MARKS- 30 Time: 45 minutes 1x24= 24
1) If the determinant
P= 1 α 3
1 3 3
2 4 4 is adjoint of a 3x3 Matrix A and |A|= 4 then α is equals to
a) 11 b) 5 c) 0 d) 4
2) The term endependent of x in expansion of
{(x +1)/(x²⁾³ - x¹⁾³+1) - (x -1)/(x - x¹⁾²)}¹⁰ is
a) 120 b) 210 c) 310 d) 4
3) If the determinants
x 1 1 & B= x 1
A=1 x 1 1 x then dA/dx=
a) 3B+1 b) 3B c) -3B d) 1- 3B
4) The value of C₀ + 2C₁ + 3C₂ +....+ (n+1) Cₙ = 576, then n is
a) 7 b) 5 c) 6 d) 9
5) The remainder when, (10¹⁰+ 1)(10¹⁰+ 2) is divided by 6 is
a) 2 b) 4 c) 0 d) 6
6) If (1+ x + x²)ⁿ = 1+ a₁x + a₂x²+ ....+ a₂ₙx²ⁿ, then 2a₁ - 3a₂+ ... -(2n +1)a₂ₙ =
a) n b) -nl c) n +1 d) -nl -1
7) The value of x satisfying the equation of determinant
Cos2x sin2x sin2x
Sin2x cos2x sin2x= 0
Sin2x sin2x cos2x
And x ∈[0,π/4] is
a) π/4 b) π/2 c) π/16 d) π/8
8) If t₅, t₁₀, t₂₅ are 5ᵗʰ, 10ᵗʰ, and 25ᵗʰ terms of an AP respectively, then the value of determinant
t₅ t₁₀ t₂₅
5 10 25
1 1 1 is equal to
a) -40 b) 1 c) -1 d) 0
9) Five dice are tossed. What is the probability that five numbers shown will be different ?
a) 5/24 b) 5/18 c) 5/27 d) 5/81
10) if the events A and B are independent and if P(A')= 2/3, P(B)= 2/7 then P(A∩B) is equals to
a) 4/21 b) 3/21 c) 5/21 d) 1/21
11) Let P= [aᵢⱼ] be a 3 x 3 matrix and let Q= [bᵢⱼ], where bᵢⱼ = 2 ᶦ⁺ʲ aᵢⱼ for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is
a) 2¹⁰ b) 2¹¹ c) 2¹² d) 2¹³
12) x(xⁿ⁻¹ - nαⁿ⁻¹) + αⁿ (n -1) is divisible by (x - α)² for
a) n> 1 b) n > 2 n ∈ N d) none
13) The sum of the series 1+ 3²/2! + 3⁴/4! + 3⁶/6!+....to ∞ is
a) e⁻³ b) e³ c) (1/2)(e³ - e⁻³) d) (1/2) (e³ + e⁻³)
14) Value of the series x/1.2 + x²/2.3 + x³/3.4 +.... is
a) 1- {(1- x)/x} log(1- x)
b) 1- {(1- x)/x} log(1+ x)
c) 1 + {(1- x)/x} log(1- x) d) none
15) Let the coefficient of powers of x in the 2ⁿᵈ, 3ʳᵈ and 4ᵗʰ terms in the expansion of (1+ x)ⁿ, where n is a positive integer, be in Arithmetic progression. The sum of the co-efficients of odd powers of x in the expansion is
a) 32 b) 64 c) 128 d) 256
16) The sum of the infinite series
1+ 1/3 + (1.3)/(3.6) + (1.3.5)/(3.6.9) + (1.3.5.7)/(3.6.9.12) + .....is equal to
a) √2 b) √3 c) √(3/2) d) √(1/√3))
17) The number of real values of K for which the system of equations
x+ 3y + 5z = Kx
5x+ y + 3z = Ky
3x+ 5y + z = Kz
has infinity number of solution is
a) 1 b) 2 c) 4 d) 6
18) Let Sₖ be the sum of an infinite GP series whose first term is K and common ratio is K/(K +1) (K> 0). Then the value of ∞ₖ₌₁∑ (-1)ᴷ/Sₖ is equal to
a) log4 b) log2 -1 c) 1- log2 d) 1- log4
19) Let A and B be two events with P(A')= 0.3, P(B)= 0.4 and P(A ∩B) = 0.5 then P(B/A U B') is equal to
a) 1/4 b) 1/3 c) 1/2 d) 2/3
20) Three numbers are choosen at random without replacement from {1, 2, 3, .....,8}. The probability that their minimum is 3, given that their maximum is 6, is
a) 1/4 b) 2/5 c) 3/8 d) 1/5
21) If C₀, C₁, C₂, C₃,.... are binomial coefficients in the expansion of (1+ x)ⁿ. Then C₀/3 - C₁/4 + C₂/5 - C₃/6+.... is equal to
a) 1/(n +1) - 2/(n+2) + 1/(n +3)
b) 1/(n +1) + 2/(n+2) - 1/(n +3)
c) 1/(n +1) - 1/(n+2) + 1/(n +3)
d) 2/(n +1) - 1/(n+2) + 2/(n +3)
22) If the matrix
A= a x
y a and xy=1. Then det(AA') is equal to
a) a²-1 b) (a²+1)² c) 1- a² d) (a²-1)²
23) Let A and B any two events. Which one of the following statements is always true ?
