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Question 1 of 30
1. Question
The function defined as
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Question 2 of 30
2. Question
If, for a positive integer n, the quadratic equation, x(x + 1) + (x + 1) (x + 2) has two consecutive integral solutions, then n is equal to
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Question 3 of 30
3. Question
Let ω be a complex number such that 2ω + 1 = z where If then k is equal to
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Question 4 of 30
4. Question
If then adj (3A^{2} + 12A) is equal to
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Question 5 of 30
5. Question
If S is the set of distinct values of b for which the following system of linear equations
x + y + z = 1
x + ay + z = 1
ax + by + z = 0
has no solution, then S is
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Question 6 of 30
6. Question
A man X has 7 friends, 4 of them are ladies and 3 are men. His wife Y also has 7 friends, 3 of them are ladies and 4 are men. Assume X and Y have no common friends. Then the total number of ways in which X and Y together can throw a party inviting 3 ladies and 3 men, so that 3 friends of each of X and Y are in this party, is
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Question 7 of 30
7. Question
The value of (^{21}C_{1} – ^{10}C_{1}) + (^{21}C_{2} – ^{10}C_{2}) + (^{21}C_{3} – ^{10}C_{3}) + (^{21}C_{4} – ^{10}C_{4}) + … + (^{21}C_{10} – ^{10}C_{10}) is
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Question 8 of 30
8. Question
For any three positive real numbers a, b and c, 9(25a^{2} + b^{2}) + 25(c^{2} – 3ac) = 15b(3a + c).
Then
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Question 9 of 30
9. Question
Let a, b, c ∈ R. If f(x) = ax^{2} + bx + c is such that a + b + c = 3 and f(x + y) = f(x) + f(y) + xy, ∀ x, y ∈ R, then is equal to
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Question 10 of 30
10. Question
equals
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Question 11 of 30
11. Question
If for the derivative of is then g(x) equals
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Question 12 of 30
12. Question
The normal to the curve y(x − 2)(x − 3) = x + 6 at the point where the curve intersects the yaxis passes through the point
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Question 13 of 30
13. Question
Twenty meters of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flowerbed, is
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Question 14 of 30
14. Question
Let If I_{4} + I_{6} = a tan^{5} x + bx^{5} + C, where C is a constant of integration, then the ordered pair (a, b) is equal to
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Question 15 of 30
15. Question
The integral is equal to
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Question 16 of 30
16. Question
The area (in sq. units) of the region {(x, y) : x ≥ 0, x + y ≤ 3, x^{2} ≤ 4y and y ≤1 + √x}
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Question 17 of 30
17. Question
If and y(0) = 1, then is equal to
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Question 18 of 30
18. Question
Let k be an integer such that the triangle with vertices (k, −3k), (5, k) and (−k, 2) has area 28 sq. units. Then the orthocenter of this triangle is at the point
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Question 19 of 30
19. Question
The radius of a circle, having minimum area, which touches the curve y = 4 – x^{2} and the lines, y = x is
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Question 20 of 30
20. Question
The eccentricity of an ellipse whose centre is at the origin is If one of its directrices is x = −4, then the equation of the normal to it at is
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Question 21 of 30
21. Question
A hyperbola passes through the point and has foci at (±2, 0). Then the tangent to this hyperbola at P also passes through the point
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Question 22 of 30
22. Question
The distance of the point (1, 3, −7) from the plane passing through the point (1, −1, −1), having normal perpendicular to both lines and is
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Question 23 of 30
23. Question
If the image of the point P(1, −2, 3) in the plane 2x + 3y – 4z + 22 = 0 measured parallel to the line, is Q, then PQ is equal to
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Question 24 of 30
24. Question
Let and Let be a vector such that and the angle between be 30°. Then is equal to
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Question 25 of 30
25. Question
A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, onebyone, with replacement, then the variance of the number of green balls drawn is
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Question 26 of 30
26. Question
For three events A, B and C, P (Exactly one of A or B occurs) = P(Exactly one of B or C occurs) = P(Exactly one of C or A occurs) = and P(All the three events occur simultaneously = Then the probability that at least one of the events occurs, is
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Question 27 of 30
27. Question
If two different numbers are taken from the set {0, 1, 2, 3, ……, 10}; then the probability that their sum as well as absolute difference are both multiple of 4, is
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Question 28 of 30
28. Question
If 5(tan^{2}x – cos^{2}x) = 2 cos 2x + 9, then the value of cos 4x is
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Question 29 of 30
29. Question
Let a vertical tower AB have its end A on the level ground. Let C be the midpoint of AB and P be a point on the ground such that AP = 2AB. If ∠BPC = β then tan β is
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Question 30 of 30
30. Question
The following statement (p → q) → [(~ p → q) → q] is
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