General Instructions :
(i) All questions are compulsory.
(ii) The question paper consists of 29 questions divided into four sections A, B, C and D. Section A comprises of 4 questions of one mark each, Section B comprises of 8 questions of two marks each, Section C comprises of 11 questions of four marks each and Section D comprises of 6 questions of six marks each.
(iii) All questions in Section A are to be answered in one word, one sentence or
as per the exact requirement of the question.
(iv) There is no overall choice. However, internal choice has been provided in
3 questions of four marks each and 3 questions of six marks each. You
have to attempt only one of the alternatives in all such questions.
(v) Use of calculators is not permitted. You may ask for logarithmic tables, if
Question numbers 1 to 4 carry 1 mark each.
2. Find the distance between the planes 2x – y + 2z = 5 and 5x – 2.5y + 5z = 20.
4. Determine the value of ‘k’ for which the following function is continuous
at x = 3 :
Question numbers 5 to 12 carry 2 marks each.
5. 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.
6. Two tailors, A and B, earn < 300 and < 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.
7. 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.
9. If A is a skew-symmetric matrix of order 3, then prove that det A = 0.
10. Find the value of c in Rolle’s theorem for the function f(x) = x3 – 3x in [– √3, 0].
11. Show that the function f(x) = x3 – 3×2 + 6x – 100 is increasing on .R
12. The length x, of a rectangle is decreasing at the rate of 5 cm/minute and the width y, is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of the area of the rectangle.
Question numbers 13 to 23 carry 4 marks each.
14. Show that the points A, B, C with position vectors 2^i –^j+^k ,^i – 3^j – 5^k and 3^i – 4^j – 4^k respectively, are the vertices of a right-angled triangle. Hence find the area of the triangle.
15. 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.
16. 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.
18. Using properties of determinants, prove that
21. Solve the following linear programming problem graphically :
Maximise Z = 34x + 45y
under the following constraints
x + y ≤ 300
2x + 3y ≤ 70
x ≥ 0, y ≥ 0
22. Find the value of x such that the points A (3, 2, 1), B (4, x, 5), C (4, 2, – 2) and D (6, 5, – 1) are coplanar.
23. Find the general solution of the differential equation y dx – (x + 2y2) dy = 0.
24. 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).
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
26. 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).
Find the area enclosed between the parabola 4y = 3×2 and the straight line 3x – 2y + 12 = 0.
28. AB is the diameter of a circle and C is any point on the circle. Show that the area of triangle ABC is maximum, when it is an isosceles triangle.