Question 3 - Jee advanced Math 2022 P2 Questions with Solutions

The greatest integer less than or equal to 

\(\int_{1}^{2}\log_{2}{(x^{3} + 1)}dx + \int_{1}^{\log_{2}{9}}(2^{x} - 1)^{\frac{1}{3}}dx\)

is ______.

Sol : 

Trick in this problem is realising that the integrands \(\log_{2}{(x^{3} + 1)}\) and \((2^{x} - 1)^{\frac{1}{3}}\) are inverse functions of each other. Here’s how.

Let \(y(x) = \log_{2}{(x^{3} + 1)}\)

\(y(x)\) exists for all \(x > -1\). It is one-one as well as onto function. Hence, its inverse exists. 

\(\implies 2^{y} = x^{3} + 1\)

\(\implies x = (2^{y} - 1)^{\frac{1}{3}}\) which is the integrand in the second term.

Rewriting the same in more familiar notations,

\(y^{-1}(x) = (2^{x} - 1)^{\frac{1}{3}}\)

So the integrands are inverse functions of each other. Now think about the definite integrals in terms of areas.  

For any non-zero invertible function \(y(x)\), the area under the curve between x = c to x = d is highlighted in red. And the blue area is equivalent to the area under the curve \(y^{-1}(x)\) between ‘y(c)’ and ‘y(d)’.  

Sum of their areas  =  area of the outer rectangle(\(d \times y(d)\))

                                       - area of the inner rectangle(\(c \times y(c)\))

In our example, \(c = 1\) and \(d = 2\) are the limits of integration of \(y(x)\).

\(\implies y(1) = \log_{2}{1^{3} + 1} = \log_{2}{2} = 1\) and

       \(y(2) = \log_{2}{2^{3} + 1} = \log_{2}{9}\)

We see that \(y(1)\) and \(y(2)\) are indeed the limits of integration of \(y^{-1}(x)\).

\(\implies \int_{1}^{2}\log_{2}{(x^{3} + 1)}dx + \int_{1}^{\log_{2}{9}}(2^{x} - 1)^{\frac{1}{3}}dx\)

  \( = d \times y(d) - c \times y(c)\)

  \( = 2\log_{2}{9} - 1\)

  \( = 4\log_{2}{3} - 1\)

  \( = 4\frac{\log_{10}{3}}{\log_{10}{2}} - 1\)

\( = 4\frac{0.477}{0.301} - 1\)

\( \approx 4(1. 05) - 1\)

\( \approx (4. 20)  - 1\)

\( \approx 3.20 \)

Therefore, the greatest integer less than or equal to the given integral is 3.


Comments

Post a Comment

Popular posts from this blog

Question 7 - Jee advanced Math 2022 P2 Questions with Solutions

Image
 Consider the hyperbola \(\frac{x^{2}}{100} - \frac{y^{2}}{64} = 1\) with foci at \(S\) and \(S_{1}\),  where \(S\) lies on positive x-axis. Let \(P\) be a point on the hyperbola, in the first quadrant. Let \(\angle{SPS_{1}} = \alpha\), with \(\alpha < \frac{\pi}{2}\). The straight line passing through the point \(S\) and having the same slope as that of the tangent at \(P\) to the hyperbola, intersects the straight line \(S_{1}P\) at \(P_{1}\). Let \(\delta\) be the distance of \(P\) from the straight line \(SP_{1}\), and \(\beta = S_{1}P\). Then the greatest integer less than or equal to \(\frac{\beta \delta}{9}\sin{\frac{\alpha}{2}}\) is ______. Sol :  There are a couple of properties of hyperbola that will help us here. (1) The tangent line at a point bisects the angle between the lines connecting the two foci with the point of tangency.  So, in the figure above, tangent at \(P\) bisects the angle \(\alpha\) between the lines joining the foci(\(S\) an...
Read more

Question 6 - Jee advanced Math 2022 P2 Questions with Solutions

 Let \(\beta\) be a real number. Consider the matrix \(\Biggl(\matrix{\beta & 0 & 1 \\ 2 & 1 & -2 \\ 3 & 1 & -2}\Biggl)\). If \(A^{7} - (\beta - 1)A^{6} - \beta A^{5}\) is a singular matrix, then the value of \(9\beta\) is _____. Sol:  \(A^{n}\) is a matrix A multiplied to itself n times. Consider \(A^{7}\)            \( = A^{6} \cdot A\)            \( = (A^{5} \cdot A) \cdot A\)            \( = A^{5} \cdot (A \cdot A)\)….. Associative law of matrix multiplication            \( = A^{5} \cdot A^{2}\) \(A^{7} - (\beta - 1)A^{6} - \beta A^{5}\)  \( = A^{5} \cdot A^{2} - (\beta - 1)[A^{5} \cdot A] - \beta A^{5}\) For any constant \(p\) and square matrices \(A\) and \(B\),  \(p (A \cdot B) = (pA) \cdot B = A \cdot (pB)\)  \(\implies = A^{5} \cdot A^{2} - A^{5} \cdot [(\beta - 1) A] - \beta A^{5}\) Also, for any square mat...
Read more

Question 3 - Jee advanced Math 2022 P1 Questions with Solutions

In a study about a pandemic, data of 900 persons was collected. It was found that 190 persons had symptom of fever, 220 persons had symptom of cough, 220 persons had symptom of breathing problem, 330 persons had symptom of fever or cough or both, 350 persons had symptom of cough or breathing problem or both, 340 persons had symptom of fever or breathing problem or both, 30 persons had all three symptoms(fever, cough and breathing problem). If a person is chosen randomly from these 900 persons, then the probability that the person has at-most one symptom is _____. Sol :  Let C be a letter for representing ‘Cough’; F for ‘Fever’; and B for ‘breathing problem’.  i) 190 = n(F) = n(F only) + n(F and C) + n(F and B) + n(all three) ii) 220 = n(C) = n(C only) + n(F and C) + n(C and B) + n(all three) iii) 220 = n(B) = n(B only) + n(F and B) + n(C and B) + n(all three) n(F or C or both) = n(F) + n(C) - n(F and C) - n(all three) (We are subtracting n(F and C) because it is already in...
Read more