KVPY-SX 2017 Maths Paper with Solutions

Question 1: Let BC be a fixed-line segment in the plane. The locus of a point A such that the triangle ABC is isosceles, is (with finitely many possible exceptional points)

a. A line
b. A circle
c. The union of a circle and a line
d. The union of two circles and a line

Answer: (d)

Let the coordinates of fixed base BC be B (0, 0), C (a, 0) and the variable point A (h, k).

Since △ABC is isosceles, three cases arises:

Case-I: AB = AC

√h² + k² = √(h – a)² + k²

h² + k² = h² + a² – 2ah + k²

h = a / 2

x = a / 2

Locus of A is straight line,

Case-II: AB = BC

√h² + k² = a

h² + k² = a²

x² + y² = a²

Circle

Case-III: AB = BC

√(h – a)² + k² = a

(h – a)² + k² = a²

(x – a)² + k² = a²

Circle

So, the locus of A is union of two circles and a line.

Question 2: The number of solution pairs (x, y) of the simultaneous equations log_1/3(x + y) + log₃(x – y) = 2 and 2y² = 512^x+1 is

a. 0
b. 1
c. 2
d. 3

Answer: (b)

Given, log_1/3(x + y) + log₃(x – y) = 2

log_3^(-1) (x + y) + log₃ (x – y) = 2

–log₃ (x + y) + log₃ (x – y) = 2 {log_b^n a = (1 / n) log_ba}

log₃ [(x – y) / (x + y)] = 2 {log m – log n = log m / n}

Taking antilog

(x – y) / (x + y) = 9

x – y = 9x + 9y

8x + 10y = 0

4x + 5y = 0 …(i)

Also, 2^{y^2} = 512^x+1

—– (ii)

From eq.(i) & (ii)

y² = 9 (-5y / 4) + 9

4y² + 45y – 36 = 0

4y² + 48y – 3y – 36 = 0

(4y – 3) (y + 12) = 0

y = 3 / 4, –12

Using (i)

x = – 15 / 16 , 15

We have (–15 / 16, 3 / 4) and (15, –12)

Since, (–15 / 16, 3 / 4) do not satisfy the original equation, rejected.

So the only answer is (15, –12).

Question 3: The value of the limit lim_x→∞ (√(4x² – x) + 2x) is

a. –∞
b. − 1 / 4
c. 0
d. 1 / 4

Answer: (b)

lim_x→∞ (√(4x² – x) + 2x)

Put x = –y; x → –∞, y → +∞

Therefore lim_y→∞ (√(4y² + y) – 2y)

Rationalizing

lim_y→∞ (y / √(4y² + y) + 2y)

Dividing numerator and denominator by y

lim_y→∞ 1 / [√(4 + [1 / y])] + 2

= 1 / [√(4 + 0) + 2] {1 / ∞ → 0}

= 1 / 4

Question 4: Let R be a relation on the set of all-natural numbers given by a R b ⇔ a divides b². Which of the following properties does R satisfy?

I. Reflexivity

II. Symmetry

III. Transitivity

a. I only
b. III only
c. I and III only
d. I and II only

Answer: (a)

R: N → N, aRb a R a ⇒ a divides b²

Reflexivity: a R a ⇒ a divides a²

True, for every natural number.

‘R’ is reflexive.

Symmetry: a R b ⇒ a divides b²

⇏ b divides a²

For example: 1 R 2 i.e. 1 divides 2², but 2 R 1 i.e. 2 does not divides 1².

‘R’ is not symmetric.

Transitivity:

Again by taking example,

8 R 4 i.e. 8 divides 4² = 16

4 R 2 i.e. 4 divides 2² = 4

But 8 R 2 i.e. 8 do not divide 2² = 4

R is not transitive.

