Science (English Medium) कक्षा ११ - CBSE Question Bank Solutions

Solve each of the following system of equations in R.

\[5x - 7 < 3\left( x + 3 \right), 1 - \frac{3x}{2} \geq x - 4\]

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x² + 1, then find f + g

If f and g are two real valued functions defined as f(x) = 2x + 1, g(x) = x² + 1, then find f – g

If [x]² – 5[x] + 6 = 0, where [ . ] denote the greatest integer function, then ______.

If f(x) = ax + b, where a and b are integers, f(–1) = – 5 and f(3) = 3, then a and b are equal to ______.

If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.

Show that 2 sin²β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α

If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and Φ is their difference, then show that sin θ = `(k + 1)/(k - 1)` sin Φ

If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.

Match each item given under column C₁ to its correct answer given under column C₂.

If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].

If tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ.

[Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]

If sinθ + cosθ = 1, then find the general value of θ.

Find the most general value of θ satisfying the equation tan θ = –1 and cos θ = `1/sqrt(2)`.

If cotθ + tanθ = 2cosecθ, then find the general value of θ.

If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a² - 2b²

[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]

If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`

[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]

Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2

[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]

If sinθ + cosecθ = 2, then sin²θ + cosec²θ is equal to ______.

If f(x) = cos²x + sec²x, then ______.

[Hint: A.M ≥ G.M.]