NIMCET Mathematics Previous Year Questions (PYQs) – Page 3 of 95
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Solution
Probability — No Black Ball is Chosen
Given:
- Yellow balls = 5
- Black balls = 4
- Green balls = 3
- Total balls = 5 + 4 + 3 = 12
We are to find:
Probability that no black ball is selected when 3 balls are drawn at random.
Step 1: Total number of ways to choose any 3 balls from 12:
\[ \text{Total ways} = \binom{12}{3} = 220 \]
Step 2: Ways to choose 3 balls such that no black ball is chosen:
Only yellow and green balls are allowed ⇒ Total = 5 (yellow) + 3 (green) = 8
\[
\text{Favorable ways} = \binom{8}{3} = 56
\]
Step 3: Probability
\[ P(\text{no black ball}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{56}{220} = \frac{14}{55} \]
\[ \boxed{\text{Probability} = \frac{14}{55}} \]
Solution
Number of Roots
Given:
\( e^x \sin x = 1 \) has two real roots → say \( x_1 \) and \( x_2 \)
Apply Rolle’s Theorem:
Since \( f(x) = e^x \sin x \) is continuous and differentiable, and \( f(x_1) = f(x_2) \), ⇒ There exists \( c \in (x_1, x_2) \) such that \( f'(c) = 0 \)
Compute:
\[ f'(x) = e^x(\sin x + \cos x) = 0 \Rightarrow \tan x = -1 \] At this point, \[ e^x \cos x = -1 \]
\[ \boxed{\text{At least one root}} \]
Solution
Correct Shortcut Method — Find \( f(5) \)
Step 1: Define a helper polynomial:
\[ g(x) = f(x) - (x + 1) \]
Given: \( f(1) = 2, f(2) = 3, f(3) = 4, f(4) = 5 \Rightarrow g(1) = g(2) = g(3) = g(4) = 0 \)
So, \[ g(x) = A(x - 1)(x - 2)(x - 3)(x - 4) \quad \Rightarrow \quad f(x) = A(x - 1)(x - 2)(x - 3)(x - 4) + (x + 1) \]
Step 2: Use \( f(0) = 25 \) to find A:
\[ f(0) = A(-1)(-2)(-3)(-4) + (0 + 1) = 24A + 1 = 25 \Rightarrow A = 1 \]
Step 3: Compute \( f(5) \):
\[ f(5) = (5 - 1)(5 - 2)(5 - 3)(5 - 4) + (5 + 1) = 4 \cdot 3 \cdot 2 \cdot 1 + 6 = 24 + 6 = \boxed{30} \]
✅ Final Answer: \( \boxed{f(5) = 30} \)
Solution
Maximum Value of \( f(x) = (x - 1)^2(x + 1)^3 \)
Step 1: Let’s define the function:
\[ f(x) = (x - 1)^2 (x + 1)^3 \]
Step 2: Take derivative to find critical points
Use product rule:
Let \( u = (x - 1)^2 \), \( v = (x + 1)^3 \)
\[
f'(x) = u'v + uv' = 2(x - 1)(x + 1)^3 + (x - 1)^2 \cdot 3(x + 1)^2
\]
\[
f'(x) = (x - 1)(x + 1)^2 [2(x + 1) + 3(x - 1)]
\]
\[
f'(x) = (x - 1)(x + 1)^2 (5x - 1)
\]
Step 3: Find critical points
Set \( f'(x) = 0 \): \[ (x - 1)(x + 1)^2 (5x - 1) = 0 \Rightarrow x = 1,\ -1,\ \frac{1}{5} \]
Step 4: Evaluate \( f(x) \) at these points
- \( f(1) = 0 \)
- \( f(-1) = 0 \)
- \( f\left(\frac{1}{5}\right) = \left(\frac{1}{5} - 1\right)^2 \left(\frac{1}{5} + 1\right)^3 = \left(-\frac{4}{5}\right)^2 \left(\frac{6}{5}\right)^3 \)
\[ f\left(\frac{1}{5}\right) = \frac{16}{25} \cdot \frac{216}{125} = \frac{3456}{3125} \]
Step 5: Compare with given form:
It is given that maximum value is \( \frac{3456}{3125} = 2^p \cdot 3^q / 3125 \)
Factor 3456: \[ 3456 = 2^7 \cdot 3^3 \Rightarrow \text{So } p = 7, \quad q = 3 \]
✅ Final Answer: \( \boxed{(p, q) = (7,\ 3)} \)
Solution
Easiest Method — Coefficient of \( x^{50} \)
Given Expression:
\[ (1 + x)^{1000} + 2x(1 + x)^{999} + 3x^2(1 + x)^{998} + \cdots + 1001x^{1000} \]
This follows a known identity that simplifies the full expression to:
\[ f(x) = (1 + x)^{1002} \]
Now: The coefficient of \( x^{50} \) in \( f(x) \) is:
\[ \boxed{\binom{1002}{50}} \]
✅ Final Answer: \( \boxed{\binom{1002}{50}} \)
Solution
Sum of Complex Roots of Unity
Given:
\[ x_k = \cos\left(\frac{2\pi k}{n}\right) + i \sin\left(\frac{2\pi k}{n}\right) = e^{2\pi i k/n} \]
Required: Find: \[ \sum_{k=1}^{n} x_k \]
This is the sum of all \( n^\text{th} \) roots of unity (from \( k = 1 \) to \( n \)).
We know: \[ \sum_{k=0}^{n-1} e^{2\pi i k/n} = 0 \] So shifting index from \( k = 1 \) to \( n \) just cycles the same roots: \[ \sum_{k=1}^{n} e^{2\pi i k/n} = 0 \]
✅ Final Answer: \( \boxed{0} \)
Solution
Points of Non-Differentiability of \( f(x) = |\cos x| + 3 \)
Step 1: \( \cos x \) is differentiable everywhere, but \( |\cos x| \) is not differentiable where \( \cos x = 0 \).
Step 2: In the interval \( [-\pi, \pi] \), we have:
\[ \cos x = 0 \Rightarrow x = -\frac{\pi}{2},\ \frac{\pi}{2} \]
So \( f(x) = |\cos x| + 3 \) is not differentiable at these two points due to sharp turns.
✅ Final Answer: \( \boxed{2 \text{ points}} \)
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