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1 Numerical
Let f(x)=3xf(x)=3-x. Find f2(7)f^{\circ 2}(7).
Correct Answer: 7
View Solution
First, f(7)=37=4\qquad f(7)=3-7=-4 Then, f2(7)=f(4)=3(4)=7\qquad f^{\circ 2}(7)=f(-4)=3-(-4)=7 Hence the answer is 7\boxed{7}.
2 Single Choice
Two elements are chosen uniformly from {1,2,3,4,5,6}\{1,2,3,4,5,6\}. What is the probability that both are even?
A
15\dfrac{1}{5}
B
13\dfrac{1}{3}
C
25\dfrac{2}{5}
D
12\dfrac{1}{2}
View Solution
There are 33 even numbers: 2,4,62,4,6. Favourable choices: (32)=3\qquad \binom{3}{2}=3 Total choices: (62)=15\qquad \binom{6}{2}=15 So the probability is 315=15\qquad \dfrac{3}{15}=\dfrac{1}{5} Hence the correct option is A\boxed{A}.
3 Single Choice
Which of the following functions from R\mathbb{R} to R\mathbb{R} is idempotent?
A
f(x)=x+1f(x)=x+1
B
f(x)=xf(x)=x
C
f(x)=2xf(x)=2x
D
f(x)=x2f(x)=x^2
View Solution
A function is idempotent if f(f(x))=f(x)\qquad f(f(x))=f(x) For f(x)=xf(x)=x, we have f(f(x))=x=f(x)\qquad f(f(x))=x=f(x) The others are not idempotent on all real numbers. Therefore the correct option is B\boxed{B}.
4 Numerical
Find the number of good paths from (0,0)(0,0) to (3,3)(3,3) that pass through (1,1)(1,1).
Correct Answer: 12
View Solution
From (0,0)(0,0) to (1,1)(1,1), the number of paths is (21)=2\qquad \binom{2}{1}=2 From (1,1)(1,1) to (3,3)(3,3), the number of paths is (42)=6\qquad \binom{4}{2}=6 Hence the required number is 26=12\qquad 2\cdot 6=12 Therefore the answer is 12\boxed{12}.
5 Numerical
If the real solutions of 1x1+1x+1=1\qquad \dfrac{1}{x-1}+\dfrac{1}{x+1}=1 are α\alpha and β\beta, then find α+β\alpha+\beta.
Correct Answer: 2
View Solution
We first note the restrictions: x1,1\qquad x \ne 1,-1 Now, 1x1+1x+1=1\qquad \dfrac{1}{x-1}+\dfrac{1}{x+1}=1 Taking LCM, (x+1)+(x1)(x1)(x+1)=1\qquad \dfrac{(x+1)+(x-1)}{(x-1)(x+1)}=1 So, 2xx21=1\qquad \dfrac{2x}{x^2-1}=1 Hence, 2x=x21\qquad 2x=x^2-1 x22x1=0\qquad x^2-2x-1=0 Let the roots be α\alpha and β\beta. For the quadratic x22x1=0\qquad x^2-2x-1=0, the sum of roots is α+β=(2)1=2\qquad \alpha+\beta = \dfrac{-(-2)}{1}=2 Also, neither root is 11 or 1-1, so both are valid. Therefore, the answer is 2\boxed{2}.

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

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