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Federal (FBISE) · Quarter book test · Free

2nd Year Mathematics 3rd Quarter Book Paper

40 marks · 1 hour · 28 questions. Covers: 07: Conic Section, 08: Inverse Trigonometric Functions and Their Graphs, 09: Solution of Trigonometric Equations.

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2nd Year Mathematics 3rd Quarter Book Paper

2nd Year · Mathematics · Federal (FBISE)

Name: ____________Roll No: ________Class/Section: ________Date: ________Total marks: 40Time: 1 hour

Q1. Multiple choice questions (Attempt all · 12 × 1 = 12)

  1. What results sec⁡−1(1)−sec⁡−1(2) \text{What results } \sec^{-1}(1) - \sec^{-1}(\sqrt{2})
    1. π6\frac{\pi}{6}
    2. π2\frac{\pi}{2}
    3. π3\frac{\pi}{3}
    4. π4\frac{\pi}{4}
  2. For what value of x∈(0,2π), sin⁡x=−12? \text{For what value of } x \in (0, 2\pi) \text{, } \sin x = -\frac{1}{2}?
    1. 7π6,π6 \frac{7\pi}{6}, \frac{\pi}{6}
    2. 7π6,11π6 \frac{7\pi}{6}, \frac{11\pi}{6}
    3. 5π6,11π6 \frac{5\pi}{6}, \frac{11\pi}{6}
    4. π6,5π6 \frac{\pi}{6}, \frac{5\pi}{6}
  3. The focus of the parabola x²= − 16y is:
    1. (4 , 0)
    2. (– 4 , 0)
    3. (0 , 4)
    4. (0 , – 4)
  4. For what value of x ∈ (0 , 2π) , Sin x =−12-\frac{1}{2}
    1. 7π6,π6\frac{7\pi}{6}, \frac{\pi}{6}
    2. 7π6,11π6\frac{7\pi}{6}, \frac{11\pi}{6}
    3. 5π6,11π6\frac{5\pi}{6}, \frac{11\pi}{6}
    4. π6,5π6\frac{\pi}{6}, \frac{5\pi}{6}
  5. If cos⁡2x=12 with x∈[0,π] then value of x is : \text{If } \cos^2 x = \frac{1}{2} \text{ with } x \in [0, \pi] \text{ then value of } x \text{ is :}
    1. π4,7π4 \frac{\pi}{4}, \frac{7\pi}{4}
    2. π4,3π4 \frac{\pi}{4}, \frac{3\pi}{4}
    3. 3π4,5π4 \frac{3\pi}{4}, \frac{5\pi}{4}
    4. 5π4,7π4 \frac{5\pi}{4}, \frac{7\pi}{4}
  6. Equation of conjugate axis of the hyperbola (x−1)24−(y+3)212=1 is: \frac{(x-1)^2}{4} - \frac{(y+3)^2}{12} = 1 \text{ is:}
    1. x = 1
    2. x = - 1
    3. y = 3
    4. y = - 3
  7. The value of cos⁡[π6+cos⁡−1(−12)] is : \text{The value of } \cos\left[\frac{\pi}{6} + \cos^{-1}\left(-\frac{1}{2}\right)\right] \text{ is :}
    1. 12\frac{1}{2}
    2. −12-\frac{1}{2}
    3. −32-\frac{\sqrt{3}}{2}
    4. 32\frac{\sqrt{3}}{2}
  8. What is the solution of sec⁡x=2, if x∈[0,π]? \text{What is the solution of } \sec x = 2 \text{, if } x \in [0, \pi]?
    1. {−π6} \left\{ -\frac{\pi}{6} \right\}
    2. {−π3} \left\{ -\frac{\pi}{3} \right\}
    3. {π3} \left\{ \frac{\pi}{3} \right\}
    4. {π6} \left\{ \frac{\pi}{6} \right\}
  9. The chord joining the two points (at12,2at1) and (at22,2at2) on the parabola y2=4ax is focal chord if: \text{The chord joining the two points } (at_1^2, 2at_1) \text{ and } (at_2^2, 2at_2) \text{ on the parabola } y^2 = 4ax \text{ is focal chord if:}
    1. t1+t2=1 t_1 + t_2 = 1
    2. t1+t2=−1 t_1 + t_2 = -1
    3. t1t2=1 t_1 t_2 = 1
    4. t1t2=−1 t_1 t_2 = -1
  10. , If sin⁡−1(12)=π2−x, then value of x is: \text{If } \sin^{-1} \left( \frac{1}{2} \right) = \frac{\pi}{2} - x \text{, then value of x is:} :
    1. π6\frac{\pi}{6}
    2. π3\frac{\pi}{3}
    3. π4\frac{\pi}{4}
    4. π2\frac{\pi}{2}
  11. One solution of a trigonometric equation cos⁡x+sin⁡x=2, if x∈[0,2π] ? \text{One solution of a trigonometric equation } \cos x + \sin x = \sqrt{2} \text{, if } x \in [0, 2\pi] \text{ ?}
    1. {π4} \left\{ \frac{\pi}{4} \right\}
    2. {−π4} \left\{ -\frac{\pi}{4} \right\}
    3. {2π4} \left\{ \frac{2\pi}{4} \right\}
    4. {3π2} \left\{ \frac{3\pi}{2} \right\}
  12. Exactly one tangent can be drawn to a circle if point lies:
    1. Outside the circle
    2. On the circle
    3. Inside the circle
    4. Centre of circle

