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The triangle formed by the tangent to the curve f(x) = x² + bx – b at the point (1, 1) and the co-ordinate axes, lies in the first quadrant. If its area is 2 then the value of b is

1) –1 2) 3 3) –3 4) 1 Solution: Given f(x) = x2 + bx - b Slope, m = dy/dx = 2x + b At (1,1) m = 2 + b The equation of the... View Article

The length of sub-tangent to the curve x²y² = a⁴ at the point (-a, a) is

1) 3a 2) 2a 3) a 4) 4a Solution: Given x2y2 = a4 y2 = a4/x2 Differentiate w.r.t.x 2y dy/dx = -2a4/x3 dy/dx = -2a4/2yx3 =... View Article

The abscissa of the points of curve y = x(x – 2) (x – 4) : where tangents are parallel to x-axis is obtained as

(1) x = 2 ± 2/√3 (2) x = 1 ± 1/√3 (3) x = 2 ± 1/√3 (4) x = ± 1 Solution: Given y = x(x - 2) (x - 4) = x(x2 - 6x + 8) = x3 - 6x2... View Article

For the curve yⁿ = a^n-1x, the subnormal at any point is constant. The value of n must be

1) 2 2) 3 3) 0 4) 1 Solution: Given yn = an-1x Differentiate w.r.t.x nyn-1dy/dx = an-1 dy/dx = an-1/nyn-1 Length of... View Article

On dropping a stone in stationary water circular ripples are observed. Rate of flow of ripples is 6 cm/sec. When radius of the circle is 10 cm, then fluid rate of increase in its area is

1) 120 sq. cm/sec 2) 12 sq. cm/sec 3) π sq. cm/sec 4) 120 π sq. cm/sec Solution: Given rate of flow of ripples is 6 cm/sec. dr/dt... View Article

The distance travelled by a particle moving in a straight line at time t is s = √(at² + 2bt + c). Acceleration of the particle is

(1) proportional to t (2) proportional to s (3) proportional to s-3 (4) none of these Solution: Given s = √(at2 + 2bt + c)... View Article

A stone, thrown vertically upward from the surface of the moon at a velocity of 24 m/sec reaches a height of s = 24 t – 0.8t² metre after t second. The acceleration due to gravity in m/sec² at the surface of the moon is

1) 0.8 2) -1.6 3) 2.4 4) -4.9 Solution: Given s = 24 t - 0.8t2 Velocity is the derivative of distance. ds/dt = 24 - 1.6 t... View Article

If x = e^t sin t, y = e^t cos t, t is a parameter, then d²y/dx² at t = π is equal to

(1) 2e-π (2) -2e-π (3) 0 (4) none of these Solution: Given x = et sin t y = et cos t dx/dt = et sin t + et cos t = et (sin t +... View Article

If x = a cos θ, y = b sin θ, then d³y/dx³ is equal to

(1) -3b/a3 cosec4 θ cot4 θ (2) -3b/a3 cosec4 θ cot3 θ (3) -3b/a3 cosec4 θ cot θ (4) none of these Solution: Given x = a cos θ... View Article

Differential coefficient of sec^-1 1/(2x² – 1) w.r.t. √(1 – x²) at x = 1/2 is

(1) 2 (2) 4 (3) 6 (4) 1 Solution: Let u = sec-1 1/(2x2 - 1) = cos-1 (2x2 - 1) Differentiate w.r.t.x du/dx = [-1/√(1 - (2x2 -... View Article

If z = x√y + y/x^1/3, then ∂z/∂y =

(1) 0 (2) 1 (3) -1 (4) x/2√y + 1/x1/3 Solution: Given z = x√y + y/x1/3 Differentiate partially w.r.t.y ∂z/∂y = x/2√y + 1/x1/3... View Article

If (x+y) sin u = x²y², then x ∂u/∂x + y ∂u/∂y is equal to

(1) sin u (2) cosec u (3) tan u (4) 3 tan u Solution: Given (x+y) sin u = x2y2 sin u = x2y2/(x+y) = x4(y2/x2)/x(1 + (y/x)) =... View Article

If [latex]u=e^{-x^{2}-y^{2}}[/latex], then

(1) xux = yuy (2) yux = xuy (3) yux + xuy = 0 (4) x2uy + y2ux = 0 Solution: Given Differentiate partially ux = Multiply by... View Article

If u = tan^-1 (x+y), then x ∂u/∂x + y ∂u/∂y =

(1) sin 2u (2) ½ sin 2u (3) 2 tan u (4) sec2 u Solution: Given u = tan-1 (x+y) Differentiate u partially w.r.t.x ∂u/∂x = 1/(1 +... View Article

If u = xy² tan^-1(y/x), then xu_x+ yu_y=

(1) u (2) 2u (3) 3u (4) u/3 Solution: Given u = xy2 tan-1(y/x) Differentiate u partially w.r.t.x ux = y2 tan-1(y/x) + xy2 [(1/(1... View Article

If x^xy^yz^z = c, then ∂z/∂x =

(1) (1 + log x)/(1 + log z) (2) -(1 + log x)/(1 + log z) (3) (1 + log y)/(1 + log z) (4) none of these Solution: Given xxyyzz = c... View Article

If u = log(x³ + y³ + z³ – 3xyz), then (∂u/∂x + ∂u/∂y + ∂u/∂z)(x + y + z) =

(1) 0 (2) 1 (3) 2 (4) 3 Solution: Given u = log(x3 + y3 + z3 - 3xyz) ∂u/∂x = (3x2 - 3yz)/(x3 + y3 + z3 - 3xyz) …(i) ∂u/∂y = (3y2... View Article

If u = sin^-1 (y/x), then ∂u/∂x is equal to

(1) -y/(x2 + y2) (2) x/√(1 - y2) (3) -y/√(x2 - y2) (4) -y/x√(x2 - y2) Solution: Given u = sin-1 (y/x) ∂u/∂x = [1/√(1-(y/x)2)] ×... View Article

Does fission release a lot of energy

During nuclear fission, a small neutron hits a uranium atom and splits it into smaller atoms. During this process, a large amount of energy gets... View Article

Is beryllium 12 stable or unstable

Beryllium has 11 isotopes but only three are measured routinely 7Be, 9Be and 10Be. Beryllium is a monoisotopic element meaning it has only one... View Article

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