CBSE Class 12 Maths Question Paper 2016 with Solutions

Updated on Jan 27, 2021 - 11:59 a.m. IST by Ravindra Pindel #CBSE Class 12th

CBSE Class 12 Maths question paper 2016 with solutions - Performance in class 12th board exams is very significant. Solving CBSE previous year question papers Class 12 Maths 2016 aids to get success in class 12th board exam. I personally suggest you solve as many CBSE previous year question papers. First, try to solve all these questions given in CBSE 12 Maths question paper 2016 on your own. You may find difficulties solving these questions. You can take help from CBSE Class 12 Maths question paper 2016 with solutions provided in this article. These questions are explained in a step-by-step manner so you will be able to understand the concept very easily. You will get analysis and detailed solutions to CBSE 12 Maths question paper 2016 for all three sets. CBSE class 12 previous year papers will help you in the quick revision of all the important concepts of class 12 maths before the exam.
Also, read CBSE class 12 Maths Preparation Tips.

CBSE Class 12 Maths Question Paper 2016 with Solutions Set-wise

Go through the following section to get solutions of CBSE previous year question papers Class 12 Maths 2016:

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CBSE 12 Class 2016 Paper: 065 set 1 S

Question 1)
If vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ are such that $|\vec{\text{a}}| = \frac{1}{2}$ , $|\vec{\text{b}}| = \frac{4}{\sqrt{3}}$ and $|\vec{\text{a}}\times \vec{\text b}| = \frac{1}{\sqrt{3}}$ , then find $| \vec{\text{a}} \cdot \vec{\text b} |$ .

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Question 2)

If $\vec{a}$ and $\vec{b}$ are unit vectors, then what is the angle between $\vec{a}$ and $\vec{b}$ for $\vec{a} - \sqrt{2}\vec{b}$ to be a unit vector?

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Question 3)

Find the distance between the planes $\vec{r}\cdot(2\hat{i}-3\hat{j}+6\hat{k}) - 4 = 0$ and $\vec{r}\cdot(6\hat{i}-9\hat{j}+18\hat{k}) + 30 = 0$ .

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Question 4)

If A is a square matrix such that $|A| = 5$ , write the value of $|AA^T|$ .

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Question 5)

If $A = \begin{pmatrix} 1 & 2 \\ 3 & - 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & -4 \\ 3 & - 2 \end{pmatrix}$ , find $|AB|$ .

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Question 6)

Differentiate $(\sin 2x)^x + \sin^{-1}\sqrt{3x}$ with respect to $x$ .

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Question 7)

Differentiate $\tan^{-1}\left(\frac{\sqrt{1 + x^2}- \sqrt{1-x^2}}{\sqrt{1 + x^2} + \sqrt{1-x^2}} \right )$ with respect to $\cos^{-1}x^2$

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Question 8)

Find equation of normal to the curve $ay^2 = x^3$ at the point whose $x$ coordinate is am².

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Question 9)

Find : $\int\frac{1-\sin x}{\sin x(1+\sin x)}dx$

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Question 10)

Find : $\int\left[(\log (\log x) + \frac{1}{(\log x)^2} \right ]dx$

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Question 11)

Evaluate: $\int^{\frac{\pi}{2}}_0\frac{\sin^2 x }{\sin x +\cos x}dx$

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Question 12)

Evaluate: $\int_0^1\cot^{-1}(1 - x + x^2)dx$

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Question 13)

Solve the differential equation : $(x + 1)\frac{dy}{dx} - y = e^{3x}(x+1)^3$

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Question 14)

Solve the differential equation: $2ye^{\frac{x}{y}}dx + \left(y-2xe^{\frac{x}{y}} \right )dy = 0$

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Question 15)

Ishan wants to donate a rectangular plot of land for a school in his village. When he was asked to give dimensions of the plot, he told that if its length is decreased by 50 m and breadth is increased by 50 m, then its area will remain same, but if the length is decreased by 10 m and breadth is decreased by 20 m, then its area will decrease by 5300 m². Using matrices, find the dimensions of the plot. Also, give the reason why he wants to donate the plot for a school.

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Question 16)

Prove that $\mathrm{2\sin^{-1}\left(\frac{3}{5} \right )- \tan^{-1}\left(\frac{17}{31} \right ) = \frac{\pi}{4}}$

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Question 17)

Solve the equation for x : $\cos(\tan^{-1}x) = \sin\left(\cot^{-1}\frac{3}{4} \right )$

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Question 18)

There are two bags A and B. Bag A contains 3 white and 4 red balls whereas bag B contains 4 white and 3 red balls. Three balls are drawn at random (without replacement) from one of the bags and are found to be two white and one red. Find the probability that these were drawn from bag B.

