Pre-Test Exam-2022_H. Maths_CQ_EV_Sample Questions
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Subject Code: 2 6 5 BAF Shaheen College Dhaka
Pre-Test Examination (Model Question), 2021 Subject: Higher Mathematics (Creative )
Time – 2hours and 30 minutes Full marks- 50
Taking at least any two from each group, 5 questions are to be answered.
Group -A 1.
.
Equation of the line OA and OB are 2𝑥 + 3𝑦 − 1 = 0 and 𝑥 − 2𝑦 + 3 = 0 and ∠𝐴𝑂𝐵 = ∠𝐵𝑂𝐷 a) Find the polar coordinates of the points (−1, −√3) and (0, −3)? 2
b) Find the coordinates 𝐵 ? 4
c) Find the equation of the line 𝑂𝐷 ? 4
2 . 𝑥2+ 𝑦2− 2𝑥 − 4𝑦 − 1 = 0 is a circle.
a) For which value of 𝜆 the straight line 3𝑥 + 4𝑦 − 𝜆 = 0 touches the circle 𝑥2+ 𝑦2− 10𝑥 = 0? 2 b) Find the equation of tangent of the circle in the stem from the point (−5,4) ? 4 c) Find the equation of circle which touches the straight line 𝑥 − 4𝑦 + 2 = 0 and passes through the point
(−5,4) and center of the circle in the stem? 4
3. Stem-I : 𝑥 = 𝑠𝑖𝑛𝑡 𝑎𝑛𝑑 𝑦 = 𝑠𝑖𝑛𝑝𝑡
Stem-ii ∶ 𝑓(𝑥) = 𝑎2𝑒𝑚𝑥+ 𝑏2𝑒−𝑚𝑥. 𝑎, 𝑏, 𝑚 > 0.
a. Find lim
𝑛→0(1 + 𝑛)𝑛1 2
b. From the stem –i, prove that, (1 − 𝑥2)𝑦2− 𝑥𝑦1+ 𝑝2𝑦 = 0. 4 c. From the stem – ii, find the extreme values of 𝑓(𝑥) 4
4. 𝑥2+ 𝑦2+ 4𝑥 − 2𝑦 + 3 = 0 and 𝑥2+ 𝑦2− 4𝑥 + 2𝑦 − 21 = 0 are equation of two circles.
a) Show that the lines 𝑥 = 𝑡, 𝑦 = 2𝑡 + 1 and 𝑥 = 2𝑡, 𝑦 = −𝑡 − 4 are perpendicular to each other. 2 b) Find the equation of bisector of the included acute angle between the common chord of given two
circles and the straight line 2𝑥 − 3𝑦 − 1 = 0? 4
c) Find the length of common chord of two circles in the stem? 4
Group- B 5. 𝑓(𝑥) = 𝑎𝑥2+ 𝑏𝑥 + 𝑐, 𝑎 ≠ 0
a) If 𝑎 + 𝑏 + 𝑐 = 0 and 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 are real and rational then prove that two roots of the equation
𝑓(𝑥) = 0 be rational. 2
b) If 3𝑏3+ 9𝑎2𝑐 + 𝑎𝑐2 = 9𝑎𝑏𝑐 then show that square of one root of the equation 𝑓(𝑥) = 0 be three times
of other. 4
c) In the equation 𝑓(𝑥) = 0, 𝑎 = 1 and sum of two roots of it is equal to the difference of two roots of the equation 𝑥2+ 𝑐𝑥 + (𝑏 + 𝑐)𝑎 = 0 then show that 𝑏 + 𝑐 = 0 𝑜𝑟 𝑐 = 𝑏 + 4𝑎 . 4
Pre-Test Exam-2022_H. Maths_CQ_EV_Sample Questions
Page 2 of 2 6. In a conic focus 𝑆(7,3) and vertex 𝐴(−1,3)
a) Find the equation of auxiliary circle of the conic 9𝑥2− 15𝑦2+ 12 = 0? 2 b) If eccentricity of the conic be 1 then find the equation of the conic? 4 c) If the length of major axis is the line segment of 𝑆𝐴 and eccentricity √3
2 then find the equation of the
conic? 4
7. 𝑓(𝑥) = cos 𝑥 𝑎𝑛𝑑 𝑔(𝑥) = tan−1𝑥 a. Find the principal value of cosec−1(−2
√3)? 2
b. Prove that 2𝑔 (√𝑎−𝑏
√𝑎+𝑏tan𝜃
2) = cos−1{𝑏+𝑎𝑓(𝜃)
𝑎+𝑏𝑓(𝜃)} 4
c. Solve : 𝑔(𝑥)𝑔(2𝑥)𝑔(3𝑥) = 1; where 0 ≤ 𝑥 ≤ 𝜋 4
8. Stem(i): sin−1𝑥 + sin−1𝑦 + sin−1𝑧 = 𝜋
Stem(ii) : 𝑎𝑥3+ 3𝑏𝑥2+ 3𝑐𝑥 + 𝑑 = 0 is a polynomial equation.
a. Find the equation whose roots be reciprocal of the roots of the equation 𝑥3− 𝑎𝑥2+ 𝑏𝑥 − 𝑐 = 0 ? 2 b) If two roots of the polynomial equation in the stem (ii) are equal then show that another root be 𝑏𝑐−𝑎𝑑
2(𝑎𝑐−𝑏2)2 4
c) From the stem (i), prove that 𝑥4+ 𝑦4+ 𝑧4+ 4𝑥2𝑦2𝑧2= 2(𝑥2𝑦2+ 𝑦2𝑧2+ 𝑧2𝑥2) 4
