question_answer3) If in a frequency distribution, the mean and median are 21 and 22 respectively, then its mode is approximately
AIEEE Solved Paper-2005
question_answer4) Let\[R=\{(3,3),(6,6),(9,9),(12,12),\]\[(6,12),\]\[(3,9),(3,12),(3,6)\}\]be a relation on the set A\[=\{3,6,9,12\}\]. The relation is
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question_answer6) If the cube roots of unity are\[1,\omega ,{{\omega }^{2}}\]then the roots of the equation \[{{(x-1)}^{3}}+8=0,\] are
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question_answer8) Area of the greatest rectangle that can be inscribed in the ellipse\[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is
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question_answer9) The differential equation representing the family of curves\[{{y}^{2}}=2c(x+\sqrt{c}),\]where\[c>0,\]is a parameter, is of order and degree as follows
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question_answer10) ABC is a triangle. Forces P,Q,R acting along \[I,A,IB\]and\[IC\]respectively are in equilibrium, where I is the incentre of\[\Delta ABC.\]Then, P : Q : R is
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question_answer11) If the coefficients of rth,\[(r+1)th\] and\[(r+2)th\]terms in the binomial expansion of\[{{(1+y)}^{m}}\]are in AP, then m and r satisfy the equation
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question_answer13) If the letters of the word SACHIN are arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
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question_answer16) If the coefficient of\[{{x}^{7}}\]in\[{{\left[ a{{x}^{2}}+\left( \frac{1}{bx} \right) \right]}^{11}}\]equals the coefficient of\[{{x}^{-7}}\]in\[{{\left[ ax-\left( \frac{1}{b{{x}^{2}}} \right) \right]}^{11}},\]then a and b satisfy the relation
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question_answer17) Let\[f:(-1,1)\to B\] be a function defined by\[f(x)={{\tan }^{-1}}\frac{2x}{1-{{x}^{2}}},\] then f is both one-one and onto when B is the interval
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question_answer18) If\[{{z}_{1}}\]and\[{{z}_{2}}\]are two non-zero complex numbers such that\[|{{z}_{1}}+{{z}_{2}}|=|{{z}_{1}}|+|{{z}_{2}}|\]then \[\arg ({{z}_{1}})-\arg ({{z}_{2}})\]is equal to
question_answer21) The system of equations \[\alpha x+y+z=\alpha -1,x+\alpha \,y+z=\alpha -l\] \[x+y+\alpha z=\alpha -1\]has no solution, if \[\alpha \] is
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question_answer22) The value of a for which the sum of the squares of the roots of the equation\[{{x}^{2}}-(a-2)x-a-1=0\]assume the least value is
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question_answer27) If\[x\]is so small that\[{{x}^{3}}\]and higher powers of\[x\] may be neglected, then \[\frac{{{(1+x)}^{3/2}}-{{\left( 1+\frac{1}{2}x \right)}^{3}}}{{{(1-x)}^{1/2}}}\]may be approximated as
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question_answer28) If\[x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}},y=\sum\limits_{n=0}^{\infty }{{{b}^{n}}},z=\sum\limits_{n=0}^{\infty }{{{c}^{n}}},\]where a, b, c are in AP and\[|a|<1,|b|<1,|c|<1,\]then\[x,\text{ }y,\text{ }z\]are in
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question_answer29) In a\[\Delta ABC,\]let \[\angle C=\pi /2,\] if r is the inradius and R is the circumradius of the\[\Delta ABC,\]then \[2(r+R)\]equals
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question_answer31) If in a\[\Delta ABC,\]the altitudes from the vertices A,B,C on opposite sides are in HP, then sin A, sin B, sin C are in
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question_answer32) The normal to the curve \[x=a(\cos \theta +\theta \sin \theta ),y=a(\sin \theta -\theta \cos \theta )\]at any point\['\theta '\]is such that
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question_answer33) A function is matched below against an interval, where it is supposed to be increasing. Which of the following pairs is incorrectly matched? Interval Function
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question_answer34) Let \[\alpha \] and \[\beta \] be the distinct roots of\[a{{x}^{2}}+bx+c=0,\]then \[\underset{x\to \alpha }{\mathop{\lim }}\,\frac{1-\cos (a{{x}^{2}}+bx+c)}{{{(x-\alpha )}^{2}}}\]is equal to
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question_answer36) The line parallel to the X-axis and passing through the intersection of the lines \[ax+2\,by+3b=0\]and\[bx-2ay-3a=0,\]where \[(a,b)\ne (0,0)\]is
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question_answer37) A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of\[50\text{ }c{{m}^{3}}/\min \]. When the thickness of ice is 15 cm, then the rate at which the thickness of ice decreases, is
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question_answer40) Let\[f(x)\]be a non-negative continuous function such that the area bounded by the curve\[y=f(x),\]X-axis and the ordinates\[x=\pi /4\] and \[x=\beta >\pi /4\] is\[\left( \beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta \right)\].Then\[f\left( \frac{\pi }{2} \right)\],is
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question_answer43) The parabolas\[{{y}^{2}}=4x\]and\[{{x}^{2}}=4y\]divide the square region bounded by the lines \[x=4,\text{ }y=4\]and the coordinate axes. If \[{{S}_{1}},{{S}_{2}},{{S}_{3}}\]are respectively the areas of these parts numbered from top to bottom, then\[{{S}_{1}}:{{S}_{2}}:{{S}_{3}}\]
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question_answer44) If the plane\[2ax-3ay+4az+6=0\]passes through the mid-point of the line joining the centres of the spheres\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}+6x-8y-2z=13\] and\[{{x}^{2}}+{{y}^{2}}+{{z}^{2}}-10x+4y-2z=8,\]then a equals