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Question 19)

Given that vectors $\vec{a}$ , $\vec{b}$ , $\vec{c}$ form a triangle such that $\vec{a} = \vec{b} + \vec{c}$ . Find p,q,r,s such that area of triangle is $5\sqrt6$ where $\vec{a} = p\hat{i} + q\hat{j} + r\hat{k}$ , $\vec{b} = s\hat{i} + 3\hat{j} + 4\hat{k}$ , $\vec{c} = 3\hat{i} + \hat{j}-2\hat{k}$ .

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Question 20)

Find the equation of plane passing through the points A (3, 2, 1), B (4, 2, 22) and C (6, 5, 21) and hence find the value of $\lambda$ for which A (3, 2, 1),B (4, 2, 22), C (6, 5, 21) and D ( $\lambda$ , 5, 5) are coplanar.

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Question 21)

Find the coordinates of the point where the line $\vec{r} =( -\hat{i}-2\hat{j} -3\hat{k}) + \lambda(3\hat{i} +4\hat{j} + 3\hat{k})$ meets the plane which is perpendicular to the vector $\vec{n} = \hat i +\hat j + 3\hat k$ and at a distance of $\frac{4}{\sqrt11}$ from origin.

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Question 22)

Let $f : \mathrm{N\rightarrow N}$ be a function defined as $f(x) = 4x^2 +12x + 15$ . Show that $f : \mathrm{N\rightarrow S}$ is invertable ( Where S is the range of $f$ ). Find the inverse of $f$ and hence find $f ^{-1}(31)$ and $f ^{-1}(87)$ .

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Question 23)

Using properties of determinants, prove that :

$\begin{vmatrix} (b+c)^2 & a^2 & bc \\ (c+a)^2 & b^2 & ca \\ (a+b)^2 & c^2 & ab \end{vmatrix} = (a-b)(b-c)(c-a)(a + b +c)(a^2 +b^2 +c^2)$

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Question 24)

Using elementary row operations, find the inverse of the following matrix :

$A = \begin{pmatrix} 2 & -1 & 3 \\ -5 & 3 & 1\\-3 & 2 & 3\end{pmatrix}$

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Question 25)

Determine the intervals in which the function $f(x) = x^4 - 8x^3 + 22x^2 - 24x +21$ is strictly increasing or strictly decreasing.

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Question 26)

Find the maximum and minimum values of $f(x) = \sec x +\log \cos^2 x,\quad 0<x < 2\pi$

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Question 27)

Using integration find the area of the region $\mathrm{\left \{ (x,y) : y^2\leq6ax {\text{ and }} x^2 + y^2\leq 16a^2\right\}}$

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Question 28)

Find the equation of the plane containing two parallel lines $\mathrm { \frac{x-1}{2} = \frac{y+1}{-1} = \frac{z}{3}}$ and $\mathrm { \frac{x-2}{3} = \frac{y-1}{1} = \frac{z-2}{6}}$ . Also, find if the plane thus obtained contains the line $\mathrm { \frac{x}{4} = \frac{y-2}{-2} = \frac{z+1}{6}}$ or not.

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Question 29)

A diet is to contain at least 80 units of Vitamin A and 100 units of minerals. Two foods F₁ and F₂ are available costing Rs 5 per unit and Rs 6 per unit respectively. One unit of food F₁ contains 4 units of vitamin A and 3 units of minerals whereas one unit of food F₂ contains 3 units of vitamin A and 6 units of minerals. Formulate this as a linear programming problem. Find the minimum cost of a diet that consists of a mixture of these two foods and also meets minimum nutritional requirements.

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Question 30)

Three numbers are selected at random (without replacement) from the first six positive integers. If X denotes the smallest of the three numbers obtained, find the probability distribution of X. Also find the mean and variance of the distribution.

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Question 31)

If $A = \begin{pmatrix}0 & 3 \\ 2 & -5 \end{pmatrix}$ and $KA = \begin{pmatrix}0 & 4a \\ -8 & 5b \end{pmatrix}$ find the values of k and a.

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Question 32)

Find k, if $f(x) = \left\{\begin{matrix} k\sin\frac{\pi}{2}(x+1), & x\leq 0\\ \frac{\tan x - \sin x}{x^3}& x> 0 \end{matrix}\right.$ is continuous at x=0.

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CBSE 12 Class 2016 Paper: 065 set 1 N

Question 1)

If $A=\begin{pmatrix} \cos \alpha & \sin \alpha \\ - \sin \alpha & \cos \alpha \end{pmatrix}$ , find $\alpha$ satisfying $0<\alpha <\frac{\pi }{2}$ when $A+A^T=\sqrt{2}I_2;$ where $A^T$ is the transpose of $A$ .

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Question 2)

If A is a $3\times 3$ matrix and $|3A|=k|A|$ , then write the value of k.