Question 5: The fractional part of a real number x is x – [x], where [x] is the greatest integer less than or equal to x. Let F₁ and F₂ be the fractional parts of (44 – √2017)²⁰¹⁷ and (44 + √2017)²⁰¹⁷ respectively. Then F₁ + F₂ lies between the numbers

a. 0 and 0.45
b. 0.45 and 0.9
c. 0.9 and 1.35
d. 1.35 and 1.8

Answer: (c)

Let x = (44 – √2017)²⁰¹⁷

x = – (√2017 – 44)²⁰¹⁷ ≅ – (0.911)²⁰¹⁷

–x = (√2017 – 44)²⁰¹⁷ {Pure fractional number}

{–x} = {(√2017 – 44)²⁰¹⁷}

1 – {x} = (√2017 – 44)²⁰¹⁷

1 – F₁ = (√2017 – 44)²⁰¹⁷ …(1)

Again, let y = (√2017 + 44)²⁰¹⁷ = I + F₂ …(2)

Where [y] = I & {y} = F₂

Now subtracting eq.(2) – eq.(1)

I + F₂ – 1 + F₁ = (√2017 + 44)²⁰¹⁷ – (√2017 – 44)²⁰¹⁷

= 2{²⁰¹⁷C₁(√2017) ²⁰¹⁶(44)¹ + ……}

= 2 (Integer)

F₁ + F₂ = 2I₁ – I + 1 ∈ Integer

= 1 (only possible value)

0.9 < F₁ + F₂ < 1.35

Question 6: The number of real solutions of the equation 2sin 3x + sin 7x – 3 = 0 which lie in the interval [–2𝛑, 2𝛑] is

a. 1
b. 2
c. 3
d. 4

Answer: (b)

2 sin3x + sin 7x – 3 = 0 ………(1)

Since maximum value of sin θ is 1, therefore only possible solution of (1) is,

sin3x = 1 and sin7x = 1

3x = (4n + 1) 𝛑 / 2, n ∈ I and 7x = (4m + 1) 𝛑 / 2, m ∈ I

x = (4n + 1) 𝛑 / 6, n ∈ I

x = (4m + 1) 𝛑 / 14, m ∈ I

x = …., – 7𝛑 / 6, – 𝛑 / 2, 𝛑 / 6, 5𝛑 / 6, 3𝛑 / 2 …..

x = …, – 3𝛑 / 14, – 𝛑 / 2, 𝛑 / 14, 5𝛑 / 14, 13𝛑 / 14, 3𝛑 / 2 …

Therefore, two solutions, x = – 𝛑 / 2 and 3𝛑 / 2.

Question 7: Suppose p, q, r are real numbers such that q = p (4 – p), r = q (4 – q), p = r (4 – r). The maximum possible value of p + q + r is

a. 0
b. 3
c. 9
d. 27

Answer: (c)

Given, q = p (4 – p) ⇒ q / p = 4 – p ..……(i)

r = q (4 – q) ⇒ r / q = 4 – q ……. (ii)

p = r (4 – r) ⇒ p / r = 4 – r ……..(iii)

Multiplying (i),(ii) and (iii)

(q / p) . (r / q) . (p / r) = (4 – p) (4 – q) (4 – r) = 1 ……..(iv)

Let x = 4 – p = 4 – 4r + r² = (r – 2)² ≥ 0

y = 4 – q = 4 – 4p + p² = (r – 2)² ≥ 0

z = 4 – r = 4 – 4q + q² = (q – 2)² ≥ 0

Now,

Using A.M. ≥ G.M.

