JEE Advance - Mathematics (2000)
- 1If $$\alpha ,\,\beta $$ are the roots of $$a{x^2} + bx + c = 0$$, $$\,\left( {a \ne 0} \right)$$ and $$\alpha + \delta ,\,\,\beta + \delta $$ are the roots of $$A{x^2} + Bx + c = 0,$$ $$\left( {A \ne 0\,} \right)\,$$ for some contant $$\delta $$, then prove that $${{{b^2} - 4ac} \over {{a^2}}} = {{{B^2} - 4Ac} \over {{A^2}}}$$.Хариулт(D)The statement is always true.
- 2A coin has probability $$p$$ of showing head when tossed. It is tossed $$n$$ times. Let $${p_n}$$ denote the probability that no two (or more) consecutive heads occur. Prove that $${p_1} = 1,{p_2} = 1 - {p^2}$$ and $${p_n} = \left( {1 - p} \right).\,\,{p_{n - 1}} + p\left( {1 - p} \right){p_{n - 2}}$$ for all $$n \ge 3.$$ХариултABC
- 3For $$x>0,$$ let $$f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.} $$ Find the function
$$f\left( x \right) + f\left( {{1 \over x}} \right)$$ and show that $$f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}.$$
Here, $$\ln t = {\log _e}t$$.Хариулт(A)$$\frac{1}{2}(\ln x)^2$$ - 4Suppose $$p\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + .......... + {a_n}{x^n}.$$ If
$$\left| {p\left( x \right)} \right| \le \left| {{e^{x - 1}} - 1} \right|$$ for all $$x \ge 0$$, prove that
$$\left| {{a_1} + 2{a_2} + ........ + n{a_n}} \right| \le 1$$.Хариулт(B)Consider the limit as x approaches 0 and apply L'Hopital's rule. - 4Let $$g\left( x \right) = \int\limits_0^x {f\left( t \right)dt,} $$ where f is such that
$${1 \over 2} \le f\left( t \right) \le 1,$$ for $$t \in \left[ {0,1} \right]$$ and $$\,0 \le f\left( t \right) \le {1 \over 2},$$ for $$t \in \left[ {1,2} \right]$$.
Then $$g(2)$$ satisfies the inequalityХариулт(B)$$0 \le g\left( 2 \right) < 2$$ - 5Let $$ABC$$ be a triangle with incentre $$I$$ and inradius $$r$$. Let $$D,E,F$$ be the feet of the perpendiculars from $$I$$ to the sides $$BC$$, $$CA$$ and $$AB$$ respectively. If $${r_1},{r_2}$$ and $${r_3}$$ are the radii of circles inscribed in the quadrilaterals $$AFIE$$, $$BDIF$$ and $$CEID$$ respectively, prove that $$${{{r_1}} \over {r - {r_1}}} + {{{r_2}} \over {r - {r_2}}} + {{{r_3}} \over {r - {r_3}}} = {{{r_1}{r_2}{r_3}} \over {\left( {e - {r_1}} \right)\left( {r - {r_2}} \right)\left( {r - {r_3}} \right)}}$$$ХариултBCDE
- 7Let $${C_1}$$ and $${C_2}$$ be respectively, the parabolas $${x^2} = y - 1$$ and $${y^2} = x - 1$$. Let $$P$$ be any point on $${C_1}$$ and $$Q$$ be any point on $${C_2}$$. Let $${P_1}$$ and $${Q_1}$$ be the reflections of $$P$$ and $$Q$$, respectively, with respect to the line $$y=x$$. Prove that $${P_1}$$ lies on $${C_2}$$, $${Q_1}$$ lies on $${C_1}$$ and $$PQ \ge $$ min $$\left\{ {P{P_1},Q{Q_1}} \right\}$$. Hence or otherwise determine points $${P_0}$$ and $${Q_0}$$ on the parabolas $${C_1}$$ and $${C_2}$$ respectively such that $${P_0}{Q_0} \le PQ$$ for all pairs of points $$(P,Q)$$ with $$P$$ on $${C_1}$$ and $$Q$$ on $${C_2}$$.Хариулт(B)$$P_0 = (0, 1), Q_0 = (1, 0)$$
- 7If the vectors $$\overrightarrow a ,\overrightarrow b $$ and $$\overrightarrow c $$ form the sides $$BC,$$ $$CA$$ and $$AB$$ respectively of a triangle $$ABC,$$ thenХариулт(B)$$\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a $$
- 8Let $$ABC$$ be an equilateral triangle inscribed in the circle $${x^2} + {y^2} = {a^2}$$. Suppose perpendiculars from $$A, B, C$$ to the major axis of the ellipse $$x.{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $$(a>b)$$ meets the ellipse respectively, at $$P, Q, R$$. so that $$P, Q, R$$ lie on the same side of the major axis as $$A, B, C$$ respectively. Prove that the normals to the ellipse drawn at the points $$P, Q$$ and $$R$$ are concurrent.Хариулт(A)The normals to the ellipse at P, Q, and R are concurrent at the origin.
