BOOK - A

Transcription

BOOK - A

Class – XIIth
Full Syllabus
Test
BOOK - A
Time : 2 hour
Maximum Marks :120
Test Code : 4007
Name : ……………………………………………….…………..
Batch : ………..………..
Roll No. …………………..……
Please read the instructions carefully. Do not tamper with/mutilate the ORS or the Booklet.

Blank papers, clipboards, log tables, slide rules, calculators, cellular phones, pagers, and electronic gadgets in
any form are not allowed to be carried in side the examination hall.
SECTION – A
Instructions for questions No. 1 to 40. Each Question has 4 choices (A), (B), (C) and (D) for its answers, out of
which only one is correct. For each question you will be awarded
3 marks if you darken the bubble
corresponding to the correct answer and zero mark if no bubble is darkened. In case of bubbling of incorrect
answer, minus one (-1) mark will be awarded.
SECTION-A
Q1 Order and degree of the differential equation
(A.) order = 4, degree = not defined
(C.) order = 4, degree = 1
⎛ d3y ⎞
d4y
⎜⎜ 3 ⎟⎟ = 0 are
sin
+
dx 4
⎝ dx ⎠
(B.) order = 4, degree = 0
(D.) order = 3, degree = 1
Q2 Let [.] denotes the greatest integer function, then the value of
(A.) 0
(B.)
3
2
2
Q3 The area bounded between the parabolas x =
(A.)
20 2
3
(B.) 10 2
(C.)
5
4
∫
15
0
x[ x 2 ]dx is
(D.)
3
4
y
and x2 = 9y and the straight line y = 2 is
4
10 2
(C.) 20 2
(D.)
3
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1
Class – XIIth
Full Syllabus
Test
∫ f (x ) dx = ψ(x ) , then ∫ x
1 3
x ψ(x3) - 3 ∫ x ψ(x3)dx + C
3
5
Q4 If
(A.)
(C.)
3
1 3
[x ψ(x3) +
3
∫x
2
ψ(x3)dx] + C
f ( x 3 )dx is equal to
(B.)
(D.)
1 3
x ψ(x3) - ∫ x 2 ψ(x3)dx + C
3
1 3
[x ψ(x3) - ∫ x 2 ψ(x3)dx] + C
3
2
π π
x
Q5 The largest interval lying in ⎛⎜ − , ⎞⎟ for which the function f (x) = 4 − x + cos −1 ⎛⎜ − 1⎞⎟ + log(cos x )
⎝ 2 2⎠
⎝2 ⎠
is defined, is
⎡ π π⎞
⎛ π π⎞
(A.) ⎢− , ⎟
(B.) ⎜ − , ⎟
⎣ 4 2⎠
⎝ 2 2⎠
⎡ π⎞
(D.) [0, π]
(C.) ⎢0, ⎟
⎣ 2⎠
Q6 The area (in square units) bounded by the curves y = √x, 2y – x + 3 = 0, x-axis and lying in the first
quadrant is
(A.) 27/4
(B.) 36
(C.) 9
(D.) 18
Q7 If the integral
(A.) - 2
5 tan x
∫ tan x − 2 dx = x + a ln | sin x – 2 cos x | + K, then
(B.) 2
(C.) 1
a is equal to
(D.) - 1
⎡5 5α α ⎤
⎥
⎢
2
Q8 Let A = ⎢0 α 5α ⎥ . If det (A ) = 25, then | α | is
⎢⎣0 0
5 ⎥⎦
2
(B.) 5
(C.) 1/5
(A.) 5
(D.) 1
dy
= y + 3 > 0 and y(0) = 2, then y(log 2) is equal to
dx
(A.) - 2
(B.) 5
(C.) 7
(D.) 13
Q9 If
2
Class – XIIth
Full Syllabus
Test
if x ≤ 2
⎧5 ,
⎪
Q10 The value of a and b. Such that the function defined by f (x) = ⎨ax + b , if 2 < x < 10 is a
⎪21,
if x ≥ 10
⎩