(x + y + z) / 3 ≥ (x . y . z)^⅓

4 – p + 4 – q + 4 – r ≥ 3 {Using eq.(iv)}

p + q + r ≤ 9 (Equality holds for p = q = r = 3)

Question 8: The parabola y² = 4x + 1 divides the disc x² + y² ≤ 1 into two regions with areas A₁ and A₂. Then |A₁ – A₂| equals

a. 1 / 3
b. 2 / 3
c. 𝛑 / 4
d. 𝛑 / 3

Answer: (b)

C₁: y² = 4x + 1; C₂: x² + y² = 1

A₁ = 2 ∫_-1/4⁰ √4x + 1 dx + 2 ∫₀¹ √1 – x² dx

= 2 [(1 / 6) – 0] + 2 [0 + (1 / 2) * (𝛑 / 2) – 0]

= 1 / 3 + π / 2

A₂ = π(1)² – A₁

Now,

|A₁ – A₂| = |2A₁ – π| = | 2 * [(1 / 3) + (π / 2)] – π | = 2 / 3

Question 9: A shooter can hit a given target with probability 1 / 4. She keeps firing a bullet at the target until she hits it successfully three times and then she stops firing. The probability that she fires exactly six bullets lies in the interval

a. (0.5272, 0.5274)
b. (0.2636, 0.2638)
c. (0.1317, 0.1319)
d. (0.0658, 0.0660)

Answer: (d)

Since she hits the target successfully three times in exactly six attempts.

So 3^rd hit must be the sixth time and in the first five attempts, there will be two hits. Now, Required Probability =

= 10 × [27 / 4⁶] = 270 / 4096 = 0.06591

Question 10: Consider the following events:

E₁: Six fair dice are rolled and at least one dice shows six.

E₂: Twelve fair dice are rolled and at least two dice show six.

Let p₁ be the probability of E₁ and p₂ be the probability of E₂.

Which of the following is true?

a. p₁ > p₂
b. p₁ = p₂ = 0.6651
c. p₁ < p₂
d. p₁ = p₂ = 0.3349

Answer: (a)

E₁: Six fair dice are rolled and at least one dice shows six.

E₂: Twelve fair dice are rolled and at least two dice show six.

P(E₁) = p₁ = 1 – {no dice shows six}

= 1 – (5 / 6)⁶ (Using Binomial Theorem)

= 0.6651

P(E₂) = p₂ = 1 – {no dice shows six + one dice shows six}

= 1 – {(5 / 6)¹² + ¹²C₁ (1 / 6) (5 / 6)¹¹} (Using Binomial Theorem)

= 0.61866

p₁ > p₂

Question 11: For how many different values of a does the following system have at least two distinct solutions? ax + y = 0

x + (a + 10) y = 0

a. 0
b. 1
c. 2
d. Infinitely many

Answer: (c)

ax + y = 0; x + (a + 10) y = 0

The given system is homogeneous system of linear equations in two variables Therefore, it will either have unique solution i.e. x = y = 0 or infinite number of solutions (when lines are coincident).

Atleast two distinct solutions → Infinite solutions

a / 1 = 1 / [a + 10] (Co-incident lines)

a² + 10a – 1 = 0 (Discriminant > 0)

Therefore, two values of ‘a’ are possible.

Question 12: Let R be the set of real numbers and f : R →R be defined by f (x) = {x} / [1 + [x]²], where [x] is the greatest integer less than or equal to x, and {x} = x – [x]. Which of the following statements are true?

I. The range of f is a closed interval

II. f is continuous on R.

III. f is one-one on R.

a. I only
b. II only
c. III only
d. None of I, II and III

Answer: (d)

f (x) = {x} / [1 + [x]²]

Since, 0 ≤ {x} < 1 & 1 ≤ 1 + [x]² < ∞

0 < {x} / [1 + [x]²] < 1

Range of f (x) = (0, 1) (I is FALSE)

Also, f (x) is discontinuous ∀ x ∈ Z. (II is TRUE)

f(x) = 0 ∀ x ∈ Z

Not one-one (III is FALSE)

Question 13: Let x_n = (2ⁿ + 3ⁿ)^1/2n for all natural numbers n. Then

a. lim_x→∞ x_n = ∞
b. lim_x→∞ x_n = √3
c. lim_x→∞ x_n = √3 + √2
d. lim_x→∞ x_n = √5

Answer: (b)

