**SECTION A**

**Question 1: The number of seven-digit integers with the sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only is:**

a. 77

b. 42

c. 35

d. 82

Answer: (a)

Case – I: 1, 1 , 1, 1, 1, 2, 3

Number of ways = 7! / 5! = 42

Case – II: 1, 1, 1, 1, 2, 2, 2

Number of ways = 7! / [4! * 3!] = 35

Total number of ways = 42 + 35 = 77

**Question 2: The maximum value of the term independent of ‘t’ in the expansion of [(tx ^{1/5} + {(1 – x)^{1/10} / t}]^{10 }where x ∈ (0, 1) is :**

a. 10! / [√3 (5!)^{2}]

b. [2 . 10!] / [3 (5!)^{2}]

c. [10!] / [3 (5!)^{2}]

d. [2 . 10!] / [3√3 (5!)^{2}]

Answer: (d)

T_{r+1} = ^{10}C_{r} (tx^{1/5})^{10-r} {[(1 – x)^{1/10}] / t}^{r}

= ^{10}C_{r} t ^{10-2r} x^{[10-r]/5} (1 – x)^{r/10}

10 – 2r = 0

r = 5

T_{6} = ^{10}C_{5} x √1 – x

dT_{6} / dx = ^{10}C_{5} [√(1 – x) – (x/2√(1 – x)] = 0

1 – x = x / 2

x = 2 / 3

Max (T_{6}) = {[10!] / [5! 5!] * [2] / [3√3]}

**Question 3: The value of ∑ _{n=1}^{100} ∫_{n-1}^{n} e^{x-[x]} dx, where [x] is the greatest integer ≤ x, is :**

a. 100 (e – 1)

b. 100e

c. 100 (1 – e)

d. 100 (1 + e)

Answer: (a)

∑_{n=1}^{100} ∫_{n-1}^{n} e^{x-[x]} dx

= ∫_{0}^{1} e^{{x} }dx + ∫_{1}^{2} e^{{x} }dx + ∫_{2}^{3} e^{{x} }dx + …….. ∫_{99}^{100} e^{{x} }dx {because {x} = x – [x]}

= e^{x} |_{0}^{1} + e^{(x-1)}|_{1}^{2} + e^{(x-2)}|_{2}^{3} + …… + e^{(x-99)}|_{99}^{100}

= (e – 1) + (e – 1) + (e – 1) + …… + (e – 1)

= 100 (e – 1)

**Question 4: The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is 1000 at initial time t = 0. The number of bacteria has increased by 20% in 2 hours. If the population of bacteria is 2000 after k / log _{e} (6 / 5) hours, then (k / log_{e} 2)^{2} is equal to :**

a. 4

b. 2

c. 16

d. 8

Answer: (a)

Let x be the number of bacteria at time t.

dx / dt ∝ x

dx / dt = λx

∫_{1000}^{x} dx / x = ∫_{0}^{t} λ dt

(ln x) – (ln 1000) = λt

ln (x / 1000) = λt

Put t = 2, x = 1200

(ln 12 / 10) = 2λ

λ = (1 / 2) ln (6 / 5)

Now, ln (x / 1000) = (t / 2) ln (6 / 5)

x = 1000e^{t/2 ln (6 / 5)}

Given, x = 2000 at t = k / log_{e} (6 / 5)

2000 = 1000 e^{[k/2 ln (6 / 5)] * [ln (6 / 5)]}

2 = e^{k/2} ⇒ ln 2 = k / 2

k / ln 2 = 2

[k / ln 2]^{2}= 4

**Question 5: If **

**\(\begin{array}{l}\vec{a}\end{array} \)**

**and\(\begin{array}{l}\vec{b}\end{array} \) are perpendicular, then \(\begin{array}{l}\vec{a}\times(\vec{a}\times(\vec{a}\times(\vec{a}\times\vec{b})))\end{array} \) is equal to :**

a. (1 / 2) |

^{4}

b.

c. |

^{4}

d.

