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QUESTION: 1

Let a_{1} = b_{1} = 0, and for each n ≥ 2, let a_{n} and b_{n} be real numbers given by

Then which one of the following is TRUE about the sequences {a_{n}} and {b_{n}}?

Solution:

QUESTION: 2

Let Let V be the subspace of defined by

Then the dimension of V is

Solution:

QUESTION: 3

Let be a twice differentiable function. Define

f(x,y,z) = g(x^{2} + y^{2} - 2z^{2}).

Solution:

**Correct Answer :- A**

**Explanation :** f(x,y,z) = g(x^{2} + y^{2} - 2z^{2}).

df'/dx = g'(x^{2} + y^{2} - 2z^{2}) (2x)

df"/dx” = g"(x^{2} + y^{2} - 2z^{2}) (4x^{2}) + g'(x^{2} + y^{2} - 2z^{2})*2.........(1)

df/dy = g'(x^{2} + y^{2} - 2z^{2}) (2y)

df"/dy” = g"(x^{2} + y^{2} - 2z^{2}) (4y^{2}) + g'(x^{2} + y^{2} - 2z^{2})*2.........(2)

df'/dz = g'(x^{2} + y^{2} - 2z^{2}) (2y)

df"/dz” = g"(x^{2} + y^{2} - 2z^{2}) (4z^{2}) + g'(x^{2} + y^{2} - 2z^{2})*2.........(3)

Adding (1), (2) and (3)

g"(x^{2} + y^{2} - 2z^{2})(4x^{2} + 4y^{2} + 16z^{2}) + g'(x^{2}+ y^{2} - 2z^{2}) (2 + 2 - 4)

= 4(x^{2} + y^{2} + 4z^{2}) g"(x^{2} + y^{2} - 2z^{2})

QUESTION: 4

be sequences of positive real numbers such that nan < bn < n^{2}a_{n} for

all n __>__ 2. If the radius of convergence of the power series then the power series

Solution:

If the radius of convergenceis 4, Then

QUESTION: 5

Let S be the set of all limit points of the set be the set of all positive

rational numbers. Then

Solution:

QUESTION: 6

If x^{h}y^{k} is an integrating factor of the differential equation y(1 + xy) dx + x(1 — xy) dy = 0, then the ordered pair (h, k) is equal to

Solution:

If x^{h} y^{k} is an I.F. of differential equation, Then given equation become exact differential equation.

x^{h} u^{k+1} (1 + xy)dx + x^{h +1} yk (1 – xy) dy = 0

So

=> (k + 1)y^{k}x^{h} + (k + 2)x^{h + 1}y^{k + 1}

= (h + 1)x^{h}y^{k} - (h + 1)x^{k + 1}y^{k + 1}

Comparing coefficients of both the sides, we have

h – k = 0

h + k – 4

⇒ h = –2, k = – 2

QUESTION: 7

If y(x) = λe^{2x} + e^{βx}, β ≠ 2, is a solution of the differential equation

satisfying dy/dx (0) = 5, then y(0) is equal to

Solution:

QUESTION: 8

The equation of the tangent plane to the surface at the point (2, 0, 1) is

Solution:

equation of the tangent plane at point (2, 0, 1) is (x – 2) fx_{(2, 0, 1)} + (y – 0) fy_{(2, 0, 1)} + (z – 1) fz_{(0, 0, 1)} = 0

Here fx_{(2,0,1)} = 3, fy_{(2,0,1)} =0, fz_{(2,0,1)} = 4

so, we have

(x – 2).3 + (y – 0).0 + (z – 1)4 = 0

⇒ 3x + 4z = 10

which is required tangent plane.

QUESTION: 9

The value of the integral is

Solution:

By the change of order of integration

Let t = (1 – x)^{2}

dt = –2(1 – x) dx

QUESTION: 10

The area of the surface generated by rotating the curve x = y^{3}, 0 ≤ y ≤ 1, about the y-axis, is

Solution:

QUESTION: 11

Let H and K be subgroups of If the order of H is 24 and the order of K is 36, then the order of the subgroup H ∩ K is

Solution:

QUESTION: 12

Let P be a 4 × 4 matrix with entries from the set of rational numbers. If with is a root of the characteristic polynomial of P and I is the 4 × 4 identity matrix, then

Solution:

Given P is a 4 × 4 matrix with rational entries.