Question 14: One of the solutions of equation 8 sin³ θ – 7 sinθ + √3 cos θ = 0 lies in the interval

a. (0°, 10°]
b. (10°, 20°]
c. (20°, 30°]
d. (30°, 40°]

Answer: (b)

Given, 8 sin³ θ – 7 sin θ + √3 cos θ = 0

2 . 4 sin³ θ – 7 sin θ + √3 cos θ = 0

2 (3 sinθ – sin 3θ) – 7 sin θ + √3 cos θ = 0

{sin 3θ = 3 sin θ – 4 sin³ θ ⇒ 4 sin³ θ = 3 sin θ – sin 3θ}

√3 cos θ – sin θ – 2 sin 3θ = 0

sin (π / 3) cos θ – cos (π / 3) sin θ = sin 3θ (sin (A – B) = sin A cos B – cos A sin B)

sin [(π / 3) – θ] = 3θ

(π / 3) – θ = nπ + (–1)ⁿ 3θ

for n = 0

(π / 3) – θ = 3θ

θ = π / 12 = 15° ∈ (10°, 20°]

Question 15: Let a, b, c, d, e, be real numbers such that a + b < c + d, b + c < d + e, c + d < e + a, d + e < a + b. Then

a. The largest is a and the smallest is b
b. The largest is a and the smallest is c
c. The largest is c and the smallest is e
d. The largest is c and the smallest is b

Answer: (a)

a + b < c + d …(i)

b + c < d + e …(ii)

c + d < e + a …(iii)

d + e < a + b …(iv)

From (i) & (iii)

a + b < e + a (x < y & y < z ⇒ x < z)

b < e

From (i) & (iv)

d + e < c + d (x < y & y < z ⇒ x < z)

e < c

From (ii) & (iv)

b + c < a + b (x < y & y < z ⇒ x < z)

c < a

b < e < c < a … (1)

Again from (i)

a + b < c + d

(a – c) + b < d

Positive real number + b < d {as a > c}

d > b … (2)

From (iii)

c + d < e + a

(c – e) + d < a

Positive real number + d < a {as c > e}

d < a … (3)

From (1), (2) & (3), we can say that a is largest and b is smallest.

Question 16: If a fair coin is tossed 5 times, the probability that heads does not occur two or more times in a row is:

a. 12 / 2⁵
b. 13 / 2⁵
c. 14 / 2⁵
d. 15 / 2⁵

Answer: (b)

The following are the cases for no two heads to come together

Case – I: There is no Head

{TTTTT}

Probability = (1 / 2)⁵

Case – II: Exactly one Head

{HTTTT, THTTT, TTHTT, TTTHT, TTTTH}

Probability = 5 * (1 / 2)⁴ (1 / 2) = 5 / 32

Case – III: Exactly Two Heads

{HTHTT, HTTHT, HTTTH, THTTH, THTHT, TTHTH}

Probability = 6 × (1 / 2)³ (1 / 2)² = 6 / 32

Case – IV: Exactly Three Heads

{HTHTH}

Probability = 1 × (1 / 2)² * (1 / 2)³ = 1 / 32

Total probability = 13 / 2⁵

Question 17: Consider the following parametric equation of a curve:

x (θ) = | cos 4 θ | cos θ

y (θ) = | cos 4 θ | sin θ

for 0 ≤ θ ≤ 2π

Which one of the following graphs represents the curve?

Answer: (a)

x (θ) = | cos 4 θ | cos θ

y (θ) = | cos 4 θ | sin θ

where, 0 ≤ θ ≤ 2π

For θ = 0°

x = 1 and y = 0

(1, 0) lies on the curve

Option (c) and (d) are rejected.

For θ = π / 4

x = y = 1 / √2

(1 / √2, 1 / √2) lies on the curve.

Option (b) is rejected.

So, the correct option is (a).