Answer: (c)

**Question 6: In an increasing geometric series, the sum of the second and the sixth term is 25 / 2 and the product of the third and fifth term is 25. Then, the sum of 4 ^{th}, 6^{th} and 8^{th} terms is equal to :**

a. 35

b. 30

c. 26

d. 32

Answer: (a)

Let a be the first term and r be the common ratio.

ar + ar^{5} = 25 / 2 and ar^{2} x ar^{4} = 25

a^{2}r^{6 }= 25

ar^{3} = 5

a = 5 / r^{3} …(1)

(5r / r^{3}) + (5r^{5} / r^{3}) = 25 / 2

(1 / r^{2}) + r^{2} = 5 / 2

Put r^{2} = t

^{2}+ 1] / t = 5 / 2

2t^{2} – 5t + 2 = 0

2t^{2} – 4t – t + 2 = 0

(2t – 1) (t – 2) = 0

t = 1 / 2, 2

r^{2 }= 1 / 2, 2

r = √2 as the G.P. is increasing.

ar^{3} + ar^{5} + ar^{7}

= ar^{3}( 1 + r^{2} + r^{4})

= 5 [1 + 2 + 4]

= 35

**Question 7: Consider the three planes**

**P _{1} : 3x + 15y + 21z = 9,**

**P _{2} : x – 3y – z = 5, and**

**P _{3} : 2x + 10 y + 14z = 5**

**Then, which one of the following is true ?**

a. P_{1} and P_{3} are parallel.

b. P_{2} and P_{3} are parallel.

c. P_{1} and P_{2} are parallel.

d. P_{1}, P_{2} and P_{3} all are parallel.

Answer: (a)

P_{1} = x + 5y + 7z = 3

P_{2} = x – 3y – z = 5

P_{3} = x + 5y + 7z = 5/2

P_{1}||P_{3}

**Question 8: The sum of the infinite series 1 + (2 / 3) + (7 / 3 ^{2}) + (12 / 3^{3}) + (17 / 3^{4}) + (22 / 3^{5}) + …… is equal to :**

a. 9 / 4

b. 15 / 4

c. 13 / 4

d. 11 / 4

Answer: (c)

s = 1 + (2 / 3) + (7 / 3^{2}) + (12 / 3^{3}) + (17 / 3^{4}) + (22 / 3^{5}) + ……

(s / 3) = (1 / 3) + (2 / 3^{2}) + (7 / 3^{3}) + ……. infinity

_________________________________

(2s / 3) = 1 + (1 / 3) + (5 / 3^{2}) + (5 / 3^{3}) + ….. infinity

2s / 3 = (4 / 3) + (5 / 3) [(1 / 3) / (1 – (1 / 3)] = (4 / 3) + (5 /6) = 13 / 6

s = 13 / 4

**Question 9: The value of **

**\(\begin{array}{l}\begin{bmatrix} (a+1)(a+2)& a+2 & 1\\ (a+2)(a+3)& a+3 & 1\\ (a+3)(a+4)& a+4 & 1 \end{bmatrix}\end{array} \)**

**is**

a. –2

b. (a + 1) (a + 2) (a + 3)

c. 0

d. (a + 2) (a + 3) (a + 4)

Answer: (a)

_{}

**Question 10: If (sin ^{-1 }x) / a = (cos^{-1} x) / b) = (tan^{-1} y) / c; 0 < x < 1, then the value of cos (πc / [a + b]) is:**

a. [1 – y^{2}] / 2y

b. [1 – y^{2}] / [1 + y^{2}]

c. 1 – y^{2}

d. [1 – y^{2}] / y √y

Answer: (b)

(sin^{-1 }x) / a = (cos^{-1} x) / b) = (tan^{-1} y) / c

(sin^{-1 }x) / a = (cos^{-1} x) / b) = [sin^{-1} x + cos^{-1} x] / (a + b) = π / [2 (a + b)]

Now, [tan^{-1} y / c] = π / [2 (a + b)]

2 tan^{-1} y = (πc / (a + b))

cos (πc / [a + b]) = cos (2 tan^{-1} y) = [1 – y^{2}] / [1 + y^{2}]