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Question 3)

For what values of k, the system of linear equations

$x+y+z=2$

$2x+y-z=3$

$3x+2y+kz=4$

has a unique solution ?

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Question 4)

Write the sum of intercepts cut off by the plane $\vec{r}\cdot (2\hat{i}+\hat{j}-\hat{k})-5=0$ on the three axes.

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Question 5)

Find $\lambda$ and $\mu$ if :

$(\hat{i}+3\hat{j}+9\hat{k})\times (3\hat{i}-\lambda \hat{j}+\mu \hat{k})=\vec{0}$

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Question 6)

If $\vec{a}=4\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$ , then find a unit vector parallel to the vector $\vec{a}+\vec{b}$ .

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Question 7)

Solve for x :

$\tan^{-1}(x-1)+\tan^{-1}x+\tan^{-1}(x+1)=\tan^{-1}3x$

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Question 8)

Prove that

$\tan ^{-1} \left ( \frac{6x-8x^3}{1-12x^2} \right )-\tan^{-1}\left ( \frac{4x}{1-4x^2} \right ) = \tan^{-1} 2x; |2x|<\frac{1}{\sqrt{3}}$

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Question 9)

A typist charges Rs 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs 180. Using matrices, find the charges of typing one English and one Hindi page separately. However typist charged only Rs 2 per page from a poor student Shyam for 5 Hindi pages. How much less was charged from this poor boy? Which values are reflected in this problem?

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Question 10)

$f(x)=\begin{cases} \frac{\sin (a+1)x+2\sin x}{x} & ,x<0 \\ 2 &,x=0 \\ \frac{\sqrt{1+bx}-1}{x}&,x>0 \end{cases}$

is continuous at $x=0$ , then find the value of a and b.

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Question 11)

If $x\cos (a+y)=\cos y$ then prove that $\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{\cos^2(a+y)}{\sin a}$ . Hence show that $\sin a\frac{\mathrm{d}^2 y}{\mathrm{d} x^2}+\sin 2(a+y)\frac{\mathrm{d} y}{\mathrm{d} x}=0$

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Question 12)

Find $\frac{\mathrm{d} y}{\mathrm{d} x}$ if $y=\sin^{-1}\left [ \frac{6x-4\sqrt{1-4x^2}}{5} \right ]$

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Question 13)

Find the equation of tangents to the curve $y=x^3+2x-4$ , which are perpendicular to line $x+14y+3=0$

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Question 14)

Find : $\int \frac{(2x-5)e^{2x}}{(2x-3)^3}dx$

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Question 15)

Find :

$\int \frac{x^2+x+1}{(x^2+1)(x+2)}dx$

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Question 16)

Evaluate :

$\int_{-2}^{2}\frac{x^2}{1+5^x}dx$

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Question 17)

If $A=\begin{pmatrix} 1 &0 &2 \\ 0 & 2 & 1\\ 2 &0 &3 \end{pmatrix}$ and $A^3-6A^2+7A+kI_3=0$ , find k.

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Question 18)

Using properties of determinants, prove that

$\begin{vmatrix} (x+y)^2 & zx &zy \\ zx & (z+y)^2 & xy \\ zy & xy & (z+x)^2 \end{vmatrix}=2xy(x+y+z)^3$

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Question 19)

A retired person wants to invest an amount of Rs 50,000. His broker recommends investing in two types of bonds 'A' and 'B' yielding 10% and 9% return respectively on the invested amount. He decides to invest at least Rs 20,000 in bond 'A' and at least Rs 10,000 in bond 'B'. He also wants to invest at least as much in bond 'A' as in bond 'B'. Solve this linear programming problem graphically to maximise his returns.

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Question 20)

Find the equation of the plane which contains the line of intersection of the planes.

$\vec{r}\cdot \left ( \hat{i}-2\hat{j}+3\hat{k} \right )-4=0$ and

$\vec{r}\cdot \left ( -2\hat{i}+\hat{j}+\hat{k} \right )+5=0$

and whose intercept on x-axis is equal to that of an y-axis.

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Question 21)

Using the method od integration, find the area of the triangular region whose vertices are (2, -2), (4,3) and (1,2)

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Question 22)

Find : $\int (x+3)\sqrt{3-4x-x^2} dx.$

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Question 23)

Find the particular solution of differential equation : $\frac{\mathrm{d} y}{\mathrm{d} x}=-\frac{x+y\cos x }{1+\sin x}$ given that y=1 and x=0.

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Question 24)

Find the particular solution of the differential equation

$2ye^\frac{x}{y}dx+(y-2x e^\frac{x}{y})dy=0$ given that x=0 when y=1.

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Question 25)

Show that the four points A(4,5,1), B(0,-1,-1), C(3,9,4) and D(-4,4,4) are coplanar.

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Question 26)

Find the coordinates of the foot of perpendicular drawn from the point A(-1,8,4) to the line joining the points B(0,-1,3) and C(2,-3,-1). Hence find the image of the point A in the line BC.

