CBSE Class 12 Mathematics Set 2- (65/3/2) Question Paper 2026 with Solution PDF

CBSE CLASS XII Questions

• 1.

  A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following: 
  
  (i) Express \(y\) as a function of \(x\) from the given equation of ellipse. 
  (ii) Integrate the function obtained in (i) with respect to \(x\). 
  (iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration. 
  OR 
  (iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\). 
   

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• 2.

  Evaluate : \[ \int_{\frac{1}{12}}^{\frac{5}{12}} \frac{dx}{1+\sqrt{\cot x}} \]

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• 3.

  Find : \[ \int \frac{2x+1}{\sqrt{x^2+6x}}\,dx \]

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• 4.

  A rectangle of perimeter \(24\) cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle. 
  

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• 5.

  Smoking increases the risk of lung problems. A study revealed that 170 in 1000 males who smoke develop lung complications, while 120 out of 1000 females who smoke develop lung related problems. In a colony, 50 people were found to be smokers of which 30 are males. A person is selected at random from these 50 people and tested for lung related problems. Based on the given information answer the following questions: 
  
  (i) What is the probability that selected person is a female? 
  (ii) If a male person is selected, what is the probability that he will not be suffering from lung problems? 
  (iii)(a) A person selected at random is detected with lung complications. Find the probability that selected person is a female. 
  OR 
  (iii)(b) A person selected at random is not having lung problems. Find the probability that the person is a male. 
   

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• 6.

  Find the domain of \(p(x)=\sin^{-1}(1-2x^2)\). Hence, find the value of \(x\) for which \(p(x)=\frac{\pi}{6}\). Also, write the range of \(2p(x)+\frac{\pi}{2}\).
  

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