**Question 11: Let A be a symmetric matrix of order 2 with integer entries. If the sum of the diagonal elements of A ^{2} is 1, then the possible number of such matrices is :**

a.6

b. 1

c. 4

d. 12

Answer: (c)

Let A =

a^{2} + 2b^{2} + c^{2 }= 1

a = 1, b = 0, c = 0

a = 0, b = 0, c = 1

a = –1, b = 0, c = 0

c = –1, b = 0, a = 0

**Question 12: The intersection of three lines x – y = 0 and x + 2y = 3 and 2x + y = 6 is a:**

a. Equilateral triangle

b. Right angled triangle

c. Isosceles triangle

d. None of the above

Answer: (c)

AB = AC = √5, BC = √2

Hence, the triangle is isosceles.

**Question 13: The maximum slope of the curve y = (1 / 2) x ^{4} – 5x^{3} + 18x^{2} – 19x occurs at the point:**

a. (2, 9)

b. (2, 2)

c. (3, 21 / 2)

d. (0, 0)

Answer: (b)

dy / dx = 2x^{3} – 15x^{2} + 36x – 19

Let f (x) = 2x^{3} – 15x^{2} + 36x – 19

f’ (x) = 6x^{2} – 30x + 36 = 0

x^{2} – 5x + 6 = 0

x = 2, 3

f’’ (x) = 12x – 30

f’’ (x) < 0 for x = 2

So, at x = 2, slope is maximum.

y = 8 – 40 + 72 – 38

= 72 – 70

= 2

Maximum slope occurs at (2, 2).

**Question 14: Let f be any function defined on R and let it satisfy the condition:**

**|f (x) – f (y)|≤ |(x – y) ^{2}|, ∀ x, y ∈ R. If f (0) = 1, then :**

a. f (x) < 0, ∀ x ∈ R

b. f (x) can take any value in R

c. f (x) = 0, ∀ x ∈ R

d. f (x) > 0, ∀ x ∈ R

Answer: (d)

|f (x) – f (y)| ≤ |(x – y)^{2}|, ∀ x, y ∈ R

|[f (x) – f (y)] / [(x – y)]| ≤ |x – y|

lim_{x→y} |[f (x) – f (y)] / [(x – y)]| ≤ 0

|f’ (y)| ≤ 0

f’ (y) = 0

f (y) = c

Since f (0) = 1, f (y) = 1, ∀ y ∈ R

**Question 15: The value of ∫ _{-π/2}^{π/2} cos^{2} x / [1 + 3^{x}] dx is: **

a. 2π

b. 4π

c. π / 2

d. π / 4

Answer: (d)

Let I = ∫_{-π/2}^{π/2} cos^{2} x / [1 + 3^{x}] dx

I = ∫_{-π/2}^{π/2} [cos^{2} x] / [1 + 3^{-x}] dx

= ∫_{-π/2}^{π/2} [3^{x} cos^{2} x] / [1 + 3^{x}] dx

2I = ∫_{-π/2}^{π/2} cos^{2} x dx

I = ∫_{0}^{π/2} cos^{2} x dx

= π / 4

**Question 16: The value of is :**

a. 3 / 4

b. 2 / √3

c. 4 / 3

d. 2 / 3

Answer: (c)

**Question 17: ****A fair coin is tossed a fixed number of times. If the probability of getting 7 heads is equal to the probability of getting 9 heads, then the probability of getting 2 heads is:**

a. 15 / 2^{12}

b. 15 / 2^{13}

c. 15 / 2^{14}

d. 15 / 2^{8}

Answer: (b)

P (x = 9) = P (x = 7)

