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Q.1. Let (x, y, z) be points with integer coordinates satisfying the system of homogeneous equations :
3x – y – z = 0
– 3x + z = 0
– 3x + 2y + z = 0
Then the number of such points for which x^{2} + y^{2 }+ z^{2} ≤ 100 is (2009)
Ans. (7)
Sol. The given system of equations is
3x  yz=0
3x +z=0
3x + 2 y +z=0
Let x = p
where p is an integer, then y = 0 and z = 3p
But x^{2 }+ y^{2} +z^{2}≤ 100
⇒ p^{2} + 9p^{2}≤ 100
⇒ p^{2} ≤ 10 ⇒ p = 0, ± 1, ± 2±3 i.e. p can take 7 different values.
∴ Number of points (x, y, z) are 7.
Q.2. The smallest value of k, for which both the roots of the equation x^{2} – 8kx + 16 (k^{2} – k + 1) = 0 are real, distinct and have values at least 4, is (2009)
Ans. (2)
Sol. The given equation is x^{2}  8kx + 16(k^{2} k + 1)=0
∵ Both the roots are real and distinct
∴ D > 0
⇒ (8k )2  4 x 16(k^{2} k + 1)>0
⇒ k > 1...(i)
∵ Both the roots are greater than or equal to 4
∴ α + β > 8 and f (4)≥ 0 ⇒ k > 1 ...(ii)
and 16  32k + 16(k^{2} k + 1)≥ 0
⇒ k^{2}  3k + 2≥0
⇒ (k  1)(k  2)≥0 ⇒ k ∈ (∞,1] ∪ [2,∞) ...(iii)
Combining (i), (ii) and (iii),
we get k ≥ 2 or the smallest value of k = 2.
Q.3. The minimum value of the sum of real numbers a^{–5}, a^{–4}, 3a^{–3}, 1, a^{8} and a^{10} where a > 0 is (2011)
Ans. (8)
Sol. ∵ a > 0, ∴ a^{–5}, a^{–4}, 3a^{–3}, 1, a^{8}, a^{10} > 0
Using AM > GM for positive real numbers we get
Q.4. The number of distinct real roots of x^{4} – 4x^{3} + 12x^{2} + x – 1 = 0 is (2011)
Ans. Sol. (2) We have x^{4} – 4x^{3} + 12x^{2} + x – 1 = 0
⇒ x^{4} – 4x^{3} + 6x^{2} – 4x + 1 + 6x^{2} + 5x – 2 = 0
⇒ (x – 1)^{4 }+ 6x^{2} + 5x – 2 = 0
⇒ (x – 1)^{4} = – 6x^{2} – 5x + 2
To solve the above polynomial, it is equivalent to find the intersection points of the curves y = (x – 1)^{4} and y = – 6x^{2} – 5x + 2 or y = (x – 1)^{4} and
The graph of above two curves as follows.
Clearly they have two points of intersection.
Hence the given polynomial has two real roots
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