TEST PAPER - 1
1) If α, β be the roots of the equation ax²+ bx + c = 0 and γ, δ those of the equation px²+ qx +r=0, show that, ac/pr = b²/q², if αδ = βγ..
2) If the sum of the first 2n terms of a GP is twice the sum of the reciprocal of the terms , then show that continued product of the terms is equal to 2ⁿ.
3) If 9α=π, find the value of sinα sin2α sin3α sin4α.
4) If tanx = (tanα - tanβ)(1- tanα tanβ), then show that
Sin2x= (sin2α - sin2β)/(1- sin2α sin2β).
5) If m(tan(α- β)/cos²β = ntanβ/cos²(α- β), show that
β = (1/2) [α - tan⁻¹{(n - m)/(n + m)} tanα].
6) If lx + my = 0 be the perpendicular bisector the statement joining the points (a,b) and (c,d), then show that
(c - a)/I = (d - b)/m = 2(la + mb)/(l²+ m²)
7) Determine the sign of the expression
(x -1)(x -2)(x -3)(x - 4)+ 5 for real values of x.
8) If cotx = 2 and cot y = 3, then find (x + y).
9) Find square root of 4ab - 2i(a²- b²).
10) If θ{x}= (x -1) eˣ +1, show that θ{x} is positive for all the values of x > 0.
11) If y= f{x}= (x +1)/(x +2), Show that, f(y)= (2x +3)/(3x +5).
12) If f(x)= tan(x - π/4), find f(x) . f(-x)
COMPLEX NUMBERS
1) Simplify: 1+ i²+ i⁴+ i⁶.
2) Write in the form of a + ib where √-1= i
a) √-144 + √441.
b) √-27 x √12 - √-125 x √-5.
3) Find the conjugate of (2+ 3i)².
4) Find x and y if (3x -7) + 5iy = 2y +3 - 4(1- x)i.
5) Find the modulus of the complex number -12 + 5i.
6) Express the reciprocal of the complex number 3+ i √5 in the form a+ ib.
19/825
SET THEORY
BOOSTER - A
1) The number of proper subset in a set consisting of four distinct elements isa) 4 b) 8 c) 16 d) 64
2) The number of proper subsets in a set consisting of five distinct elements is
a) 5 b) 10 c) 32 d) 31
3) If x ∈A=> x ∈ B then
a) A= B b) A ⊂B C) A ⊆B d) B ⊆A
4) If A ⊆ B and B ⊆ A then
a) A= ∅ b) A ∩B = ∅ c) A= B d) none
5) For two sets if A ∪B = A ∩B then
a) A ⊆B b) B ⊆ A c) A= B d) none
6) A - B = ∅ iff
a) A≠ B b) A ⊂B c) B ⊂ A d) A ∩B = ∅
7) If A ∩ B = B then
a) A ⊆B b) B ⊆ A c) A= B d) A= ∅
8) If A and B are two disjoint sets then n(A ∪B)=
a) n(A)+ n(B) b) n(A) - n(B) c) 0 d) none
9) For any two set A and B, n(A)+ n(B) - n(A ∩B)=
a) n(A ∪B) b) n(A) - n(B) c) ∅ d) none
10) The dual of A ∪ U= U is
a) A ∪ U= U b) A ∪∅= ∅ c) A ∪∅ = A d) A ∩∅= ∅
11) The dual of A ∪(B ∩C) = (A ∪B) ∩ (A ∪ C) is
a) (A ∩ B) ∪ (A∩ C)
b) (A ∪B) ∪(A ∪C)
c) (A∩B) ∩ (A ∩ C)
d) (A∪B)∪ (A ∪C)
12) State which of the following statements is true?
a) Subset of an infinite set is so an infinite set
b) The set of even integers greater that 889 is an infinite set.
c) The set of odd negative integers greater than (-150) is an infinite set.
d) A={x : x is real and 0< x ≤1) is a singleton set.
13) State which of the following statements is not true?
a) If a ∈ A and a ∈ B then A ⊆B.
b) If A⊆B and B ⊆C then A ⊆ C.
c) If A ⊆ B and B ⊆A, then A= B.
d) For any set A, if A ∪∅ = ∅(∅ being the null set) then A= ∅.