⇒ ^{n}C_{9} (1 / 2)^{n-9 }(1 / 2)^{9 }= ^{n}C_{7 }(1 / 2)^{n-7 }(1 / 2)^{7}

⇒ ^{n}C_{9} (1 / 2)^{n }= (1 / 2)^{n }x ^{n}C_{7}

⇒ n = 9 + 7 = 16

P (x = 2) = ^{16}C_{2 }(1 / 2)^{14 }(1 / 2)^{2}

= ^{16}C_{2 }(1 / 2)^{16}

= 15 / 2^{13}

**Question 18: If (1, 5, 35), (7, 5, 5), (1, λ, 7) and (2 λ, 1, 2) are coplanar, then the sum of all possible values of λ is:**

a. − 44 / 5

a. 39 / 5

a. − 39 / 5

a. 44 / 5

Answer: (d)

Let P (1, 5, 35), Q (7, 5, 5), R (1, λ, 7) and S (2 λ, 1, 2)

⇒ {−33λ + 165 − 112} + 5(λ − 5)(2λ − 1) = 0

⇒ 53 − 33λ + 5 {2λ^{2} − 11λ + 5} = 0

⇒ 10λ^{2} − 88λ + 78 = 0

5λ^{2} − 44λ + 39 = 0

∴ λ_{1} + λ_{2} = 44 / 5

**Question 19: Let R = {P,Q)|P and Q are at the same distance from the origin} be a relation, then the equivalence class of (1,–1) is the set :**

a. S = {(x, y)|x^{2 }+ y^{2 }= 1}

b. S = {(x, y)|x^{2 }+ y^{2 }= 4}

c. S = {(x, y)|x^{2 }+ y^{2 }= √2}

d. S = {(x, y)|x^{2 }+ y^{2 }= 2}

Answer: (d)

P(a, b) , Q(c, d), OP = OQ

a^{2} + b^{2} = c^{2} + d^{2}

R(x ,y), S = (1, –1) OR = OS ( equivalence class)

x^{2} + y^{2} = 2

**Question 20: In the circle given below, let OA = 1 unit, OB = 13 unit and PQ perpendicular to OB. Then, the area of the triangle PQB (in square units) is :**

a. 26 √3

b. 24 √2

c. 24 √3

d. 26 √2

Answer: (c)

OC = 13 / 2 = 6.5

AC = CO – AO

= 6.5 – 1

= 5.5

In △PAC

PA = √(6.5^{2} – 5.5^{2})

PA = √12

PQ = 2PA = 2 √12

Now, area of △PQB = (1 / 2) * PQ * AB

= (1 / 2) * 2 √12 * 12

= 12 √12

= 24 √3

**Section-B**

**Question 1: The area bounded by the lines y = |x – 1| – 2 is ______.**

Answer: 4

NTA Ans. (8)

Required area area of △PQR

Area = (1 / 2) * 4 * 2 = 4

This is a bonus question as the second curve is also not given.

If the second curve is x – axis, then answer will be 4

**Question 2: The number of integral values of ‘k’ for which the equation 3 sin x + 4 cos x = k + 1 has a solution, k ∈ R is ______.**

Answer: 11

3 sin x + 4 cos x = k + 1

⇒ −5 ≤ k + 1 ≤ 5

⇒ −6 ≤ k ≤ 4

−6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4 → 11 integral values

**Question 3: Let m, n ∈ N and gcd (2, n) = 1. If 30 ^{30}C_{0} + 29 ^{30}C_{1} + ….. + 2 ^{30}C_{28} + 1 ^{30}C_{29} = n . 2^{m}, then n + m =**

Answer: 45

Let S = ∑_{r=0}^{30} (30 – r) ^{30}C_{r}

= 30 ∑_{r=0}^{30}^{30}C_{r} – ∑_{r=0}^{30} r ^{30}C_{r}

= 30 * 2^{30} – ∑_{r=1}^{30} r (30 / r) ^{29}C_{r-1}

= 30 * 2^{30} – 30 * 2^{29}

= 30 * 2^{29} (2 – 1)

= 15 * 2^{30}

n = 15 and m = 30

n + m = 45

**Question 4: If y = y (x) is the solution of the equation e ^{siny} cosy (dy / dx) + e^{siny} cosx = cos x, y (0) = 0; then 1 + y (π / 6) + (√3 / 2) y (π / 3) + (1 / √2) y (π / 4) is equal to ______.**

