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SECTION 'B' (Short-Answer Questions) (40)
NOTE: Attempt any TEN part questions in all. All questions carry equal marks. (i.e. 4 marks of each part).
2.
(i) Find the real and imaginary parts of (1−√5i)−1.
ii) Using the properties of determinant show that:
|
|= 0
| 1 | x | y+z |
| 1 | y | z+x |
| 1 | z | x+y |
iii) Find the rank of the matrix A =
| 1 | 2 | 3 |
| 2 | 3 | 4 |
| 0 | 2 | 2 |
iv) For what value of λ if the vector î + ĵ + 2k̂, λî − ĵ + k̂ and 3î − 2ĵ − k̂ are coplanar.
v) Find the 13th term of an A.P. where first term and the common difference are 3 and −4 respectively. Also write its first four terms.
vi) Find the 8th term of the H.P. 45, 25, 415 ⋯
vii) Find the sum of series ∑k=1∞ 14k2−1
viii) There are 11 men and 9 women members of a club. How many communities of 8 members can be formed, having exactly five men?
ix) The principle of mathematical induction, prove that 2 + 4 + 6 + ⋯ + 2n = n(n + 1), ∀n∈N.
x) Find the inverse of the real valued function defined by f(x) = 5x + 7 and verify it.
xi) Show that: sin 6θ + sin 4θcos 6θ + cos 4θ = tan 5θ (by using trigonometric identities)
xii) The measure of two sides of a triangle are 4 and 5 units. Find the third side so that area of triangle is 6 square units.
xiii) Prove that: 1r2 − 1r12 + 1r22 + 1r32 = a2+b2+c2Δ2.
xiv) Find the maximum and minimum values of the function: y = 14−5sin(7θ−8°).
SECTION "C" (Detailed Answer Questions) (40)
NOTE: Attempt any FIVE question from the question. All questions carry equal marks.
3.
Use Gauss-Jordan method to solve the system of linear equations: x + 5y + 2z = 9 , x + y + 7z = 6 , −3y + 4z = −2.
4.
A, B, C are the points of ā, b⃗ and 2ā − b⃗ respectively. D divides \vec{AC} in 2:3 and E divides \vec{BD} in 4:1, Find the position vector of E⃗.
5.
The king, queens and jack of clubs are removed from a deck of 52 playing cards and then shuffled. A card is drawn from remaining cards. Find the probability of getting:
(a) a queen
(b) a club
(c) '9' of red colour
6.
If x = 13 + 1·33·6 + 1·3·53·6·9 + ⋯ then prove that x2 + 2x − 2 = 0.
7.
Using formulae of trigonometric function, prove that: sin−135 + sin−11213 = cos−17785.
8.
If sin α = 1213 and sin β = 35 , where α and β lie in the first quadrant then find the values of cos(α + β) and sin(α + β)
9.
Find the points of intersection of the functions y = f(x) = 2x + 1 and y = g(x) = x2 − 4x + 6; ∀x ∈ R graphically.
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