CBSE Previous Year Question Papers Class 12 Mathematics with solutions

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1/2.

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2.

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A team of 8 couples(husband and wife) attended a lucky draw in which 4 persons were picked up for a prize. The
probability that at least one couple is selected for the prize is

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A bag contains 16 coins of which two are counterfeit with heads on both sides. The rest are fair coins. One is
selected at random from the bag and tossed. The probability of getting a head is

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The objective function of a linear programming problem is a

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linear equation.

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The vertices of a triangle ABC are A(–1, 2, –3), B(5, 0, –6) and C(0, 4, –1). The direction cosines of the bisector of the angle BAC are

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The direction cosines of the normal to the plane 2x + 2y + z = 5 are

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2/3, 2/3, 1/3.

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Let A = {2, 3, 4}, B = {5, 6, 7, 9} and let f = {(1, 4), (3, 6), (4, 7)} be a function from A to B, then function is ______________.

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Since, every element of set A has only one image in set B i.e., 2→5, 3→6 and 4→7. Therefore, f is one-one function.

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If A and B are two square matrices and K is a scalar quantity then K(A+B) = ______.

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K(A+B) = KA+KB

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$The function f given by f (x) = x^{3} - {6x}^{2} + 14x, x \in R is________on R .$

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\begin{array}{l} The given function, \\ f (x) = x^{3} - 6 x^{2} + 14 x \\ Differentiating w .r .t . x, we get \\ f' (x) = 3 x^{2} - 12 x + 14 \\ = 3 x^{2} - 12 x + 12 + 2 \\ = 3 (x^{2} - 4 x + 4) + 2 \\ = 3 {(x - 2)}^{2} + 2 > 0 in every interval of R . \\ Therefore, the function f is \underline{strictly increasing} on R . \end{array} MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2Daebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@F369@

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If x = 4at and y = 3at², then the equation of tangent at t = 2 is _______________.

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\begin{array}{l} Given, x = 4at and y = {3at}^{2} \\ Differentiating w .r .t . x, we get \\ \frac{d x}{d t} = 4 a and \frac{d y}{d t} = 6 a t \\ \frac{d y}{d x} = \frac{(\frac{d y}{d t})}{(\frac{d x}{d t})} \\ = \frac{6 a t}{4 a} \\ = \frac{3 t}{2} \\ {(\frac{d y}{d x})}_{t = 2} = \frac{3 (2)}{2} \\ = 3 \\ At t = 2, \\ x = 4 a (2) and y = 3 a {(2)}^{2} \\ = 8 a = 12 a \\ The equation of tangent is, \\ y - y_{1} = {(\frac{d y}{d x})}_{t = 2} (x - x_{1}) \\ y - 12 a = 3 (x - 8 a) \\ y - 12 a = 3 x - 24 a \\ \Rightarrow 3 x - y = 12 a \\ Therefore, the required equation of tangent is \underline{3 x - y = 12 a} . \end{array} MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2Daebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@7E82@

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$\begin{array}{l} If \hat{a} is a unit vector and (\vec{x} - \hat{a}) (\vec{x} + \hat{a}) = 15, then the value of \\ | \vec{x} | is ____. \end{array} MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2Daebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@7F17@$

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\begin{array}{l} Since \hat{a} is a unit vector, so | \hat{a} | = 1. \\ ∴ (\vec{x} - \hat{a}) (\vec{x} + \hat{a}) = 15 \\ \vec{x} . \vec{x} + \vec{x} . \hat{a} - \hat{a} . \vec{x} - \hat{a} . \hat{a} = 15 \\ \Rightarrow {| \vec{x} |}^{2} - 1 = 15 \\ [∵ \vec{x} . \hat{a} = \hat{a} . \vec{x} and \hat{a} . \hat{a} = 1] \\ \Rightarrow {| \vec{x} |}^{2} = 16 \\ \Rightarrow | \vec{x} | = 4 \\ ∴ If \hat{a} is a unit vector and (\vec{x} - \hat{a}) (\vec{x} + \hat{a}) = 15, then the \\ value of | \vec{x} | is \underline{4} . \end{array} MathType@MTEF@5@5@+=feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2Daebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@64EA@

Q:

$\begin{array}{l} The scalar projection of the vector \vec{a} = 4 \hat{i} - 3 \hat{j} + 2 \hat{k} on the \\ vector \vec{b} = 4 \hat{i} - 4 \hat{j} + 7 \hat{k} is _______. \end{array} MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2Daebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@8D0C@$