14) State which of the following is the set of factors of the number 12
a) {2,3,4,6} b) {2,3,4,6,12} c) {2,3,4,8,6} d) {1,2,3,4,6,12}
15) State which of the following is a null set?
a) {0} b) {∅}
c) {x: x is an integer and 1< x <2}
d) {x: x is a real number and 1< x <2}
16) If B be power set of A, state which of the following is true?
a) A ⊃B b) B ⊃A c) A ∈B d) A= B
17) If x ∈ A ∪B, State which of the following is true?
a) x ∈A b) x ∈B c) x ∈ A∀ x ∈B d) x ∈A ∧ x ∈B
18) If x ∈ A ∩B, state which of the following is true?
a) x ∈ A ∧ x ∈B b) x ∈B c) x ∈A ∨ x ∈B d) x ∉ A
19) If A= {2,4,6,8}, state which of the following is true?
a) {2,4} ∈ A b) {2,4} ⊆A c) {2,4} ⊂ A d) {2,4} ∈ Aᶜ
20) State which of the following statements is true?
a) {a} ∈ {a, b,c}
b) a ∉ {a,b,c}
c) a ⊂ {a,b,c}
d) {a} ⊂ {a,b,c}
21) State which of the following four sets are equal?
a) A={0} b) B={∅}
c) C={x : x is a perfect square and 2≤ x ≤6}
d) D={x : x is an integer and -1< x < 1}
22) Some well defined sets are given below. Identify the null set:
a) A==x: x is the cube of an integer and 2≤ x≤7}
b) B={0} c) C={∅} d) D={x: x is an integer and 2< x ≤3}
23) State which of the following sets is an infinite set?
a) A={x : x is an integer and -1≤ x < 1}
b) B= set of negative even integers greater than (-100)
c) C= set of positive integers less than 100
d) D= {x: x is real and -1≤ x <1.
TEST PAPER -5
1) Answer the following questions (alternatives are to be noted): 1x10
a) Fill in the gap :
The quadratic equation with real co-efficients having 5i as one of roots is ____.
b) If A and B are two sets containing m and n elements respectively, how many different relations can be defined from A to B?
a) m+ n b) mn c) 2ᵐⁿ d) 2ᵐ⁺ⁿ
OR
Which one of the following is the value of n when ⁿC₄ = ⁿC₃ ?
a) 0 b) 2 c) 5 d) 7
c) If sinθ= 3/5 and θ is in the second quardant, the value of sin2θ is
a) 24/25 b) -24/25 c) 7/25 d) -7/25
Which one is correct?
d) Fill in the gap:
(Sin²60 + sin²30)/(Sec²50- tan²50)= ____
OR
Value of cos(-2220°)
a) 1 b) -1 c) 1/2 d) -1/2
e) Value of cos(π/4 + x) + cos(π/4 - x) is
a) √2 b) √2 cosx c) cosx d) -√2 cosx
f) The gradient of a line parallel to x-axis is
a) -1 b) 0 c) 1 d) undefined
OR
The radius of the circle x = 2 cost +3, y= 2sin t + 5, t being the parameter is
a) 2 b) 3 c) 4 d) 5
g) 3 points (-2,-5)(2,-2) and (8,a) are collinear. The value of a is
a) 2.5 b) 1.5 c) -2.5 d) -1.5
h) Which one of the following is the value of
lim ₓ→ₐ (√x - √a)/(x - a) ?
a) √a/2 b) 2/√a c) 1/2√a d) 2√a
i) State whether the following relation is true or false :
(A U A's)' = ∅ (A' denotes the complement of the set A)
j) State whether the following statement is true or false :
The set A is a null set, when A={x: x is a real and x²+9=0}
OR
Which one of the following is the set of odd integer divisible by 2 ?
a) ∅ b) U c) {0} d) {∅}
2) a) Answer any two questions (2 x2)
i) If the roots of the equation x²- px +q=0 are in the ratio 2:1, Express q as a fraction of p.
ii) If z₁ = (-i +1)/√2 and z₂ = (1+ i√3)/2, find the principal argument of z₁z₂.