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Question 27)

A bag X contains 4 white balls and 2 black balls, while another bag Y contains 3 white balls and 3 black balls. Two balls are drawn (without replacement ) at random from one of the bags and were found to be white and one black. Find the probability that the balls were drawn from bag Y.

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Question 28)

A and B throw a pair of dice alternately, till one of them gets a total of 10 and wins the game. Find their respective probabilities of winning, if A starts first.

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Question 29)

Three numbers are selected at random (without replacement ) from first six positive integers. Let X denote the largest of the three numbers obtained. Find the probability distribution of X. Also, find the mean and variance of the distribution.

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Question 30)

Let $A=R\times R$ and * be a binary operation on A defined by $(a,b)*(c,d)=(a+c,b+d)$ . Show that * is commutative and associative. Find the identity element for * on A. Also find the inverse of every element $(a,b)\in A$ .

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Question 31)

Prove that $y=\frac{4\sin \theta }{2+\cos \theta }-\theta$ is an increasing function of $\theta$ on $\left [ 0,\frac{\pi}{2} \right ]$ .

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Question 32)

Show that semi-verticle angle of a cone of maximum volume and given slant height is $\cos^{-1}\left ( \frac{1}{\sqrt{3}} \right )$ .

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CBSE 12 Class 2016 Paper: 065 set 1 E

Question 1)

Write the value of

$\left|\begin{array}{lll} a-b & b-c & c-a \\ b-c & c-a & a-b \\ c-a & a-b & b-c \end{array}\right|$

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Question 2)

If $A=\left(\begin{array}{lll} 1 & -2 & 3 \\ -4 & 2 & 5 \end{array}\right)$ and $B=\left(\begin{array}{ll} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{array}\right)$ and $BA=(b_{ij})$ , find $b_{12}+b_{23}$ .

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Question 3)

Write the number of all possible matrices of order $2\times 3$ with each entry 1 or 2.

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Question 4)

Write the coordinates of the point which is the reflection of the point $(\alpha, \beta, \gamma)$ in the XZ-plane.

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Question 5)

Find the position vector of the point which divides the join of points with position vectors $\overrightarrow{\mathrm{a}}+3 \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}$ internally in the ratio 1: 3

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Question 6)

If $|\overrightarrow{\mathrm{a}}|=4,|\overrightarrow{\mathrm{b}}|=3$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}=6 \sqrt{3},$ then find the value of $|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|$ .

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Question 7)

Solve for x : $\tan ^{-1}\left(\frac{2-x}{2+x}\right)=\frac{1}{2} \tan ^{-1} \frac{x}{2}, x>0$

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Question 8)

Prove that $2\sin^{-1}\left ( \frac{3}{5} \right )-\tan^{-1}\left ( \frac{17}{31} \right )=\frac{\pi }{4}$

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Question 9)

On her birthday Seema decided to donate some money to children of an orphanage home. If there were 8 children less, every one would have got Rs 10 more. However, if there were 16 children more, every one would have got rs 10 less. Using matrix method, find out the number of children and the amount distributed by Seema. What values are reflected by Seema's decision ?

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Question 10)

If $x=\mathrm{e}^{\cos2t}$ and $y=\mathrm{e}^{\sin2t}$ , then prove that $\frac{\mathrm{d} y}{\mathrm{d} x}=-\frac{y \log x}{x \log y}$ .

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Question 11)

Verify the Mean Value theorem for the function $f(x)=2\sin x +\sin2x$ on $[0,\pi ]$

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Question 12)

Show that the function $f$ given by :

$f(x)=\begin{cases} \frac{e^{\frac{1}{x}}-1}{e^{\frac{1}{x}}+1},& \text{if}\, \, x\neq 0 \\-1, & \text{if}\, \, x=0 \end{cases}$ is discontinous at $x=0$

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Question 13)

Find the equation of the tangent line to the curve $y=\sqrt{5x-3}-5$ , which is parallel to the line $4x-2y+5=0$ .

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Question 14)

Evaluate : $\int_{1}^{5}\left \{ \left | x-1 \right |+\left | x-2 \right | \left | x-3 \right |\right \}dx$

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Question 15)

Evaluate :

$\int_{0}^{\pi }\frac{x\sin x}{1+3\cos^2x}dx$

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Question 16)

Find :

$I=\int\frac{2x+1}{(x^2+1)(x^2+4)}dx$

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Question 17)

Find :

$\int (3x+5)\sqrt{5+4x-2x^2}dx$

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Question 18)

Solve the differential equation :

$x\frac{\mathrm{d} y}{\mathrm{d} x}+y-x+xy\cot x =0; x \ne 0$

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Question 19)

Solve the differential equation :

$(x^2+3xy+y^2)dx-x^2dy=0$ given that $y=0$ , when $x=1$

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Question 20)

Show that the lines $\frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $\frac{x-4}{2}=\frac{y}{0}=\frac{z+1}{3}$ intersect. Find their point of intersection.

