Date & Time: 14th March 2016, 10:30 am

Duration: 3h

Write the number of vectors of unit length perpendicular to both the vectors `veca=2hati+hatj+2hatk and vecb=hatj+hatk`

Chapter: [4.02] Vectors

Write the number of all possible matrices of order 2 × 2 with each entry 1, 2 or 3.

Chapter: [2.02] Matrices

If ` x in N and |[x+3,-2],[-3x,2x]|=8` , then find the value of *x*.

Chapter: [2.01] Determinants

Write the position vector of the point which divides the join of points with position vectors `3veca-2vecb and 2veca+3vecb` in the ratio 2 : 1.

Chapter: [4.02] Vectors

Find the vector equation of the plane with intercepts 3, –4 and 2 on *x*, *y* and *z*-axis respectively.

Chapter: [4.01] Three - Dimensional Geometry

Use elementary column operation C_{2} → C_{2} + 2C_{1} in the following matrix equation :

`[[2,1],[2,0]] = [[3,1],[2,0]] [[1,0],[-1,1]]`

Chapter: [2.02] Matrices

The equation of tangent at (2, 3) on the curve y^{2} = ax^{3} + b is y = 4x – 5. Find the values of a and b.

Chapter: [3.02] Applications of Derivatives

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

Chapter: [4.01] Three - Dimensional Geometry

find : `int(3x+1)sqrt(4-3x-2x^2)dx`

Chapter: [3.05] Integrals

The two adjacent sides of a parallelogram are `2hati-4hatj-5hatk and 2 hati+2hatj+3hatj` . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

Chapter: [4.02] Vectors

Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

Chapter: [3.04] Differential Equations

In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses

Chapter: [6.01] Probability

A bag contains 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?

Chapter: [6.01] Probability

A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. Using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question?

Chapter: [2.02] Matrices

Differentiate x^{sinx}+(sinx)^{cosx} with respect to x.

Chapter: [3.01] Continuity and Differentiability

If y=2 cos(logx)+3 sin(logx), prove that `x^2(d^2y)/(dx2)+x dy/dx+y=0`

Chapter: [3.01] Continuity and Differentiability

Solve the equation for x:sin^{−1}x+sin^{−1}(1−x)=cos^{−1}x

Chapter: [1.01] Inverse Trigonometric Functions

If `cos^-1( x/a) +cos^-1 (y/b)=alpha` , prove that `x^2/a^2-2(xy)/(ab) cos alpha +y^2/b^2=sin^2alpha`

Chapter: [1.01] Inverse Trigonometric Functions

If x=a sin 2t(1+cos 2t) and y=b cos 2t(1−cos 2t), find `dy/dx `

Chapter: [3.01] Continuity and Differentiability

Solve the differential equation :

`y+x dy/dx=x−y dy/dx`

Chapter: [3.04] Differential Equations

Evaluate : `∫_0^(π/2)(sin^2 x)/(sinx+cosx)dx`

Chapter: [3.05] Integrals

Evaluate `∫_0^(3/2)|x cosπx|dx`

Chapter: [3.05] Integrals

Find : `int x^2/(x^4+x^2-2) dx`

Chapter: [3.05] Integrals

Using properties of determinants, show that ΔABC is isosceles if:`|[1,1,1],[1+cosA,1+cosB,1+cosC],[cos^2A+cosA,cos^B+cosB,cos^2C+cosC]|=0`

Chapter: [2.01] Determinants

A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety, 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.

Chapter: [2.02] Matrices

There are two types of fertilisers 'A' and 'B'. 'A' consists of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid. After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs Rs 10 per kg and 'B' cost Rs 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost

Chapter: [5.01] Linear Programming

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is `6sqrt3` r.

Chapter: [3.02] Applications of Derivatives

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is π/3.

Chapter: [3.02] Applications of Derivatives

Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.

Chapter: [6.01] Probability

Prove that the curves y^{2 }= 4x and x^{2} = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

Chapter: [3.03] Applications of the Integrals

Show that the binary operation * on A = **R** – { – 1} defined as a*b = a + b + ab for all a, b ∈ A is commutative and associative on A. Also find the identity element of * in A and prove that every element of A is invertible.

Chapter: [1.02] Relations and Functions

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector

`2hati+3hatj+4hatk` to the plane `vecr` . `(2hati+hatj+3hatk)−26=0` . Also find image of P in the plane.

Chapter: [4.02] Vectors

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