Page 29 - ISC-12

6Jai Baba Ki ISC Mathematics – Class XII by Gupta–Bansal

23. Let a, λ, µ ∈ R. Consider the system of linear equations
ax + 2y = λ
3x − 2y = µ
Which of the following statement(s) is(are) correct?

(A) If a = −3, then the system has infinitely many solutions for all values of λ and µ.

(B) If a 6= −3, then the system has a unique solution for all values of λ and µ.
(C) If λ + µ = 0, then the system has infinitely many solutions for a = −3.
(D) If λ + µ 6= 0, then the system has no solution for a = −3.

[JEE 2016]

1
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
3
24. Let ˆu = u 1 i + u 2 j + u 3 k be a unit vector in R and ˆw = √ (i + j + 2k). Given that there exists a
6
3
vector ~v in R such that |ˆu × ~v| = 1 and ˆw · (ˆu × ~v) = 1. Which of the following statement(s) is(are)
correct?
(A) There is exactly one choice for such ~v.
(B) There are infinitely many choices for such ~v.
(C) If ˆu lies in the xy-plane then |u 1 | = |u 2 |.
(D) If ˆu lies in the xz-plane then 2|u 1 | = |u 3 |.
[JEE 2016]

25. Let X and Y be two arbitrary, 3 × 3, non-zero, skew-symmetric matrices and Z be an arbitrary 3 × 3,
non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric?

44
4
23
3
4
3
4
3
3
4
(A) Y Z − Z Y . (B) X 44 + Y . (C) X Z − Z X . (D) X 23 + Y .
[JEE 2015]
26. Which of the following values of α satisfy the equation

(1 + α) 2 (1 + 2α) 2 (1 + 3α) 2

2 2 2
(2 + α) (2 + 2α) (2 + 3α) = −648α?
2 2
(3 + α) (3 + 2α) (3 + 3α)
2
(A) −4. (B) 9. (C) −9. (D) 4.
[JEE 2015]
3
27. In R , consider the planes P 1 : y = 0 and P 2 : x + z = 1. Let P 3 be a plane, different from P 1 and P 2 ,
which passes through the intersection of P 1 and P 2 . If the distance of the point (0, 1, 0) from P 3 is 1
and the distance of a point (α, β, γ) from P 3 is 2, then which of the following relations is (are) true?

(A) 2α + β + 2γ + 2 = 0. (B) 2α − β + 2γ + 4 = 0.
(C) 2α + β − 2γ − 10 = 0. (D) 2α − β + 2γ − 8 = 0.

[JEE 2015]