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question_answer45) The distance between the line\[r=2\hat{i}-2\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})\]and the plane\[r.(\hat{i}+5\hat{j}+\hat{k})=5\]is
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question_answer46) For any vector a, the value of\[{{(a\times \hat{i})}^{2}}+{{(a\times \hat{j})}^{2}}+{{(a\times \hat{k})}^{2}}\]is equal to
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question_answer47) If non-zero numbers a, b, c are in HP, then the straight line \[\frac{x}{a}+\frac{y}{b}+\frac{1}{c}=0\]always passes through a fixed point. That point is
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question_answer48) It a vertex of a triangle is (1, 1) and the mid-points of two sides through this vertex are (-1, 2) and (3, 2), then the centroid of the triangle is
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question_answer49) If the circles\[{{x}^{2}}+{{y}^{2}}+2ax+cy+a=0\]and \[{{x}^{2}}+{{y}^{2}}-3\text{ }ax+dy-1=0\]intersect in two distinct points P and 0, then the line \[5x+by-a=0\]passes through P and Q for
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question_answer50) A circle touches the X-axis and also touches the circle with centre at (0, 3) and radius 2. The locus of the centre of the circle is
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question_answer51) If a circle passes through the point (a, b) and cuts the circle\[{{x}^{2}}+{{y}^{2}}={{p}^{2}}\]orthogonally, then the equation of the locus of its centre is
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question_answer52) An ellipse has OB as semi-minor axis, F and F' its foci and the angle FBF' is a right angle. Then, the eccentricity of the ellipse is
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question_answer53) The locus of a point \[P(\alpha ,\beta )\] moving under the condition that the line\[y=\alpha x+\beta \]is a tangent to the hyperbola\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1\]is
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question_answer54) If the angle\[\theta \]between the line\[\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}\]and the plane \[2x-y+\sqrt{\lambda }z+4=0\]is such that \[\sin \theta =\frac{1}{3}\]The value of\[\lambda \]is
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question_answer56) Let A and B be two events such that \[P\overline{(A\cup B)}=\frac{1}{6},P(A\cap B)=\frac{1}{4}\]and\[P(\overline{A})=\frac{1}{4},\]where \[\overline{A}\] stands for complement of event A. Then, events A and B are
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question_answer57) Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house, is
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question_answer59) Two points A and B move from rest along a straight line with constant acceleration f and f?, respectively. If A takes m second more than B and describes n unit more than B in acquiring the same speed, then
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question_answer60) A lizard, at an initial distance of 21 cm behind an insect, moves from rest with an acceleration of\[2\text{ }cm/{{s}^{2}}\]and pursues the insect which is crawling uniformly along a straight line at a speed of 20 cm/s. Then, the lizard will catch the insect after
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question_answer61) The resultant R of two forces acting on a particle is at right angles to one of them and its magnitude is one-third of the other force. The ratio of larger force to smaller one is
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question_answer63) Let a, b and c be distinct non-negative numbers. If the vectors\[a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}\]and \[c\hat{i}+c\hat{j}+b\hat{k}\] lie in a plane, then c is
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question_answer64) If a, b, b are non-coplanar vectors and \[\lambda \], is a real number, then\[[\lambda (a+b){{\lambda }^{2}}\,b\,\,\lambda c]=[ab+cd]\]for
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question_answer65) A and B are two like parallel forces. A couple of moment H lies in the plane of A and B and is contained with them. The resultant of A and B after combining is displaced through a distance
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question_answer67) Let\[{{x}_{1}},{{x}_{2}},.....,{{x}_{n}}\]be n observations such that \[\Sigma x_{i}^{2}=400\]and\[\Sigma {{x}_{i}}=80\]. Then, a possible value of n among the following is
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question_answer68) A particle is projected from a point O with velocity u at an angle of\[60{}^\circ \]with the horizontal. When it is moving in a direction at right angle to its direction at O, then its velocity is given by
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question_answer69) If both the roots of the quadratic equation\[{{x}^{2}}-2kx+{{k}^{2}}+k-5=0\]are less than 5, then k lies in the interval
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question_answer71) A real valued function\[f(x)\]satisfies the functional equation \[f(x-y)=f(x)\text{ }f(y)-f(a-x)\text{ }f(a+y)\] where, a is a given constant and\[f(0)=1\]\[f(2\text{ }a-x)\]is equal to
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question_answer72) If the equation \[{{a}_{n}}{{X}^{n}}+{{a}_{n-1}}{{X}^{n-1}}+....+{{a}_{1}}x=0,\] \[{{a}_{1}}\ne 0,n\ge 2,\]has a positive root \[x=\alpha ,\] then the equation \[n{{a}_{n}}{{x}^{n-1}}+(n-1){{a}_{n-1}}{{X}^{n-2}}+....+{{a}_{1}}=0\] has a positive root, which is
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question_answer75) If the pair of linesa\[{{x}^{2}}+2(a+b)xy+b{{y}^{2}}=0\] lie along diameters of a circle and divide the circle into four sectors such that the area of one of the sector is thrice the area of another sector, then
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