=\frac{x^2+bx+c}{x^2+b_{1}x+c_1},where\alpha ,\beta$ are th roots of the equation $x^2+bx+c=0$ and ${\alpha }_1,{\beta }_1$ are the roots of $x^2+b_{1}x+c_1=0$. Now, answer the following question for f(x). A combination of graphical and analytical approach may be helpful in solving these problems. If ${\alpha }_{1}and{\beta }_1$ are real, then f(x) has vertical asymptote at $x=({\alpha }_1,{\beta }_1$).

If ${\alpha }_1<{\beta }_1<\alpha <\beta$, then