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Class – XIIth
Full Syllabus
Test
BOOK - A
Time : 2 hour
Maximum Marks :120
Test Code : 4007
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Batch : ………..………..
Roll No. …………………..……
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™
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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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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
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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
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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
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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
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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
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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
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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
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