Secant of a Curve y =f(x)

Q. Let $S_1$ and $S_2$ be two circles touching externally and having radius as $2$ and $3$ respectively. $S_1$ and $S_2$ touch a variable circle $S_3$ internally at $A$ and $B$ respectively. If the tangents to $S_3$ at $A$ and $B$ meet at $T$ and $TA=4\text{units}$, then which of the following is/are correct?
(Here, $C_i$ represents the centre of circle $S_i$)

1. The radius of circle $S_3$ is $8$ units.
2. The area of circle circumscribing $△TAB$ is $20\pi$ sq. units.
3. $C_3{C}_1+C_3{C}_2=5$
4. $C_3{C}_1–C_3{C}_2=1$

Q.

The line $x+y=2$ is tangent to the curve $x^2=3-2y$ at its point

1. $\left(1,1\right)$

2. $\left(-1,1\right)$

3. $\left(\sqrt{3},0\right)$

4. $\left(3,-3\right)$

Q.

If the normal to the curve $y=f\left(x\right)$ at the point $(3,4)$ make an angle $\frac{3\mathrm{\pi }}{4}$​ with the positive $x-$axis, then $f\prime \left(3\right)$ is

1. $-1$

2. $\frac{-3}{4}$

3. $\frac{4}{3}$

4. $1$

Q.

If the slope of the tangent to the curve $y=x^3$ at a point is equal to the ordinate of a point, then the point is:

1. $\left(27,3\right)$

2. $\left(3,27\right)$

3. $\left(1,2\right)$

4. $\left(-1,3\right)$

Q.

The equation of the tangent to the curve $x^2-2xy+y^2+2x+y–6=0$ at $(2,2)$ is

1. $2x+y–6=0$

2. $x+3y–8=0$

3. $3x+y–8=0$

4. $x+y–4=0$

Q.

The focus and directrix of a parabola are (1, 2) and 2x-3y+1=0. Then the equation of the tangent at vertex is

1. 4x-6y+5=0

2. 4x-6y+9=0

3. 4x-6y+11=0

4. 4x-6y+7=0

Q.

The slope of the tangent to the curve $x=3t^2+1$, $y=t^3-1$ at $x=1$ is

1. $0$

2. $\frac{1}{2}$

3. $\infty$

4. $-2$

Q.

The equations of the lines through the point $\left(3,2\right)$ which makes an angle of $45°$ with the line $x-2y=3$ are

1. $3x-y=7\&x+3y=9$

2. $x-3y=7\&3x+y=9$

3. $x-y=3\&x+y=2$

4. $2x+y=7\&x-2y=9$

Q.

The acute angle between the line x + y = 3 and the line joining the points (1, 1) and (–3, 4) is

1. 

2. 

3. 

4. 

Q. A variable plane intersects the coordinate axes at $A,B,C$ and is at a constant distance $p$ from $O(0,0,0).$ Then the locus of the centroid of the tetrahedron $OABC$ is

1. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{1}{p^2}$
2. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{4}{p^2}$
3. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=\frac{16}{p^2}$
4. $\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=16p^2$

Q.

The point on the curve$x^2+y^2=a^2,y\ge 0$ at which the tangent is parallel to the $x$-axis is

1. $(a,0)$

2. $(-a,0)$

3. $\left(\frac{a}{2},\sqrt{\frac{3}{2}}a\right)$

4. $(0,a)$

5. $(0,a^2)$

Q.

If the straight line x cos $\alpha$ + y sin $\alpha$ = p touches the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then prove that $a^{2}co{s}^2\alpha +b^{2}si{n}^2\alpha =p^2$.

Q. The equation of the curve passing through the point (1, 1) such that the slope of the tangent at any point (x, y) is equal to the product of its co-ordinates is

1. $2\log y=x^2-1$
2. $2\log x=y^2-1$
3. $2\log x=y^2+1$
4. $2\log y=x^2+1$

Q.

If $\mathrm{sin\theta }=\frac{\mathrm{a}}{\mathrm{b}}$then $\mathrm{cos\theta }$ is equal to.

1. $\frac{\mathrm{b}}{\sqrt{{\mathrm{b}}^2-{\mathrm{a}}^2}}$

2. $\frac{\mathrm{b}}{\mathrm{a}}$

3. $\frac{\sqrt{{\mathrm{b}}^2-{\mathrm{a}}^2}}{\mathrm{b}}$

4. $\frac{\mathrm{a}}{\sqrt{{\mathrm{b}}^2-{\mathrm{a}}^2}}$

Q.

