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Maths | CBSE Board | Grade 12 | 2012
Q. An open box with a square base is to be made out of a given quantity of cardboard of area c2 square units. Show that the maximum volume of the box is c363 cubic units.
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Q. Find the coordinates of the point where line through the points A=(3, 4, 1) and B=(5, 1, 6) crosses the xy-plane.
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Q. Find the particular solution of the differential equation x(x21)dydx=1, y=0 when x=2.
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Q. If x=asin1t, y=acos1t, show that dydx=yx.
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Q. Consider the binary operations :R×RR and o:R×RR defined as ab=|ab| and aob=a for all a, bR. Show that is commutative but not associative, 'o' is associative but not commutative.
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Q. Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.
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Q. Find the principal value of tan13sec1(2).
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Q. Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
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Q. Evaluate: 21|x3x|dx.
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Q. Evaluate: 10xsin1x1x2dx.
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Q. Write the value of (^i×^j)^k+^i^j.
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Q.
Show that f:NN, given by
f(x)={x+1, if x is oddx1, if x is even
is both one-one and onto.
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Q. Find the scalar components of the vector AB with initial point A(2, 1) and terminal point B(5, 7).
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Q. Let A be a square matrix of order 3×3. Write the value of |2A|, where |A|=4.
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Q. Using matrices, solve the following system of equations:
2x+3y+3z=5, x2y+z=4, 3xy2z=3.
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Q. Two cards are drawn simultaneously (without replacement) from a well-shuffled pack of 52 cards. Find the mean and variance of the number of red cards.
  1. Mean =0.6 and Variance =0.3
  2. Mean =0.1 and Variance =0.7
  3. Mean =0.49 and Variance =0.37
  4. Mean =0 and Variance =0.45
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Q. A 13m long ladder is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is it's height on the wall decreasing when the foot of the ladder is 5 m away from the wall?
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Q. Evaluate: 204x2dx.
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Q. Prove the following:
cos(sin135+cot132)=6513.
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Q. Solve the following differential equation: (1+x2)dy+2xy dx=cotx dx;x0
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Q. The binary operation :R×RR is defined as ab=2a+b. Find (23)4.
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Q. Using properties of determinants, show that ∣ ∣b+caabc+abcca+b∣ ∣=4abc.
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Q. A dietician wishes to mix two types of foods in such away that the vitamin contents of the mixture contains at least 8 units of vitamin A and 10 units of vitamin C. Food I contains 2 units/kg of vitamin A and 1 unit/kg of vitamin C while Food II contains 1 unit/kg of vitamin A and 2 units/kg of vitamin 1 unit/kg of vitamin C. It costs Rs.5 per kg to purchase food I and Rs.7 per kg to purchase Food II. Determine the maximum cost of such a mixture. Formulate the above as a LPP and solve it graphically.
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Q. Find the value of x+y from the following equation:
2[x57y3]+[3412]=[761514]
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Q. Let a=^i+4^j+2^k, b=3^i2^j+7^k and c=2^i^j+4^k. Find a vector p which is perpendicular to both a and b and pc=18.
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Q. Find the distance of a point 3x4y+122=3 from the origin.
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Q. Differentiate tan1[1+x21x] with respect to x.
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Q. Given ex(tanx+1)secxdx=exf(x)+c. Find f(x).
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Q. If AT=341201 and B=[121123], then find ATBT.
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Q. If x=a(cost+tsint) and y=a(sinttcost), 0<t<π2, find d2xdt2, d2ydt2 and d2ydx2.
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