Answer: 1

e^{siny} cosy (dy / dx) + e^{siny} cosx = cos x

Put e^{siny} = t

e^{siny} * cos y (dy / dx) = dt / dx

Then, dt / dx + t cos x = cos x

IF = e^{∫cosx dx} = e^{sinx}

Solution of differential equation

t . e^{sinx} = ∫e^{sinx} cosx dx

e^{siny }. e^{sinx} = e^{sinx} + c

At x = 0, y = 0

1 = 1 + c ⇒ c = 0

sin y + sin x = sin x

y = 0

y (π / 6) = 0, y (π / 3) = 0, y (π / 4) = 0

The required answer is 1 + 0 + 0 + 0 = 1.

**Question 5: The number of solutions of the equation log _{4} (x – 1) = log_{2} (x – 3) is ______.**

Answer: 1

(1 / 2) log_{2 }(x − 1) = log_{2 }(x − 3)

⇒ x − 1 = (x − 3)^{2}

⇒ x^{2} − 6𝑥 + 9 = x − 1

⇒ x^{2} − 7𝑥 + 10 = 0

⇒ x = 2, 5

x = 2 Not possible as log_{2 }(x − 3) is not defined.

∴ Number of solution = 1

**Question 6: If √3 (cos ^{2} x) = (√3 – 1) cos x + 1 the number of solutions of the given equation when x ∈ [0, π / 2] is _______.**

Answer: 1

√3 t^{2} – (√3 – 1) t – 1 = 0, where t = cos x

Now, t = {(√3 – 1) ± [√4 + 2 √3]} / 2 √3

t = cos x = 1 or – 1 / √3 (rejected as x ∈ [0, π / 2])

cos x = 1

Number of solution = 1

**Question 7: Let (λ, 2, 1) be a point on the plane which passes through the point (4, – 2, 2). If the plane is perpendicular to the line joining the points (– 2, – 21, 29) and (– 1, – 16, 23), then (λ / 11) ^{2} – (4λ / 11) – 4 is equal to ______.**

Answer: 8

AB is perpendicular to PQ.

[(4 – λ) i – 4j + k] . [i + 5j – 6k] = 04 – λ – 20 – 6 = 0

λ = – 22

Now, (λ / 11) = – 2

(λ / 11)^{2} – (4λ / 11) – 4 = 4 + 8 – 4 = 8

**Question 8: The difference between degree and order of a differential equation that represents the family of curves given by y ^{2} = a (x + (√a / 2)), a > 0 is ______.**

Answer: 2

y^{2} = a (x + (√a / 2))

Differentiating w.r.t.

2yy’ = a

y^{2} = 2yy’ [x + (√2yy’ / 2)]

y = 2y’ [(x + √yy’ / 2)]

y – 2xy’ = √2y’ √yy’

[y – 2x (dy / dx)]^{2}= 2y (dy / dx)

^{3}

Degree = 3 and Order = 1

Degree – Order = 3 – 1 = 2

**Question 9: The sum of 162 ^{th} power of the roots of the equation x^{3} – 2x^{2} + 2x – 1 = 0 is _________.**

Answer: 3

Let roots of x^{3} – 2x^{2} + 2x – 1 = 0 be ɑ, β, γ.

(x^{3} – 1) – (2x^{2} – 2x) = 0

(x – 1) (x^{2} – x + 1) = 0

x = 1, – 𝞈, – 𝞈^{2}

Now, ɑ^{162} + β^{162} + γ^{162 }

= 1 + (𝞈)^{162 }+ (𝞈^{2})^{162}

= 1 + (𝞈^{3})^{54} + (𝞈^{3})^{108} = 3

**Question 10: The value of the integral ∫ _{0}^{π} |sin 2x| dx is _______.**

Answer: 2

I = ∫_{0}^{π} |sin 2x| dx

= 2 ∫_{0}^{π/2} |sin 2x| dx = 2 ∫_{0}^{π/2} sin 2x dx

= 2 [- cos (2x) / 2]_{0}^{π/2}

= 2

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