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General Instructions:
(i) All questions are compulsory.
(ii) This question paper contains 29 questions.
(iii) Questions 1- 4 in Section A are very short-answer type questions carrying 1 mark each.
(iv) Questions 5-12 in Section B are short-answer type questions carrying 2 marks each.
(v) Questions 13-23 in Section C are long-answer I type questions carrying 4 marks each.
(vi) Questions 24-29 in Section D are long-answer II type questions carrying 6 marks each.
Question 1
• Q1

Determine the value of 'k' for which the following function is continuous at x = 3:

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• Q2

If for any 2 × 2 square matrix A, A(adj A) = $\left[\begin{array}{cc}8& 0\\ 0& 8\end{array}\right]$, then write the value of |A|.

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• Q3

Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20.

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• Q4
• Q5

Find :

$\int \frac{\mathrm{dx}}{5-8\mathrm{x}-{\mathrm{x}}^{2}}$

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• Q6

Two tailors, A and B earn Rs 300 and Rs 400 per day respectively. A can stitch 6 shirts and 4 pairs of trousers while B can stitch 10 shirts and 4 pairs of trousers per day. To find how many days should each of them work and if it is desired to produce at least 60 shirts and 32 pairs of trousers at a minimum labour cost, formulate this as an LPP.

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• Q7

A die, whose faces are marked 1, 2, 3, in red and 4, 5, 6 in green, is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.

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• Q8

The x-coordinate of a point on the line joining the points P(2, 2, 1) and Q(5, 1, –2) is 4. Find its z-coordinate.

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• Q9

Show that the function f(x) = x3 – 3x2 + 6x – 100 is increasing on ℝ.

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• Q10

Find the value of c in Rolle's theorem for the function f(x) = x3 – 3x in  .

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• Q11

If A is a skew-symmetric matrix of order 3, then prove that det A = 0.

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• Q12

The volume of a sphere is increasing at the rate of 8 cm3/s. Find the rate at which its surface area is increasing when the radius of the sphere is 12 cm.

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• Q13

There are 4 cards numbered 1, 3, 5 and 7, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.

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• Q14

Show that the points A, B, C with position vectors $2\stackrel{^}{\mathrm{i}}-\stackrel{^}{\mathrm{j}}+\stackrel{^}{\mathrm{k}}$, $\stackrel{^}{\mathrm{i}}-3\stackrel{^}{\mathrm{j}}-5\stackrel{^}{\mathrm{k}}$ and $3\stackrel{^}{\mathrm{i}}-4\stackrel{^}{\mathrm{j}}-4\stackrel{^}{\mathrm{k}}$ respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle.

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• Q15

Of the students in a school, it is known that 30% have 100% attendance and 70% students are irregular. Previous year results report that 70% of all students who have 100% attendance attain A grade and 10% irregular students attain A grade in their annual examination. At the end of the year, one student is chosen at random from the school and he was found to have an A grade. What is the probability that the student has 100% attendance? Is regularity required only in school? Justify your answer.

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• Q16

If , then find the value of x.

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• Q17

Using properties of determinants, prove that

$\left|\begin{array}{ccc}{a}^{2}+2a& 2a+1& 1\\ 2a+1& a+2& 1\\ 3& 3& 1\end{array}\right|={\left(a-1\right)}^{3}$

OR

Find matrix A such that

$\left(\begin{array}{cc}2& -1\\ 1& 0\\ -3& 4\end{array}\right)A=\left(\begin{array}{cc}-1& -8\\ 1& -2\\ 9& 22\end{array}\right)$

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• Q18

If xy + yx = ab, then find $\frac{\mathrm{dy}}{\mathrm{dx}}.$

OR

If ey(x + 1) = 1, then show that $\frac{{\mathrm{d}}^{2}\mathrm{y}}{{\mathrm{dx}}^{2}}={\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{2}.$

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• Q19

Evaluate :

OR

Evaluate :

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• Q20

Solve the following linear programming problem graphically :
Maximise Z = 7x + 10y
subject to the constraints
4x + 6y ≤ 240
6x + 3y ≤ 240
x ≥ 10
x ≥ 0, y ≥ 0

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• Q21

Find :

$\int \frac{{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}}{{\left({\mathrm{e}}^{\mathrm{x}}-1\right)}^{2}\left({\mathrm{e}}^{\mathrm{x}}+2\right)}$

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• Q22

If , then express $\stackrel{\to }{b}$ in the form of $\stackrel{\to }{b}=\stackrel{\to }{{b}_{1}}+\stackrel{\to }{{b}_{2}}$, where $\stackrel{\to }{{b}_{1}}$ is parallel to is perpendicular to $\stackrel{\to }{a}$.

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• Q23

Find the general solution of the differential equation .

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• Q24

Using the method of integration, find the area of the triangle ABC, coordinates of whose vertices are A(4, 1), B(6, 6) and C(8, 4).

OR

Find the area enclosed between the parabola 4y = 3x2 and the straight line 3x – 2y + 12 = 0.

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• Q25

Find the particular solution of the differential equation $\left(x-y\right)\frac{dy}{dx}=\left(x+2y\right)$, given that y = 0 when x = 1.

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• Q26

Find the coordinates of the point where the line through the points (3, –4, –5) and (2, –3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2, –3) and (0, 4, 3).

OR

A variable plane which remains at a constant distance 3p from the origin cuts the coordinate axes at A, B, C. Show that the locus of the centroid of triangle ABC is $\frac{1}{{x}^{2}}+\frac{1}{{y}^{2}}+\frac{1}{{z}^{2}}=\frac{1}{{p}^{2}}$.

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• Q27

Consider . Show that f is bijective. Find the inverse of f and hence find f–1 (0) and x such that f–1 (x) = 2.

OR

Let $\mathrm{A}=\mathrm{ℚ}×\mathrm{ℚ}$ and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for (a, b), (c, d) ∊ A. Determine, whether * is commutative and associative. Then, with respect to * on A
(i) Find the identity element in A.
(ii) Find the invertible elements of A.

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• Q28

If , then find A–1 and hence solve the system of linear equations .

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• Q29

A window is in the form of a rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

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