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SUBJECT : MATHEMATICS
a
x- x

òe

1
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Find a ?

0

1

Sol
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Tangent and normal are drawn to parabola y2 = 2x at A(2, 2)
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Area of DPAQ in

Sol
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lim 2 – cos x
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=?

æ x +2 ö
lim ç 2 ÷ 1- cos x cos 2x
x ø
e
x ®0 è

lim

e

(x + 2) 1-cos2 x
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(1) ax2 – 6x + 15 is minimum at x =

æ3 ö
3
) and decreasing in ç , ¥ ÷ , then
è4 ø
4

3
4

(2) ax2 – 6x + 15 is minimum at x = –

3
4

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(3) ax2 – 6x + 15 is maximum at x =

3
4

(4) ax2 – 6x + 15 is maximum at x = –
Ans
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-

3
4

b
3
-6 3
=
Þ
=
2a 4
2a 4

a = –4
\ for ax2 – 6x + 15
vertex is at

6
-3
=
2a
4

Þ Maximum at x = –

3
4

Topic : Monotonocity
Subtopic :
Level : Easy

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r
r
r r r
r r r
ˆ b = ˆi + ˆj, cr is a vector such that a
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r r r
| (a ´ b) ´ c | is
Sol
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b =| c | Þ | c |= 2 + 1 = 3

r r r
r r r
\ | (a ´ b) ´ c | = | a ´ b || c | sinq
ˆi ˆj kˆ
r r
Also | a ´ b |= 2 1 –2 = 2iˆ – 2ˆj + kˆ
1 1 0
r r r
9
\ | (a ´ b) ´ c |= 3 ´ 3 ´ sin30º =
2
Topic : Vector
Subtopic :
Level : Medium
If y(x) satisfy

6
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3

Sol
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= t +C

1- y2 + C

Q y(1) = –1 ® C = 0
\ (x – 1) ex =

1- y2

\ At x = 3
2e3 = 1 - y 2
4e6 = 1 – y2
(y(3))2 = 1 – 4e6
Topic : Differential Equation
Subtopic : Differential Equation
Level : Medium

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10
ways
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least 4, bowlers, 5 betsman, and 1 wicket keepers ?
Sol
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Solve for x :

tan-1 x 2 + x + sin-1 x 2 + x + 1 =
Sol
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In the expansion of (1 –x)101 (1 + x + x2)100, find coefficient of x 256
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1

Sol
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Sol
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rrr
r r r
If a,b,c are mutually perpendicualr unit vector, equally inclined to a + b + c at angle q
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r r r r
a
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1
æ 41
ö
ç 4 + 56 ÷
è
ø

Sol
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120 -r
4

r

56

\ r = 4l & 6 l
Þ r = 12 l
\ r = 0, 12,
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It the shortest distance between
r
ˆ ; l Î R, a>0
r1 = a ˆi - 2jˆ + 2kˆ + l(iˆ – 2jˆ + 2k)

r
ˆ ;
r 2 = -4iˆ - kˆ + m(3iˆ – 2ˆj - 2k)

mÎR

is 9, find 'a' = ?

Sol
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x2 + 31/4x +

96 12
96
12
3 = 0 has roots a, b then a (a –1) + b (b – 1) equals

Sol
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A lne y = mx + c pass through focus of parabola y2 = –64x and touches the circle (x + 10)2 + y2 = 4
Find 4 2 m + c
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F(–16, 0) Þ 0 = –16m + c
c = 16m
Centre (–10, 0)

r=2

Þp=r
-10m + c

=2

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m2 + 1

Þ (–10m + c)2 = 4(m2 + 1)

Þ 36m 2 = 4m 2 + 4

m2 =

m=

c=

4 1
=
32 8

1
2 2
8
2

æ 1
8 ö
+
4 2 m+c = 4 2ç
÷
2
2
2ø
è
= 34
Topic : Conic Section
Subtopic : Conic Section
Level : Hard

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At x = 0
RHL = LHL = f(0)
Þ a + (–1) = –1
Þa=0
At x = 1
RHL = LHL = f(1)
Þ2–b=a–1
Þb=3
Qa+b=3

Topic : Continuity
Subtopic :
Level : Medium

18
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æ 3p ö
içq +
÷
2 ø
è

z = r
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If the domach of
| x | -2
f(x) =

| x | -3

is

-¥,a È b,c È 4, ¥
Find a + b + c
Sol
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