iii) Out of 14 footballers, two are goalkeepers, in how many ways a team of 11 footballers, containing only one goalkeeper, can be selected ?
b) Answer any two questions (2x2)
i) Find x if cosèx = cos2x.
ii) What is the sum of the series:
sinx + sin(π+ x)+ sin(2π+ x)+ ......+ Sin(nπ+ x)? n being a positive integer.
iii) prove that : sin(7π/12) = (1/4) (√6+ √2).
c) Answer any two questions (2x2)
i) If two opposite angular points of a square are (3,5) and (1,-3). Find the area of the square.
ii) If the straight lines ax - 3y + 5 = 0 and 3x + 2y - 7 = 0 intersects at (1,2), find the value of a.
iii) if (a,0) and (0,b) are two vertices of a triangle, find the third vertex of it so that the centroid is at origin.
d) Answer any one question: (2x1)
i) if a = 5i - 2j, b= i + 3j, find the magnitude of 2a - b.
ii) The position vector of the points A and B are the respectively i+ 2j and -3i + 6j. Find the position vector of midpoint of AB.
e) Answer any two questions (2x2)
i) Find the limit: lim ₓ→₀ (eˢᶦⁿˣ -1)/x.
ii) Find the domain of the real valued function f(x)= (x²+ 2x +3)/(x²- 5x +6).
iii) If y= xeˣ, prove x(dy/dx)= (1+ x)y.
3) a) Answer any three questions: (4x3)
i) If a,b,c are in GP such that a + b + c = bx, then show that either x< -1 or x = 3.
ii) In how many ways can the letters of the word PENCIL be arranged so that N is always next to E?
iii) Find the square root of √(-8i).
iv) If the roots of the equation ax²+ bx + c =0 are in the ratio m: n then show that √(m/n) + √(n/m) = √(b²/ac).
v) Find the 4th term from the end in the expansion of (4x/5 - 5/2x)⁹.
b) Answer any three questions (4x3)
i) prove that: tan(π/3 + θ) tan(π/3 - θ)= (2 cos2θ+1)/(2cos2θ -1).
ii) show that the solution of the equation tanax = tanbx, a²+ b²≠ 0 are in AP. Find the command difference of the series.
iii) Prove : cosx/(1- sinx)= tan(π/4 + x/2).
iv) Solve: 2cos²x + 3sinx =0.
v) (sinx - siny)/(cosx + cosy)= tan{(x - y)/2}.
c) Answer any two questions (4x2)
i) A(1,2) and B(5,-2) are two points P is a point moving in such a way that area of the triangle ABP is 12 sq units . Find the locus of P.
ii) A(a,0) and A'(-a,0) are two points. P is a point such that AP makes an angle 45° with A'P. Find the locus of P.
iii) Find the equation the circle passing through the points of intersection of the circles x²+ y²= 9 and x²+ y² - 4x + y - 6 = 0 and through the origin.
d) Answer any two questions (4x2)
i) A, B, C are three sets such that A U B = A U C and A' U B = A' U C. prove that B = C, (A' denote the complement of A).
ii) If f(x)= x -1 and g(x)= x²+1, find
(f+ g), (f - g), f/g
iii) Let A={1,2,3,4,6} and let R={(a,b): a, b ∈ A and a divides b}
A) write R in the roaster form
B) Find domain(R) and range (R).
e) Answer any one question (4x1)
i) A card is drawn from a well shuffled pack of 52 cards. Find
A) the odds in favour of getting a face card.
B) the odds against getting a spade.
ii) Find the variance and standard deviations for the following data:
X: 10 15 18 20 25
f: 3 2 5 8 2
f) Answer any one question 4x1)
i) Find the mean deviation about mean for the following data:
Height(in cm) Number of boys
95-105 9
105-115 13
115-125 25
125-135 30
135-145 13
145-155 10
If is a relation on the set of natural number given by the function defined by let the function define continuous at find the value of exam in the existence
Find out the 99 term of the series 27 14 23 34 the sum of the three numbers in GP 7 some other square is 21 what is the sum of their series are positive numbers are in GP so that they are log exams to a fixed base are in AP prove that complex cube root of unity if one root of a equation square of the other show that how many numbers line between 101 can be found with the digit 034689 if x y z 3 real number show that prove that solve find the range of the function find the limit find out the
4/11/24
A. P
1) If the third and the 6th terms of an AP are 7 and 13 respectively, find the first term and the common difference.
