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JEE Main 2006 · 30 Apr 2006 · Offline

149 questions in the paper's printed order. Open any question for its complete solution.

Physics

Chemistry

Mathematics

Q1.ABC is a triangle, right angled at A. The resultant of the forces acting along $AB$ , $AC$ with magnitudes $\frac{1}{A B}$ and $\frac{1}{A C}$ respectively is the force along $A D$ , where $D$ is the foot …
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Q2.Suppose a population $A$ has 100 observations $101, 102, \dots, 200$ , and another population $B$ has 100 observations $151, 152, \dots, 250$ . If $V_{A}$ and $V_{B}$ represent the variances of the two populations, respectively, then $\frac{V _{A}}{V _{B}}$ is
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Q3.If the roots of the quadratic equation $x^{2} + p x + q = 0$ are $tan 3 0^{\circ}$ and $tan 1 5^{\circ}$ , respectively then the value of $2 + q - p$ is
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Q4.The value of the integral, $\int_{3}^{6} \frac{x}{9 - x + x} d x$ is
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Q5.The number of values of $x$ in the interval $[0, 3 π]$ satisfying the equation $2 sin^{2} x + 5 sin x - 3 = 0$ is
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Q6.If $(\overline{a} \times \overline{b}) \times \overline{c} = \overline{a} \times (\overline{b} \times \overline{c})$ , where $\overline{a}, \overline{b}$ and $\overline{c}$ are any three vectors su …
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Q7.Let W denote the words in the English dictionary. Define the relation R by:
$R = {(x, y) \in W \times W ∣$ the words x and y have at least one letter in common $}$ . Then R is
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Q8.If $A$ and $B$ are square matrices of size $n \times n$ such that $A^{2} - B^{2} = (A - B) (A + B)$ , then which of the following will be always true?
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Q9.The value of $\sum_{k = 1}^{10} (sin \frac{2 k π}{11} + i cos \frac{2 k π}{11})$ is
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Q10.All the values of $m$ for which both roots of the equations $x^{2} - 2 m x + m^{2} - 1 = 0$ are greater than -2 but less than 4, lie in the interval
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Q11.A particle has two velocities of equal magnitude inclined to each other at an angle $θ$ . If one of them is halved, the angle between the other and the original resultant velocity is bisected by the new resultant. Then $θ$ is
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Q12.At a telephone enquiry system the number of phone cells regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10-minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is
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Q13.A body falling from rest under gravity passes a certain point $P$ . It was at a distance of 400 m from $P$ , 4 $s$ prior to passing through P. If $g = 10 m / s^{2}$ , then the height above the point P from where the body began to fall is
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Q14. $\int_{0}^{π} x f (sin x) d x$ is equal to
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Q15.A straight line through the point $A (3, 4)$ is such that its intercept between the axes is bisected at A. Its equation is
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Q16.The two lines $x = a y + b, z = cy + d$ ; and $x = a^{'} y + b^{'}, z = c^{'} y + d^{'}$ are perpendicular to each other if
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Q17.The locus of the vertices of the family of parabolas $y = \frac{a ^{3} x ^{2}}{3} + \frac{a ^{2} x}{2} - 2 a$ is
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Q18.The values of $a$ , for which the points $A, B, C$ with position vectors $2 \hat{i} - \hat{j} + \hat{k}, \hat{i} - 3 \hat{j} - 5 \hat{k}$ and $a \hat{i} - 3 \hat{j} + \hat{k}$ respectively are the vertices of a right-angled triangle with $C = \frac{π}{2}$ are
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Q19. $\int_{- 3 π /2}^{- π /2} [(x + π)^{3} + cos^{2} (x + 3 π)] d x$ is equal to
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Q20.If $x$ is real, the maximum value of $\frac{3 x ^{2} + 9 x + 17}{3 x ^{2} + 9 x + 7}$ is
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Q21.In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is
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Q22.Let $A = (1324)$ and $B = (a 0 0 b), a, b \in N$ . Then
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Q23.The function $f (x) = \frac{x}{2} + \frac{2}{x}$ has a local minimum at
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Q24.Angle between the tangents to the curve $y = x^{2} - 5 x + 6$ at the points $(2, 0)$ and $(3, 0)$ is
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Q25.Let $a_{1}, a_{2}, a_{3}, \dots$ be terms of an A.P. If $\frac{a _{1} + a _{2} + \dots a _{p}}{a _{1} + a _{2} + \dots + a _{q}} = \frac{p ^{2}}{q ^{2}}, p \neq = q$ , then $\frac{a _{6}}{a _{21}}$ equals
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Q26.The set of points where $f (x) = \frac{x}{1 + ∣ x ∣}$ is differentiable is
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Q27.A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length x. The maximum area enclosed by the park is
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Q28.At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are 10 candidates and 4 are of be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is
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Q29.If the expansion in powers of $x$ of the function $\frac{1}{( 1 - a x ) ( 1 - b x )}$ is $a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + \dots$ , then $a_{n}$ is
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Q30.For natural numbers $m, n$ if $(1 - y)^{m} (1 + y)^{n} = 1 + a_{1} y + a_{2} y^{2} + \dots$ , and $a_{1} = a_{2} = 10$ , then $(m, n)$ is
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Q31.The value of $\int_{1}^{a} [x] f^{'} (x) d x, a > 1$ , where $[x]$ denotes the greatest integer not exceeding $x$ is
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Q32.If the lines $3 x - 4 y - 7 = 0$ and $2 x - 3 y - 5 = 0$ are two diameters of a circle of area $49 π$ square units, the equation of the circle is
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Q33.The differential equation whose solution is $A x^{2} + B y^{2} = 1$ , where $A$ and $B$ are arbitrary constants is of
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Q34.Let C be the circle with centre $(0, 0)$ and radius 3 units. The equation of the locus of the mid points of the chords of the circle $C$ that subtend an angle of $\frac{2 π}{3}$ at its centre is
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Q35.If (a, $a^{2}$ ) falls inside the angle made by the lines $y = \frac{x}{2}, x > 0$ and $y = 3 x, x > 0$ , then a belongs to
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Q36.If $z^{2} + z + 1 = 0$ , where $z$ is a complex number, then the value of $(z + \frac{1}{z})^{2} + (z^{2} + \frac{1}{z ^{2}})^{2} + (z^{3} + \frac{1}{z ^{3}})^{2} + \dots + (z^{6} + \frac{1}{z ^{6}})^{2}$ is
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Q37.If $0 < x < π$ and $cos x + sin x = \frac{1}{2}$ , then $tan x$ is
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Q38.If $a_{1}, a_{2}, \dots, a_{n}$ are in H.P., then the expression $a_{1} a_{2} + a_{2} a_{3} + \dots + a_{n - 1} a_{n}$ is equal to
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Q39.If $x^{m} \cdot y^{n} = (x + y)^{m + n}$ , then $\frac{d y}{d x}$ is
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