JEE Math Previous Year Questions and Solutions

Considering only the principal values of the inverse trigonometric function, the value of \(\frac{3}{2} \cos^{-1}{\sqrt{\frac{2}{2+\pi^{2}}}} + \frac{1}{4}\sin^{-1}{\frac{2\sqrt{2}\pi}{2 + \pi^{2}}} + \tan^{-1}{\frac{\sqrt{2}}{\pi}}\) is ____. Sol : Convert \(\cos^{-1}\) and \(\sin^{-1}\) terms into \(\tan^{-1}\). To Convert \(\cos^{-1}\) term into \(\tan^{-1}\) : Let \(t = \cos^{-1}{\sqrt{\frac{2}{2+\pi^{2}}}} = \cos^{-1}{\frac{\sqrt{2}}{\sqrt{2+\pi^{2}}}}\) …(1) \(\implies\)\(\cos{t} = \frac{\sqrt{2}}{\sqrt{2+\pi^{2}}}\) Use right angle triangle to determine  ‘\(\tan{t}\)’. Consider a right triangle as shown above, with side adjacent to angle ‘t’ equal to \(\sqrt{2}\), and hypotenuse equal to \(\sqrt{2+\pi^{2}}\).  Using Pythagoras’ theorem,  side opposite to ‘t’ \(= \sqrt{2+\pi^{2} - 2} = \pi\) Hence,  \(\sin{t} = \frac{\pi}{\sqrt{2+\pi^{2}}}\) And,  \(\tan{t} = \frac{\sin{t}}{\cos{t}}\)         \(  = \fra...

Let \(P_{1}\) and \(P_{2}\) be two planes given by  \(P_{1} : 10x + 15y + 12z - 60 = 0\). \(P_{2} : -2x + 5y + 4z - 20 = 0\). Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on \(P_{1}\) and \(P_{2}\)? A) \(\frac{x - 1}{0} = \frac{y - 1}{0} = \frac{z - 1}{5}\) B) \(\frac{x - 6}{-5} = \frac{y}{2} = \frac{z}{3}\) C) \(\frac{x}{-2} = \frac{y - 4}{5} = \frac{z}{4}\) D) \(\frac{x}{1} = \frac{y - 4}{-2} = \frac{z}{3}\) Sol :  The two faces of a tetrahedron lie on the planes \(P_{1}\) and \(P_{2}\). Let L be the line of intersection of the planes. L is one of the six edges of the tetrahedron. In the figure above, the other five edges of the tetrahedron are \(l_{1}\), \(l_{2}\), \(l_{3}\), \(l_{4}\) and \(l_{5}\), out of which, \(l_{1}\), \(l_{2}\), \(l_{3}\) and \(l_{4}\), all four of them completely lie in either plane \(P_{1}\) or \(P_{2}\) …AND… they each intersect the line L at a single point(A and B in the figure).  So the li...

Consider the parabola \(y^{2} = 4x\). Let \(S\) be the focus of the parabola. A pair of tangents drawn to the parabola from the point \(P(-2, 1)\) meet the parabola at \(P_{1}\) and \(P_{2}\). Let \(Q_{1}\) and \(Q_{2}\) be points on the lines \(SP_{1}\) and \(SP_{2}\) respectively such that \(PQ_{1}\) is perpendicular to \(SP_{1}\) and \(PQ_{2}\) is perpendicular to \(SP_{2}\). Then, which of the following is/are TRUE? A) \(SQ_{1} = 2\) B) \(Q_{2}Q_{1} = \frac{3\sqrt{10}}{5}\) C)  \(PQ_{1} = 3\) D) \(SQ_{2} = 1\) Sol :  \(y^{2} = 4x\) is a standard parabola \(y^{2} = 4ax\) with the vertex at the origin. \(\implies 4a = 4 \implies a = 1(> 0)\) So this is a parabola which opens to the right(since \(a > 0\)) in the Cartesian plane. \(\implies\) Focus of the parabola is S(1, 0).  \(PP_{1}\) and \(PP_{2}\) are tangents to the parabola from \(P(-2, 1)\). And \(PQ_{1}\), \(PQ_{2}\) are perpendiculars to \(SP_{1}\) and \(SP_{2}\).  We are looking for the coordinate...