a) P(A'/B) = P(A/B)
b) P(A'/B) = P(B'/A)
c) P(A'/B) = 1- P(A/B)
d) P(A'/B) = 1- P(A/B')
24) The inverse of a symmetric matrix is
a) skew symmetric
b) symmetric
c) diagonal matrix d) none
AIEEE-21
1) Let X be the universal set for sets A and B. If n(A)= 200, n(B)= 300 and n(A∩B)= 100, then n(A' ∩ B') is equal to 300 provided n(X) is equal to
a) 600 b) 700 c) 800 d) 900
2) The sum of n terms of the infinite series 1.3²+ 2.5²+ 3.7²+...... ∞ is
a) (n/6)(n+1)(6n²+14n+7) b) (n/6)(2n+1)(3n+1) c) 4n³+ 4n²+ n d) none
3) If x²ᵏ occurs in the expansion of (x + 1/x²)ⁿ⁻³, then
a) n - 2k is a multiple of 2.
b) n - 2k is a multiple of 3.
c) n = 0 d) none
4) If y= 1- x + x²/2! - x³/3! + x⁴/4! - ........, then d²y/dx² is
a) x b) - x c) - y d) y
5) The value of ∫ secx dx/√{sin(2x+θ)+ sinθ} is
a) √{(tanx + tanθ) secθ}+ c
b) √{2(tanx + tanθ) secθ}+ c
c) √{2(sinx + tanθ) secθ}+ c d) none
6) The solution of the equation x² d²y/dx² = log x when x=1, y=0 and dy/dx = -1 is
a) y= (1/2)(logx)²+ logx
b) y= (1/2)(logx)²- logx
c) y= -(1/2)(logx)²+ logx
d) y= - (1/2)(logx)²- logx
7) If C²+ S²= 1 then (1+ C + iS)/(1+ C - iS) is equal to
a) C+ iS b) C- iS c) S+ iC d) S- iC
8) The number of real roots of (x + 1/x)³+ (x + 1/x)= 0 is
a) 4 b) 6 c) 2 d) 0
9) If Tₚ , Tq , Tᵣ are pᵗʰ , qᵗʰ and rᵗʰ terms of an AP, then
Tₚ Tq Tᵣ
p q r
1 1 1 is equal to
a) p+ q+ r b) 0 c) 1 d) -1
10) If A is a square Matrix of order n x n and k is a scalar then adj.(kA) is equal to
a) kⁿ⁻¹ adj. A b) kⁿ⁺¹ adj. A c) kⁿ adj. A d) k adj. A
11) 6 persons A, B, C, D, E and F are to be seated at a circular table. The number of ways this can be done if A must have either B or C on is right and B must have either C or D on his right is
a) 24 b) 12 c) 18 d) 36
12) For what values of α
lim ₓ→∞ √(2α²x²+ αx+7) - √(2α²x²+7) will be 1/2√2
a) any value of α b) α ≠ 0 c)α = 1 d) α = -1
13) A window is in the form of a rectangle with the semicircular bend on the top. If the perimeter of the window is 10m, the radius in metres of the semicircular bend that maximize the amount of light admitted is
a) 20/(4+π) b) 10/(4+π) c) (10 -π) d) none
14) There is line with positive slop λ through origin which cuts off a segment of length √10 between the parallel line 2x - y + 5 = 0 and 2 x - y + 10 = 0. Then λ should be
a) 1/2 b) 1/3 c) 1/5 d) none
15) Let C₁ and C₂ be the circles given by equations x²+ y²- 4x -5= 0 and x²+ y² + 8y +7 = 0. Then the circle having the common chord of C₁ and C₂ as its diameter has
a) centre at (- 1, - 1) and radius 2
b) centre at (1, - 2) and radius √5
c) centre at ( 1, - 2) and radius 2
16) The equation of common tangent to the parabola y²= 16x and the circle x²+ y²= 8 are
a) y= x+ 4; y= - x -4 b) y= 2x+ 4; 2y= - x +9 c) y= x+ 9; y= - x -4 d) none
17) An ellipse has OB as semi minor axis. F and F' its foci and the angle FBF' is a right angle. Then the eccentricity of the ellipse is