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Question 21)

A committee of 4 students is selected at random from a group consisting of 7 boys and 4 girls. Find the probablity that there are exactly 2 boys in the committee, given that at least one girl must be there in the committee.

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Question 22)

A random variable X has the following probablity distribution :

X	0	1	2	3	4	5	6
P(X)	C	2C	2C	3C	$C^2$	$2C^2$	$7C^2+C$

Find the value of C and also calculate mean of the distribution.

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Question 23)

Show that the relation R defined by $(a,b)R(c,d)\Rightarrow a+d=b+c$ on $A\times A$ where $A=\left \{ 1,2,3...10 \right \}$ is an equivalence relation. Hence write the equivalence [(3,4)]; $a,b,c,d\in A.$

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Question 24)

Solve for x :

$\begin{vmatrix} a+x &a-x &a-x \\ a-x &a+x &a-x \\ a-x &a-x &a+x \end{vmatrix}=0$ , using properties of determinants.

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Question 25)

Using elementary row operations find the inverse of matrix $A=\begin{pmatrix} 3 &-3 &4 \\ 2 &-3 &4 \\ 0 &-1 &1 \end{pmatrix}$ and hence solve the following system of equations $3x-3y+4z=21,2x-3y+4z=20,-y+z=5$

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Question 26)

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle $\alpha$ is one-third that of the cone and the greatest volume of the cylinder is $\frac{4}{27}\pi h^3\tan^2\alpha$ .

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Question 27)

Find the intervals in which the function $f(x)=\frac{4\sin x}{2+\cos x}-x;0\leq x\leq 2\pi$ is strictly increasing or decreasing.

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Question 28)

Using integration, find the area of the triangle formed by negative x- axis and tangent and normal to the circle $x^2+y^2=9$ at $(-1,2\sqrt{2})$

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Question 29)

Find the coordinates of the foot of perpendicular and perpendicular distance from the point $P(4,3,2)$ to the plane $x+2y+3z=2$ . Also find the image of P in the plane.

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Question 30)

A,B and C throw a pair of dice in that order alternately till one of them gets a total of 9 and wins the game. Find their respective probabilities of winning, if A starts first.

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Question 31)

A company manufactures two types of cardigans : type A and type B. It costs Rs 360 to make type A cardigan and Rs 120 to make type B cardigan. The company can make at most 300 cardigans and spend at most Rs 72,000 a day. The number of cardigans of type B cannot exceed the number of cardigans of type A by more than 200. The company makes a profit of Rs 100 for each cardigan of type A and Rs 50 for every cardigan of type B.

Formulate this problem as a linear programming problem to maximize the profit to the company. Solve it graphically and find the maximum profit.

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Question 32)

Find the angles between the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ if $\vec{a}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{b}=3\hat{i}+\hat{j}-2\hat{k}$ , and hence find a vector perpendicular to both $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ .

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CBSE 12 Class 2016 Paper: 065 set 1 C

Question 1)
If $x\, \varepsilon\, N\: and\: \begin{vmatrix} x+3 &-2 \\ -3x&2x \end{vmatrix}= 8$ , then find the value of x.

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Question 2)

Use elementary column operation $C_{2}\rightarrow C_{2}+2C_{1}$ in the following matrix equation:
$\begin{pmatrix} 2 &1 \\ 2& 0 \end{pmatrix}= \begin{pmatrix} 3 &1 \\ 2& 0 \end{pmatrix}\begin{pmatrix} 1 &0 \\ -1& 1 \end{pmatrix}$

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Question 3)

Write the number of all possible matrices of order $2\times 2$ with each entry 1,2 or 3.

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Question 4)

Write the operation vector of the point which divides the join of points with position vectors $3\vec{a}-2\vec{b}$ and $2\vec{a}+3\vec{b}$ in the ratio 2:1.

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Question 5)

Write the number of vectors of unit length perpendicular to both the vectors $\vec{a}= 2\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}= \hat{j}+\hat{k}$ .

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Question 6)

Find the vector equation of the plane with intercepts 3,-4 and 2 on x,y and z-axis respectively.

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Question 7)

Find the coordinates of the point where the line through the points A(3,4,1) and B(5,1,6) crosses the XZ plane, Also find the angle which this line makes with the XZ plane.

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Question 8)

The two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}-5\hat{k}$ and $2\hat{i}+2\hat{j}+3\hat{k}\cdot$ Find the two unit vectors parallel to its diagonals.Using the diagonal vectors, find the area of the parallelogram.

