Laws of Motion: Detailed Chapter-wise Tests with Solutions. Score High in JEE Main Physics

JEE Main Questions

• 1.

  The reaction $A(g) \rightleftharpoons B(g) + C(g)$ was initiated with the amount 'a' of $A(g)$. At equilibrium it is found that the amount of $A(g)$ remaining is $(a - x)$ at a total pressure of p.
  The equilibrium constant $K_p$ of the reaction can be calculated from the expression :

  • $\dfrac{x^2}{a^2 + x^2} \times p$
  • $\dfrac{x^2}{a^2 - x^2} \times p$
  • $\dfrac{a + x^2}{x^2} \times p$
  • $\dfrac{a^2 - x^2}{x^2} \times p$
  View Solution

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• 2.

  If $f: \mathbb{N} \to \mathbb{Z}$ is defined by \[ f(n) = \begin{vmatrix} n & -1 & -5 \\ -2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{vmatrix}, k \in \mathbb{N}, \] and $\sum_{n=1}^k f(n) = 98$, then $k$ is equal to :

  • 3
  • 4
  • 5
  • 6
  View Solution

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• 3.

  Let \[ A= \begin{bmatrix} 1 & 2\\ 1 & \alpha \end{bmatrix} \quad \text{and} \quad B= \begin{bmatrix} 3 & 3\\ \beta & 2 \end{bmatrix}. \] If \(A^2-4A+I=O\) and \(B^2-5B-6I=O\), then among the following statements: (S1): \[ [(B-A)(B+A)]^T= \begin{bmatrix} 13 & 15\\ 7 & 10 \end{bmatrix} \] (S2): \[ \det(\operatorname{adj}(A+B))=-5 \] Choose the correct option:

  • only (S1) is correct
    
  • only (S2) is correct
    
  • both (S1) and (S2) are correct
    
  • both (S1) and (S2) are wrong
  View Solution

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• 4.

  Consider the system of linear equations in x, y, z:
  x+2y+tz = 0,
  6x + y + 5t z = 0,
  3x + y + f(t) z = 0,
  where f: R$\rightarrow$ R is a differentiable function. If this system has infinitely many solutions for all t $\in$ R, then f

  • is a constant function
  • is strictly increasing on R
  • is strictly decreasing on R
  • has two critical points
    
  View Solution

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• 5.

  Consider the matrices} \[ A = \begin{bmatrix} 2 & -2 \\ 4 & -2 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & 9 \\ 1 & 3 \end{bmatrix}. \] If matrices \( P \) and \( Q \) are such that \( PA = B \) and \( AQ = B \), then the absolute value of the sum of the diagonal elements of \( 2(P + Q) \) is _______.}

  View Solution

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• 6.

  The shortest distance between the lines
  \(\vec{r} = (\frac{1}{3}\hat{i} + \frac{8}{3}\hat{j} - \frac{1}{3}\hat{k}) + \lambda(2\hat{i} - 5\hat{j} + 6\hat{k})\)
  and \(\vec{r} = (-\frac{2}{3}\hat{i} - \frac{1}{3}\hat{k}) + \mu(\hat{j} - \hat{k}), \lambda, \mu \in \mathbb{R}\), is:
  

  • \(\sqrt{5}\)
    
  • 3
    
  • \(2\sqrt{3}\)
    
  • \(\sqrt{15}\)
  View Solution