SEE 2081_RE2021B_OPT Math

Group 'A'[10*1=10]

1. Write the definition of quadratic function.
2. Write the formula to find the sum of a geometric series having n terms, where common ratio (r) is more than 1.
3. Write lim x→a f(x) in sentence.
4. If the inverse matrix of a square matrix A of order 2 × 2 is A⁻¹, what is equal to A·A⁻¹? Write it.
5. The slopes of two straight lines are m₁ and m₂ respectively. Write the formula to find the angle between them.
6. In which condition a plane surface intersects a cone to form a parabola? Write it.
7. Express cos2A in terms of cosA.
8. Write sinC − sinD in terms of product of sine and cosine.
9. If a⃗ and b⃗ are perpendicular, then write the value of a⃗ · b⃗.
10. If inversion point of a point P is P′ in a circle with centre O and radius r, what is equal to OP × OP′? Write it.

Group 'B'[8*2=16]

1. If one factor of polynomial f(x) = x³ − 5x² + (k + 1)x + 8 is (x − 2), find the value of k.
2. Draw the graph of the inequality 2x + y ≤ 3.
3. Find the values of Dx and Dy using Crammer's rule from the equations 2x − y = 5 and x − 2y = 1.
4. If two lines having equations cx + dy + e = 0 and fx + gy + h = 0 are parallel to each other then prove that cg − df = 0.
5. Prove that: (sinθ + sin(θ/2)) / (cosθ + cos(θ/2)) = tan(θ/2).
6. Solve: √3 tanA − 1 = 0, [0° ≤ A ≤ 180°].
7. If |a⃗| = 4, |b⃗| = 6 and a⃗ · b⃗ = 12 then find angle between a⃗ and b⃗.
8. In a continuous series, third quartile is two times of the first quartile. If the sum of the first quartile and third quartile is 90, find the quartile deviation.

Group 'C'[11*3=33]

1. If g(x) = 4x − 17, f(x) = \frac{2x + 8}{5} and gog(x) = f⁻¹(x), find the value of x.
2. Solve graphically: x² + x − 2 = 0.
3. If the function f(x) = {2x + 4 for x < 3, 4x − 2 for x ≥ 3} is defined, is the function f(x) continuous at x = 3? Give reason.
4. Solve by matrix method: x − y = 2 and 4x − 3y = 1.
5. Find the equation of the circle whose centre is (2, 3) and passes through the centre of the circle x² + y² − 10x + 4y + 13 = 0.
6. Prove that: \cot\left(\frac{A}{2}+\frac{\pi}{4}\right)-\tan\left(\frac{A}{2}-\frac{\pi}{4}\right)=\frac{2\cos A}{1+\sin A}.
7. If A + B + C = π then prove that: cosA + cosB − cosC = 4cos(A/2)cos(B/2)sin(C/2) − 1.
8. The angles of depression and elevation of the pinnacle of a temple 10 m high with pinnacle are found to be 60° and 30° respectively from the top and bottom of a tower. Find the height of the tower.
9. If the matrix [[a, 2], [b, 2]] transforms a unit square to the parallelogram [[0, 4, c, 2], [0, 1, 3, d]] then find the values of a, b, c and d.
10. Find the mean deviation from median of the given data.

Marks obtained	0–10	10–20	20–30	30–40	40–50
No. of students	3	8	5	6	4

11. Find the standard deviation from the given data.

Age (in years)	0–4	4–8	8–12	12–16	16–20	20–24
No. of students	7	8	10	12	9	6

Group 'D'[4*4=16]

1. The sum of three numbers in an arithmetic series is 18. If the geometric mean between the first and third numbers is 4√2 then find the numbers.
2. Find the separate equations of a pair of lines represented by the equation x² − 2xy cosecθ + y² = 0. Also find the angle between them.
3. PQ is the diameter of semi-circle with centre O and M is a point on the circumference of the semi-circle. Prove by vector method that ∠PMQ = 90°.
4. A triangle PQR with vertices P (4, 3), Q ( 2, 0) and R (5, 2) is translated by T = and the image so obtained is rotated through 90° in negative direction about origin. Find the co-ordinates of the vertices of the obtained images. Plot the given triangle and the images in the same graph paper.