- 8Let the vectors $$\overrightarrow a ,\overrightarrow b ,\overrightarrow c $$ and $$\overrightarrow d $$ be such that
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d } \right) = 0.$$ Let $${P_1}$$ and $${P_2}$$ be planes determined
by the pairs of vectors $$\overrightarrow a .\overrightarrow b $$ and $$\overrightarrow c .\overrightarrow d $$ respectively. Then the angle between $${P_1}$$ and $${P_2}$$ isХариулт(A)$$0$$ - 9Let $$ABC$$ and $$PQR$$ be any two triangles in the same plane. Assume that the prependiculars from the points $$A, B, C$$ to the sides $$QR, RP, PQ$$ respectively are concurrent. Using vector methods or otherwise, prove that the prependiculars from $$P, Q, R $$ to $$BC,$$ $$CA$$, $$AB$$ respectively are also concurrent.ХариултABD
- 10For points $$P\,\,\, = \left( {{x_1},\,{y_1}} \right)$$ and $$Q\,\,\, = \left( {{x_2},\,{y_2}} \right)$$ of the co-ordinate plane, a new distance $$d\left( {P,\,Q} \right)$$ is defined by $$d\left( {P,\,Q} \right)$$$$ = \left( {{x_2},\,{y_2}} \right)\left| {{x_1} - {x_2}} \right| + \left| {{y_1} - {y_2}} \right|.$$ Let $$O = (0, 0)$$ and $$A = (3, 2)$$. Prove that the set of points in the first quadrant which are equidistant (with respect to the new distance) from $$O$$ and $$A$$ consists of the union of a line segment of finite length and an infinite ray. Sketch this set in a labelled diagram.ХариултAD
- 12Prove by induction on, that $${p_n} = A{\alpha ^n} + B{\beta ^n}$$ for all $$n \ge 1,$$ where $$\alpha $$ and $$\beta $$ are the roots of quadratic equation $${x^2} - \left( {1 - p} \right)x - p\left( {1 - p} \right) = 0$$ and $$A = {{{p^2} + \beta - 1} \over {\alpha \beta - {\alpha ^2}}},B = {{{p^2} + \alpha - 1} \over {\alpha \beta - {\beta ^2}}}.$$Хариулт(A)The problem asks to prove a formula for the probability of no consecutive heads in n coin tosses using induction, given a recurrence relation and initial values. The formula involves roots of a quadratic equation and constants A and B. The task is essentially to verify that the given formula satisfies the recurrence and initial conditions.
- 13Let $$a,\,b,\,c$$ be possitive real numbers such that $${b^2} - 4ac > 0$$ and let $${\alpha _1} = c.$$ Prove by induction that $${\alpha _{n + 1}} = {{a\alpha _n^2} \over {\left( {{b^2} - 2a\left( {{\alpha _1} + {\alpha _2} + ... + {\alpha _n}} \right)} \right)}}$$ is well-defined and
$${\alpha _{n + 1}} < {{{\alpha _n}} \over 2}$$ for all $$n = 1,2,....$$ (Here, 'well-defined' means that the denominator in the expression for $${\alpha _{n + 1}}$$ is not zero.)ХариултABCDE - 14For every possitive integer $$n$$, prove that
$$\sqrt {\left( {4n + 1} \right)} < \sqrt n + \sqrt {n + 1} < \sqrt {4n + 2}.$$
Hence or otherwise, prove that $$\left[ {\sqrt n + \sqrt {\left( {n + 1} \right)} } \right] = \left[ {\sqrt {4n + 1} \,\,} \right],$$
where $$\left[ x \right]$$ denotes the gratest integer not exceeding $$x$$.Хариулт(B)Squaring all terms in the inequality is a key step to remove square roots and simplify the expressions, facilitating a comparison of integer parts. - 15For any positive integer $$m$$, $$n$$ (with $$n \ge m$$), let $$\left( {\matrix{ n \cr m \cr } } \right) = {}^n{C_m}$$
Prove that $$\left( {\matrix{ n \cr m \cr } } \right) + \left( {\matrix{ {n - 1} \cr m \cr } } \right) + \left( {\matrix{ {n - 2} \cr m \cr } } \right) + ........ + \left( {\matrix{ m \cr m \cr } } \right) = \left( {\matrix{ {n + 1} \cr {m + 2} \cr } } \right)$$Hence or otherwise, prove that $$\left( {\matrix{ n \cr m \cr } } \right) + 2\left( {\matrix{ {n - 1} \cr m \cr } } \right) + 3\left( {\matrix{ {n - 2} \cr m \cr } } \right) + ........ + \left( {n - m + 1} \right)\left( {\matrix{ m \cr m \cr } } \right) = \left( {\matrix{ {n + 2} \cr {m + 2} \cr } } \right).$$.
ХариултABC - 16In any triangle $$ABC,$$ prove that $$$\cot {A \over 2} + \cot {B \over 2} + \cot {C \over 2} = \cot {A \over 2}\cot {B \over 2}\cot {C \over 2}.$$$Хариулт(B)The identity holds true because A + B + C = 180 degrees, and using trigonometric identities for cotangent of half-angles and the fact that cot(90-x) = tan(x).
- 30Consider the following statements in $$S$$ and $$R$$
$$S:$$ $$\,\,\,$$$ Both $$\sin \,\,x$$ and $$\cos \,\,x$$ are decreasing functions in the interval $$\left( {{\pi \over 2},\pi } \right)$$
$$R:$$$$\,\,\,$$ If a differentiable function decreases in an interval $$(a, b)$$, then its derivative also decreases in $$(a, b)$$.
Which of the following is true ?Хариулт(D)$$S$$ is correct and $$R$$ is wrong