Question 18: Let A = (a₁, a₂) and B = (b₁, b₂) be two points in the plane with integer coordinates. Which one of the following is not a possible value of the distance between A and B?

a. √65
b. √74
c. √83
d. √97

Answer: (c)

Distance between A (a₁, b₁) and B (a₂, b₂)

AB = √(a₁ – b₁)² + (a₂ – b₂)²{Using Distance Formula}

Given a₁, a₂, b₁, b₂ ∈ I ⇒ (a₁ – b₁), (a₂ – b₂) ∈ I

Now, √65 = √8² + 1² → Possible

√74 = √7² + 5² → Possible

√97 = √9² + 4² → Possible

√83 → Not Possible

Question 19: Let f (x) = max {3, x², (1 / x²)} for (1 / 2) ≤ x ≤ 2. Then the value of the integral ∫_1/2² f(x)dx is:

a. 11 / 3
b. 13 / 3
c. 14 / 3
d. 16 / 3

Answer: (c)

f (x) = max {3, x², (1 / x²)} for (1 / 2) ≤ x ≤ 2

Question 20: Let a_i = I + (1 / i) for I = 1, 2,………., 20. Put p = (1 / 20) (a₁ + a₂ + …….. a₂₀) and q = (1 / 20) [(1 / a₁) + (1 / a₂) + …… (1 / a₂₀). Then

Answer: (a)

Clearly, q > 0

We will try to contra prove that q < [22 – p] / 21 OR q + (p / 21) < (22 / 21)

Question 21: Let x, y, z be positive integers such that HCF (x, y, z) = 1 and x² + y² = 2z². Which of the following statements are true?

I. 4 divides x or 4 divides y.

II. 3 divides x + y or 3 divides x – y.

III. 5 divides z (x² – y²)

a. I and II only
b. II and III only
c. II only
d. III only

Answer: (b)

Given x, y, z ∈ Z⁺

HCF (x, y, z) = 1 ⇒ x, y and z are co-prime numbers

x² + y² = 2z² = z² + z² ⇒ Sum of the squares of 2 co-prime numbers.

Thus, x = 7, y = 1, z = 5 or x = 1, y = 7, z = 5 are possible combinations.

i.e. 2z² = 2 . 5² = 50 and x² + y² = 7² + 1² = 49 + 1 = 50

Now, x – y is divisible by 3 and z (x² – y²) is divisible by 5.

Statement II and III are true.

Question 22: How many different (mutually non-congruent) trapeziums can be constructed using four distinct side lengths from the set {1, 3, 4, 5, 6}?

a. 5
b. 11
c. 15
d. 30

Answer: (b)

Here, the sides a, b, c, d ∈ {1, 3, 4, 5, 6}

Condition of trapezium,

|a – c| < |b – d| < (a + c)

Cases:

Total number of trapezium = 11

Question 23: A solid hemisphere is mounted on a solid cylinder, both having equal radii. If the whole solid is to have a fixed surface area and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

a. 1 : 1
b. 1 : 2
c. 2 : 1
d. √2 : 1

Answer: (a)

Let Radius of hemisphere and cylinder is R height of cylinder be h.

The whole surface area, S = 2πR² + 2πRh + πR²

S = 2πRh + 3πR² …(1)

S – 3πR² = 2πRh

h = S – 3πR² / 2πR …(2)

Now, V = (2 / 3) πR³ + πR²h

V = (2 / 3) πR³ + πR² [S – 3πR² / 2πR] {from eq.(2)}

Differentiate with respect to R

dV / dR = 2πR² + (S / 2) – (9 / 2) πR²

For maximum and minimum dV / dR = 0

0 = 2πR² + (S / 2) – (9 / 2) πR²

0 = –5πR² + S

5πR² = S

5πR² = 3πR² + 2πRh {from eq.(1)}

5R = 3R + 2h

2R = 2h

h : R = 1 : 1

Question 24: Let ABC be an acute scalene triangle, and O and H be its circumcentre and orthocenter respectively. Further, let N be the midpoint of OH. The value of the vector sum NA + NB + NC is:

Answer: (c)

Let circumcenter O of triangle ABC be at the origin.

