Question Paper - Mathematics 2014 - 2015-CBSE 12th-Class 12 CBSE (Central Board of Secondary Education) (CBSE)



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Year2014 - 2015 (March)

Alternate Sets


Marks: 100
Q: 1[1]

If `vec a=7hati+hatj-4hatk and vecb=2hati+6hatj+3hatk` , then find the projection of `vec a and vecb`

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Q: 2[1]

Find λ, if the vectors `veca=hati+3hatj+hatk,vecb=2hati−hatj−hatk and vecc=λhatj+3hatk`  are coplanar.

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Q: 3[1]

If a line makes angles 90°, 60° and θ with x, y and z-axis respectively, where θ is acute, then find θ.

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Q: 4[1]

rite the element a23 of a 3 ✕ 3 matrix A = (aij) whose elements aij are given by `a_(ij)=∣(i−j)/2∣`

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Q: 5[1]

Find the differential equation representing the family of curves v=A/r+ B, where A and B are arbitrary constants.


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Q: 6[1]

Find the integrating factor of the differential equation.


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Q: 7 | Attempt any one[4]
Q: 7.1[4]

If `A=([2,0,1],[2,1,3],[1,-1,0])` find A2-5A+4I and hence find a matrix X such that  A2-5A+4I+X=O

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Q: 7.2[4]

If A = `[[1,-2,3],[0,-1,4],[-2,2,1]]` ,find (A')-1

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Q: 8[4]

If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).

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Q: 9 | Attempt any one[4]
Q: 9.1[4]

Find: `I=intdx/(sinx+sin2x)`

Find: `I=int (d theta)/(sintheta+sin2theta)`

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Q: 9.2[4]

Integrate the following w.r.t. x `(x^3-3x+1)/sqrt(1-x^2)`

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Q: 10[4]

Evaluate :

`∫_(-pi)^pi (cos ax−sin bx)^2 dx`

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Q: 11 | Attempt any one:[4]
Q: 11.1[4]

A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B, If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.

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Q: 11.2[4]

An unbiased coin is tossed 4 times. Find the mean and variance of the number of heads obtained.

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Q: 12[4]

If `vecr=xhati+yhatj+zhatk` ,find `(vecrxxhati).(vecrxxhatj)+xy`

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Q: 13[4]

Find the distance between the point (−1, −5, −10) and the point of intersection of the line `(x-2)/3=(y+1)/4=(z-2)/12` and the plane x-y+z=5

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Q: 14 | Attempt any one [4]
Q: 14.1[4]

If sin [cot−1 (x+1)] = cos(tan1x), then find x.

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Q: 14.2[4]

If (tan1x)2 + (cot−1x)2 = 5π2/8, then find x.

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Q: 15[4]

If `y=tan^(−1) ((sqrt(1+x^2)+sqrt(1−x^2))/(sqrt(1+x^2)−sqrt(1−x^2)))` , x21, then find dy/dx.

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Q: 16[4]

If x = a cos θ + b sin θ, y = a sin θ − b cos θ, show that `y^2 (d^2y)/(dx^2)-xdy/dx+y=0`

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Q: 17[4]

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

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Q: 18[4]

Find : `int(x+3)sqrt(3-4x-x^2dx)`

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Q: 19[4]

Three schools A, B and C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs 25, Rs 100 and Rs 50 each. The number of articles sold are given below:


Find the funds collected by each school separately by selling the above articles. Also find the total funds collected for the purpose.

Write one value generated by the above situation.

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Q: 20[6]

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

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Q: 21 | Attempt any one :[6]
Q: 21.1[6]

Using integration find the area of the triangle formed by positive x-axis and tangent and normal of the circle

`x^2+y^2=4 at (1, sqrt3)`

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Q: 21.2[6]

Evaluate `int_1^3(e^(2-3x)+x^2+1)dx`  as a limit of sum.

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Q: 22 | Attempt any one [6]
Q: 22.1[6]

Solve the differential equation :


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Q: 22.2[6]

Find the particular solution of the differential equation `dy/dx=(xy)/(x^2+y^2)` given that y = 1, when x = 0.

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Q: 23[6]

If lines `(x−1)/2=(y+1)/3=(z−1)/4 and  (x−3)/1=(y−k)/2=z/1` intersect, then find the value of k and hence find the equation of the plane containing these lines.

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Q: 24[6]

If A and B are two independent events such that `P(barA∩ B) =2/15 and P(A ∩ barB) = 1/6`, then find P(A) and P(B).

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Q: 25[6]

Find the local maxima and local minima, of the function f(x) = sin x − cos x, 0 < x < 2π.

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Q: 26[6]

Find graphically, the maximum value of z = 2x + 5y, subject to constraints given below :

2x + 4y  83

x + y  6

x + y  4

x  0, y 0

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