2) Find the sum of all natural numbers between 100 and 1000 which are multiple of 5.
3) How many terms of the AP -6, -11/2, -5, .... are needed to give the sum -25?
4) Determine the sum of the first 25 terms of an AP if a₂= 2 and a₇= 22.
5) If the first term of an AP is 2 and the sum of first five terms is equals to the one fourth of the sum of the next 5 terms , show that the 20th term is -112.
6) Insert 3 Arithmetic mean between 2 and 10.
7) The sum of all odd numbers between 1 and 100 divisible by 3, is
a) 83667 b) 90000 c) 83660 d) none
2/11/24
1) lim ₓ→₁/₂ (4x²-1)/(2x -1).
2) lim ₓ→₀ sin3x/sin2x.
3) lim ₓ→₀ (sin2x + sin6x)/(sin5x - sin3x).
4)
lim ₓ→₀ (tan3x - 2x)/(3x - sin²x).
5) lim ₓ→∞ (2x²+ 7x +5)/(4x²+ 3x -1).
6) lim ₓ→₀ (2sinx - sin2x)/x³.
7) lim ₓ→ₐ {√(1+ ax) - √(1- ax)}/x.
8) lim ₓ→π sinx/(π - x).
14/11/23
Find dy/dx of following:
1) y= (Secx+ tanx)/
(Secx - tanx).
2) y= sinx°
3) x = sint, y= cospt.
4) √(sin√x).
5) If f(x)= logₓ(logₑx), then find f'(e).
11/11/23
2) out of 20 members of a family 15 prefer tea and 9 prefer tea but not cofee. Each member of the family prefers atleast one drink. Find the number of members who
A) prefer both drinks. 6
B) prefer cofee but not tea. 5
3) out of 50 students of a school 35 students prefer Bengali and 25 students prefer both Bengali and English. Each student prefer atleast one subject. Find the number of students who
A) prefer English. 40, 15
B) prefer English but not Bengali.
4) Out of 1600 students in a school, 390 played cricket, 580 played football, 450 played hockey, 90 played cricket and hockey, 125 played hockey and football and 155 played cricket and football, 50 played all three games. How many students did not play any game? 500
7/11/23
1) If A={5,6,9,10}, B={7,8,9,11} and C=
{9,10,11,12} then show that AU(BUC)= (AUB)UC
3) With the help of set operation find the H.C.F. of 35, 77, 119. 7
4) In an examination, out of 100 students 70 passed in Mathematics, 65 passed in Physics and 55 passed in chemistry. Of these students 50 passed in Mathematics and physics. 45 passed in Mathematics and Chemistry, 40 passed in Physics and chemistry and 35 passed in all three subjects.
A) how many students passed in exactly two of the three subjects?
B) how many students passed in exactly one of the three subjects?
C) how many students failed in three subjects? 30, 25, 10
5/11/23
1) Write down the first five terms of the sequence, whose nth term is (-1)ⁿ⁻¹. 5ⁿ⁺¹. 25,-125,625,-3125, 15625
2) If the 3rd and 6th terms of an AP are 7 and 13 respectively, find the first term and the common difference. 3 and 2
3) find the sum of all natural numbers between 100 and 1000 which are multiple of 5. 98450
4) how many terms of the AP -6, -11/2, -5,.... are needed to give the sum -25 ? 5 or 20.
6) If the first term of an AP is 2 and the sum of first five terms is equal to one fourth of the sum of the next five terms, show that the 20th term is --112
7) Insert 3 arithmetic mean between 2 and 10. 4,6,8
8) The sum of three decreasing numbers in AP is 27. If -1, -1, 3 are added to them respectively, the resulting series is in GP. The numbers are
A) 5,8,13 B)15,9,3 C)13,9,5 D) 17,9,1
9) The sum of all odd numbers between 1 and 100 which are divisible by 3, is..
A) 83667 B) 90000 C) 83660 D) n
10) If 7th and 13th terms of an AP be 34 and 64 respectively, then its 18th term is.