a) 1/√2 b) 1/2 c) 1/4 d) 1/√3
18) The angle between the lines 2 x= 3y= - z and 6x = - y = -4z is
a) 0° b) 30° c) 45° d) 90°
19) A vector parapendicular to the plane containing the vectors i - 2j - k and 3i - 2j - k is inclined to the vector i + j + k at an angle
a) tan⁻¹√14 b) sec⁻¹√14 c) tan⁻¹√15 d) none
20) Three integers are chosen at random from the first 20 integers. The probability that their product is even, is
a) 2/19 b) 3/29 c) 17/19 d) 4/29
21) If cotθ + tanθ = m and secθ - cosθ = n, then which of the following is correct?
a) m(mn²)¹⁾³ - n(nm²)¹⁾³ = 1
b) m(m²n)¹⁾³ - n(mn²)¹⁾³ = 1
c) n(mn²)¹⁾³ - m(nm²)¹⁾³ = 1
d) n(m²n)¹⁾³ - m(mn²)¹⁾³ = 1
22) If 3 sin⁻¹{2x/(1+ x²)} - 4 cos⁻¹{(1- x²)/(1+ x²)} + 2 tan⁻¹{2x/(1- x²)} =π/3, then value of x is
a) √3 b) 1/√3 c) 1 d) none
23) The A. M of ²ⁿ⁺¹C₀ , ²ⁿ⁺¹C₁ , ²ⁿ⁺¹C₂ , .... ²ⁿ⁺¹Cₙ is
a) 2ⁿ/n b) 2ⁿ/(n +1) c) 2²ⁿ/n d) 2²ⁿ/(n +1)
24) Let p and q be two statements, then (p∪q) ∪ - p is
a) tautology b) contradiction c) Both a and b d) none
25) Let f(x)= x - [x], for every real number of x, where [x] is the integral parts of x. Then ¹₋₁∫ f(x) dx is equal to
a) 1 b) 2 c) 0 d) -1/2
Directions (26-30): This section contains 5 questions numbered 26 to 30. Each question contains statement-1 (Assertion ) and statement-2 (Reason). Each question has four choices (a), (b), (c) and (d) out of which ONLY ONE is correct.
a) Statement-1 is True , Statement-2 is True , Statement-2 is a correct explanation for Statement -1.
b) statement-1 is true , statement-2 is true; statement-2 is not a correct explanation for Statement -1.
c) Statement -1 is True , Statement-2 is false .
d) statement-1 is false, statement -2 is true.
26) statement -1: 5/3 and 5/4 are the eccentricity of two conjugate hyperbolas.
Statement -2: If e and e₁ are the eccentricities of two conjugate hyperbolas, then ee₁ > 1.
27) Statement -1: The maximum area of triangle formed by the point (0,0), (a cosθ , b sinθ), (a cosθ , - b sinθ) is (1/2) |ab|.
Statement -2: Maximum value of sinθ is 2.
28) Statement -1: The Coefficient of xⁿ in the binomial expansion of (1- x)⁻² is (n +1).
Statement-2: The Coefficient of xʳ in (1- x)⁻ⁿ when n ∈N is ⁿ⁺ʳ⁻¹C ᵣ.
29) Statement -1: 20 persons are sitting in a row. Two of these persons are selected at random, The probability that two selected person are not together is 0.7.
Statement-2: if A is an event, then
P(not A)= 1- P(A).
30) Statement -1: A flagstaff of length 100m stands on tower of height h. if at a point on the ground the angle of elevation of the tower and top of the flagstaff be 30°, 45°, then h= 50(√3 +1)m.
Statement -2: A flagstaff of length 'd' stands on tower of height h. If at a point on the ground the angle of elevation of the tower and top of the flagstaff be α, β then h= d cotβ/(cot α - cotβ).
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