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A block of mass m = 1 kg slides with velocity v = 6 m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks to it as shown

A block of mass m = 1 kg slides with velocity v = 6 m/s on a frictionless horizontal surface and collides with a uniform vertical rod and sticks... View Article

Let omega not equal 1 be a cube root of unity Then the minimum of the set a + b ohm + c ohm 22 ; Where, a, b, c distinct non-zero integers equals

Solution:| a + bω + cω2|2= (a + bω + cω2) (a + cω + bω2)= (a2 + b2 + c2 - ab – bc – ca)[because, conjugate of (a + bω + cω2) = (a + cω + bω2)]=... View Article

Find the value of (6Δ)² equals

Three lines are given by Let the lines cut the plane x + y + z 1 at the points A, B and C respectively. If the area of the triangle ABC is Δ... View Article

Let S be the sample space of all 3 × 3 matrices with entries from the set…..

Let S be the sample space of all 3 × 3 matrices with entries from the set {0, 1}. Let the events E1 and E2 be given by E1 = {A ϵ S : det A = 0}... View Article

Let AP (a; d) denote the set of all the terms of an infinite arithmetic progression with first term a and common difference d > 0. If AP(1; 3) ∩ AP (2; 5) ∩ AP (3; 7) = AP (a; d) then a + d equals

Solution:I: 1, 4, 7, 10, 13, 16 ….. 32II: 2, 7, 12, 17, 22, 27, 32, 37, 42, 47, 52III: 3, 10, 17, 24, 31, 38, 45, 5252 <-- a + d --> LCM of... View Article

Let the point B be the reflection of the point A(2, 3) with respect to the line 8x – 6y – 23 = 0. Let _A and _B….

Let the point B be the reflection of the point A(2, 3) with respect to the line 8x – 6y – 23 = 0. Let A and B be circle of radii 2 and 1 with... View Article

If [latex]I = \frac{2}{\pi} \int_{-\pi/4}^{\pi/4} \frac{dx}{(1 + e^{sin\ x})(2-cos 2x)}[/latex] then 27I² equals

Solution:27I2 = 4... View Article

Let L₁ and L₂ denotes the lines….

Let L1 and L2 denotes the lines respectively. If L3 is a line which is perpendicular to both L1 and L2 and cuts both of them, then which of... View Article

In a non-right-angle triangle ΔPQR, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively

In a non-right-angle triangle ΔPQR, let p, q, r denote the lengths of the sides opposite to the angles at P, Q, R respectively. The median from R... View Article

Let denote a curve y = y(x) which is in the first quadrant and let the point (1, 0) lie on it. Let the tangent to at a point P intersect the y-axis at Y_P. If PY_P has length 1 for each point P on , then which of the following is options is/are correct?

a) d) xy' + √[1-x2] = 0Solution: y - y1 = m(x – x1)yp - y1 = -mx1 => 1 = x12 + m2 x12 1 = x2[1 + (dy/dx)2]Or (dy/dx)2 = 1/x2 – 1dy/dx =... View Article

Let α and β be the roots of x² – x – 1 = 0….

Let α and β be the roots of x2 – x – 1 = 0, with α > β. For all positive integers n, define As, b1 = 1 and bn = an-1 + an+1, n ≥ 2 a1 +... View Article

Which of the following options is/are correct?

Then which of the following options is/are correct?a) f’ has a local maximum at x = 1b) f is ontoc) f is increasing on (-∞, 0)d) f’ is NOT... View Article

Let [latex]M = \begin{bmatrix} 0 &1 & a \\ 1& 2 &3 \\ 3 &b &1 \end{bmatrix}[/latex] and adj M = [latex]\begin{bmatrix} -1 &1 & -1 \\ 8& -6 &2 \\ -5 &3 &-1 \end{bmatrix}[/latex] where a and b are real numbers. Which of the following options is/are correct?

a) a + b = 3b) det(adj M2) = 81c) (adj M)-1 + adj M-1 = -Md) If |M| = -2And a + b = 2(b) |adj M2|= |M2|2 = |M|4 = 16(c) (adj M)-1 + adj M-1 = 2... View Article

There are three bags B₁, B₂ and B₃. The bag B₁ contains 5 red and 5 green balls

There are three bags B1, B2 and B3. The bag B1 contains 5 red and 5 green balls, B2 contains 3 red and 5 green balls, and B3 contains 5 red and... View Article

The area of the region {(x, y) : xy ≤ 8, 1 ≤ y ≤ x²} is

The area of the region {(x, y) : xy ≤ 8, 1 ≤ y ≤ x2} isSolution:{(x, y) : xy ≤ 8, 1 ≤ y ≤ x2}8/x = x2 = 16 loge 2 – 14/3... View Article

Let S be the set of all complex numbers z satisfying |z – 2 + i| ≥ √5. If the complex number z₀ is such that 1/|z₀ – 1| is the maximum of the set [latex]begin{Bmatrix} frac{1}{|z-1|}; z in S end{Bmatrix}[/latex] then the principle argument of [latex]frac{4-z_0 – bar{z_0}}{z_0- bar{z_0} +2i}[/latex]

a) π/4b) -π/2c) 3π/4d) π/2Solution:|z - 2 + i| ≥ √5P is along but at √5 distance from C.... View Article

A line y = mx + 1 intersects the circle (x – 3)² + (y + 2)² = 25 at the points P and Q. If the midpoint of the line segment PQ has x-coordinate -3/5, then which one of the following options is correct?

a) 6 ≤ m < 8b) 2 ≤ m < 4c) 4 ≤ m < 6d) -3 ≤ m < -1Solution:y = mx + 1(x - 3)2 + ((mx +1) + 2)2 = 25=> x2 (1+ m2 ) + 6(m -1) x - 7... View Article

Find the value of α* + β*

Let M = M2 = αM + βIa11 = sin θ – 1 – sin2 θ – cos2 θ – cos2 θ sin2 θ = β + α sin4 θ sin8 θ – 2 – cos2 θsin2 θ= β + α sin4 θa21 = sin4 θ + cos2... View Article

Four resistors of resistance 15 Ω, 12 Ω, 4Ω and 10Ω respectively in cyclic order to form a wheatstone’s network. The resistance that is to be connected in parallel with the resistance of 10 to balance the network is

[(10R)/(10+R) ]×12 = 15 ×4 R = 10Ω... View Article

A one metre long (both ends open) organ pipe is kept in a gas that has double the density of air at STP. Assuming the speed of sound in air at STP is 300 m/s, the frequency difference between the fundamental and second harmonic of this pipe is __Hz.

V = √(B/ρ) Vpipe/ Vair = Vpipe = Vair/√2 fn = Vpipe(n+1)/2l f1-f0 = Vpipe/ 2l = 300/2√2 = 106.06 Hz ( if √2 = 1.414) ≈ 106 Hz... View Article