Let S be the reflection of point Q with respect to the plane given by \(\vec{r} = -(t + p)\hat{i} + t\hat{j} + (1 + p)\hat{k}\) where t, p are real parameters and \(\hat{i}, \hat{j}, \hat{k}\) are the unit vectors along the three positive coordinate axes. If the position vectors of Q and S are \(10\hat{i}+ 15\hat{j} + 20\hat{k}\) and \(\alpha \hat{i} + \beta \hat{j} + \gamma \hat{k}\) respectively, then which of the following is/are TRUE? A) \(3(\alpha + \beta) = -101\) B) \(3(\beta + \gamma) = -71\) C) \(3(\gamma + \alpha) = -86\) D) \(3(\alpha + \beta + \gamma) = -121\) Sol :  S is the reflection of Q. \(\implies\) Both are same distance(perpendicular) from the given plane. Drop perpendiculars from S and Q onto the plane; let A be the point on the plane where they meet. Let \((x_{1}, y_{1}, z_{1})\) be the coordinates of A. \(\implies\) \(\overrightarrow{OA} = x_{1}\hat{i} + y_{1}\hat{j} + z_{1}\hat{k}\) is position vector of A. Also, let \(\overrightarrow{OQ} = 10\hat{i}+ 15\...

Let ABC be the triangle with \(AB = 1, AC = 3\) and \(\angle{BAC} = \frac{\pi}{2}\). If a circle of radius \(r > 0\) touches the sides AB, AC and also touches internally the circumcircle of the triangle ABC, then the value of \(r\) is _____.  Sol : Let \(C_{1}\) and \(r_{1}\) be the centre and the radius of the circumcircle(circle through the three vertices) of triangle ABC. Let \(C_{2}\) be the centre of the circle which is touching sides AB and AC of triangle ABC and touching the circumcircle at P(say). From the question, letter \(r\) denotes the radius of this circle.  The first result that we will use here is : If two circles touch each other internally(or externally) then their centres and the point of contact of circles are aligned, i.e., points \(C_{1}, C_{2}\) and P are collinear.  \(\implies r = r_{1} - d(C_{1}, C_{2})\) …..(1) where \(d(C_{1}, C_{2})\) is the distance between \(C_{1}\) and \(C_{2}\). The second result that we require here is related to the...

Let \(a_{1}, a_{2}, a_{3}, ….\) be an arithmetic progression with \(a_{1}=7\) and common difference \(8\). Let \( T_{1}, T_{2}, T_{3},…\) be such that \(T_{1} = 3\) and \(T_{n+1} - T_{n} = a_{n}\) for \(n \geq 1\). Then, which of the following is/are TRUE? A) \(T_{20} =1604\) B) \(\sum_{k=1}^{20} T_{k} = 10510\) C) \(T_{30} = 3454\) D)  \(\sum_{k=1}^{30} T_{k} = 35610\) Sol :  \(a_{1}, a_{2}, a_{3}, ….\) is an arithmetic progression. \(\implies a_{n} = a_{1} + (n - 1)d\) Also, \(T_{n+1} - T_{n} = a_{n}\) \(\implies T_{n+1} = T_{n} + a_{n}\)  i.e., \(T_{2} = T_{1} + a_{1}; \:  T_{3} = T_{2} + a_{2}; \: T_{4} = T_{3} + a_{3}; \:\) and so on….. Let’s investigate the four options. A)  \(T_{20} = 1604\) \(\implies T_{19} + a_{19} = 1604\) \(\implies T_{18} + a_{18} + a_{19} = 1604\) …… \(\implies T_{1} + a_{1} + a_{2} + a_{3} + ….+ a_{19} = 1604\)  \(\implies 3 + a_{1} + (a_{1} + d) + (a_{1} + 2d) + ….+ (a_{1} + 18d) = 1604\) \(\implies 3 + (19 \times a_{1}) ...

 Consider the equation \(\int_{1}^{e} \frac{(\log_{e}{x})^{\frac{1}{2}}}{x(a - (\log_{e}{x})^\frac{3}{2})^{2}}dx = 1, \: \: a \in (-\infty, 0) U (1, \infty)\). Which of the following statements is/are true? (A) No \(a\) satisfies the above equation (B) An integer \(a\) satisfies the above equation (C) An irrational number \(a\) satisfies the above equation (D) More than one \(a\) satisfy the above equation Sol :  \(\int_{1}^{e} \frac{(\log_{e}{x})^{\frac{1}{2}}}{x(a - (\log_{e}{x})^\frac{3}{2})^{2}}dx = 1\) \(\implies \int_{1}^{e} \frac{(\log_{e}{x})^{\frac{1}{2}}\times \frac{1}{x}}{(a - (\log_{e}{x})^\frac{3}{2})^{2}}dx = 1\) Let \((\log_{e}{x})^\frac{3}{2} = t\) \(\implies (\frac{3}{2}(\log_{e}{x})^\frac{1}{2} \times \frac{1}{x})dx = dt\) \(\implies ((\log_{e}{x})^\frac{1}{2} \times \frac{1}{x})dx = \frac{2}{3}dt\) \(x = 1 \implies (\log_{e}{1})^\frac{3}{2} = 0 = t\) \(x = e \implies (\log_{e}{e})^\frac{3}{2} = 1 = t\) \(\implies \int_{0}^{1} \frac{\frac{2}{3}dt}{(a - t)^...