1. The equation of the tangent to the curve y = 9 − 2x² at the point where the ordinate and the abscissa are equal, is
(a) 2x + y – 3√3 = 0
(b) 2x + y + √3 = 0
(c) 2x + y – √3 = 0
(d) None of these 

2. 

(a) 0, 1
(b) 1, 2
(c) 1, 3
(d) None of these   

3. The approximate value of f(5.001), where
f(x) = x³ – 7x² + 15, is
(a) –34.995
(b) –33.995
(c) –33.335
(d) –35.993 

4. 

decreasing function of x for all x ∈ R and b ∈ R, b being independent of x, then
(a) a ∈ (0, √6)
(b) a ∈ (− √6, √6)
(c) a ∈ (− √6,0)
(d) None of these 

5. The minimum intercepts made by the axes on the tangent to the ellipse x²/16 + y²/9 + = 1 is 
(a) 25
(b) 7
(c) 1
(d) None of these 

6. A foot of the normal from the point (4, 3) to a circle is (2, 1) and a diameter of the circle has equation 2x – y = 2. Then the equation of the circle is
(a) x² + y² + 2x – 1 = 0
(b) x² + y² – 2x – 1 = 0
(c) x² + y² – 2y – 1 = 0
(d) None of these 

7. If the chord of contact of tangents from a point P to the parabola y² = 4ax, touches the parabola x² = 4by, then the locus of P is a/an
(a) circle
(b) parabola
(c) ellipse
(d) hyperbola 

8.

will represent the ellipse, if r lies in the interval
(a) (– ∞, 2)
(b) (3, ∞)
(c) (5, ∞)
(d) (1, ∞) 

9. The equation of the asymptotes of the hyperbola 2x² + 5xy + 2y² – 11x – 7y – 4 = 0, are
(a) 2x² + 5xy + 2y² – 11x – 7y – 5 = 0
(b) 2x² + 4xy + 2y² – 7x – 11y + 5 = 0
(c) 2x² + 5xy + 2y² – 11x – 7y + 5 = 0
(d) None of the above

10.

(a) 1
(b) –1
(c) 1/2
(d) None of these 

11. The probability of getting qualified in IIT/JEE and EAM/CET by a student are respectivley 1/5 and 3/5 The probability that the student gets qualified for at least one of these test, is 
(a) 3/25
(b) 8/25
(c) 17/25
(d) 22/25 

12. If the mean of a poisson distribution is 1/2 ,then one ratio of P(X = 3) to P(X = 2) is
(a) 1 : 2
(b) 1 : 4
(c) 1 : 6
(d) 1 : 8 

13. In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is 1/3 . The probability that he copies is 1/6 and the probability that his answer is correct given that he copied it is 1/8. The probability that he knew the answer to the question given that he correctly answered it, is
(a) 24/29
(b) 1/4
(c) 3/4
(d) 1/2 

14. Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P ?
(a) There is a rational number x ∈ S such that x ≤ 0.
(b) There is no rational number x ∈ S such that x ≤ 0.
(c) Every rational number x ∈ S satisfies x ≤ 0.
(d) x ∈ S and x ≤ 0 ⇒ x is not rational.   

15. Consider the following statements
P : Suman is brilliant
Q : Suman is rich
R : Suman is honest
The negation of the statement “Suman is brilliant and dishonest if and only if Suman is rich” can be expressed as
(a) ~ Q ↔ ~ P ∧ R
(b) ~ (P ∧ ~ R) ↔ Q
(c) ~ P ∧ (Q ↔ ~ R)
(d) ~ (Q ↔ (P ∧ ~ R)) 

16. If a̅, b̅ and c̅ are unit vectors, then  |a̅ − b̅| ² + |b̅ − c̅| ² + |c̅ − a̅| ² does not exceed 
(a) 4
(b) 9
(c) 8
(d) 6 

17. The diagonals of a parallelogram are given by d̅₁ = 2î + 3j − 6k^ and d̅₂ = 3î + 4j − k^ then area is
(a) 1/2 √50 sq. units
(b) 1/2 √1005 sq. units
(c) 1/2 √1105 sq. units
(d) None of these 

18. Equation of the sphere for which the circle x² + y² + z² + 7y – 2z + 2 = 0, on the plane 2x + 3y + 4z – 8 = 0 be great circle, must be
(a) x² + y² + z² + 4x + 2y – 6z + 10 = 0
(b) x² + y² + z² – 4x + 2y – 6z + 10 = 0
(c) x² + y² + z² – 6x – 4y – 2z – 10 = 0
(d) x² + y² + z² – 2x + 4y – 6z + 10 = 0   

