Question 1:
Match the given functions in Column A with the equations of their tangents at the origin $(0,0)$ in column B and the plotted graphs and the tangents in Column C, given in Table M2W2G1.
Function (Column A)
Tangent at $(0,0)$ (Column B)
Graph (Column C)
i)
$f(x)=x2^{x}$
a) $y=-4x$
1)
ii)
$f(x)=x(x-2)(x+2)$
b) $y=x$
2)
iii)
$f(x)=-x(x-2)(x+2)$
c) $y=4x$
3)
Options:
Option 1: ii) $\rightarrow$ a) $\rightarrow 1$
Option 2: i) $\rightarrow$ b) $\rightarrow 3$
Option 3: iii) $\rightarrow$ b) $\rightarrow 1$
Option 4: iii) $\rightarrow$ c) $\rightarrow 2$
Option 5: i) $\rightarrow$ a) $\rightarrow 1$
Solution:
i): $f(x)=x2^{x}$, $f’(0)=1$ → tangent: $y=x$ → matches with b) and graph 3 i) $\rightarrow$ b $$\rightarrow 3
ii): $f(x)=x(x-2)(x+2)$, $f’(0)=-4$ → tangent: $y=-4x$ → matches with a) and graph 1 ii) $\rightarrow$ a) $$\rightarrow 1
iii): $f(x)=-x(x-2)(x+2)$, $f’(0)=4$ → tangent: $y=4x$ → matches with c) and graph 2
iii) $\rightarrow$ c) $$\rightarrow 2
Question 2:
Consider the following two functions $f(x)$ and $g(x)$:
Options:
Option 1: $f(x)$ is discontinuous at both $x=0$ and $x=3$
Option 2: $f(x)$ is discontinuous only at $x=0$
Option 3: $f(x)$ is discontinuous only at $x=3$
Option 4: $g(x)$ is discontinuous at $x=2$
Option 5: $g(x)$ is discontinuous at $x=3$
Solution:
f(x): Continuous at $x=0$, discontinuous at $x=3$ → Option 3
g(x): Continuous at $x=2$, discontinuous at $x=3$ → Option 5
Question 3:
Consider the graphs given below (Curve 1, Curve 2, Curve 3, Curve 4).
Options:
Option 1: Curve 1 is both continuous and differentiable at the origin.
Option 2: Curve 2 is continuous but not differentiable at the origin.
Option 3: Curve 2 has derivative 0 at $x=0$.
Option 4: Curve 3 is continuous but not differentiable at the origin.
Option 5: Curve 4 is not differentiable anywhere.
Option 6: Curve 4 has derivative 0 at $x=0$.
Solution:
Curve 1: Continuous and differentiable at origin → Option 1
Curve 2: Continuous and differentiable at origin, derivative 0 → Option 2, 3 (but note: Option 2 is false as per solution, but solution says Curve 2 is differentiable at origin with derivative 0. Likely only Option 3 is correct, but solution text is ambiguous. Most likely: Option 3.)
Curve 3: Continuous but not differentiable at origin → Option 4
Curve 4: Differentiable at $x=1$, not everywhere, but not differentiable at all points is false; option 5 is false. Option 6 is also false.
Clarification: Based on solution, only Option 1 and Option 4 are correct, but solution text is unclear about Curve 2; likely Option 3 is also correct, but solution says Curve 2 is differentiable. User should verify with figure.However, as per solution: Curve 2 is differentiable at origin with derivative 0, so Option 3 is correct. Option 2 is incorrect (since Curve 2 is differentiable at origin). Option 4 is correct.
