Thursday, 7 April 2022

XI Model Test paper -3

Full Marks -80

Question -1.                        1x10

i) A relation R is defined from A={1,2,3,4,5} to B={1,3,4} in such a way that, (x,y) belongs to R => x > y. Express R as a ordered pairs.

ii) If 16Cr = 16C 2r+1, then value of r. 
A) 5 B) 4 C) 3 D) 2

iii) If w be the imaginary cube root of 1, then the value of (3+ w + 3w²)⁴. 
A) 16 B) w C) 16w D) 1

iv) If the difference between the roots of the quadratic equation x² + px +8= 0 be 2, then the value of p be.
A) ±3 B) ±6 C) ±4 D) none

v) If P-th term of an arithmetic progression be Q and Q-th term be P, then(P+ Q)-th term is 
A) 0 B) 1 C) P D) Q

vi) the value of lim ₓ→₀ (eˣ - e⁻ˣ)/x. 
A) 0 B) 1 C) 2 D) 3

vii) If y= cos² x/2 find dy/dx.
A) -1/2 sinx B) 1/2 sinx C) 1/2 cosx D) - 1/2 cosx

viii) The probability that a non leap year selected at random will have are 53 Sundays is. 
A) 1/2 B) 1)3 C) 1/5 D) 1/7

ix) If the variance of a distribution is 4 and coefficient of variation is 5%, then find mean. 
A) 40 B) 60 C) 30 D) 70

x) The domain of x² +3x +2 is
A) 1 B) 0 C) 5 D) none


Question -2                  2x13= 26

i) y= f(x)= (px +q)/(rx - p), then show that x= f(y).

ii) Show 1/sin 10 - √3/cos 10 = 4

iii) If nPr= 504 and nCr= 84, then find the value of n and r. 

iv) Find the coefficient of 1/x² in the expansion of (2x - 1/x²)⁶.

v) solve the following in equation: {|x +2| +2x}/(x+2) > 2.

vi) lim ₓ→₀ (cos 5x - cos 7x)/(cosx - cos 5x). 

vii) If y= sec 2x find dy/dx at x=π/8

viii) If P(A∩ B)= 5/13, then find the value of P(A' U B'). 

ix) d/dx of sin(logx)

x) the number of all numbers having 5 digits, with the distinct digits is..

xi) Two finite sets A and B consist of m and n elements respectively. The number of subsets in A exceeds that of B by 112. Find the values of m and n.                     

xii) If (a,b) and (b,c) are elements of A x A, find the set A and other elements of A x A.

xiii) If f(x + 1/x)= x² + 1/x², find f(x).        

Question -3.               3x10= 30

i) For any three sets A, B and C, prove that A - (B U C)= (A - B)∩ (A - C).

ii) If z be a complex number and |z+5|≤ 6, then find the maximum and minimum value of |z +2|. 

iii) Solve: 4 sin x sin 2x sin 4x= sin 3x. 

iv) if tan x/2 = √{(1- e)/(1+ e)} tan y/2 , prove cos y = (cos x - e)/(1- e cos x).

v) If n belongs to N, then prove by mathematical induction that 7²ⁿ + 2 ³⁽ⁿ⁻¹⁾. 3 ⁿ⁻¹ is always a multiple of 25.

vi) If z= x+ iy and (z -i)/(z+ 1) is purely imaginary, then show that the point z always lies on a circle.

vii) how many odd numbers of 5 digits can be found with the digits 3,6,7,2,0 when no digit is repeated. 

viii) Show that the mid term in the expansion of (1+ x)²ⁿ is {1.3.5. ....(2n-1) 2ⁿ. xⁿ})n! (Where n belongs to N)
OR
If the ratio of the sum of first n terms of two arithmetic series is (4n -13): (3n +10), then find the ratio of their ninth terms. 55:61

ix) solve applying formula: 3x² - (2- i)x + 10 - 4i= 0. 
OR
 Out of 14 marbles 10 are red in colour and remaining 4 are of different colours. How many ways can you select 10 marbles out of these 14 marbles. 

x)  A survey shows that 75% of the Students of a school like Mathematics and 65% like Physics. If x% of the students like both Mathematics and Physics. find the maximum and minimum values of x.                    
OR
 In a survey of 35 students of a class it was found that 17 students like Mathematics and 10 like Mathematics but not Biology. Find the number of students who like 
A) Biology.                        
B) Biology but not Mathematics, it being given that each students takes at least one of the two subjects.                      


Question 4.                     3x3= 6

i) Find the equation of the directrix of the parabola x² - 4x - iy +12 = 0. y= 0
OR
 Find the equation of the lines passing through the point (4,5) making equal angles with the lines 3x = 4y +7 and 5y = 12x +6.

ii) A circle in the first quadrant touches both the axes and its centre lies in the straight line lx + my + n= 0, show that the equation of that circle is (l+m)² (x² + y²)+ 2n(l+m) (x+y) + n²= 0.

OR
 The ellipse x²/a² + y²/b² = 1 passes through the point of intersection of the lines 7x = -13y + 87 and 5x = 8y +7 and the length of its latus rectum is 32√2/5 units. Find the values of a and b. 5√2, 4√2


Question 5.             4+4

i) The standard deviation of 32 numbers is 5. If the sum of the numbers is 80, then find the sum of the squares of the numbers.      (4)

ii) calculate the mean deviation of median for the following data:
Marks.      No of students
00-10            6
10-20            5 
20-30            8 
30-40           15 
40-50             7 
50-60             6 
60-70              3                          (4)




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