19. The equations of the perpendicular from the origin to the 2x + 3y + 4z + 5 = 0 and x + 2y + 3z + 4 = 0 must be
(a) x + 2y – z = 0 = 3x – 2y – z
(b) 2x + y + z = 0 = x – 2y – z
(c) x + 2y – z = 0 = 3x + 2y + z
(d) x – 2y + z = 0 = 3x + 2y + z 

20. Find the value of l if the following equations are consistent
x + y – 3 = 0, (1 + λ)x + (2 + λ)y – 8 = 0,
x – (1 + λ)y + (2 + λ) = 0.
(a) 0,−1/2
(b) 1,−5/3
(c) 2,9/5
(d) 4, –3 

21. Evaluate

22. Evaluate 

where k (≠ 0) is a function and n ∈ N.
(a) 1/ke⁻¹
(b) − 1/ke⁻¹
(c) ke⁻¹
(d) – k e 

23. If f(x) be a continuous function such that f(a – x) + f(x) = 0 for all x ∈ [0, a], then evaluate

(a) −1/2 a
(b) 1/2 a 
(c) 3a
(d) 2a 

24. If A₁ is the area of the parabola y² = 4ax lying between vertex and the latus rectum and A₂ is the area between the latus rectum and the double ordinate x = 2a, then A₁/A₂ = 
(a) 2√2 − 1
(b) 1/7 (2√2 + 1)
(c) 1/7 (2√2 − 1)
(d) none of these 

25. Solution of the differential equation tan y
sec²x dx + tan x sec²y dy = 0 is
(a) tanx/tany = K
(b) tanx tany = K
(c) tanx + tany = K
(d) tanx – tany = K

26. Which of the following functions is a solution of the differential equation?

(a) y = 2x² – 4
(b) y = 2x – 4
(c) y = 2x
(d) y = 2 

27. For two independent events A and B, P(A∩B) = 3/25, P(A′ ∩B) = 8/25, then P(B) =
(a) 3/11
(b) 7/25
(c) 11/25
(d) none of these 

28. If the line ax + by + c = 0 is a normal to the curve xy = 1, then
(a) a > 0, b > 0
(b) a > 0, b < 0
(c) a < 0, b < 0
(d) none of these 

29. 

30. 

(a) x – a, x – b and x + a + b.
(b) x + a, x + b and x + a + b.
(c) x + a, x + b and x – a – b.
(d) x – a, x – b and x – a – b.

31. If A_ij is the cofactor of the element aij of the determinant

,then write the value of a₃₂·A₃₂.
(a) 200
(b) 150
(c) 110
(d) 90 

32. A balloon, which always remains spherical on inflation is being inflated by pumping in 900 cm³/s of gas. Find the rate at which theradius of the balloon increases when the radius is 15 cm.
(a) 11 cm/s
(b) 2π cm/s
(c) 1/π cm/s
(d) π² cm/s 

33. Find the point on the curve y = x³ – 11x + 5, at which the tangent is y = x – 11.
(a) (4, –7)
(b) (0, 3)
(c) (–2, –13)
(d) (2, –9)   

34.

(a) 0
(b) –2
(c) –1
(d) 2 

35. Sketch the region lying in the first quadrant and bounded by y = 9x², x = 0, y = 1 and y = 4. Find the area of region using integration. 
(a) 5/3 sq. units
(b) 10 sq. units
(c) 14/9 sq. units
(d) 9 sq. units 

36. Solve the following differential equation

37. Let p be real and |p|≥ 2. If A, B and C are variable angles such that 

then the minimum value of tan²A + tan²B + tan²C is
(a) 8
(b) 12
(c) 18
(d) 6 

38. Let a̅ and b̅ be two non-collinear unit vectors. 
If α¯ = a̅ − (a̅ ⋅ b̅)b̅ and β¯ = a̅ × b̅, then |β¯| is
(a) |α¯| 
(b) |α¯|+|α¯⋅ a̅|
(c) |α¯|+|α¯⋅ b̅|
(d) |α¯|+ α¯ .(α¯+ b̅) 

39. The odds in favour of a book reviewed by three independent critics are, respectively, 5 : 2, 4 : 3 and 3 : 4. The probability that majority of the critics give favourable remark is 
(a) 210/343
(b) 209/343
(c) 211/343
(d) 205/343 

40. Bag A contains 5 white and 3 black balls. Bag B is empty. Four balls are taken at random from A and transferred to empty bag B. From B, a ball is drawn at random and is found to be black. Then, the probability that among the transferred balls three are black and one is white is
(a) 1/8
(b) 7/8
(c) 6/7
(d) 1/7