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Question 9)

In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown, The man decided to throw a die thrice but to quit as and he gets a number greater than 4. Find the expected value of the amount he Wins/loses.

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Question 10)

A bag contain 4 balls. Two balls are drawn at random (without replacement) and are found to be white. What is the probability that all balls in the bag are white?

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Question 11)

Differentiate $x^{\sin x}+\left ( \sin x \right )^{\cos x}$ with respect to x.

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Question 12)

If $y= 2\cos \left ( \log x \right )+3\sin \left ( \log x \right ),$ prove that $x^{2}\frac{d^{2}y}{dx^{2}}+x\frac{dy}{dx}+y= 0\cdot$

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Question 13)

if $x= a\sin 2t\left ( 1+\cos 2t \right )\: and\: y= b\cos 2t\left ( 1-\cos 2t \right ),$ find $\frac{dy}{dx}\, \ at \ \, t= \frac{\pi }{4}\cdot$

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Question 14)

The equation of tangent at (2,3) on the curve $y^{2}= ax^{3}+b$ is $y= 4x-5.$ Find the values of a and b.

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Question 15)

find: $\int \frac{x^{2}}{x^{4}+x^{2}-2}dx$

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Question 16)

Evaluate: $\int_{0}^{\frac{\pi }{2}}\frac{\sin ^{2}x}{\sin x+\cos x}dx$

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Question 17)

Evaluate:
$\int_{0}^{\frac{3}{2}}|x\cos \pi x|dx$

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Question 18)

Find: $\int \left ( 3x+1 \right )\sqrt{4-3x-2x^{2}}dx$

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Question 19)

Solve the differential equation:
$y+x\frac{dy}{dx}= x-y\frac{dy}{dx}$

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Question 20)

From the differential equation of the family of circles in the second quadrant and touching the coordinates axes.

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Question 21)

Solve the equation for $x:\sin^{-1}x+\sin^{-1}\left ( 1-x \right )= \cos^{-1}x$

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Question 22)

If $\cos^{-1}\frac{x}{a}+\cos^{-1}\frac{y}{b}= \alpha , \ prove\ \, that \ \frac{x^{2}}{a^{2}}-2\frac{xy}{ab}\cos \alpha +\frac{y^{2}}{b^{2}}= \sin ^{2}\alpha$

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Question 23)

A trust invested some money in two type of bonds. The first bond pays 10% interest and second bond pays 12% interest. The trust received Rs 2,800 as interest. However, if trust had interchanged money in bonds, they would have got Rs 100 less as interest. using matrix method, find the amount invested by the trust. Interest received on this amount will be given to Helpage India as donation. Which value is reflected in this question ?

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Question 24)

There are two types of fertiliser 'A' and 'B'. 'A' consist of 12% nitrogen and 5% phosphoric acid whereas 'B' consists of 4% nitrogen and 5% phosphoric acid .After testing the soil conditions, farmer finds that he needs at least 12 kg of nitrogen and 12 kg of phosphoric acid for his crops. If 'A' costs Rs 10 per kg and 'B' cost Rs 8 per kg, then graphically determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost.

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Question 25)

Five had oranges are accidently mixed with 20 good ones. if four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.

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Question 26)

find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i}+3\hat{j}+4\hat{k}$ to the plane $\vec{r}\cdot \left ( 2\hat{i}+\hat{j}+3\hat{k} \right )-26= 0.$ Also find image of P in the plane.

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Question 27)

Show that the binary operation $\ast$ on $A= \mathbb{R}-\left \{ -1 \right \}$ defined as $a\ast b= a+b+ab$ for all $a,b\, \epsilon\, A$ is commutative and associative on A, Also find the identity element of $\ast$ in A and prove that every element of A is invertible.

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Question 28)

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is $6\sqrt{3}r\cdot$

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Question 29)

If the sum of lengths of hypotenuse and a side of a right angled triangle is given, show that area of triangle is maximum, when the angle between them is $\frac{\pi }{3}\cdot$

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Question 30)

Prove that the curves $y^{2}= 4x$ and $x^{2}= 4y$ divide the area of square bounded by $x= 0,x= 4,y= 4$ and $y= 0$ into three equal parts.

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Question 31)

Using properties of determinants, show that $\Delta ABC$ is isosceles if:
$\begin{vmatrix} 1 & 1 & 1\\ 1+\cos A & 1+\cos B &1+\cos C \\ \cos^{2}A+\cos A&\cos^{2}B+\cos B & \cos^{2}C+\cos C \end{vmatrix}= 0$

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Question 32)

A shopkeeper has 3 varieties of pens 'A', 'B' and 'C'. Meenu purchased 1 pen of each variety for a total of Rs 21. Jeevan purchased 4 pens of 'A' variety, 3 pens of 'B' variety and 2 pens of 'C' variety for Rs 60. While Shikha purchased 6 pens of 'A' variety. 2 pens of 'B' variety and 3 pens of 'C' variety for Rs 70. Using matrix method, find cost of each variety of pen.