A:

\begin{array}{l} Scalar projection of \vec{a} on \vec{b} is given by \\ \frac{\vec{a} . \vec{b}}{| \vec{b} |} = \frac{(4 \hat{i} - 3 \hat{j} + 2 \hat{k}) . (4 \hat{i} - 4 \hat{j} + 7 \hat{k})}{| 4 \hat{i} - 4 \hat{j} + 7 \hat{k} |} \\ = \frac{16 + 12 + 14}{\sqrt{4^{2} + 4^{2} + 7^{2}}} \\ = \frac{42}{\sqrt{81}} \\ = \frac{42}{9} \\ = \frac{14}{3} \\ ∴ The scalar projection of the vector \vec{a} = 4 \hat{i} - 3 \hat{j} + 2 \hat{k} on \\ the vector \vec{b} = 4 \hat{i} - 4 \hat{j} + 7 \hat{k} is \underline{\frac{14}{3}} . \end{array} MathType@MTEF@5@5@+=feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLnhiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2Daebbfv3ySLgzGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=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@2731@

Q:

The value of determinant will _______ if its rows and columns are interchanged.

A:

remain unchanged

Q:

Find the area of a triangle whose vertices are given by (x₁, y₁), (x₂, y₂) and (x₃, y₃).

A:

Q:

Evaluate

A:

Q:

Evaluate:

A:

Q:

Evaluate

A:

Q:

Evaluate

A:

Q:

A:

Q:

Evaluate

A:

Q:

A:

Q:

Construct a 2 2 matrix A = [ a_ij] whose elements are given by

A:

Q:

A:

Q:

A:

Q:

Write in the simplest form.

A:

Q:

Find the value of .

A:

Q:

If A, B, C have position vectors (2, 0, 0), (0, 1, 0) and (0, 0, 2), show that ABC is isosceles.

A:

Q:

Prove that the angle in a semi-circle is a right angle.

A:

Let O be the centre of the semi-circle and r be its radius.

Let C be any point on the semi-circle.

Now,

Q:

A:

Q:

Evaluate .

A:

Q:

A:

Q:

A:

Q:

A:

Q:

For the curve y = 4x³ - 2x⁵, find all the points at which the tangent passes through the origin.

A:

Let (x₁, y₁) be the required point on the given curve y = 4x³ - 2x⁵.
Slope of the curve dy/dx = 12x² - 10x⁴

Slope of the curve at (x₁, y₁) = 12x₁² - 10x₁⁴

Thus, the equation of tangent is

y - y₁ = (12x₁² - 10x₁⁴)(x - x₁)

As tangent passes through the origin

y₁ = (12x₁² - 10x₁⁴)(x₁)

4x₁³ - 2x₁⁵= 12x₁³ - 10x₁⁵ or 8x₁⁵ - 8x₁³ =0

8x₁³ ( x₁²- 1) =0

x₁ = 0, - 1, 1

at x₁ = 0 y₁ = 0

at x₁ = - 1 y₁ = - 2, and

at x₁ = 1 y₁ = 2

Hence, the required points are (0, 0), (-1, -2) and (1, 2).

Q:

Solve the following linear programing problem graphically:
Maximise Z = 60x + 15y

Subject to constraints

x + y 50
3x + y 90
x, y 0

A:

Maximise Z = 60x + 15y (1)

Subject to constraints

x + y 50 (2)
3x + y 90 (3)
x, y 0 (4)

Graph the inequalities (2), (3) and (4). The feasible region (shaded) is shown below:

Corner Point	Z = 60x +15y
A(0, 50)	750
B(20, 30)	1650
C(30, 0)	1800Maximum

The maximum value of Z on the feasible region occurs at x = 30 and y = 0.

Q:

Using integration, find the area of ABC where A is (2, 3), B is (4, 7) and C is (6, 2).

A:

Q:

Using the method of integration find the area bounded by the curve |x| + |y| = 1.

A:

Q:

In a class of 60 students, 32 like Maths, 30 like Biology and 24 like both Maths and Biology. If one of these students is selected at random, find the probability that the selected student

(a) likes Maths or Biology
(b) likes neither Maths nor Biology
(c) likes Maths but not Biology.

A:

Let the probability that a student likes Maths be denoted by P(M) and Biology be denoted by P(B).It is given that
P(M) = 32/60 = 8/15, P(B) = 30/60 = 1/2, P(MB) = 24/60 = 2/5

(a) Probability that the selected student likes Maths or Biology
P(MB) = P(M) + P(B) – P(MB)

= 8/15 + 1/2 - 2/5 = 19/30.

(b) Probability that the student likes neither Maths nor Biology = 1 - P(MB)

= 1 - 19/30 = 11/30

(c) Probability that a student likes Maths but not Biology = P(M) – P(MB)

= 8/15 - 2/5 = 2/15.

Q:

P,Q,R shot to hit a target. If P hits it 3 times in 4 trials, Q hits it 2 times in 3 trials and R hits it 4 times in 5 trials, what is the probability that the target is hit by at least two persons?