Position vector of Centroid, G = [a + b + c] / 3

Now, we know that centroid(G) divides the distance from the orthocenter (H) to the circumcenter(O) in the ratio 2:1.

Question 25: The quotient when 1 + x² + x⁴ + x⁶ + ………..x³⁴ is divided by 1 + x + x² + x³ +……+ x¹⁷ is

a. x¹⁷ – x¹⁵ + x¹³ – x¹¹…… + x
b. x¹⁷ + x¹⁵ + x¹³ + x¹¹…… + x
c. x¹⁷ + x¹⁶ + x¹⁵ + x¹⁴…… + x
d. x¹⁷ – x¹⁶ + x¹⁵ – x¹⁴…… – x

Answer: (d)

1 + x² + x⁴ + x⁶ + ………..x³⁴ (Forms a G.P with common ratio x²)

Question 26: Let R be the region of the disc x² + y² ≤ 1 in the first quadrant. Then the area of the largest possible circle contained in R is

a. π (3 − 2√2)
b. π (4 − 3√2)
c. π / 6
d. π (2√2 – 2)

Answer: (a)

Let the radius of inner circle = r

This inner circle touches both the axes,

Centre of circle = (r, r)

Now, OP = OC + CP

1 = √(r² + r²) + r {Using Distance Formula}

1 = √2r + r

r = 1 / √(2) + 1

r = [1 / √(2) + 1] * [√(2) – 1 / √(2) – 1]

r = √(2) – 1 / 1 = √(2) – 1

Area of Circle = πr²

= π (√(2) – 1)²

= π (2 + 1 – 2√2)

= π (3 – 2√2)

Question 27: Let R be the set of real numbers and f : R → R be given by f (x) = √|x| – log (1 + |x|). We now make the following assertions:

I. There exists a real number A such that f (x) ≤ A for all x.

II. There exists a real number B such that f (x) ≥ B for all x.

a. I is true and II is false
b. I is false and II is true
c. I and II both are true
d. I and II both are false

Answer: (b)

f (x) = √|x| – log (1 + |x|)

For x > 0

f (x) = x – log (1 + x)

On differentiating both sides with respect to x

f (x) is increasing when x > 0 …(i)

Also, f (0) = √|0| – log (1 + |0|) = 0 …(ii) &

f (–x) = √|–x| – log (1 + |–x|) = f (x)

f(x) is Even Function …(iii)

Therefore, using (i), (ii) and (iii) graph of f (x) can be drawn as

Clearly, f (x) is continuous everywhere but non-differentiable at x = 0.

Question 28: Define g(x) = ∫_-3³ f (x – y) f (y) dy, for all real x, where

Then

a. g (x) is not continuous everywhere
b. g (x) is continuous everywhere but differentiable nowhere
c. g (x) is continuous everywhere and differentiable everywhere except at x = 0, 1
d. g (x) is continuous everywhere and differentiable everywhere except at x = 0, 1, 2

Answer: (d)

Given, ∫_-3³ f (x – y) f (y) dy and f (y) =

g (x) = ∫₀¹ f (x – y) dy

Now, put x – y = t

dy = –dt

Lower limit, y = 0 ; t = x

Upper limit, y = 1 ; t = x – 1

g (x) = – ∫_x^x-1 f (t) dt {Using property: – ∫_a^b f (x) dx = ∫_b^a f (x) dx}

g (x) = ∫_x-1^x f (t) dt

g (x) is continuous everywhere, but non-diff. at x = 0, 1, 2 (see figure)

Question 29: The integer part of the number

is

a. 50
b. 52
c. 57
d. 59

Answer: (c)

Question 30: The number of continuous functions f : [0, 1] → R that satisfy

a. 0
b. 1
c. 2
d. infinity

Answer: (b)

Therefore, the number of such continuous functions is 1.