VITEEE PYQs for Differential equations with Solutions: Practice VITEEE Previous Year Questions

1.

If \[ \sin x + \cos x = \frac{1}{5} \] then \( \tan 2x \) is: 

• \( \frac{25}{17} \)
  
• \( \frac{7}{25} \)
  
• \( \sqrt{\frac{25}{7}} \)
  
• \( \frac{24}{7} \)
  

2.

Negation of the Boolean expression \[ p \Leftrightarrow (q \Rightarrow p) \] is:

• \( \sim p \wedge q \)
  
• \( p \wedge \sim q \)
  
• \( \sim p \vee \sim q \)
  
• \( \sim p \wedge \sim q \)
  

3.

The function \( f(x) \) is given by:

\[ f(x) = \begin{cases} x \sin \left( \frac{1}{x} \right), & \text{for } x \neq 0 \\ 0, & \text{for } x = 0 \end{cases} \]

• continuous as well as differentiable
  
• differentiable but not continuous
  
• continuous but not differentiable
  
• neither continuous nor differentiable
  

4.

The length of the perpendicular from the point \( (1, -2, 5) \) on the line passing through \( (1, 2, 4) \) and parallel to the line given by \( x + y - z = 0 \) and \( x - 2y + 3z - 5 = 0 \) is:

• \( \frac{\sqrt{21}}{2} \)
  
• \( \frac{\sqrt{9}}{2} \)
  
• \( \frac{\sqrt{73}}{2} \)
  
• 1

5.

In $\triangle ABC$, the midpoints of the sides $AB$, $BC$, and $CA$ are respectively $(1, 0, 0)$, $(0, m, 0)$, and $(0, 0, n)$. Then, $$ \frac{AB^2 + BC^2 + CA^2}{1^2 + m^2 + n^2} \text{ is equal to:} $$ 

• 8
  
• 16
  
• 9
  
• 25
  

6.

The coordinates of the point which divides the line segment joining the points \( (2, -1, 3) \) and \( (4, 3, 1) \) internally in the ratio \( 3:4 \) are:

• \( \left( \frac{2}{7}, \frac{20}{7}, \frac{10}{7} \right) \)
  
• \( \left( \frac{10}{7}, \frac{15}{7}, \frac{2}{7} \right) \)
  
• \( \left( \frac{20}{7}, \frac{5}{7}, \frac{15}{7} \right) \)
  
• \( \left( \frac{15}{7}, \frac{20}{7}, \frac{3}{7} \right) \)

7.

The area of the region bounded by the ellipse $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$ is: 

• \( 12\pi \)
  
• \( 3\pi \)
  
• \( 24\pi \)
  
• \( \pi \)
  

8.

The number of distinct real roots of the equation: \[ x^7 - 7x - 2 = 0 \] is:

• 5
  
• 7
  
• 1
  
• 3
  

9.

The minimum value of the function \[ y = x^4 - 2x^2 + 1 \] in the interval \( \left[\frac{1}{2}, 2 \right] \) is:

• \( 0 \)
  
• \( 2 \)
  
• \( 8 \)
  
• \( 9 \)
  

10.

Evaluate: \[ \tan (\cos^{-1} \frac{4}{5}) + \tan^{-1} \frac{2}{3} \]

• \( \frac{6}{17} \)
  
• \( \frac{7}{16} \)
  
• \( \frac{16}{7} \)
  
• None of these
  

11.

If \[ R = \{ (x, y) : x \text{ is exactly } 7\text{cm taller than } y \} \] then \( R \) is:

• Not symmetric
  
• Reflexive
  
• Symmetric but not transitive
  
• An equivalence relation
  

12.

If \[ \frac{a^n + b^n}{a^{n-1} + b^{n-1}} \] is the arithmetic mean (A.M.) between \( a \) and \( b \), then the value of \( n \) is:

• 1
  
• 2
  
• 3
  
• 4
  

13.

If 

then \( {Adj} (A) \) is equal to:

14.

The point \( (t^2 + 2t + 5, 2t^2 + t - 2) \) { lies on the line} \( x + y = 2 \) { for:}

• All real values of \( t \)
  
• Some real values of \( t \)
  
• \( t = -3 \pm \frac{\sqrt{3}}{6} \)
  
• None of these

15.

The integral \[ \int x^n (1 + \log x) \, dx \] is equal to: 

• \( x^n + C \)
  
• \( x^{2x} + C \)
  
• \( x^n \log x + C \)
  
• \( \frac{1}{2}(1 + \log x)^2 + C \)