Date: March 2014

If A is a square matrix such that A^{2} = I, then find the simplified value of (A – I)^{3} + (A + I)^{3} – 7A.

Chapter: [2.02] Matrices

If `[[x-y,z],[2x-y,w]]=[[-1,4],[0,5]]` find the value of x+y.

Chapter: [2.02] Matrices

If tan^{-1}x+tan^{-1}y=π/4,xy<1, then write the value of x+y+xy.

Chapter: [1.01] Inverse Trigonometric Functions

If `[[3x,7],[-2,4]]=[[8,7],[6,4]]`, find the value of x

Chapter: [2.02] Matrices

If `f(x) =∫_0^xt sin t dt` , then write the value of *f *' (x).

Chapter: [3.05] Integrals

Find the value of 'p' for which the vectors `3hati+2hatj+9hatk and hati-2phatj+3hatk` are parallel

Chapter: [4.02] Vectors

If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.

Chapter: [1.02] Relations and Functions

If the Cartesian equations of a line are ` (3-x)/5=(y+4)/7=(2z-6)/4` , write the vector equation for the line.

Chapter: [4.01] Three - Dimensional Geometry

If `int_0^a1/(4+x^2)dx=pi/8` , find the value of a.

Chapter: [3.05] Integrals

If `veca and vecb` are perpendicular vectors, `|veca+vecb| = 13 and |veca| = 5` ,find the value of `|vecb|.`

Chapter: [4.02] Vectors

Solve the differential equation ` (1 + x2) dy/dx+y=e^(tan^(−1))x.`

Chapter: [3.04] Differential Equations

Show that four points A, B, C and D whose position vectors are

`4hati+5hatj+hatk,-hatj-hatk-hatk, 3hati+9hatj+4hatk and 4(-hati+hatj+hatk)` respectively are coplanar.

Chapter: [4.01] Three - Dimensional Geometry

The scalar product of the vector `veca=hati+hatj+hatk` with a unit vector along the sum of vectors `vecb=2hati+4hatj−5hatk and vecc=λhati+2hatj+3hatk` is equal to one. Find the value of λ and hence, find the unit vector along `vecb +vecc`

Chapter: [4.02] Vectors

Evaluate :

`∫_0^π(4x sin x)/(1+cos^2 x) dx`

Chapter: [3.05] Integrals

Evaluate :

`∫(x+2)/sqrt(x^2+5x+6)dx`

Chapter: [3.05] Integrals

Find the value(s) of x for which y = [x(x − 2)]^{2} is an increasing function.

Chapter: [3.02] Applications of Derivatives

Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .

Chapter: [3.02] Applications of Derivatives

If the function* f* : R → R be given by *f*[*x*] = *x*^{2} + 2 and *g* : R → R be given by `g(x)=x/(x−1)` , x≠1, find fog and gof and hence find fog (2) and gof (−3).

Chapter: [1.02] Relations and Functions

If `tan^(-1)((x-2)/(x-4)) +tan^(-1)((x+2)/(x+4))=pi/4` ,find the value of x

Chapter: [1.01] Inverse Trigonometric Functions

Prove that

`tan^(-1) [(sqrt(1+x)-sqrt(1-x))/(sqrt(1+x)+sqrt(1-x))]=pi/4-1/2 cos^(-1)x,-1/sqrt2<=x<=1`

Chapter: [1.01] Inverse Trigonometric Functions

An experiment succeeds thrice as often as it fails. Find the probability that in the next five trials, there will be at least 3 successes.

Chapter: [6.01] Probability

If y = P e^{ax} + Q e^{bx}, show that

`(d^y)/(dx^2)=(a+b)dy/dx+aby=0`

Chapter: [3.04] Differential Equations

Using properties of determinants, prove that :

`|[1+a,1,1],[1,1+b,1],[1,1,1+c]|=abc + bc + ca + ab`

Chapter: [2.01] Determinants

If x = cos t (3 – 2 cos^{2} t) and y = sin t (3 – 2 sin^{2} t), find the value of dx/dy at t =4/π.

Chapter: [3.01] Continuity and Differentiability

Find the particular solution of the differential equation log(dy/dx)= 3x + 4y, given that y = 0 when x = 0.

Chapter: [3.04] Differential Equations

Find the value of p, so that the lines `l_1:(1-x)/3=(7y-14)/p=(z-3)/2 and l_2=(7-7x)/3p=(y-5)/1=(6-z)/5 ` are perpendicular to each other. Also find the equations of a line passing through a point (3, 2, – 4) and parallel to line l_{1}.

Chapter: [4.01] Three - Dimensional Geometry

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

Chapter: [4.01] Three - Dimensional Geometry

Find the distance of the point (2, 12, 5) from the point of intersection of the line

`vecr=2hati-4hat+2hatk+lambda(3hati+4hatj+2hatk) `

Chapter: [4.01] Three - Dimensional Geometry

Using integration, find the area of the region bounded by the triangle whose vertices are (−1, 2), (1, 5) and (3, 4).

Chapter: [3.03] Applications of the Integrals

A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours for fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30, respectively. The company makes a profit of Rs 80 on each piece of type A and Rs 120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?

Chapter: [5.01] Linear Programming

There are three coins. One is a two-headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the times and the third is also a biased coin that comes up tails 40% of the time. One of the three coins is chosen at random and tossed and it shows heads. What is the probability that it was the two-headed coin?

Chapter: [6.01] Probability

Two the numbers are selected at random (without replacement) from first six positive integers. Let X denote the larger of the two numbers obtained. Find the probability distribution of X. Find the mean and variance of this distribution.

Chapter: [6.01] Probability

Two schools A and B want to award their selected students on the values of sincerity, truthfulness and helpfulness. School A wants to award Rs x each, Rs y each and Rs z each for the three respective values to 3, 2 and 1 students, respectively with a total award money of Rs 1,600. School B wants to spend Rs 2,300 to award 4, 1 and 3 students on the respective values (by giving the same award money to the three values as before). If the total amount of award for one prize on each value is Rs 900, using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for an award.

Chapter: [2.01] Determinants

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum, when the angle between them is 60º.

Chapter: [3.02] Applications of Derivatives

Evaluate :

`int1/(sin^4x+sin^2xcos^2x+cos^4x)dx`

Chapter: [3.05] Integrals

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