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If θ₁ and θ₂be respectively the smallest and the largest values of θ in (0,2π) − {π} which satisfy the equation [latex]2cot ^{2}theta -frac{5}{sin theta }+4 = 0[/latex] then [latex]int_{theta _{1}}^{theta _{2}}cos ^{2}3theta dtheta[/latex] is equal to:

a. 2π/3 b. π/3 c. (π/3) + (1/6) d. π/9 Solution: , θ∈[0,2π) ⇒ 2cosec2θ - 2 - 5cosec θ + 4 = 0 ⇒2 cosec2θ - 4cosec θ - cosec θ + 2 = 0 ⇒... View Article

Let a, b,c be three unit vectors such that

Let a, b,c be three unit vectors such that and ) is equal to: a. (3/2, 3a×c) b. (-3/2, 3c×b) c. (-3/2, 3a×b) d. (3/2, 3b×c) Solution: Given... View Article

Let a₁, a₂, a₃, … be a G.P. such that a₁ < 0, a₁ + a₂ = 4 and a₃ + a₄ = 16. If [latex]sum_{i=1}^{9}a_{i} = 4lambda[/latex] then λ is equal to

a. 171 b. 511/3 c. -171 d. -513 Solution: a1 + a2 = 4 ⇒ a + ar = 4 ⇒ a(1 + r) = 4 a3 + a4 = 16 ⇒ ar2 + ar3 = 16 ⇒ ar2(1 + r) = 16 ⇒ 4r2 =... View Article

The number of ordered pairs (r, k) for which 6 ⋅ (35)^C_r = (k_2 − 3) ⋅ (36)^C_(r+1), where k is an integer, is

a. 4 b. 6 c. 2 d. 3 Solution: Using 36Cr+1 = (36/r+1)×35Cr (36/r+1)×35Cr×(k2-3) = 35Cr×6 k2 - 3 = (r+1)/6 k2 = ((r+1)/6) +3 k∈I r → Non-negative... View Article

Let f(x) be a polynomial of degree 5 such that x = ± 1 are its critical points

Let f(x) be a polynomial of degree 5 such that x = ± 1 are its critical points. If then which one of the following is not true? a. f(1)... View Article

Let = [ ] and = [ ] be two 3 × 3 real matrices such that = (3^{)( + −2)} , where , = 1, 2, 3. If the determinant of is 81, then the determinant of is :

a. 1/9 b. 1/81 c. 1/3 d. 3 Solution: 𝑏𝑖 = (3)(𝑖+𝑗−2)𝑎𝑗𝑖 B = and Taking 32 common each from C3 and R3 Taking 3 common each from C2... View Article

If [latex]\frac{3+i\sin \theta }{4-i\cos \theta }, \theta \in [0,2\pi ][/latex] is a real number, then the argument of sin 𝜃 + 𝑖 cos 𝜃 is :

a. π- tan-1 (4/3) b. - tan-1 (3/4) c. π- tan-1 (4/3) d. tan-1 (4/3) Solution: Let 𝑧 is real. 4 sin 𝜃 + 3 cos 𝜃 = 0 ⇒ tan 𝜃 = -3/4 [𝜃 lies in... View Article

If the sum of the first 40 terms of the series, 3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + ….. is (102)m, then m is equal to

a. 10 b. 25 c. 5 d. 20 Solution: S = 3 + 4 + 8 + 9 + 13 + 14 +…..40 terms S = (3+4)+(8+9)+(13+14)+…..40 terms S = 7 + 17 + 27 + 37 +….20... View Article

The value of α for which [latex]4alpha int_{-1}^{2}e^{-alpha left | x right |}dx = 5[/latex] is:

a. loge 2 b. loge √2 c. loge (4/3) d. loge (3/2) Solution: ⇒ 4[1 − e−2α − e−α + 1] = 5 Let e−α = t ⇒ 4t2 + 4t − 3 = 0 ⇒ t = 1/2 = e−α ⇒... View Article

The locus of mid points of the perpendiculars drawn from points on the line x = 2y to the line x = y is

a. 2x − 3y = 0 b. 3x − 2y = 0 c. 5x − 7y = 0 d. 7x − 5y = 0 Solution: Let R be the midpoint of PQ PQ is perpendicular on line y = x Equation... View Article

in a workshop there are five machines and the probability of any one of them to be out of service on a day is

a. 17/2 b. 4 c. 17/4 d. 17/8 Solution: P(Machine being faulty) = p = (1/4) q = 3/4 P(at most two machines being faulty)=P(zero machine... View Article

The coefficient of x^7 in the expression (1 + x)^10 + x(1 + x)^9 + x^2(1 + x)^8 + ⋯ +x^10 is

a. 420 b. 330 c. 210 d. 120 Solution: Coefficient of x7 in (1+x)10+ x(1+x)9+ x2(1+x)8+⋯ +x10 Applying sum of terms of G.P = = (1 + x)11 − x11... View Article

Let A, B, C and D be four non-empty sets. The contrapositive statement of “If A ⊆ B and B ⊆ D then A ⊆ C ” is

a. If A ⊆ C, then B ⊂ A or D ⊂ B b. If A ⊈ C, then A ⊆ B and B ⊆ D c. If A ⊈ C, then A ⊈ B and B ⊆ D d. If A ⊈ C, then A ⊈ B or B ⊈ D Solution:... View Article

If 3x+4y =12√2 is a tangent to the ellipse (x²/a²) + (y²/9) = 1 for some a∈R then the distance between the foci of the ellipse is :

a. 2√5 b. 2√7 c. 2√2 d. 4 Solution: 3x+4y =12√2 is a tangent to the ellipse (x2/a2) + (y2/9) = 1 Equation of tangent to ellipse (x2/a2) + (y2/9)... View Article

A particle of mass m is fixed to one end of a light spring having force constant k and unstretched length l

A particle of mass is fixed to one end of a light spring having force constant k and unstretched length l. The other end is fixed. The system... View Article

Which of the following set of points are non-collinear

1) (1, – 1, 1), (–1, 1, 1), (0, 0, 1) 2) (1, 2, 3), (3, 2, 1), (2, 2, 2) 3) (–2, 4, – 3), (4, – 3, – 2), (–3, – 2, 4) 4) (2, 0,– 1), (3, 2, –... View Article

If the centroid of a triangle whose vertices are (a, 1, 3), (–2, b, – 5) and (4, 7, c) be the origin, then the values of a, b, c are

1) –2, – 8, – 2 2) 2, 8, – 2 3) –2, – 8, 2 4) 7, – 1, 0 Solution: (3) –2, – 8, 2... View Article