Find the equation of a plane which is at a distance $3\sqrt{3}$ units from origin and the normal to which is equally inclined to coordinate axis.

Q.

If a straight line is tangent to one point and normal to another point on the curve $y=8t^3-1,x=4t^2+3$, then the equation of such line is

1. $x-y=\sqrt{2}$

2. $\sqrt{2}x-y=\left[\frac{\left(89\sqrt{2}\right)}{27}\right]+1$

3. $x-y=89\sqrt{2}$

4. None of these

Q.

A point on the curve $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ is

1. None of these

2. 

3. 

4. 

Q.

At each point $(x,y)$of a curve, the intercept of the tangent on the$y-axis$ is equal to $2x{y}^2$. Find the equation of the curve.

1. $\frac{x}{y}=x^2+c$

2. $\frac{y}{x}=y^2+c$

3. $\frac{x}{y}=y^2+c$

4. $\frac{y}{x}=x^2+c$

Q. Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=5\hat{i}+\hat{j}-\hat{k}$ be three vectors. The area of the region formed by the set of points whose position vector $\overrightarrow{r}$ satisfy the equations $\overrightarrow{r}.\overrightarrow{a}=5$ and $|\overrightarrow{r}-\overrightarrow{b}|+|\overrightarrow{r}-\overrightarrow{c}|=4$ is closest to the integer

1. $4$
2. $9$
3. $14$
4. $19$

Q. Find the equation of one of the sides of an isosceles right-angled triangle whose hypotenuse is given by $3x+4y=4$ and the opposite vertex of the hypotenuse is $(2,2)$.

Q. The slope of the tangent to the curve $y=3x^2+3\sin x$ at $x=0$ is

1. $3$
2. $2$
3. $1$
4. $-1$

Q.

Equation of the tangent to $y^2=6x$ at the positive end of the latusrectum is

1. 2x-2y+3=0

2. 2x+2y+3=0

3. y=4

4. y+4=0

Q. Suppose the parabola $(y–k{)}^2=4(x–h)$, with vertex $A$, passes through $O=(0,0)$ and $L=(0,2)$. Let $D$ be an end point of the latus rectum. Let the $y$-axis intersect the axis of the parabola at $P$. Then $\angle PDA$ is equal to

1. ${\tan }^{-1}\frac{2}{19}$
2. ${\tan }^{-1}\frac{4}{19}$
3. ${\tan }^{-1}\frac{1}{19}$
4. ${\tan }^{-1}\frac{8}{19}$

Q. Which of the following is(are) correct for concavity of $f\left(x\right)=\sin x+\cos x$ on $[0,2\pi ]?$

1. Concave downward on $(0,\frac{3\pi }{4})$
2. Concave downward on $(\frac{7\pi }{4},2\pi )$
3. Concave upward on $(\frac{7\pi }{4},2\pi )$
4. Concave upward on $(\frac{3\pi }{4},\frac{7\pi }{4})$

Q. The area of the triangle formed by the tangents from the point $(3,2)$ to the hyperbola $x^2-9y^2=9$ and the chord of contact with respect to the point $(3,2)$ is

Q. Normals are drawn from the external point (2, 4) to the rectangular hyperbola xy=64. If circles are drawn through the feet of these normals taken three at a time then centre of circle lies on another hyperbola whose centre and recentricity is

1. (1, 2)
   
2. (2, 4)
   
3. $\sqrt{2}$
4. $\sqrt{2}+1$

Q. Find the slope of the tangent to the curve $y=\frac{x-1}{x-2},x\ne 2$ at $x=10$.

Q. If $x=\sqrt{2^{{\text{cosec}}^{-1}t}}$ and $y=\sqrt{2^{{\sec }^{-1}t}}$ $(|t|\ge 1),$ then $\frac{dy}{dx}$ is equal to

1. $\frac{y}{x}$
2. $-\frac{y}{x}$
3. $-\frac{x}{y}$
4. $\frac{x}{y}$

Q. The slope of the normal to the curve $y=x^2-\frac{1}{x^2}$ at $(-1,0)$ is

1. $-\frac{1}{4}$
2. $\frac{1}{4}$
3. $4$
4. $-4$
5. $0$

Q. The slope of the tangent to the curve $(y-x^5{)}^2=x(1+x^2{)}^2$ at the point $(1,3)$ is ..................