JEE Advance - Mathematics (1996)
- 4For the three events $$A, B,$$ and $$C,P$$ (exactly one of the events $$A$$ or $$B$$ occurs) $$=P$$ (exactly one of the two events $$B$$ or $$C$$ occurs)$$=P$$ (exactly one of the events $$C$$ or $$A$$ occurs)$$=p$$ and $$P$$ (all the three events occur simultaneously) $$ = {p^2},$$ where $$0 < p < 1/2.$$ Then the probability of at least one of the three events $$A,B$$ and $$C$$ occurring isотговор(A)$${{3p + 2{p^2}} \over 2}$$
- 5In how many ways three girls and nine boys can be seated in two vans, each having numbered seats, $$3$$ in the front and $$4$$ at the back? How many seating arrangements are possible if $$3$$ girls should sit together in a back row on adjacent seats? Now, if all the seating arrangements are equally likely, what is the probability of $$3$$ girls sitting together in a back row on adjacent seats?отговор(A)The total number of seating arrangements is $$7 \times 13!$$, the number of seating arrangements with the three girls sitting together in the back row on adjacent seats is $$12!$$ and the probability is $$1/9!$$.
- 6A nonzero vector $$\overrightarrow a $$ is parallel to the line of intersection of the plane determined by the vectors $$\widehat i,\widehat i + \widehat j$$ and the plane determined by the vectors $$\widehat i - \widehat j,\widehat i + \widehat k.$$ The angle between $$\overrightarrow a $$ and the vector $$\widehat i - 2\widehat j + 2\widehat k$$ is ................отговорCD
- 7If $$\overrightarrow b \,$$ and $$\overrightarrow c \,$$ are two non-collinear unit vectors and $$\overrightarrow a \,$$ is any vector, then $$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b + \left( {\overrightarrow a .\overrightarrow c } \right)\overrightarrow c + {{\overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)} \over {\left| {\overrightarrow b \times \overrightarrow c } \right|}}\left( {\overrightarrow b \times \overrightarrow c } \right) = $$ ..............отговор(C)$$\overrightarrow a$$
- 8The position vectors of the vertices $$A, B$$ and $$C$$ of a tetrahedron $$ABCD$$ are $$\widehat i + \widehat j + \widehat k,\,\widehat i$$ and $$3\widehat i\,,$$ respectively. The altitude from vertex $$D$$ to the opposite face $$ABC$$ meets the median line through $$A$$ of the triangle $$ABC$$ at a point $$E.$$ If the length of the side $$AD$$ is $$4$$ and the volume of the tetrahedron is $${{2\sqrt 2 } \over 3},$$ find the position vector of the point $$E$$ for all its possible positions.отговор(C)$$-\hat i + 3\hat j + 3\hat k$$
- 10The angle between a pair of tangents drawn from a point P to the circle $${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\, + \,9\,{\sin ^2}\,\alpha \, + \,13\,{\cos ^2}\,\alpha \, = \,0$$ is $$2\,\alpha $$.
The equation of the locus of the point P isотговор(D)$${x^2}\, + \,{y^2}\, + \,\,4x\, - \,6\,y\,\, + \,9\,\, = \,0$$ - 12The value of the expression
$$1 \bullet \left( {2 - \omega } \right)\left( {2 - {\omega ^2}} \right) + 2 \bullet \left( {3 - \omega } \right)\left( {3 - {\omega ^2}} \right) + \,....... + \left( {n - 1} \right).\left( {n - \omega } \right)\left( {n - {\omega ^2}} \right),$$where $$\omega $$ is an imaginary cube root of unity, is..........
отговор(A)$$\frac{1}{4}n(n-1)(n^2 + 3n + 4)$$ - 13For positive integers $${n_1},\,{n_2}$$ the value of the expression $${\left( {1 + i} \right)^{^{{n_1}}}} + {\left( {1 + {i^3}} \right)^{{n_1}}} + {\left( {1 + {i^5}} \right)^{{n_2}}} + {\left( {1 + {i^7}} \right)^{{n_2}}},$$
where $$i = \sqrt { - 1} $$ is real number if and only ifотговор(D)$${n_1} > 0,\,{n_2} > 0$$ - 16Let n and k be positive such that $$n \ge {{k(k + 1)} \over 2}$$ . The number of solutions $$\,({x_1},\,{x_2},\,.....{x_k}),\,{x_1}\,\, \ge \,1,\,{x_2}\, \ge \,2,.......,{x_k} \ge k$$, all integers, satisfying $${x_1} + {x_2} + \,..... + {x_k} = n,\,$$ is......................................отговор(A)$${{\left[ {k + \left( {n - {{k(k + 1)} \over 2}} \right) - 1} \right]!} \over {\left[ {n - {{k(k + 1)} \over 2}} \right]!\,(k - 1)\,!}}$$
- 19The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are in AP. Find the intervals in which $$\beta \,\,and\,\gamma $$ lie.отговор(A)$$\beta \, \in \left( { - \infty ,\,{1 \over 3}} \right],\,\gamma \, \in \,\left[ { - {1 \over {27}},\infty } \right)$$
- 23A circle passes through three points A, B and C with the line segment AC as its diameter. A line passing through A angles DAB and CAB are $$\,\alpha \,\,and\,\,\beta $$ respectively and the distance between the point A and the mid point of the line segment DC is d, prove that the area of the circle is $$${{\pi \,{d^2}\,\,{{\cos }^2}\,\,\alpha } \over {{{\cos }^2}\,\alpha \, + \,{{\cos }^2}\,\beta \, + \,\,2\,\cos \,\,\alpha \,\,\cos \,\beta \,\cos \,\,(\beta - \alpha )\,}}$$$отговор(A)$${{pi ,{d^2},,{{cos }^2},,alpha } over {{{cos }^2},alpha , + ,{{cos }^2},eta , + ,,2,cos ,,alpha ,,cos ,eta ,cos ,,(eta - alpha ),}}$$
- 24Find the intervals of value of a for which the line y + x = 0 bisects two chords drawn from a point $$\left( {{{1\, + \,\sqrt 2 a} \over 2},\,{{1\, - \,\sqrt 2 a} \over 2}} \right)$$ to the circle $$\,\,2{x^2}\, + \,2{y^2} - (\,1\, + \sqrt 2 a)\,x - (1 - \sqrt 2 a)\,y = 0$$.отговор(A)a ∈ (-∞, -2) ∪ (2, ∞)
- 25An ellipse has eccentricity $${1 \over 2}$$ and one focus at the point $$P\left( {{1 \over 2},1} \right)$$. Its one directrix is the common tangent, nearer to the point $$P$$, to the circle $${x^2} + {y^2} = 1$$ and the hyperbol;a $${x^2} - {y^2} = 1$$. The equation of the ellipse, in the standard form, is ............отговор(A)${{{{\left( {x - {1 \over 3}} \right)}^2}} \over {{{\left( {{1 \over 3}} \right)}^2}}} + {{{{\left( {y - 1} \right)}^2}} \over {{{\left( {{1 \over {2\sqrt 3 }}} \right)}^2}}} = 1$
- 30A curve $$y=f(x)$$ passes through the point $$P(1, 1)$$. The normal to the curve at $$P$$ is $$a(y-1)+(x-1)=0$$. If the slope of the tangent at any point on the curve is proportional to the ordinate of the point, determine the equation of the curve. Also obtain the area bounded by the $$y$$-axis, the curve and the normal to the curve at $$P$$.отговор(A)y = e^(a(x-1))