Let characteristic polynomial of P is

hp(x) = x^{4} + ax^{3} + bx^{2} + cx + d, ...(i)

where a, b, c and d are rational.

Since √2 + i is a root of (1), Then √2 − i is also a root of (1)

⇒ (x^{2} − 22x + 3) is a factor of (1)

Since x^{2} + 3 − 2√2x is factor of (1), Then

x^{2} + 3 + 2√2x is also a factor of (1)

⇒ (x^{2} + 3 − 2√2x) (x^{2} + 3 + 2√2x) is a factor of (1)

x^{4} + 6x^{2} + 9 – 8x^{2}.

= x^{4} – 2x^{2} + 9 ...(ii)

From (1) & (2), we have

P^{4} – 2P^{2} + 9I = 0

⇒ P^{4} = 2P^{2} – 9I

QUESTION: 13

The set as a subset of

Solution:

QUESTION: 14

The set as a subset of

Solution:

QUESTION: 15

For −1 < x < 1, the sum of the power series

Solution:

QUESTION: 16

Let f(x) = (ln x)^{2} , x > 0. Then

Solution:

QUESTION: 17

Let be a differentiable function such that f'(x) > f(x) for all and f(0) = 1. Then f( 1) lies in the interval

Solution:

Let be defined as

f(x) = e^{ax}, a > 1 x ← R

f′(x) = ae^{ax}.

f(o) = ae^{ao} = 1 & f′(x) = ae^{ax }> e^{ax} = f(x) ∀ x ∈ R.

hence f(1) = e^{a}

.

e < ea < ∞

⇒ f(1). lies in the interval (e, ∞)

QUESTION: 18

For which one of the following values of k, the equation 2x^{3} + 3x^{2} − 12x − k = 0 has three distinct real roots?

Solution:

Let g(x) = 2x^{3} + 3x^{2} – 12x.

For the roots of g(x) = 0 x(2x^{2} + 3x – 12) = 0

for the graph of g(x):

g′(x) = 6x^{2} + 6x – 12

g′(x) = 0 ⇒ x = 1, –2

function has maximum at x = –2 & minimum at x = 1 in between the roots of g(x) = 0. The graph of function shown as.

g(–2) = 2(–2)^{3} + 3(–2)^{2} – 12(–2) = 20

If y = K intersect at three distinct points with g(x), then K should be less than 20, So K = 16

QUESTION: 19

Which one of the following series is divergent?

Solution:

**For option (b);**

Let vn = 1 / n3

Then,

Asis convergent, sois convergent.

**For option (c);**

Let vn = 1 / n3

Then

Thenis also convergent.

**For option (d)**

Sois also convergent.

Hence option (a) is divergent

(By Cauchy-condensation test)

QUESTION: 20

Let S be the family of orthogonal trajectories of the family of curves 2x^{2} + y^{2} = k, for and k > 0. If passes through the point (1, 2), then passes through

Solution:

For the orthogonal trajectories of the family of curves,

integrating, we have

log_{e} y = log_{e}√x + log c_{1} ⇒ y = c_{1}√x

The curve passes through the point (1, 2), then c_{1} = 2

Now, orthogonal trajectories in y = 2√x

At point (2, 2√2) satisfy the given condition.

QUESTION: 21

Let x, x + e^{x} and 1 + x + e^{x} be solutions of a linear second order ordinary differential equation with constant coefficients. If y(x) is the solution of the same equation satisfying y(0) = 3 and y'(0) = 4, then y(1) is equal to

Solution:

Here x, x + e^{x} are two linear independent

solution. so general solution can be written as

y = c_{1}x + c_{2} (x + e^{x})

= (c_{1}+ c_{2}) x + c_{2}e^{x}

y′= (c_{1}+ c_{2}) + c_{2}e^{x}

.

y(0) = c_{2} = 3

y′ (0) = 4 = c_{1}+ 2c_{2}

⇒ c_{1} = 4 – 6 = –2

so, y(x) (–2 + 3) x + 3e^{x }= 3e^{x}+ x

y(1) = 3e + 1

QUESTION: 22

The function f(x,y) = x^{3} + 2xy + y^{3} has a saddle point at

Solution:

Here

f_{x} = 3x^{2} + 2y ⇒ f_{xx} = 6x, f_{xy} = 2

f_{y} = 2x + 3y^{2} ⇒ f_{yy} = 6y

then f_{xx} f_{yy} – (f_{xy})^{2} = 36xy – 4

so at the point (0, 0) f_{xx} f_{yy} – (f_{xy})^{2} < 0

⇒ (0, 0) is a saddle point.

