question_answer2) Let \[\hat{a}\] and \[\hat{b}\] be two unit vectors. If the vectors \[\vec{c}=\hat{a}+2\hat{b}\] and \[\vec{d}=5\hat{a}-4\hat{b}\] are perpendicular to each other, then the angle between \[\hat{a}\] and \[\hat{b}\] is:
AIEEE Solved Paper-2012
question_answer3) A spherical balloon is filled with \[4500\pi \] cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \[72\pi \] cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is:
AIEEE Solved Paper-2012
question_answer4) Statement-1: The sum of the series 1 + (1 + 2 + 4) + (4 + 6 + 9) + (9 + 12 + 16) + .... + (361 + 380 + 400) is 8000. Statement-2: \[\sum\limits_{k=1}^{n}{({{k}^{3}}-{{(k-1)}^{3}}={{n}^{3}}}\], for any natural number n.
AIEEE Solved Paper-2012
A)
Statement-1 is false, Statement-2 is true.
doneclear
B)
Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
doneclear
C)
Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
question_answer6) If the integral \[\int{\frac{5\tan x}{\tan x-2}dx=x+a\,\ell n\left| \sin x-2\cos x \right|+k}\], then a is equal to:
AIEEE Solved Paper-2012
question_answer7) Statement-1: An equation of a common tangent to the parabola \[{{y}^{2}}=16\sqrt{3}x\] and the ellipse \[2{{x}^{2}}+{{y}^{2}}=4\] is \[y=2x+2\sqrt{3}\]. Statement-2: If the line \[mx+\frac{4\sqrt{3}}{m},(m\ne 0)\] is a common tangent to the parabola \[{{y}^{2}}=16\sqrt{3}x\] and the ellipse \[2{{x}^{2}}+{{y}^{2}}=4\], then m satisfies\[{{m}^{4}}+2{{m}^{2}}=24\].
AIEEE Solved Paper-2012
A)
Statement-1 is false, Statement-2 is true.
doneclear
B)
Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
doneclear
C)
Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
question_answer9) If n is a positive integer, then \[{{\left( \sqrt{3}+1 \right)}^{2n}}-{{\left( \sqrt{3}-1 \right)}^{2n}}\] is:
AIEEE Solved Paper-2012
question_answer10) If 100 times the 100th term of an AP with non zero common difference equals the 50 times its 50th term, then the 150th term of this AP is :
AIEEE Solved Paper-2012
question_answer13) If the line \[2x+y=k\] passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals :
AIEEE Solved Paper-2012
question_answer14) Let \[{{x}_{1}},{{x}_{2}},\,....\,,{{x}_{n}}\] be n observations, and let \[\overline{x}\] be their arithmetic mean and \[{{\sigma }^{2}}\] be the variance Statement-1: Variance of \[2{{x}_{1}},2{{x}_{2}},\,......,\,2{{x}_{n}}\] is \[4{{\sigma }^{2}}\]. Statement-2: Arithmetic mean \[2{{x}_{1}},2{{x}_{2}},\,......,\,2{{x}_{n}}\] is \[4\overline{x}\].
AIEEE Solved Paper-2012
A)
Statement-1 is false, Statement-2 is true.
doneclear
B)
Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
doneclear
C)
Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
question_answer15) The population p(t) at time t of a certain mouse species satisfies the differential equation\[\frac{dp(t)}{dt}=0.5\,p(t)-450\]If \[p(0)=850\], then the time at which the population becomes zero is :
AIEEE Solved Paper-2012
question_answer16) Let a, \[b\in R\] be such that the function f given by \[f(x)=\ell n\left| x \right|+b{{x}^{2}}+ax,\,x\ne 0\] has extreme values at \[x=-1\] and \[x=2\]. Statement-1: f has local maximum at \[x=-1\] and at \[x=2\]. Statement-2: \[a=\frac{1}{2}\] and \[b=\frac{-1}{4}\].
AIEEE Solved Paper-2012
A)
Statement-1 is false, Statement-2 is true.
doneclear
B)
Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.
doneclear
C)
Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.
question_answer17) The area bounded between the parabolas \[{{x}^{2}}=\frac{y}{4}\] and \[{{x}^{2}}=9y\] and the straight line \[y=2\]is:
AIEEE Solved Paper-2012
question_answer18) Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is:
AIEEE Solved Paper-2012
question_answer19) If f : \[R\to R\] is a function defined by \[f(x)=[x]\cos \left( \frac{2x-1)}{2} \right)\pi \], where \[[x]\] denotes the greatest integer function, then f is:
AIEEE Solved Paper-2012
A)
continuous for every real \[x\].
doneclear
B)
discontinuous only at \[x=0\].
doneclear
C)
discontinuous only at non-zero integral values of \[x\].
question_answer20) If the line \[\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}\] and \[\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}\] intersect, then k is equal to :
AIEEE Solved Paper-2012
question_answer21) Three numbers are chosen at random without replacement from {1, 2, 3, ..., 8}. The probability that their minimum is 3, given that their maximum is 6, is :
AIEEE Solved Paper-2012
question_answer22) If \[z\ne 1\] and \[\frac{{{z}^{2}}}{z-1}\] is real, then the point represented by the complex number z lies :
AIEEE Solved Paper-2012
A)
either on the real axis or on a circle passing through the origin.
doneclear
B)
on a circle with centre at the origin.
doneclear
C)
either on the real axis or on a circle not passing through the origin.
question_answer23) Let P and Q be \[3\times 3\] matrices \[P\ne Q\]. If \[{{P}^{3}}={{Q}^{3}}\] and \[{{P}^{2}}={{Q}^{2}}\], then determinant of \[({{P}^{2}}={{Q}^{2}})\] is equal to :
AIEEE Solved Paper-2012
question_answer25) The length of the diameter of the circle which touches the x-axis at the point (1, 0) and passes through the point (2, 3) is :
AIEEE Solved Paper-2012
question_answer26) Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that \[Y\subseteq X,Z\subseteq X\] and \[Y\cap Z\] is empty, is :
AIEEE Solved Paper-2012
question_answer27) An ellipse is drawn by taking a diameter of the circle \[{{(x-1)}^{2}}+{{y}^{2}}=1\] as its semi-minor axis and a diameter of the circle \[{{x}^{2}}+{{(y-2)}^{2}}=4\] is semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is :
AIEEE Solved Paper-2012
question_answer29) A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a triangle OPQ, where O is the origin. if the area of the triangle OPQ is least, then the slope of the line PQ is :
AIEEE Solved Paper-2012
question_answer30) Let ABCD be a parallelogram such that \[\overrightarrow{AB}=\vec{q},\,\,\overrightarrow{AD}=\vec{p}\] and \[\angle BAD\] be an acute angle. If \[\vec{r}\] is the vector that coincides with the altitude directed from the vertex B to the side AD, then \[\vec{r}\] is given by :
AIEEE Solved Paper-2012