Options:
Option 1: $f(x)$ is not continuous at $x=0$
Option 2: $f(x)$ is continuous at $x=0$
Option 3: $f(x)$ is not differentiable at $x=0$
Option 4: $f(x)$ is differentiable at $x=0$
Option 5: The derivative of $f(x)$ at $x=0$ (if exists) is 0
Option 6: The derivative of $f(x)$ at $x=0$ (if exists) is 1
Solution:
f(x): Continuous at $x=0$ → Option 2
Differentiable at $x=0$, derivative is 0 → Option 4, 5
Question 6:
Let $f$ be a differentiable function at $x=3$. The tangent line to the graph of the function $f$ at the point $(3,0)$, passes through the point $(5,4)$. What will be the value of $f’(3)$?
Solution:
Slope of tangent = $\frac{4-0}{5-3}=2$.
Answer: $f’(3)=2$
Question 7:
Let $f$ and $g$ be two functions which are differentiable at each $x \in \mathbb{R}$. Suppose that, $f(x)=g(x^{2}+5x)$, and $f’(0)=10$. Find the value of $g’(0)$.
Solution:
$f’(x)=g’(x^{2}+5x)(2x+5)$
At $x=0$, $f’(0)=g’(0)\times5=10$
Answer: $g’(0)=2$
3. Comprehension Type Questions
Question 8:
Consider the following statements:
Statement P: Both the functions $p(t)$ and $q(t)$ are continuous.
Statement Q: $p(t)$ is continuous, but $q(t)$ is not.
Statement R: $q(t)$ is continuous, but $p(t)$ is not.
Statement S: Neither $p(t)$ nor $q(t)$ is continuous.
q(t): Not continuous at $t=2$ (left limit $e^5$, right limit $e^4$)
Only Statement Q is correct.Answer: 1
Question 9:
If $L_p(t)=At+B$ denotes the best linear approximation of the function $p(t)$ at the point $t=1$, then find the value of $2A+B$.
Solution:
$p(t)=\frac{t^{3}-27}{t-3}$ for $0 \leq t < 3$
$p(1)=13$, $p’(1)=5$
Linear approximation: $L_p(t)=5t+8$
So, $A=5$, $B=8$
Answer: $2A+B=18$
Question 10:
If $L_q(t)=e^{4}(At+B)+Ce^{5}$ denotes the best linear approximation of the function $q(t)$ at the point $t=3$, then find the value of $A+B+C$.
Solution:
$q(3)=e^{5}-e^{4}$, $q’(3)=e^{4}$
Linear approximation: $L_q(t)=e^{4}(t-4)+e^{5}$
So, $A=1$, $B=-4$, $C=1$
Answer: $A+B+C=-2$
Question 11:
Consider a function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as:
$$
f(x)= \begin{cases}
\frac{\sin 14x + A \sin x}{19x^{3}} & \text{if } x \neq 0 \\
B & \text{if } x=0
\end{cases}
$$
If $f(x)$ is continuous at $x=0$, then find the value of $114B-A$.
Solution:
For continuity, $A=-14$
$B=\frac{-2730}{114}$
Answer: $114B-A=-2716$
Question 12:
The distance (in meters) traveled by a car after $t$ minutes is given by the function $d(t)=g(4t^{3}+2t^{2}+5t+2)$, where $g$ is a differentiable function with domain $\mathbb{R}$. Find the instantaneous speed of the car after 5 min, where $g’(577)=2$.
Solution:
$d’(t)=g’(4t^{3}+2t^{2}+5t+2)(12t^{2}+4t+5)$
At $t=5$, $d’(5)=g’(577)\times325=2\times325=650$
Answer: 650
Statements:
P: Both $p(t)$ and $q(t)$ are continuous
Q: Both are not differentiable
R: $p(t)$ is continuous, $q(t)$ is differentiable
S: $q(t)$ is continuous, $p(t)$ is not differentiable
T: Neither is continuous
Find the number of correct statements.
Solution:
p(t): Not continuous at $t=2$
q(t): Continuous, but not differentiable at $x=0,7,8$
Only S is correct.Answer: 1
Question 14:
If linear function $L_p(t)=At+B$ denotes the best linear approximation of the function $p(t)$ at the point $t=1$, find the value of $\frac{-2}{e^{-1}-1}(A+B)$.