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CBSE 12 Maths question paper 2016 pattern

CBSE previous year question papers Class 12 Maths 2016 contains 26 questions divided into three sections A, B, and C. All questions are compulsory where no overall choice has been given.

Section A of CBSE class 12 maths question paper 2016 comprises six questions of one mark each where you need to answer in one word, one sentence or as per the exact requirement of the question. The internal choice has been provided in one question of this section.
CBSE 12 Maths question paper 2016 section B comprises 13 questions of four marks each where you need to provide stepwise solutions for each question. The internal choice has been provided in the four questions of this section.
Section D of CBSE class 12 maths question paper 2016 comprises six long answer type questions of six marks each where you need to write a step-by-step complete solution for each question. The internal choice has been provided in two questions of this section and you have to attempt only one of the alternatives in all such questions.

The marks distribution, number, and type of question:-

Section	Types of Questions	No. of Questions	Marks per Question	Total marks
A	Very Short Answer Type Questions	6	1	6
B	Short Answer Type Question - II	13	4	52
C	Long Answer Type Questions	7	6	42
Total		26		100

Analysis of CBSE Class 12 Maths Question Paper 2016 with Solutions

CBSE class 12 maths question paper 2016 was tough and lengthy. Although the question paper was NCERT based, some questions were from the outside of the NECRT. As students are prepared for NCERT, they found it difficult to solve questions, seeing them for the first time. Students who had conceptual clarity were able to solve this paper. As new questions take time to think and solve, some students weren’t able to solve it on time and found it lengthy. Overall the question paper was tough and lengthy as compared to last year’s paper. The chapter-wise and unit-wise marks distribution is given below.

CBSE 12 Maths Question Paper 2016 - Chapter-wise weightage

S. No	Unit	No. of Questions	Total Marks
1	Relation and Function	1	6
2	Inverse Trigonometric Functions	1	4
3	Matrices	3	6
4	Determinants	2	7
5	Continuity and Differentiability	2	8
6	Applications of Derivatives	2	10
7	Integrals	3	12
8	Applications of Integrals	1	6
9	Differential Equations	2	8
10	Vector Algebra	3	6
11	Three Dimensional Geometry	3	11
12	Linear Programming	1	6
13	Probability	2	10
Total		26	100

Unit wise weightage in terms of total marks:-

1	Relation and Functions	10
2	Algebra	13
3	Calculus	44
4	Vector and 3-D Geometry	17
5	Linear Programming	6
6	Probability	10
	Total	100

The unit marks distribution is more important as CBSE follows this weightage system to prepare the CBSE question paper.

Conclusion of CBSE Class 12 Maths Question Paper 2016 with Solutions

The CBSE class 12 maths 2016 question paper was based on the NCERT topics. Students who had the conceptual clarity won’t find it difficult and who have prepared for NCERT were found it difficult to solve. It is advised to solve more questions from different NCERT based sources and previous year papers. It will give you conceptual clarity and you will be able to solve the final paper on time.

Previous Year Question Paper

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Questions related to CBSE Class 12th

Showing 85 out of 85 Questions

28 Views

I m in 12th PCM (CBSE) I have not given the pre board exam, and my school is threatening me that they will not give me those marks which are going from school. But i think the marks are given by the external examiner only on the basis of practicals which carry 30 marks.Am I correct?

anirudha vats 23rd Feb, 2021

Hey,

In every school pre boards are a mandatory thing to give before your boards. It is a kind of sample test to check your preparation before your actual board exams.. Some schools also have rules for student to pass in the board exam in order to get the Admit Card.

Yes the marks are given by the external. But it's no loss if you talk to the school authorities calmly and come up with a solution to cope with problem.

All the very best for your Board exams.

Thank you.

17 Views

I am in 12th PCM from CBSE board, and I am not giving the pre boards which have been started from tofay in my school because my JEE is on 25th. And they are saying that, they will not send those 20 (each subject) marks which will be going from school. What should I do.

Ayush 22nd Feb, 2021

Hello candidate,

If your JEE Mains examination is on 25th, definitely go and attend examination with full preparation.

Now, regarding the pre Board examinations in your school you can tell them in advance and ask for their permission and the School authority cannot restrict you from attending any examination.

Hope that this information was helpful for you.

All the best for your examinations!!

14 Views

I passed I 11 in 2020 and want to take admission in class 12 as a regular candidate and Im from CBSE... can I take admission AS a regular candidate??

Soham sengupta 21st Feb, 2021

Hello student

Your admission in 12th class in other school as a regular candidate will be possible if your parents transfer from one City to another city due to service or business. Your health problem is bound to change your city for you have to get regular treatment for unavailable reasons of ailment.