continuous function is
(A.) a, b ∈ R
(C.) a = 1 and b = 2
(B.) a = 2 and b = 1
(D.) None of these
1
1
1
1 for xy ≠ 0, then D is divisible by
Q11 If D = 1 1 + x
1
1
1+ y
(A.) neither x nor y
(C.) x but not y
(B.) both x and y
(D.) y but not x
Q12 For real x, let f (x) = x3 + 5x + 1, then
(A.) f is neither one one nor onto R
(C.) f is onto R but not one one
(B.) f is one one and onto R
(D.) f is one one but no onto R
x
⎛ 5π ⎞
⎟ , define f(x) = ∫ 0 t sin t dt . Then, f has
Q13 For x ∈ ⎜ 0,
⎝ 2 ⎠
(A.) local maximum at π and local minimum at 2π
(B.) local minimum at π and 2π
(C.) local maximum at π and 2π
(D.) local minimum at π and local maximum at 2π
Q14 Let I be any interval disjoint from (-1, 1). Prove that the function f given by f (x) = x +
(A.) neither increasing nor decreasing on I
(C.) None of these
1
is
x
(B.) strictly decreasing on I
(D.) strictly increasing on I
Q15 The real number k for which the equation, 2x3 + 3x + k = 0 has two distinct real roots in [0, 1]
(A.) does not exist
(B.) lies between 2 and 3
(C.) Lies between – 1 and 0
(D.) lies between 1 and 2
3
Class – XIIth
Full Syllabus
Test
Q16
2 x + 3x
∫ 5 x dx is equal to
(A.) None of these
(C.)
2/5
3/5
+
+C
x log e (2 / 5) x log e (3 / 5)
(B.)
2x
3x
5x
+
−
+C
log 2 log 3 log 5
(D.)
(2 / 5) x
(3 / 5) x
+
+C
log e (2 / 5) log e (3 / 5)
Q17 Suppose the cubic x3 – px + q has three distinct real roots, where p > 0 and q > 0. Then, which one of
the following holds?
p
p
(A.) The cubic has minima at and maxima at
3
3
(B.) The cubic has minima at both
(C.) The cubic has minima at
(A.)
y
x
(B.)
p
3
p
and maxima at 3
(D.) The cubic has maxima at both
m n
m+n
, then
Q18 If x y = (x + y)
p
and 3
p
3
p
p
and −
3
3
dy
is
dx
x
y
(C.) xy
(D.)
x+y
xy
Q19 The value of a for which ax2 + sin-1 (x2 – 2x + 2) + cos-1 (x2 – 2x + 2) = 0 has a real solution, is
(A.) - 2/π
(B.) 2/π
(C.) - π/2
(D.) π/2
-1
-1
Q20 If cos x – cos
(A.) 4
y
= α, then 4x2 – 4xy cos α + y2 is equal to
2
(B.) 4 sin2 α
(C.) - 4 sin2 α
(D.) 2 sin2 α
4
Class – XIIth
Full Syllabus
Test
2
⎧ (log x − 1) ⎫
dx is equal to
Q21 ∫ ⎨
2 ⎬
⎩1 + (log x ) ⎭
log x
(A.)
(log x ) 2 + C
(C.)
xe x
+C
1+ x2
(B.)
(D.)
dy
at x = 1 is equal to
dx
1
(C.)
(B.)
2
x
+C
(log x ) 2 + 1
x
+C
2
x +1
-1
Q22 If y = sec (tan x), then
(A.) 1
1
2
Q23 The function f (x) = tan-1 (sin x + cos x) is an increasing function in
⎛π π⎞
⎛ π π⎞
⎛ π π⎞
(A.) ⎜ , ⎟
(B.) ⎜ − , ⎟
(C.) ⎜ − , ⎟
⎝4 2⎠
⎝ 2 4⎠
⎝ 2 2⎠
Q24
(D.)
(D.)
2
⎛ π⎞
⎜ 0, ⎟
⎝ 2⎠
e 2x − 1
∫ e 2 x + 1 dx is equal to
e x + e −x
+C
e x − x −x
e x + e −x
+C
e x − e −x
