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CMI BS Hons Mock Tests

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1 Numerical

Let

f(x)=3-x

. Find

f^{\circ 2}(7)

.

Correct Answer: 7

View Solution

First,

\qquad f(7)=3-7=-4

Then,

\qquad f^{\circ 2}(7)=f(-4)=3-(-4)=7

Hence the answer is

\boxed{7}

.

2 Single Choice

Two elements are chosen uniformly from

\{1,2,3,4,5,6\}

. What is the probability that both are even?

A

\dfrac{1}{5}

B

\dfrac{1}{3}

C

\dfrac{2}{5}

D

\dfrac{1}{2}

View Solution

There are

3

even numbers:

2,4,6

. Favourable choices:

\qquad \binom{3}{2}=3

Total choices:

\qquad \binom{6}{2}=15

So the probability is

\qquad \dfrac{3}{15}=\dfrac{1}{5}

Hence the correct option is

\boxed{A}

.

3 Single Choice

Which of the following functions from

\mathbb{R}

to

\mathbb{R}

is idempotent?

A

f(x)=x+1

B

f(x)=x

C

f(x)=2x

D

f(x)=x^2

View Solution

A function is idempotent if

\qquad f(f(x))=f(x)

For

f(x)=x

, we have

\qquad f(f(x))=x=f(x)

The others are not idempotent on all real numbers. Therefore the correct option is

\boxed{B}

.

4 Numerical

Find the number of good paths from

(0,0)

to

(3,3)

that pass through

(1,1)

.

Correct Answer: 12

View Solution

From

(0,0)

to

(1,1)

, the number of paths is

\qquad \binom{2}{1}=2

From

(1,1)

to

(3,3)

, the number of paths is

\qquad \binom{4}{2}=6

Hence the required number is

\qquad 2\cdot 6=12

Therefore the answer is

\boxed{12}

.

5 Numerical

If the real solutions of

\qquad \dfrac{1}{x-1}+\dfrac{1}{x+1}=1

are

\alpha

and

\beta

, then find

\alpha+\beta

.

Correct Answer: 2

View Solution

We first note the restrictions:

\qquad x \ne 1,-1

Now,

\qquad \dfrac{1}{x-1}+\dfrac{1}{x+1}=1

Taking LCM,

\qquad \dfrac{(x+1)+(x-1)}{(x-1)(x+1)}=1

So,

\qquad \dfrac{2x}{x^2-1}=1

Hence,

\qquad 2x=x^2-1

\qquad x^2-2x-1=0

Let the roots be

\alpha

and

\beta

. For the quadratic

\qquad x^2-2x-1=0

, the sum of roots is

\qquad \alpha+\beta = \dfrac{-(-2)}{1}=2

Also, neither root is

1

or

-1

, so both are valid. Therefore, the answer is

\boxed{2}

.

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Frequently Asked Questions

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We offer 10 full-length mock tests with 226+ questions, designed to simulate the actual CMI exam experience.

Is the first mock test free?

Yes! Mock Test 1 is completely FREE. You can attempt it with full features including detailed solutions and performance analytics.

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