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Section - (I)

a. This section contains Five questions.
b. Each question has FOUR options (A), (B), (C) and (D). ONE OR More of these four options is/are correct.

1. Let S₁ & S₂ are two circles of same radius 2 and their centres C₁ & C₂ lie on line y = x. Line 4x – 3y = 5 is one common tangent of two circles which touches S₁ at point A & S₂ at point B. Identify the correct statement(s)...

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2. Consider ƒ(a) = b²sin²a – bcos2a + 1, where a, b ∈ R, then which f the following is/are correct...

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3. P(x) is a least degree polynomial such that (P(x) – 1) is divisible by (x – 1)² & (P(x) – 3) is divisible by (x + 1)² , then...

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4. Let ƒ(x) be a biquadratic function of x such that $\lim _{x\rightarrow 0}\left( \dfrac{f\left( -x\right) }{2x^{3}}\right) ^{1/x}=\dfrac{1}{e^{3}}$, then...

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5. Let y = ƒ(x) be a non negative function such that area of quadrilateral formed by tangent at any point P on the curve, co-ordinate axes & ordinate of point P is equal to abscissa of point P. If ƒ(1) = 2, then...

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6. Normals are drawn from a point P(h, k) with slopes m₁, m₂, m₃ to the parabola C₁: y² = 4x. If curve C is the locus of point P with m₁m₂ = 2, then number of common tangents to curve C₁ and curve C is/are...

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SECTION – (II)

a. This section contains Two paragraphs.
b. Based on each paragraph, there are TWO questions.
Each question has FOUR options (A), (B), (C) and (D) ONLY ONE of these four options is correct.

Paragraph for Questions 7 and 8

Consider set S = {(a, b) : (a + 3)t² = 3 – a and bt² – 4t + b = 0}, where t is are real parameter. Let C is curve which is formed by all elements of set S where (a, b) is a point in R². Tangents are drawn from the point P(3, 4) to the curve C touching the curve C at point Q and R.

7. The circumcentre of triangle PQR is (u, v), then value of u + 3v is

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8. Curve E is equation of ellipse whose foci are Q, R and the curve E is passing through point P, then the eccentricity of ellipse E is...

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Paragraph for Questions 9 and 10

If ƒ(x,y) = 0 be the solution of differential equation (2y cosec2x + ln(coty))dx + (ln(tanx) – 2x cosec2y)dy = 0 such that f(π/4, π/2) = 0, then...

9. ƒ(x,y) is...

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10. If $\int ^{\infty }_{0}\dfrac{f\left( \dfrac{5\pi }{4},\dfrac{2017\pi }{4}\right) \cdot cosx}{x}dx=\dfrac{\pi }{2}$, then the value of $\int ^{\infty }_{0}\dfrac{\left( 1-\sin ^{2}x\right) ^{3/2}}{x}dx$ is equal to...

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SECTION - (III)

a. This section contains One question.
b. Each question has Matching List-I (P, Q, R, S) with Matching List-II (1, 2, 3, 4, 5). The answer codes out of which ONLY ONE is correct

11. Match Matching List-I with Matching List-II and select the correct answer using the code given below the list.

Matching List - I

(P) The eccenticity of ellipse x²/(a² + 1) + y²/(a² + 2) = 1 be 1/√3. If its latus rectum is 4/√b, then the value of 2b is...
(Q) Least prime value of K, such that K||x|ln|x|| = 1 has exactly 6 distinct solutions, is...
(R) If A(cosa, sina), B(sina, – cosa) and C(2, 1) are the vertices of Triangle ABC. If centre of locus of centroid is (p, q), then the value of p + q is...
(S) If (xʸ)ˣ = x/y, where x, y ∈ R⁺, then the slope of tangent at the point (e,1) on the curve is...

Matching List - II

(1) 0
(2) 1
(3) 3
(4) an even integer
(5) an odd integer

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Section - (IV)

a. This section contains THREE questions.
b. The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9.

12. Number of solution(s) of equation (cos⁵x) + (sinx)(cos⁴x) – (sin⁴x) + (sin³x) + (cosx)(sin²x) – (cos²x) = 0, x ∈ [0, π] is...

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13. The sum of integral values of k for which the equation sin⁻¹x + tan⁻¹x = sin⁻¹(sinx) + 2k – 1, has a real solution is...

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14. If x,y ∈ R, x² + y² + xy = 1, then minimum value of x³y + xy³ + 7 is...