Determine Whether the Given Values Are Solutions of the Equation Question: Determine whether the given values are solution of the given equation or not: \[ x^2+x+1=0 \] (i) \(x=0\) (ii) \(x=1\) Solution A value of \(x\) is a solution of an equation if it makes the left-hand side equal to zero. Checking \(x=0\) Substitute \(x=0\) […]

Determine Whether the Given Values Are Solutions of the Equation Question: Determine whether the given values are solution of the given equation or not: \[ x^2-3x+2=0 \] (i) \(x=2\) (ii) \(x=-1\) Solution A value of \(x\) is a solution of an equation if it makes the left-hand side equal to the right-hand side. Checking \(x=2\)

Which of the Following Are Quadratic Equations? Question: Which of the following are quadratic equations? (i) \(x^2+6x-4=0\) (ii) \(\sqrt3x^2-2x+\frac12=0\) (iii) \(x^2+\frac1{x^2}=5\) (iv) \(x-\frac3x=x^2\) (v) \(2x^2-\sqrt3x+9=0\) (vi) \(x^2-2x-\sqrt x-5=0\) (vii) \(3x^2-5x+9=x^2-7x+3\) (viii) \(x+\frac1x=1\) (ix) \(x^2-3x=0\) (x) \(\left(x+\frac1x\right)^2=3\left(x+\frac1x\right)+4\) (xi) \((2x+1)(3x+2)=6(x-1)(x-2)\) (xii) \(x+\frac1x=x^2,\;x\ne0\) (xiii) \(16x^2-3=(2x+5)(5x-3)\) (xiv) \((x+2)^3=x^3-4\) (xv) \(x(x+1)+8=(x+2)(x-2)\) Solution A quadratic equation can be written in the

If sin x + cos x = a, Find the Value of |sin x − cos x| Question If \[ \sin x+\cos x=a, \] find the value of \[ |\sin x-\cos x|. \] Solution Given \[ \sin x+\cos x=a. \] Squaring both sides, \[ a^2 = (\sin x+\cos x)^2 \] \[ = \sin^2x+\cos^2x+2\sin x\cos x

RD Sharma Chapter 4 : Quadratic Equation – Exercise 4.1 Solutions Which of the following are quadratic equations?    (i) x² + 6x – 4 = 0              (ii) √3x² – 2x + 1/2 = 0   (iii) x² + 1/x² = 5               (iv)

If sin x + cos x = a, Find the Value of sin⁶x + cos⁶x Question If \[ \sin x+\cos x=a, \] find the value of \[ \sin^6x+\cos^6x. \] Solution Given \[ \sin x+\cos x=a. \] Squaring both sides, \[ a^2 = \sin^2x+\cos^2x+2\sin x\cos x \] \[ a^2 = 1+2\sin x\cos x \] \[ \sin

If tan A = (1 − cos B)/sin B, Then Find tan 2A Question If \[ \tan A=\frac{1-\cos B}{\sin B}, \] then find the value of \[ \tan 2A. \] Solution Using the standard half-angle identity \[ \tan\frac{\theta}{2} = \frac{1-\cos\theta}{\sin\theta}, \] we get \[ \tan A = \tan\frac{B}{2}. \] Therefore, \[ A=\frac{B}{2}. \] Multiplying both

Find the Value of cos(π/7) cos(2π/7) cos(4π/7) Question Find the value of \[ \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}. \] Solution Use the standard identity \[ \cos x \cos 2x \cos 4x = \frac{\sin 8x}{8\sin x}. \] Putting \[ x=\frac{\pi}{7}, \] we get \[ \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7} = \frac{\sin\frac{8\pi}{7}} {8\sin\frac{\pi}{7}}. \] Since \[ \sin\frac{8\pi}{7} = \sin\left(\pi+\frac{\pi}{7}\right) = -\sin\frac{\pi}{7}, \] therefore \[ \cos\frac{\pi}{7}\cos\frac{2\pi}{7}\cos\frac{4\pi}{7}

If π/4 < x < π/2, Then Find the Value of √(1 − sin 2x) Question If \[ \frac{\pi}{4}

Find the Value of cos²76° + cos²16° − cos76° cos16° Question Find the value of \[ \cos^2 76^\circ+\cos^2 16^\circ-\cos76^\circ\cos16^\circ. \] Solution Since \[ 76^\circ+14^\circ=90^\circ, \] we have \[ \cos76^\circ=\sin14^\circ. \] Let \[ a=\cos76^\circ,\qquad b=\cos16^\circ. \] Using the identity \[ a^2+b^2-ab=(a-b)^2+ab, \] and the product-to-sum formula, \[ \cos76^\circ\cos16^\circ = \frac{\cos60^\circ+\cos92^\circ}{2}. \] \[ = \frac{\frac12-\sin2^\circ}{2}. \] Also,