CBSE
Let A be a 2 × 2 matrix with non-zero entries and let A2 = I, where I is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.
Statement-1: Tr(A) = 0.
Statement-2: |A| = 1.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
C.
Statement-1 is true, Statement-2 is false.
A satisfies A_{2} -Tr(A). A + (det A) l = 0
on comparing with A2-I = 0,
we get
Tr (A) = 0, |A| = - 1
If two tangents drawn from a point P to the parabola y^{2}= 4x are at right angles, then the locus of P is
X = 1
2x +1 = 0
x = -1
2x-1 = 0
C.
x = -1
We know that the locus of point P from which two perpendicular tangents are drawn to the parabola is the directrix of the parabola.
Hence, the required locus is x = -1
let f : (-1, 1) → R be a differentiable function
with f(0) = -1 and f'(0) = 1.
Let g(x) = [f(2f(x) + 2)]^{2}. Then g'(0) =
4
-4
0
-2
B.
-4
g(x) = (f(2(f(x) + 2))^{2}
g'(x) 2f (2f (x) 2) f '(2f (x) 2) 2f '(x)
g'(0) 2f (2f (0) 2) f '(2f (0) 2) 2f '(0)
= 4f(0) × (f '(0))^{2}– 4
The number of 3 × 3 non-singular matrices, with four entries as 1 and all other entries as 0, is
less than 4
5
6
atleast 7
D.
atleast 7
Let f : R → R be a positive increasing function with
1
2/3
3/2
3
A.
1
Since f(x) is a positive increasing function.
⇒ 0< f(x)<f(2x)<f(3x)
⇒ 0<1<
⇒
Let f : R → R be defined by
If f has a local minimum at x = - 1 then a possible value of k is
1
0
-1/2
-1
C.
-1/2
k – 2x > 1 k + 2 = 1
k > 1 + 2x k = -1
k > 1 + 2(-1)
k > -1
The equation of the tangent to the curve, that is parallel to the x-axis, is
y = 0
y = 1
y = 3
y =2
C.
y = 3
Let f : R → R be a continuous function defined
by f(x) = 1/e^{x} + 2e^{-x}
Statement - 1: f(c) = 1/3, for some c ∈ R.
Statement-2: 0 < f(x)≤ , for all x ∈ R.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
A.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Statement-1: S_{3} = 55 × 2^{9}.
Statement-2: S_{1} = 90 × 2^{8} and S2 = 10 × 2^{8}.
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1.
Statement-1 is true, Statement-2 is false.
Statement-1 is false, Statement-2 is true.
C.
Statement-1 is true, Statement-2 is false.
Consider the system of linear equation
x^{1} + 2x^{2} + x^{3} = 3
2x^{1} + 3x^{2} + x^{3} = 3
3x^{1} + 5x^{2} + 2x^{3} = 1
The system has
infinite number of solutions
exactly 3 solutions
a unique solution
no solution
D.
no solution
Subtracting the Eq. (ii) – Eq. (i)
We get x_{1} + x_{2} = 0
Subtract equations
Eq. (iii) – 2 × eq. (ii)
x_{1} + x_{2} = 5
Therefore, no solutions