(A.) log | ex + e-x | + C
(B.) log
(C.) log | ex – e- x | + C
(D.)
Q25 If f (x) =
(A.) 1
cos 2 x + sin 4 x
for x ∈ R, then f (2002) is equal to
sin 2 x + cos 4 x
(B.) 4
(C.) 3
(D.) 2
5
Class – XIIth
Full Syllabus
Test
Q26 If x is real, the maximum value of
(A.) 1/4
(B.) 1
3x 2 + 9 x + 17
is
3x 2 + 9 x + 7
(C.) 17/7
(D.) 41
⎡1 α 3 ⎤
⎢
⎥
Q27 If P = ⎢1 3 3⎥ is the adjoint of 3 × 3 matrix A and | A | = 4, then α is equal to
⎢⎣2 4 4⎥⎦
(A.) 11
(B.) 5
(C.) 0
(D.) 4
Q28 How many real solutions does the equation x7 + 14x5 + 16x3 + 30x – 560 = 0 have?
(A.) 7
(B.) 1
(C.) 3
(D.) 5
Q29 The value of
(A.) π/a
cos 2 x
∫ −π 1 + a x dx , a > 0, is
(B.) π/2
π
(C.) aπ
(D.) 2π
⎡1 0 0 ⎤
⎡1 ⎤
⎡0 ⎤
⎥
⎥
⎢
⎢
Q30 Let A = 2 1 0 . If u1 and u2 are column matrices such that Au1 = 0 and Au2 = ⎢1⎥ , then
⎥
⎢
⎢ ⎥
⎢ ⎥
⎢⎣3 2 1⎥⎦
⎢⎣0⎥⎦
⎢⎣0⎥⎦
u1 + u2 is equal to
⎡− 1⎤
⎡− 1⎤
⎡− 1⎤
⎡1⎤
⎢− 1⎥
⎢1⎥
⎢1⎥
⎢ ⎥
(B.) ⎢ ⎥
(C.) ⎢ ⎥
(D.) ⎢− 1⎥
(A.) ⎢ ⎥
⎢⎣ 0 ⎥⎦
⎢⎣ 0 ⎥⎦
⎢⎣− 1⎥⎦
⎢⎣− 1⎥⎦
⎛1⎞
Q31 Let f (x) = f (x) + f ⎜ ⎟ , where f (x) =
⎝x⎠
(A.) 2
(B.) 1
∫
x
1
log t
dt . Then, f (e) is equal to
1+ t
1
(D.) 0
(C.)
2
6
Class – XIIth
Full Syllabus
Test
1 + a 2 x (1 + b 2 ) x (1 + c 2 ) x
2
2
2
2
2
2
Q32 If a + b + c = - 2 and f (x) = (1 + a ) x 1 + b x (1 + c ) x , then f (x) is a polynomial of degree
(1 + a 2 ) x (1 + b 2 ) x 1 + c 2 x
(A.) 0
Q33
(B.) 2
(D.) 1
dx
∫ cos x +
⎛x
(A.) log tan⎜
⎝2
⎛x
(C.) log tan⎜
⎝2
is equal to
3 sin x
π⎞
+ ⎟+C
12 ⎠
π⎞
− ⎟+C
12 ⎠
Q34 If A = sin2 x + cos4 x, then for all real x
3
13
≤A≤
(A.)
4
16
13
≤ A ≤1
(C.)
16
Q35 lim
x →0
(A.) -
(C.) 3
(B.)
(D.)
1
⎛x π ⎞
log tan⎜ − ⎟ + C
2
⎝ 2 12 ⎠
1
⎛x π ⎞
log tan⎜ + ⎟ + C
2
⎝ 2 12 ⎠
(B.) 1 ≤ A ≤ 2
(D.)
3
≤A≤1
4
(1 − cos 2x )(3 + cos x )
is equal to
x tan 4x
1
4
(B.) 1
Q36 If f (x) = xn + 4, then the value of f (1) +
(A.) None of these
(C.) 2n – 1
(C.) 2
(D.)
1
2
f ' (1) f ' ' (1)
f n (1)
+
+ ....... +
is
1!
2!
n!
1
1 1
(B.) 1 + + + ......... +
n!
1! 2!
(D.) 2n + 4
7
Class – XIIth
Full Syllabus
Test
-1
-1
Q37 sin x + sin
(A.) π
1
1
+ cos-1 x + cos-1 , x ∉ ± 1 is equal to
x
x
(B.) None of these
(C.) 3π/2
2⎞
⎛
-1 5
+ tan −1 ⎟ is
Q38 The value of cot ⎜ cosec
3⎠
3
⎝
5
6
(A.)
(B.)
17
17
(C.)
3
17
(D.) π/2
(D.)
4
17
Q39 The number of arbitrary constants in the general solution of a differential equation of fourth order is
(A.) 4
(B.) 3
(C.) 2
(D.) zero
Q40 The intercepts on x-axis made by tangents to the curve,
y=
(A.) ± 3
∫
x
0
| t | dt , x ∈ R, which are parallel to the line y = 2x, are equal to
(B.) ± 4
(C.) ± 1
(D.) ± 2
8

BOOK - A

Transcription

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