A) 87 B) 88 C) 89 D) 90
11) If the sum of p terms of an AP is q and the sum of q terms is q, then the sum of the p + q terms will be..
A) 0 B) p-q C) p+q D) -(p-q)
12) If the sum of n terms of AP be n² - n and its common difference is 6, then its first term is..
A) 2 B) 3 C) 1 D) 4
13) Sum of all two digit numbers which when divided by 4 yield Unity as reminder is..
A) 1200 B)1210. C)1250. D) n
14) In n AM's introduced between 3 and 17 such that the ratio of the last mean to the first mean is 3:1, then the value of n is..
A) 6 B) 8 C) 4 D) n
15) The 1st and last terms of an AP are 1 and 11. If the sum of its terms is 36, then the number of terms will be.
A) 5 B) 6 C) 7 D) 8
16) Find the sum of all odd integers from 1 to 1001. 251001
17) If the ratio between the sums of n terms of two AP is (7n+1):(4n+27) find the ratio of their 11th term. 148: 111
18) If the sum of m terms of an AP be n and the sum of n terms be m, show that the sum of m+n terms is -(m+n).
19) If the sum of n terms of an AP is (pn+ qn²), where p and q are constants, find the common difference. 2q
20) In an AP, the first term is 2 and the sum of first five terms is one-fourth of the sum of next terms. Show that the 20th term is - 112 and the sum of first 20 term is -1100.
21) If the sum of n terms of an AP is given by (3n²+ 4n), find its rth term. 6r +1
22) The digits of a three-digit numbers are in AP and their sum is 15. The number obtained by reversing the digits is 594 less than the original number. Find the number. 852
23) Between 1 and 31, m numbers have been inserted in such a way that the ratio of 7th and (m-1)th numbers is 5:9. Find the value of m. 14
24) In the arithmetic progression whose common difference is non zero, the sum of the first 3n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2n terms the next to 2n terms is
A) 1/5. B) 2/3 C) 3/4 D) none
25) If four numbers in AP are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are:
A) 5,10,15,20 B) 4,10,16,22
C) 3,7,11,15 D) none
26) The first and the last term of an AP are a and l respectively. if S is the sum of all the terms of the AP. and the common difference is given by (l²-a²)/{k -(l+a)}, then k is
A) S B) 2S C) 3S D) none
27) If the sum of the first n even natural number is equal to K times the sum of the first n odd natural numbers, then k is..
A) 1/n B) (n-1)/n C)(n+1)/2n D)(n+1)/n
28) If the first, second and last term of an AP are a,b and 2a respectively, then its sum is
A) ab/{2(b-a)} B) ab/(b-a)
C) 3ab/{2(b-a)} D) none
29) If x is the sum of an arithmetic progression of n odd number of terms and y the sum of the terms of the series in odd places, then x/y is
A) 2n/(n+1) B) n/(n+1)
C) (n+1)/2n D) (n+1)/n
30) If the first term of an AP is 2 and common difference is 4, then the sum of its 40 terms is
A) 3200 B) 1600 C) 200 D) 2800
31) The number of terms of the AP 3, 7, 11, 15, ... to be so that the sum is 406 is...
A) 5 B) 10 C) 12 D) 14 E) 20
32) If a(1/b+ 1/c), b(1/c + 1/a), c(1/a + 1/b) are in AP , then
A) a, b, c are in AP
B) 1/a, 1/b, 1/c are in AP
C) a, b, c are in HP
D) 1/a, 1/b, 1/c are in GP.
33) If the sum of the three numbers in AP be 18 then what is the middle term ? 6
34) The fifth term and the 11th term of an AP are 41 and 20 respectively. Find the first term. What will be the sum of first 11 terms of the AP. ? 425/2
36) The middle term, of an AP having 11th term is 12. Find the sum of the 11 terms of that progression. 132
37) There are n arithmetic means between 4 and 31. If the second mean : last mean=5: 14 then find the value of n. 8
38) If the sum of the first P terms of an AP be equal to the sum of the first Q terms then show that the sum of the first P +Q terms is zero.
