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Question 1 of 30
1. Question
The equation e^{sin x} − e^{−sin x} – 4 = 0 has
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Question 2 of 30
2. Question
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Question 3 of 30
3. Question
A spherical balloon is filled with 4500π cu m of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π cu m/min, then the rate (in m/min) at which the radius of the balloon decreases 49 min after the leakage began is
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Question 4 of 30
4. Question
Statement 1: The sum of the series 1+(1 + 2+ 4) + (4 + 6 + 9) + (9 + 12 +16)+……….+(361 + 380 + 400) is 8000.
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Question 5 of 30
5. Question
The negation of the statement “If I become a teacher, then I will open a school”, is
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Question 6 of 30
6. Question
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Question 7 of 30
7. Question
Statement 1: An equation of a common tangent to the parabola y^{2} = 16√3x and the ellipse 2x^{2} + y^{2} = 4 is y = 2x + 2√3.
Statement 2: If the line , (m≠0) is a common tangent to the parabola y^{2} = 16√3x and ellipse 2x^{2} + y^{2} = 4, then m satisfies m^{4} + 2m^{2}= 24.
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Question 8 of 30
8. Question
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Question 9 of 30
9. Question
If n is a positive integer, then (√3 + 1)^{2n} – (√3 – 1)^{2n} is
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Question 10 of 30
10. Question
If 100 times the 100^{th} term of an AP with nonzero common difference equals the 50 times its 50^{th} term, then the 150^{th} term of this AP is
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Question 11 of 30
11. Question
In a ∆PQR , if 3 sin P + 4 cos Q = 6 and 4 sin Q + 3 cos P = 1, then the angle R is equal to
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Question 12 of 30
12. Question
An equation of a plane parallel to the plane x – 2y + 2z – 5 = 0 and at a unit distance from the origin is
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Question 13 of 30
13. Question
If the line 2x + y = k passes through the point which divides the line segment joining the points (1, 1) and (2, 4) in the ratio 3 : 2, then k equals
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Question 14 of 30
14. Question
Let x_{1}, x_{2}….., x_{n} be n observations and let be their arithmetic mean and σ^{2} be the variance.
Statement 1: Variance of 2x_{1}, 2x_{2}, . . . . , 2x_{n} is 4σ^{2}.
Statement 2: Arithmetic mean 2x_{1}, 2x_{2} …….., 2x_{n} is 4
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Question 15 of 30
15. Question
The population p(t) at time t of a certain mouse species satisfies the differential equation . If p(0) = 850, then the time at which the population becomes zero is
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Question 16 of 30
16. Question
Let a, b ∈R be such that the function f given by f(x) = logx + bx^{2} + ax, x ≠ 0 has extreme values x = −1 and x = 2.
Statement 1: f has local maximum at x = −1 and at x = 2
Statement 2: a = ½ and b = −1/4
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Question 17 of 30
17. Question
The area bounded between the parabolas x^{2} = y/4 and x^{2}= 9y and the straight line y = 2 is
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Question 18 of 30
18. Question
Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is
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Question 19 of 30
19. Question
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Question 20 of 30
20. Question
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Question 21 of 30
21. Question
Three numbers are chosen at random without replacement from {1, 2, 3, ….8}. The probability that their minimum is 3, given that their maximum is 6, is
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Question 22 of 30
22. Question
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Question 23 of 30
23. Question
Let P and Q be 3 × 3 matrices P ≠ Q. If P^{3} = Q^{3} and P^{2}Q = Q^{2}P, then determinant of (P^{2} + Q^{2}) is equal to
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Question 24 of 30
24. Question
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Question 25 of 30
25. Question
The length of the diameter of the circle which touches the xaxis at the point (1, 0) and passes through the point (2, 3) is
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Question 26 of 30
26. Question
Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that Y ⊆ X, Z ⊆ X and Y ∩ Z is empty, is
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Question 27 of 30
27. Question
An ellipse is drawn by taking a diameter of the circle (x – 1)^{2} + y^{2} = 1 as its semiminor axis and a diameter of the circle x^{2} + (y – 2)^{2} = 4 is semimajor axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is
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Question 28 of 30
28. Question
Consider the function f(x) = x – 2 x – 5, x ∈ R.
Statement 1: f'(4) = 0
Statement 2: f is continuous in [2, 5], differentiable in (2, 5) and f(2) = f(5).
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Question 29 of 30
29. Question
A line is drawn through the point (1, 2) to meet the coordinate axes at P and Q such that it forms a ∆PQR, where O is the origin, if the area of the ∆PQR is least, then the slope of the line PQ is
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Question 30 of 30
30. Question
Let ABCD be a parallelogram such that AB = q, AD = p and ∠BAD be an acute angle. If r is the vector that coincides with the altitude directed from the vertex B to the side AD, then r is given by
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