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General Instructions :
(i) All questions are compulsory.
(ii) Please check that this Question Paper contains 26 Questions.
(iii) Marks for each question are indicated against it.
(iv) Questions 1 to 6 in Section-A are Very Short Answer Type Questions carrying one mark each.
(v) Questions 7 to 19 in Section-B are Long Answer I Type Questions carrying 4 marks each.
(vi) Questions 20 to 26 in Section-C are Long Answer II Type Questions carrying 6 marks each.
(vii) Please write down the serial number of the Question before attempting it.
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Question 1
• Q1

Write the number of vectors of unit length perpendicular to both the vectors .

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• Q2

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

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• Q3

If x ∈ N and = 8, then find the value of x.

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• Q4

Write the position vector of the point which divides the join of points with position vectors in the ratio 2 : 1.

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• Q5

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

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• Q6

Use elementary column operation C2 → C2 + 2C1 in the following matrix equation :

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• Q7

The equation of tangent at (2, 3) on the curve y2 = ax3 + b is y = 4x – 5. Find the values of a and b.

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• Q8

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.

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• Q9

Find : $\int \left(3x+1\right)\sqrt{4-3x-2{x}^{2}}dx$

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• Q10

The two adjacent sides of a parallelogram are . Find the two unit vectors parallel to its diagonals. Using the diagonal vectors, find the area of the parallelogram.

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• Q11

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

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• Q12

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.

OR

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?

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• Q13

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?

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• Q14

Differentiate ${x}^{\mathrm{sin}x}+{\left(\mathrm{sin}x\right)}^{\mathrm{cos}x}$ with respect to x.

OR

If , prove that ${x}^{2}\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^{2}}+x\frac{\mathrm{d}y}{\mathrm{d}x}+y=0$.

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• Q15

Solve the equation for $x:{\mathrm{sin}}^{-1}x+{\mathrm{sin}}^{-1}\left(1-x\right)={\mathrm{cos}}^{-1}x$

OR

If ${\mathrm{cos}}^{-1}\frac{x}{a}+{\mathrm{cos}}^{-1}\frac{y}{\mathrm{b}}=\alpha$, prove that

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• Q16
• Q17

Solve the differential equation :

$y+x\frac{\mathrm{d}y}{\mathrm{d}x}=x-y\frac{\mathrm{d}y}{\mathrm{d}x}$

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• Q18

Evaluate : $\underset{0}{\overset{\frac{\mathrm{\pi }}{2}}{\int }}\frac{{\mathrm{sin}}^{2}x}{\mathrm{sin}x+\mathrm{cos}x}\mathrm{d}x$

OR

Evaluate :

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• Q19

Find : $\int \frac{{x}^{2}}{{x}^{4}+{x}^{2}-2}\mathrm{d}x$

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• Q20

Using properties of determinants, show that ΔABC is isosceles if:

$\left|\begin{array}{ccc}1& 1& 1\\ 1+\mathrm{cosA}& 1+\mathrm{cosB}& 1+\mathrm{cosC}\\ {\mathrm{cos}}^{2}\mathrm{A}+\mathrm{cosA}& {\mathrm{cos}}^{2}\mathrm{B}+\mathrm{cosB}& {\mathrm{cos}}^{2}\mathrm{C}+\mathrm{cosC}\end{array}\right|=0$

OR

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.

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• Q21

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.

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• Q22

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}$r.

OR

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 $\frac{\mathrm{\pi }}{3}.$

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• Q23

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.

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• Q24

Prove that the curves y2 = 4x and x2 = 4y divide the area of square bounded by x = 0, x = 4, y = 4 and y = 0 into three equal parts.

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• Q25

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.

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• Q26

Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\stackrel{^}{i}+3\stackrel{^}{j}+4\stackrel{^}{k}$ to the plane $\stackrel{\to }{r}.\left(2\stackrel{^}{i}+\stackrel{^}{j}+3\stackrel{^}{k}\right)-26=0$. Also find image of P in the plane.

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