Tangent of a Curve y =f(x)

Q. If the line $3x-4y=0$ is tangent in the first quadrant to the curve $y=x^3+k,$ then value of $k$ is

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

Q. The tangent to the curve $y=x{e}^{x^2}$ is passing through the point $(1,e)$ and also passes through the point :

1. $(2,3e)$
2. $(3,6e)$
3. $(\frac{4}{3},2e)$
4. $(\frac{5}{3},2e)$

Q. If the curve $y=a{x}^2+bx+c,x\in R,$ passes through the point $(1,2)$ and the tangent line to this curve at origin is $y=x,$ then the possible values of $a,b,c$ are

1. $a=1,b=1,c=0$
2. $a=-1,b=1,c=1$
3. $a=1,b=0,c=1$
4. $a=\frac{1}{2},b=\frac{1}{2},c=1$

Q.

Tangents are drawn from the origin to the curve y = sin x. Their points of contact lie on the curve

1. $x^2{y}^2=x^2+y^2$

2. $x^2{y}^2=x^2-y^2$

3. $x^2{y}^2=y+x$

4. $XY=X+Y$

Q. If m is slope of common tangents y = $x^2$ - x + 1, y = $x^2$ - 3x + 1 then m is

1. 16
2. 7
3. 9
4. -2

Q. If the tangent to the curve $y=\frac{x}{x^2-3},x\in R,(x\ne \pm \sqrt{3},)$ at a point $(\alpha ,\beta )\ne (0,0)$ on it is parallel to the line $2x+6y-11=0,$ then :

1. $|6\alpha +2\beta |=19$
2. $|2\alpha +6\beta |=11$
3. $|6\alpha +2\beta |=9$
4. $|2\alpha +6\beta |=19$

Q.

If the slope of the curve $y=\frac{ax}{b-x}$ at the point (1, 1) is 2 then values of a and b respectively

1. 1, -2

2. -1, 2

3. 1, 2

4. 2, 7

Q. The tangent at a point $P$ on the curve $x^m{y}^n=a^{m+n}$ meets the $x$-axis at $A$ and $y$-axis at $B$. The triangle formed by $AOB$ has centroid at $Q$, If $O$ be the origin, then

1. $AP:PB=n:m$
2. $AP:PB=m:n$
3. locus of $Q$ is $\left(3xm{)}^m\right(3yn{)}^n=\left[a\right(m+n){]}^{m+n}$
4. locus of $Q$ is $\left(3xm{)}^n\right(3yn{)}^m=\left[a\right(m+n){]}^{m+n}$

Q. A curve is represented parametrically by $x=t+e^{at}$ and $y=-t+e^{at},t\in R$ and $a>0$. Then, the curve touches the $x-$ axis at

1. $(1,0)$
2. $(\frac{1}{e},0)$
3. $(e,0)$
4. $(2e,0)$

Q. At what point, the slope of the tangent to the curve $x^2+y^2-2x-3=0$ is zero :

1. (3, 0); (-1, 0)
2. (3, 0) ( 1, 2)
3. (-1, 0) (1, 2)
4. (1, 2) (1, -2)

Q.

If m be the slope of a tangent to the curve $e^{2y}=1+4x^2$, then

1. $m<1$

2. $|M|\le 1$

3. $|M|>1$

4. $m=2$

Q. For the curve y = 3sin $\theta cos\theta ,x=e^{\theta }sin\theta ,0\le \theta \le \pi$; tangent is parallel to x-axis, then $\theta$ =

1. \N
2. $\frac{\pi }{2}$
3. $\frac{\pi }{4}$
4. $\frac{\pi }{6}$

Q.

Number of possible tangents to the curve $y=cos(x+y),-3\pi \le x\le 3\pi$ that are parallel to the line x+2y = 0, is

1. 1

2. 2

3. 3

4. 4

Q.

The chord joining the points where x =p and x=q on the curve $y=a{x}^2+bx+c$ is parallel to the tangent at the point on the curve whose abscissa is

1. $\frac{p+q}{2}$

2. $\frac{p-q}{2}$

3. $\frac{pq}{2}$

4. $p+q$

Q.

The chord joining the points where x =p and x=q on the curve $y=a{x}^2+bx+c$ is parallel to the tangent at the point on the curve whose abscissa is

1. $\frac{p+q}{2}$

2. $\frac{p-q}{2}$

3. $\frac{pq}{2}$

4. $p+q$

Q.

If m be the slope of a tangent to the curve $e^{2y}=1+4x^2$, then

1. $m<1$

2. $|M|\le 1$

3. $|M|>1$

4. $m=2$

Q.

The angle between the tangents to the curve $y^2=2ax$ at the points where $x=\frac{a}{2}$, is

1. $\frac{\pi }{6}$

2. $\frac{\pi }{4}$

3. $\frac{\pi }{3}$

4. $\frac{\pi }{2}$

Q. Consider $f\left(x\right)=\frac{x^2}{1+x^2}$ and $g\left(x\right)=px$ where $p\in R.$ Then which of the following statements is (are) CORRECT?

