Commerce (English Medium)
Science (English Medium)
Arts (English Medium)
Academic Year: 2019-2020
Date & Time: 17th March 2020, 10:30 am
Duration: 3h
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General Instructions:
Read the following instructions very carefully and strictly follow them:
- This question paper comprises four sections – A, B, C and D.
This question paper carries 36 questions. All questions are compulsory. - Section A – Question no. 1 to 20 comprises of 20 questions of one mark each.
- Section B – Question no. 21 to 26 comprises of 6 questions of two marks each.
- Section C – Question no. 27 to 32 comprises of 6 questions of four marks each.
- Section D – Question no. 33 to 36 comprises of 4 questions of six marks each.
- There is no overall choice in the question paper. However, an internal choice has been provided in 3 questions of one mark, 2 questions of two marks, 2 questions of four marks and 2 questions of six marks. Only one of the choices in such questions have to be attempted.
- In addition to this, separate instructions are given with each section and question, wherever necessary.
- Use of calculators is not permitted.
The value of `sin^-1 (cos (3π)/5)` is ______.
`π/10`
`(3π)/5`
`- π/10`
`(-3π)/5`
Chapter:
If A = `[(2, -3, 4)]`, B = `[(3),(2),(2)]`, X = `[(1, 2, 3)]` and Y = `[(2),(3),(4)]`, then AB + XY equals
[28]
[24]
28
24
Chapter:
If `|(2, 3, 2),(x, x, x),(4, 9, 1)| + 3 = 0`, then the value of x is ______.
3
0
–1
1
Chapter:
`int_0^(π/8) tan^2 (2x)` is equal to ______.
`(4 - π)/8`
`(4 + π)/8`
`(4 - π)/4`
`(4 - π)/2`
Chapter:
If `veca * vecb = 1/2 |veca| |vecb|`, then the angle between `veca` and `vecb` is ______.
0°
30°
60°
90°
Chapter:
The two lines x = ay + b, z = cy + d; and x = a'y + b', z = c'y + d' are perpendicular to each other, if
`a/a^' + c/c^' = 1`
`a/a^' + c/c^' = -1`
aa' + cc' = 1
aa' + cc' = –1
Chapter:
The two planes x – 2y + 4z = 10 and 18x + 17y + kz = 50 are perpendicular, if k is equal to ______.
– 4
4
2
– 2
Chapter:
The value of `tan [1/2 cos^-1 (sqrt(5)/3)]` is ______.
`(3 + sqrt(5))/2`
`(3 - sqrt(5))/2`
`(-3 + sqrt(5))/2`
`(-3 - sqrt(5))/2`
Chapter:
If A = `[(a, 0, 0),(0, a, 0),(0, 0, a)]`, then det (adj A) equals
a27
a9
a6
a2
Chapter:
The line `(x - 2)/3 = (y - 3)/4 = (z - 4)/5` is parallel to the plane
2x + 3y + 4z = 0
3x + 4y – 5z = 7
2x + y – 2z = 0
x – y + z = 2
Chapter:
If f : R → R be given by f(x) = (3 – x3)1/3, then fof(x) = ______.
Chapter:
If `[(x + y, 7),(9, x - y)] = [(2, 7),(9, 4)]`, then x · y = ______.
Chapter:
The number of points of discontinuity of f defined by f(x) = |x| – |x + 1| is ______.
Chapter:
The slope of the tangent to the curve y = x3 – x at the point (2, 6) is ______.
Chapter:
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The rate of change of the area of a circle with respect to its radius r, when r = 3 cm, is ______.
Chapter:
If [x] denotes the greatest integer function, then find `int_0^(3/2) [x^2] dx`
Chapter:
Check if the relation R on the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y) : y is divisible by x} is (i) symmetric (ii) transitive.
Chapter:
Prove that: `(9π)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt(2))/3)`
Chapter:
Find the value of `dy/dx` at `θ = π/3`, if x = cos θ – cos 2θ, y = sin θ – sin 2θ.
Chapter:
Show that the function f defined by f(x) = (x – 1)ex + 1 is an increasing function for all x > 0.
Chapter:
Find `|veca|` and `|vecb|`, if `|veca| = 2|vecb|` and `(veca + vecb).(veca - vecb) = 12`.
Chapter:
Find the unit vector perpendicular to each of the vectors `veca = 4hati + 3hatj + hatk` and `vecb = 2hati - hatj + 2hatk`.
Chapter:
Show that the function `f(x) = x/3 + 3/x` decreases in the intervals (–3, 0) ∪ (0, 3).
Chapter:
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Three distinct numbers are chosen randomly from the first 50 natural numbers. Find the probability that all the three numbers are divisible by both 2 and 3.
Chapter:
Prove that the relation R on Z, defined by R = {(x, y) : (x – y) is divisible by 5} is an equivalence relation.
Chapter:
If `y = sin^-1 ((sqrt(1 + x) + sqrt(1 - x))/2)`, then show that `dy/dx = (-1)/(2sqrt(1 - x^2)`.
Chapter:
Verify the Rolle’s Theorem for the function f(x) = ex cos x in `[- π/2, π/2]`
Chapter:
Evaluate: `int_0^π (x sin x)/(1 + cos^2x) dx`.
Chapter: [7] Integrals
For the differential equation given below, find a particular solution satisfying the given condition `(x + 1) dy/dx = 2e^-y + 1; y = 0` when x = 0.
Chapter:
Find the general solution of the differential equation `dy/dx + 1/x = e^y/x`.
Chapter:
Using properties of determinants prove that:
`|(a - b, b + c, a),(b - c, c + a, b),(c - a, a + b, c)| = a^3 + b^3 + c^3 - 3abc`.
Chapter:
If A = `[(1, 3, 2),(2, 0, -1),(1, 2, 3)]`, then show that A3 – 4A2 – 3A + 11I = 0, Hence find A–1 .
Chapter:
Find the intervals on which the function f(x) = (x – 1)3 (x – 2)2 is (a) strictly increasing (b) strictly decreasing.
Chapter:
Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.
Chapter: [6] Applications of Derivatives
Find the area of the region lying in the first quadrant and enclosed by the x-axis, the line y = x and the circle x2 + y2 = 32.
Chapter:
Find the image of the point (–1, 3, 4) in the plane x – 2y = 0.
Chapter:
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