CBSE Class 12 Mathematics Question Paper 2011

CBSE CLASS XII Questions

• 1.

  A function \[ f:\mathbb{R}-\left\{\frac{3}{5}\right\} \to \mathbb{R}-\left\{\frac{3}{5}\right\} \] is defined as \[ f(x)=\frac{3x+2}{5x-3}. \] Show that \(f\) is one-one and onto.

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• 2.

  If \[ \frac{d}{dx}(F(x))=\frac{1}{e^x+1}, \] then find \(F(x)\), given that \[ F(0)=\log\left(\frac{1}{2}\right). \]

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• 3.

  Find: \[ \int \frac{x^2}{(x^2+9)(x^2+16)}\,dx \]

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• 4.

  For two vectors \(\vec{a}\) and \(\vec{b}\):  
  
  Assertion (A): \[ |\vec{a}\times\vec{b}|^2+(\vec{a}\cdot\vec{b})^2 = |\vec{a}|^2|\vec{b}|^2 \] Reason (R): \[ |\vec{a}\times\vec{b}| = (\vec{a}\cdot\vec{b})\tan\theta, \quad \theta\neq\frac{\pi}{2}. \]

  • Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
    
  • Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
    
  • Assertion (A) is true, but Reason (R) is false.
    
  • Assertion (A) is false, but Reason (R) is true.
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• 5.

  Find the domain of \[ q(x)=\cos^{-1}(4x^2-3). \] Hence, find the value of \(x\) for which \[ q(x)=0. \] Also, write the range of \[ 3q(x)-\pi. \] 

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• 6.

  Evaluate: \[ \cos\left[\sin^{-1}(-1)-\tan^{-1}(-\sqrt{3})\right]. \]

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