QUESTION: 23

The area of the part of the surface of the paraboloid x^{2 }+ y^{2} + z = 8 lying inside the cylinder x^{2} + y^{2} = 4 is

Solution:

QUESTION: 24

be the circle (x − 1)^{2} + y^{2} = 1, oriented counterclockwise. Then the value of the line integral

is

Solution:

By Green's Theorem,

Now by using polar x = rcosθ, y = r sinθ. r = 0 to 2 cosθ, θ = 0 to 2θ.

we obtain the solution.

QUESTION: 25

be the curve of intersection of the plane x + y + z = 1 and the cylinder x^{2} + y^{2} = 1. Then the value of

is

Solution:

By stoke theorem,

Here

QUESTION: 26

The tangent line to the curve of intersection of the surface x^{2} + y^{2} − z = 0 and the plane x + y = 3 at the point (1, 1, 2) passes through

Solution:

QUESTION: 27

The set of eigenvalues of which one of the following matrices is NOT equal to the set of eigenvalues of

Solution:

QUESTION: 28

For , define

Then, at (0, 0), the function f is

Solution:

QUESTION: 29

Let {a_{n}} be a sequence of positive real numbers such that a_{1} = 1, for all n ≥ 1.

Then the sum of the series lies in the interval

Solution:

QUESTION: 30

Let {a_{n}} be a sequence of positive real numbers. The series converges if the series

Solution:

Here, {an} is a sequence of a positive real number.

The series converges if the series converges.

*Multiple options can be correct

QUESTION: 31

Let G be a noncyclic group of order 4. Consider the statements I and II:

I. There is NO injective (one-one) homomorphism from

II. There is NO surjective (onto) homomorphism from

Then

Solution:

*Multiple options can be correct

QUESTION: 32

Let G be a nonabelian group, y ∈ G, and let the maps f, g, h from G to itself be defined by f(x) = yxy^{-1}, g(x) = x^{-1} and h = g _{° }g.

Then

Solution:

*Multiple options can be correct

QUESTION: 33

Let S and T be linear transformations from a finite dimensional vector space V to itself such that S(T(v)) = 0 for all v ∈ V. Then

Solution:

*Multiple options can be correct

QUESTION: 34

Let be differentiable vector fields and let g be a differentiable scalar function. Then

Solution:

*Multiple options can be correct

QUESTION: 35

Consider the intervals S = (0, 2] and T = [1, 3). Let S° and T° be the sets of interior points of S and T, respectively. Then the set of interior points of S\T is equal to

Solution:

QUESTION: 36

Let {a_{n}} be the sequence given by

Then which of the following statements is/are TRUE about the subsequences {a_{6n−1}} and {a_{6n+4}}?

Solution:

*Multiple options can be correct

QUESTION: 37

Let f(x) = cos(|π - x|) + (x - n) sin |x| and g(x) = x^{2} If h(x) = f(g(x)), then

Solution:

*Multiple options can be correct

QUESTION: 38

be given by f(x) = (sin x)^{π} − π sin x + π.

Then which of the following statements is/are TRUE?

Solution:

*Multiple options can be correct

QUESTION: 39

Let

Then at (0, 0),

Solution:

*Multiple options can be correct

QUESTION: 40

Let {a_{n}} be the sequence of real numbers such that a_{1} = 1 and a_{n+1 }= a_{n} + a^{2}_{n} for all n __>__ 1.