Q2. Short questions (Attempt any 8 of 12 · 8 × 2 = 16)

  1. Solve graphically the trigonometric equation : sin⁡θ=θ2\sin \theta = \frac{\theta}{2} where θ∈[−π2,π2]\theta \in \left[ -\frac{\pi}{2}, \frac{\pi}{2} \right]
  2. Draw the graph of each of the following functions against the intervals mentioned: y = sin⁻¹ (2x) for x ∈ [−12,12].\begin{bmatrix} -\frac{1}{2}, \frac{1}{2} \end{bmatrix}..
  3. Check whether the point P(2 , 4) lies outside, on or inside the circle: 2x² + 2y² - 6x + 8x + 1 = 0
  4. Solve the following trigonometric equations graphically. sin⁡(θ2)=θ;θ∈[−π2,π2]\sin\left(\frac{\theta}{2}\right) = \theta; \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]
  5. Solve inverse trigonometric equation algebraically. 4 ArcTan² (x) - 3π ArcTan(x) - π² = 0
  6. Check whether the line x + 2y - 3 = 0 is tangent to circle x² + y² = r² or not.
  7. Solve tan⁡(θ2)=1\tan\left( \frac{\theta}{2}\right) = 1 for θ\theta
  8. Find the unknown angle and use a calculator to evaluate as real number to three decimal place.sin⁡−1(−15)=−cos⁡−1() \sin^{-1} \left( - \frac{1}{\sqrt{5}} \right) = - \cos^{-1} ( )
  9. From the given information, find the equation of ellipse in each of the following. Eccentricity is 154 \frac{\sqrt{15}}{4} ; centre (- 5 , 5) ; co-vertex (- 8 , 5)
  10. Solve the following trigonometric equations for the values of θ in the range 0∘≤ θ ≤ 360∘: cos5θ + cosθ = cos3θ
  11. Find the principal value of the following without using a calculator.r. cos⁡−1(32) \cos^{-1} \left( \frac{\sqrt{3}}{2} \right)
  12. Find focus, vertex, axis of symmetry, directrix, length of latus rectum, and endpoints of the latus rectum of parabola with the given equations. Also draw the graph of the parabola.y2=−32x y^2 = -\frac{3}{2}x

Q3. Long questions (Attempt any 3 of 4 · 3 × 4 = 12)

  1. Find the general solution, in radians, of the following equations: cos4x + cos6x = cos5x
  2. Find the exact real number value of without using a calculator.sin⁡[sin⁡−123+cos⁡−112]\sin \left[ \sin^{-1} \frac{2}{3} + \cos^{-1} \frac{1}{2} \right]
  3. Find the equations of tangent and normal to the circle x²+ y²+ 3x + 2y + 3 = 0 at the point (2 , −1).
  4. Solve the following trigonometric equations when Ø ∈ [0 , 2π]: cos² Ø - 2 cosØ + 1 = 0