A:

Let E₁, E₂ and E₃be independent events that P, Q and R hit the target respectively. Then
P(E₁) = 3/4 , P(E₂) = 2/3 and P(E₃) = 4/5
P(E₁^c) = 1 - 3/4 = 1/4 , P(E₂^c) = 1 - 2/3 = 1/3 and P(E₃^c) = 1 - 4/5 = 1/5 .

The target is hit by atleast two persons in the following mutually exclusive ways:

(a) P hits, Q hits and R does not hit i.e., E₁E₂E₃^c
(b) P hits, Q does not hit and R hits i.e., E₁E₂^cE₃
(c) P does not hit, Q hits and R hits i.e., E₁^cE₂E₃
(d) P hits, Q hits and R hits i.e., E₁E₂E₃
Required Probability = P[(a) (b) (c) (d)] = P(a) + P(b) + P(c) + P(d)

= (3/4 x 2/3 x 1/5) + (1/4 x 2/3 x 4/5) + (3/4 x 2/3 x 4/5)

= 25/30 = 5/6.

Q:

Solve the following system of equations by matrix method.

A:

Q:

A:

Solved Board Paper For CBSE Class 12 Mathematics

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Sample Paper

Exam Weightage

Solved Board Paper For CBSE Class 12 Mathematics

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A team of 8 couples(husband and wife) attended a lucky draw in which 4 persons were picked up for a prize. The probability that at least one couple is selected for the prize is

A bag contains 16 coins of which two are counterfeit with heads on both sides. The rest are fair coins. One is selected at random from the bag and tossed. The probability of getting a head is

The objective function of a linear programming problem is a

The vertices of a triangle ABC are A(–1, 2, –3), B(5, 0, –6) and C(0, 4, –1). The direction cosines of the bisector of the angle BAC are

The direction cosines of the normal to the plane 2x + 2y + z = 5 are

Let A = {2, 3, 4}, B = {5, 6, 7, 9} and let f = {(1, 4), (3, 6), (4, 7)} be a function from A to B, then function is ______________.

If A and B are two square matrices and K is a scalar quantity then K(A+B) = ______.

The function f given by f( x ) = x 3 – 6x 2 + 14x, x∈R is ________on R.

If x = 4at and y = 3at2, then the equation of tangent at t = 2 is _______________.

The value of determinant will _______ if its rows and columns are interchanged.

Find the area of a triangle whose vertices are given by (x1, y1), (x2, y2) and (x3, y3).

Evaluate

Evaluate:

Evaluate

Evaluate

Evaluate

Construct a 2 2 matrix A = [ aij ] whose elements are given by

Write in the simplest form.

Find the value of .

If A, B, C have position vectors (2, 0, 0), (0, 1, 0) and (0, 0, 2), show that ABC is isosceles.

Prove that the angle in a semi-circle is a right angle.

Evaluate .

For the curve y = 4x3 - 2x5, find all the points at which the tangent passes through the origin.

Solve the following linear programing problem graphically: Maximise Z = 60x + 15y Subject to constraints x + y 50 3x + y 90 x, y 0

Using integration, find the area of ABC where A is (2, 3), B is (4, 7) and C is (6, 2).

Using the method of integration find the area bounded by the curve |x| + |y| = 1.

In a class of 60 students, 32 like Maths, 30 like Biology and 24 like both Maths and Biology. If one of these students is selected at random, find the probability that the selected student (a) likes Maths or Biology (b) likes neither Maths nor Biology (c) likes Maths but not Biology.

P,Q,R shot to hit a target. If P hits it 3 times in 4 trials, Q hits it 2 times in 3 trials and R hits it 4 times in 5 trials, what is the probability that the target is hit by at least two persons?

Solve the following system of equations by matrix method.

Download Extramarks – The Learning App

Sample Paper

Exam Weightage

A team of 8 couples(husband and wife) attended a lucky draw in which 4 persons were picked up for a prize. The
probability that at least one couple is selected for the prize is

A bag contains 16 coins of which two are counterfeit with heads on both sides. The rest are fair coins. One is
selected at random from the bag and tossed. The probability of getting a head is

$The function f given by f (x) = x^{3} - {6x}^{2} + 14x, x \in R is________on R .$

If x = 4at and y = 3at², then the equation of tangent at t = 2 is _______________.

Find the area of a triangle whose vertices are given by (x₁, y₁), (x₂, y₂) and (x₃, y₃).

Construct a 2 2 matrix A = [ a_ij] whose elements are given by

For the curve y = 4x³ - 2x⁵, find all the points at which the tangent passes through the origin.

Solve the following linear programing problem graphically:
Maximise Z = 60x + 15y

Subject to constraints

x + y 50
3x + y 90
x, y 0

In a class of 60 students, 32 like Maths, 30 like Biology and 24 like both Maths and Biology. If one of these students is selected at random, find the probability that the selected student

(a) likes Maths or Biology
(b) likes neither Maths nor Biology
(c) likes Maths but not Biology.