Your admission will be provincial and will be subject to the approval of CBSE board.

For this reason you have to submit a written affidavit on a 20 rupees stamp paper by your parents showing all the valid reasons for it. You have to submit documents like 11th result card ,11th registration card , document of your parents transfer due to service transfer or business transfer.

Hope this will be helpful.

53 Views

hii, i have passed cbse class 12th examination in 2020 with 49%(pcb) .want to give neet exam,so i have applied for improvement in 2021 for chemestry and biology if my improved no. will less than previous one, then which will be valid. In case new one is less than previous one then what will happen?

Suchayita Saha 20th Feb, 2021

Hello,

Dont think too much dear. Improvement number is always higher than your previous year number. So just don't think negatively. You just focused on your study. And give the improvement examination. You will be passed with well number. You can get above 55% markd minimum. Thats your target and you will achieve it. Just prepare yourself.

All the best.

22 Views

I want to know about distribution of those 20 marks which are given on the basis of Practicals and all in CBSE class 12th PCM

Guljabin Quadri 18th Feb, 2021

Hello,

The partial questions for CBSE class 12th PCM is consists of :-

Very Short Answer (VSA) – 1 marks (No. of question 2)
Short Answer (SA) – 2 marks (No. of question 2)
Long Answer – I (LA-I) – 4 marks (No. of question 2)
Long Answer – II (LA-II) – 6 marks (No. of question 1)

Which ,makes the total mark of 20(marks).

View All Questions

Related E-books and Sample Papers

Central Board of Secondary Education Class 12th Examination (CBSE Class 12th)

Chapter 1	NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
Chapter 2	NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions
Chapter 3	NCERT solutions for class 12 maths chapter 3 Matrices
Chapter 4	NCERT solutions for class 12 maths chapter 4 Determinants
Chapter 5	NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability
Chapter 6	NCERT solutions for class 12 maths chapter 6 Application of Derivatives
Chapter 7	NCERT solutions for class 12 maths chapter 7 Integrals
Chapter 8	NCERT solutions for class 12 maths chapter 8 Application of Integrals
Chapter 9	NCERT solutions for class 12 maths chapter 9 Differential Equations
Chapter 10	NCERT solutions for class 12 maths chapter 10 Vector Algebra
Chapter 11	NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry
Chapter 12	NCERT solutions for class 12 maths chapter 12 Linear Programming
Chapter 13	NCERT solutions for class 12 maths chapter 13 Probability

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CBSE Class 12 Maths Question Paper 2016 with Solutions

CBSE Class 12 Maths Question Paper 2016 with Solutions Set-wise

CBSE 12 Class 2016 Paper: 065 set 1 S

Question 1)If vectors and are such that , and , then find .View Solutions

CBSE 12 Class 2016 Paper: 065 set 1 N

CBSE 12 Class 2016 Paper: 065 set 1 E

CBSE 12 Class 2016 Paper: 065 set 1 C

Question 1)If , then find the value of x.View Solutions

More About CBSE Class 12 Maths question paper 2016 with solutions

CBSE 12 Maths question paper 2016 pattern

Analysis of CBSE Class 12 Maths Question Paper 2016 with Solutions

Latest Articles

Questions related to CBSE Class 12th

I m in 12th PCM (CBSE) I have not given the pre board exam, and my school is threatening me that they will not give me those marks which are going from school. But i think the marks are given by the external examiner only on the basis of practicals which carry 30 marks.Am I correct?

I am in 12th PCM from CBSE board, and I am not giving the pre boards which have been started from tofay in my school because my JEE is on 25th. And they are saying that, they will not send those 20 (each subject) marks which will be going from school. What should I do.

I passed I 11 in 2020 and want to take admission in class 12 as a regular candidate and Im from CBSE... can I take admission AS a regular candidate??

hii, i have passed cbse class 12th examination in 2020 with 49%(pcb) .want to give neet exam,so i have applied for improvement in 2021 for chemestry and biology if my improved no. will less than previous one, then which will be valid. In case new one is less than previous one then what will happen?

I want to know about distribution of those 20 marks which are given on the basis of Practicals and all in CBSE class 12th PCM

Related E-books and Sample Papers

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Question 1)
If vectors $\vec{\text{a}}$ and $\vec{\text{b}}$ are such that $|\vec{\text{a}}| = \frac{1}{2}$ , $|\vec{\text{b}}| = \frac{4}{\sqrt{3}}$ and $|\vec{\text{a}}\times \vec{\text b}| = \frac{1}{\sqrt{3}}$ , then find $| \vec{\text{a}} \cdot \vec{\text b} |$ .

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Question 1)
If $x\, \varepsilon\, N\: and\: \begin{vmatrix} x+3 &-2 \\ -3x&2x \end{vmatrix}= 8$ , then find the value of x.

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