Practice Paper
TERM II (2021 – 2022)
Class – XI
Mathematics (041)
Time: 2 hours Maximum Marks: 40
General Instructions:
1. This question paper contains three sections – A, B and C. Each part is compulsory.
2. Section – A has 6 short answer type (SA1) questions of 2 marks each.
3. Section – B has 4 short answer type (SA2) questions of 3 marks each.
4. Section – C has 4 long answer type questions (LA) of 4 marks each.
5. There is an internal choice in some of the questions.
6. Q14 is a case-based problem.
SECTION – A
1. If in two circles arcs of the same length subtend angles 600 and 750at the centre, find the ratio of their radii.
R= arc/angle
R1/R2 = (l1/θ1)(θ 2/l2)
l1=l2
ratio of radii = 5:4
OR
If cot x = -5/12, x lies in second quadrant, find the value of other five trigonometric functions.
Sin x = 12/13 cos x = -5/13, tan x = -12/5
Other ratios
2. Solve the inequation:
- -1/ (x + 2) ≥ 0
- (x+2) ≤ 0
- X ≤ -2
3. Solve the system of inequations graphically.
2x+y ≥ 8. x+2y ≥8, x+y ≤ 6
4. Find the lengths of the medians of the triangle with vertices A (0,0,6), B (0,4,0) and C (6,0,0)
Mid points of sides, mid-point of AB is F(0,2,3), BC is D (3,2,0) and CA is E(3,0,3)
Lengths of sides, AD=7, BE = 34 CF= 7
5. Find the derivative of :
Correct application of quotient formula
Correct derivative is (tan x – sin x tan x – x sec2x – sec x)/ tan2x
6. A fair coin with 1 marked on one face and 6 on other and a fair die are both tossed, find the probability that the sum of numbers that turns up is (i) 3 (ii) 12
Total outcomes = {(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)}
=12
(i) favourable outcomes = {(1,2)} so probability = 1/12
(ii) favourable outcomes = {(6,6)} so probability = 1/12
SECTION – B
7. The minute hand of a watch is 35 cm long. How far does it move in 9 minutes?
Angle made by minute hand in 9 minute is 540 =3π/10
use θ= l/r
l = 33 cm
OR
Prove that: – (cosA – cosB)2 +(sinA – sinB)2 = 4 sin2(A-B)/2
8. How many different words can be formed with the letters of the word “EQUATION” so that
(i) The words begin with E?
(ii) The words begin with E and end with N?
(iii) The words begin and end with a consonant?
Total letters = 8
(i) words starting with E, fixing E and remaining 7 will be formed 7! Words
(ii) starting with E and end with N, fixing E and N remaining 6 letters will form 6! Words.
(ii) starting and end with a consonant, No. of words = 3P2 . 6P6 = 6.6!
9. Find the area of the triangle formed by the lines joining the vertex of the parabola, x2 = 12y to the ends of the latus rectum.
Value of a = 3
Co-ordinates of the ends of latus rectum L(6,3) and L’(-6,3)
Area of triangle = 18 sq
OR
If foci of a hyperbola are (0, ±5) and length of semi transverse axis is 3 units, then find the equation of hyperbola.
Given, a = 3 and c = 5 => b2 = c2 – a2 = 52 – 32 = 42 => b = 4.
Equation of hyperbola with transverse axis along y-axis is (y/a)2−(x/b)2=1.
So, equation of given hyperbola is (y/3)2−(x/4)2=1.
10. If x lies in the first quadrant and cosx = 8/17 then prove that: –
SECTION – C
11.
12. Find the number of words with or without meaning which can be made using all the letters of the word “AGAIN”. If these words are written as in a dictionary, what will be the 50th word?
Total words taken all = 5!/2! = 60
Number of words starting with A = 4! = 24
Number of words starting with G = 4!/2! = 12
Number of words starting with I 4!/2! = 12
Total words = 48, so 49 th word NAAGI
50 th word is NAAIG
OR
How many numbers greater than 1000000 can be formed by using the digits 1, 2, 0, 2, 4, 2, 4?
13. In a relay race there are five teams A, B, C, D and E.
(a) What is the probability that A, B and C finish first, second and third, respectively.
(b) What is the probability that A, B and C are first three to finish (in any order)
(Assume that all finishing orders are equally likely)
total outcomes = {out of 5 teams any three come first three positions
= 5P3 = 60
(a) A, B and finish first, second and third respectively. There is only one finishing order i.e., ABC so probability (ABC) = 1/60
(b) A, B and C are the first three finishers there will be 3! Arrangements for A, B and C so
Probability (A, B and C in any order) 3!/60= 1/10
CASE BASED
Rakesh wishes to install 2 handpumps in his field for watering. He moves in the field while watering in such a way that sum of distances between the Rakesh and each handpump is always 26 metres. Also, the distance between handpumps is 10 metres.
Based on the above information, answer the following questions:
(i) Name the curve along which Rakesh moves.
Rakesh moves along an ellipse
(ii) Find the equation of curve traced by Rakesh.
according to question the sum of distances Rakesh and each handpump
= 26 so 2a = 26 therefore a= 13,
distance between handpumps is 2c = 10 so c = 5
b = 12 so equation of an ellipse is x2/169 + y2/144 = 1
(iii) Find the eccentricity of the curve along which Rakesh moves.
eccentricity = c/a = 5/13
(iv) Find the co-ordinates of handpumps.
since handpumps lie on foci so co-ordinates of handpumps (5,0) and (-5,0)