Test Quiz 1

 2 mark

| -0.25 mark |

 1 minutes

00:00:00

Question 1:

If \( f(x)=(x+1)^{\cot x} \) is continuous at \( x =0 \), then what is \( f(0) \) equal to?

Question 2:

Let \[
f(x)=
\begin{cases}
3x-4, & 0 \le x \le 2 \\
2x+l, & x>2
\end{cases}
\]

 

 

if \( f\) is continuous at \(x=2\), then what is the value of \(l\)?

Question 3:

If the function \( f(x)=\frac{x(x-2)}{x^{2}-4},\;x\neq\pm2 \) is continuous at \( x=2 \), then what is \( f(2) \) equal to?

Question 4:

Which one of the following is correct in respect of the function \( f(x)=\frac{x^{2}}{|x|} \) for \( x \neq 0 \) and \( f(0)=0 \).

Question 5:

Directions (5) : Given a function
\[
f(x)=\left\{\begin{array}{clc}
-1 & \text{ if } & x \leq 0 \\
a x + b & \text{ if } & 01 & \text{ if } & x \geq 1\end{array}\right.\]
where \(a, b\) are constants. The function is continuous everywhere.
What is the value of \(a\)?

Question 6:

Consider the function
\[
\left\{\begin{array}{l}
\frac{\tan k x}{x}, x<0
3 x+2 k^{2}, x \geq 0
\end{array} .\right.
\]

What is the non-zero value of \( k \) for which the function is continuous at \( x=0 \) ?

Question 7:

Directions (7) : Given a function
\[
f(x)=\left\{\begin{array}{clc}
-1 & \text{ if } & x \leq 0 \\
a x + b & \text{ if } & 01 & \text{ if } & x \geq 1\end{array}\right.\]
where \(a, b\) are constants. The function is continuous everywhere

What is the value of \( b \)?

Question 8:

Directions (8-9) : Consider the function
\[
f(x)=\left\{\begin{array}{ccc}
\frac{\alpha \cos x}{\pi-2 x} & \text { if } & x \neq \frac{\pi}{2} \\
3 & \text { if } & x=\frac{\pi}{2}
\end{array}\right.
\]
which is continuous at \( x=\frac{\pi}{2} \), where \( \alpha \) is a constant.
What is the value of \( \alpha \)?

Question 9:

Directions (8-9) : Consider the function
\[
f(x)=\left\{\begin{array}{ccc}
\frac{\alpha \cos x}{\pi-2 x} & \text { if } & x \neq \frac{\pi}{2} \\
3 & \text { if } & x=\frac{\pi}{2}
\end{array}\right.
\]
which is continuous at \( x=\frac{\pi}{2} \), where \( \alpha \) is a constant.

What is \( \lim_{x \rightarrow 0} f(x) \) equal to?

Question 10:

Directions (10-11) : Consider the function
\[
f(x)=\left\{\begin{array}{cc}
-2 \sin x & \text{ if } x \leq -\frac{\pi}{2} \\
A \sin x + B & \text{ if } -\frac{\pi}{2} < x < \frac{\pi}{2} \\
\cos x & \text{ if } x \geq \frac{\pi}{2}\end{array}\right.\]
which is continuous everywhere.
The value of A is-

Question 11:

Directions (10-11) : Consider the function
\[
f(x)=\left\{\begin{array}{cc}
-2 \sin x & \text{ if } x \leq -\frac{\pi}{2} \\
A \sin x + B & \text{ if } -\frac{\pi}{2} < x < \frac{\pi}{2} \\
\cos x & \text{ if } x \geq \frac{\pi}{2}\end{array}\right.\]
which is continuous everywhere.

The value of \( B \) is-

Question 12:

Consider the function
\[
f(x)=\begin{cases} a x - 2, & \text{for } -2 < x < -1 \\
-1, & \text{for } -1 \le x \le 1 \\
a + 2(x-1)^2, & \text{for } 1 < x < 2 \end{cases}
\]
What is the value of \( a \) for which \( f(x) \) is continuous at \( x=-1 \) and \( x=1 \)?

Question 13:

If the function \( f(x)=\frac{2 x-\sin^{-1} x}{2 x+\tan^{-1} x} \) is continuous at each point in its domain, then what is the value of \( f(0) \)?

Question 14:

The value of \( k \) which makes
\[
f(x)=\left\{\begin{array}{rr}
\sin x, & x \neq 0 \\
k, & x=0
\end{array}\right.
\]
continuous at \( x=0 \) is

Question 15:

For what value of k is the function \(f(x)=\begin{cases}2x+\frac{1}{4}, & x<0\\k, & x=0\\(x+\frac{1}{x})^{2}, & x>0\end{cases}\) continuous at \(x=0\)?

Question 16:

If the function \( f(x)=\left\{\begin{array}{cr}a+b x, & x<1\\5, & x=1\\b - a x, & x>1\end{array}\right. \) is continuous, then what is the value of \(a+b\)?

Question 17:

Let \[
f(x)=
\begin{cases}
1+\frac{x}{2k}, & 0 \le x \le k \\
\text{(next expression)}, & x>k
\end{cases}
\]
\[
\text{If } \lim_{x \to 2} f(x) \text{ exists, then what is the value of } k\text{?}
\]

Question 18:

What is the value of \( q \)?

Question 19:

Let \( f(x) \) be defined as follows
\[
f(x)=\left\{\begin{array}{cc}
2 x+1, & -3x-1, & -2 \leq x<0 \\\
x+2, & 0 \leq x<1
\end{array}\right.
\]
Which one of the following statements is correct in respect of the above function?

Question 20:

If \( f(x)=\frac{x^{2}-9}{x^{2}-2 x-3}, x \neq 3 \) is continuous at \( x=3 \), then which one of the following is correct?

Question 21:

Directions (22-23): Let
\[
f(x)=\left\{\begin{array}{ll}
\frac{x-3}{|x-3|}+a,&x<3\\
b,&x=3\\
\frac{x-3}{|x-3|}+b,&x>3
\end{array}\right. \]
and \(f(x)\) be continuous at \(x+3\).
What is the value of \(a\)?

Question 22:

Directions (22-23): Let
\[
f(x)=\left\{\begin{array}{ll}
\frac{x-3}{|x-3|}+a,&x<3\\
b,&x=3\\
\frac{x-3}{|x-3|}+b,&x>3
\end{array}\right. \]
and \(f(x)\) be continuous at \(x+3\)

What is the value of \( b \)?

Question 23:

The function \(f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\) is not defined at \(x=\pi\). The value of \(f(\pi)\) so that \(f(x)\) is continuous at \(x=\pi\) is:

Question 24:

Consider the following
I. \( \lim _{x \rightarrow 0} \frac{1}{x} \) exists
II. \( \lim _{x \rightarrow 0} e^{1 / x} \) does not exist
Which of the above is/are correct
serach-cds
x