) Find the sum upto n terms of the series 1²- 2²+ 3²- 4²+ 5²- 6²+.. .. -n/2 (n+1) (n= 2r)
39) if the sum of p terms of an AP is to the sum of q terms as p²:q², show that (pth term)/(qth term)= (2p-1)/(2q-1).
40) The first term of an AP is a, the second term is b and the last term is c. Show that the sum is {(a+c)(b+c-2a)}/{2(b-a)}.
41) The sides of a right angled triangle are in AP. if the smallest side is 5cm then find the largest side. 25/3
42) find the sum of natural numbers from 1 to 200 excluding those divisible by 5. 16000
43) Show that the sum of all odd numbers between 2 and 1000 which are divisible by 3 is 83667 and of those not divisible by 3 is 166332.
44) Find the 14 A. M which can be inserted between 5 and 8 and show that their sum is 14 times the Arithmetic mean between 5 and 8.
45) Divide 25/2 into five parts in AP, such that the first and the last parts are in the ratio 2: 3. 2,9/4,5/2, 11/3, 3.
46) For what value of m, the sequence 2(4m+7), 6m + 1/2, 12m-7 forms an AP. -3/4
47) Find the 20th term of the AP 80, 75, 70,... Calculate the number of terms required to make the sum equal to zero. 35
48) Prove that if unity is added to the sum of any number of terms of the AP 3, 5,7,9...the resulting sum is a perfect square.
49) The sum of n terms of the series 25, 22, 19, 16,.. is 116. Find the number of terms and the last term. The given series is AP. 18405
50) Find the sum of all natural numbers from 100 to 300:
a) which is divisible by 4. 10200
b) excluding those which are divisible by 4. 30000
c) which are exactly divisible by 5.
d) which are exactly divisible by 4 and 5. 8200, 2200
e) which are exactly divisible by 4 or 5. 16200
4/11/23
1) If x= 2+ 3i and y= 2- 3i, find the value of (x³- y³)/(x³+ y³).
2) If w be an imaginary cube root of 1, show that (1- w²)(1- w⁴)(1- w⁸)(1- w¹⁰)= 9.
3) If 7 cosx + 5 Sinx =5, find the value of 5 cosx - 7 Sinx.
4) If x =π/19, show that (sin 23x -sin 3x)/(sin 16x + sin 4x) =-1.
5) Evaluate: {cot570 + sin(-330)}/{tan(-210)+ cosec(-750)}.
31/10/23
1) Find the value of:
a) cos(-1170°).
b) cos(-870°).
c) tan(-1755°).
d) cot 660° + tan(-1050°).
e) sec(-945°).
f) cosec(-840°).
28/10/23
Find dy/dx of the following:
a) 1 b) x°. x c) x d) none
b) y= f(x)
a) f(x) b) f'(x) c) x d) 1 e) none
3) y= c
a) cx b) c c) 1 d) 0 e)
none
4) dy/dx is the the derivative of
a) y respect to x
b) x respect to y
c) y respect to c
d) x respect to c
e) none
5) y= 2ˡᵒᵍ ˣ
a) 2ˡᵒᵍ ˣ b) 2ˡᵒᵍ ˣ. 2 c) log 2 d) none
6) y= logₑx
a) 1/x b) 1/logₑx c) 1/x logₑe d) none
7) y= logₐx
a) 1 b) 0 c) 1/x d) none
8) y= logₓx.
a) 1 b) 0 c) 1/x d) none
9) y= log ₓe.
a) x b) 1/e c) 1/x d) none
10) log₇7.
a) 0 b) 1 c) x d) none
11) y= xᵐ
a) mx b) mxᵐ⁻¹ c) 0 d) none
12) y= mˣ.
a) mˣlog m b) mˣ c) m logx d) none
12) y= mˣ + xᵐ + mᵐ + mxᵐ
13) y= (x²+3)⁵.
14) ₂3x²
15) log(x²-5)
16) (2x³+5)¹⁰
17) √(2x²+ 9).
18) √(x²+ a²).
19) The derivative of an even function is always an odd function. T/F
20) The derivative of an odd function is always an even function. T/F
21) y= √(x²-1) at x=1.
22) y= 1/(3x -1) at x=0
23) log(ax + b)³
24) (2x²+3)⁴.
25) log(logx).
26) √(logx)³.