Then

Solution:

*Answer can only contain numeric values

QUESTION: 41

Let x be the 100-cycle (1 2 3 ⋯ 100) and let y be the transposition (49 50) in the permutation group S_{100}. Then the order of xy is ______

Solution:

= (1 2 ------ 48, 50, 51 ------ 100) So the order of xy is 99

*Answer can only contain numeric values

QUESTION: 42

Let W_{1} and W_{2} be subspaces of the real vector space defined by

W_{1} = {(x_{1,}x_{2}, ...,x_{100}) : x_{i} = 0 if i is divisible by 4},

W_{2} = { (x_{1};x_{2}, ....x_{100}) : x_{i} = 0 if i is divisible by 5}.

Then the dimension of W_{1} ∩ W_{2} is ____

Solution:

*Answer can only contain numeric values

QUESTION: 43

Consider the following system of three linear equations in four unknowns x_{1}, x_{2}, x_{3} and x_{4}

x_{1} + x_{2} + x_{3} + x_{4} = 4,

x_{1} + 2x_{2} + 3x_{3} + 4x_{4} = 5,

x_{1} + 3x_{2} + 5x_{3} + kx_{4} = 5.

If the system has no solutions, then k = _____

Solution:

*Answer can only contain numeric values

QUESTION: 44

Let and be the ellipse

oriented counter clockwise. Then the value of (round off to 2 decimal places) is_______________

Solution:

*Answer can only contain numeric values

QUESTION: 45

The coefficient of in the Taylor series expansion of the function about x = π/2, is ____________

Solution:

*Answer can only contain numeric values

QUESTION: 46

Let be given by

Then

max {f(x): x ∈ [0,1]} - min [f(x):x ∈ [0,1]}

is ___________

Solution:

*Answer can only contain numeric values

QUESTION: 47

If then g′(1) = _______

Solution:

*Answer can only contain numeric values

QUESTION: 48

Let

Then the directional derivative of f at (0, 0) in the direction of

Solution:

*Answer can only contain numeric values

QUESTION: 49

The value of the integral (round off to 2 decimal places) is ___________

Solution:

*Answer can only contain numeric values

QUESTION: 50

The volume of the solid bounded by the surfaces x = 1 - y^{2} and x = y^{2} - 1, and the planes z = 0 and z = 2 (round off to 2 decimal places) is

Solution:

*Answer can only contain numeric values

QUESTION: 51

The volume of the solid of revolution of the loop of the curve y^{2} = x^{4} (x + 2) about the x-axis (round off to 2 decimal places) is ___________

Solution:

*Answer can only contain numeric values

QUESTION: 52

The greatest lower bound of the set (round off to 2 decimal places) is ______________

Solution:

*Answer can only contain numeric values

QUESTION: 53

Let G = : n __<__ 55, gcd(n, 55) = 1} be the group under multiplication modulo 55. Let x ∈ G be such that x^{2} = 26 and x > 30. Then x is equal to _______

Solution:

*Answer can only contain numeric values

QUESTION: 54

The number of critical points of the function is ___________

Solution:

*Answer can only contain numeric values

QUESTION: 55

The number of elements in the set {x ∈ S_{3}: x^{4} = e), where e is the identity element of the permutation group S_{3}, is

Solution:

*Answer can only contain numeric values

QUESTION: 56

If is an eigenvector corresponding to a real eigenvalue of the matrix then z − y is equal to__________

Solution:

*Answer can only contain numeric values

QUESTION: 57

Let M and N be any two 4 × 4 matrices with integer entries satisfying

Then the maximum value of det(M) + det(N) is ___________

Solution:

*Answer can only contain numeric values

QUESTION: 58

Let M be a 3 × 3 matrix with real entries such that M^{2} = M + 2I, where I denotes the 3 × 3 identity matrix. If α, β and γ are eigenvalues of M such that αβγ = -4, then α+β+γ is equal to ______

Solution:

*Answer can only contain numeric values

QUESTION: 59

Let y(x) = xv(x) be a solution of the differential equation If v(0) = 0 and v(1) = 1, then v(-2) is equal to ______

Solution:

*Answer can only contain numeric values

QUESTION: 60

If y(x) is the solution of the initial value problem then y(ln 2) is (round off to 2 decimal places) equal to ____________

Solution:

### Math Past Year Paper with Solution - 2019, Class 10

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