1. If $f\left(x\right)=g\left(|x|\right)$ has exactly three distinct solutions, then number of all possible real values of $p$ is $1.$
2. If $f\left(x\right)=g\left(|x|\right)$ has exactly five distinct solutions, then range of all possible real values of $p$ is $(0,\frac{1}{2}).$
3. Number of distinct tangents that can be drawn to the curve $y=f\left(x\right)$ from the origin is $3.$
4. Number of distinct tangents that can be drawn to the curve $y=f\left(x\right)$ from the origin is $1.$

Q. $\begin{array}{cc}\text{Column I}& \text{Column II}\\ \text{a. The sides of a triangle vary slightly in such a way that its circum-radius}& \text{p.}1\\ \text{remains constant. If}\frac{da}{\cos A}+\frac{db}{\cos B}+\frac{dc}{\cos C}+1=|m|,\text{then the value}& \\ \text{of}m\text{is}& \\ \text{b. The length of sub-tangent to the curve}{x}^2{y}^2=16\text{at the point (-2, 2) is |k|.}& \text{q.}-1\\ \text{Then the value of k is}& \\ \text{c. The curve}y=2e^{2x}\text{intersects the y-axis at an angle}{\cot }^{-1}\left|\frac{8n-4}{3}\right|.& \text{r.}2\\ \text{Then the value of}n\text{is}& \\ \text{d. The area of a triangle formed by normal at the point (1, 0) on the curve}& \text{s.}-2\\ x=e^{\sin y}\text{with axes is}\frac{|2t+1|}{6}\text{sq. units. Then the value of}t\text{is}& \end{array}$

Which of the following is correct ?

1. $a\to p,q;b\to r,s;c\to q,r;d\to p,s$
2. $a\to p,r;b\to q,s;c\to q,r;d\to p,s$
3. $a\to p,q;b\to r,s;c\to q,r,s;d\to p,q$
4. $a\to p,q;b\to q,s;c\to p,r;d\to p,s$

Q. Consider $f\left(x\right)=\frac{x^2}{1+x^2}$ and $g\left(x\right)=px$ where $p\in R.$ Then which of the following statements is (are) CORRECT?

1. If $f\left(x\right)=g\left(|x|\right)$ has exactly three distinct solutions, then number of all possible real values of $p$ is $1.$
2. If $f\left(x\right)=g\left(|x|\right)$ has exactly five distinct solutions, then range of all possible real values of $p$ is $(0,\frac{1}{2}).$
3. Number of distinct tangents that can be drawn to the curve $y=f\left(x\right)$ from the origin is $3.$
4. Number of distinct tangents that can be drawn to the curve $y=f\left(x\right)$ from the origin is $1.$

Q.

The slope of the tangent to the curve $x=t^2+3t-8,y=2t^2-2t-5$ at the point t = 2 is

1. $\frac{7}{6}$

2. $\frac{5}{6}$

3. $\frac{6}{7}$

4. $1$

Q. If the tangent at any point $P(4m^2,8m^3)$ on the curve $x^3-y^2=0$ is a normal to itself at another point $Q,$ then the value(s) of $m$ is/are

1. $\sqrt{2}$
2. $-\sqrt{2}$
3. $\sqrt{\frac{2}{9}}$
4. $-\sqrt{\frac{2}{9}}$

Q. Tangent is drawn to a rectangular hyperbola $xy=c^2$.
The area of the triangle formed by tangents
and its intercepts is $72$ sq. units.

1. $c^2=36$
2. $xy=36$
3. Equation of tangent at $(6,6)$ is $x+y=6$
4. Equation of tangent at $(6,6)$ is $2x-y=6$

Q. The number of points in the rectangle $\left\{\right(x,y)|-12\le x\le 12\text{and}-3\le y\le 3\}$ which lie on the curve $y=x+\sin x$ and at which the tangent to the curve is parallel to the $x$ - axis is

1. $0$
2. $2$
3. $4$
4. $8$

Q.

A horse runs along a circle with a speed of 20 km/hr. A lantern is at the centre of the circle. A fence is along the tangent to the circle at the point at which the horse starts. The speed with which the shadow of the horse move along the fence at the moment when it covers $\frac{1}{8}$ of the circle in km/hr is

1. 20

2. 40

3. 30

4. 60

Q. A function $y=f\left(x\right)$ has a second-order derivative $f^{\prime \prime }\left(x\right)=6(x-1)$. If its graph passes through the point $(2,1)$ and at that point tangent to the graph is $y=3x-5$, then the value of $f\left(0\right)$ is

1. $1$
2. $-1$
3. $2$
4. $0$

Q. If m is slope of common tangents y = $x^2$ - x + 1, y = $x^2$ - 3x + 1 then m is

1. 16
2. 7
3. 9
4. -2

Q. Find the equation of tangent to curve $x^2-9y^2=9$ at the point whose eccentric angle is $\frac{\pi }{3}$ .

1. $2x-3\sqrt{3}y=3$
2. $x-\sqrt{3}y=6$
3. $x-3\sqrt{3}y=3$
4. $2x-3\sqrt{3}y=6$