Recent questions in Model Papers

Evaluate : $\int e^{2-5x}\: dx.$

asked Jan 31, 2013 by thanvigandhi_1

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Without expanding, show that : $\begin{vmatrix} 42 & 1 & 6 \\ 28 & 7 & 4 \\ 14 & 3 & 2 \end{vmatrix} = 0$

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Show that the vectors $\hat i + \hat j + \hat k$ is equally inclined to the axes.

asked Jan 31, 2013 by thanvigandhi_1

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If vectors $2\hat i - 5\hat j + b\hat k \: and \: 2\hat i + a\hat j+4\hat k$ are parallel, find the values of a and b.

asked Jan 31, 2013 by thanvigandhi_1

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A line makes an angles $30^{\circ}, 90^{\circ}, 60^{\circ}$ with co-ordinate axes. Find its direction cosines.

asked Jan 31, 2013 by thanvigandhi_1

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Given relation R in the set Z of all integers is given by $R = {(a,b) \in Z * Z\: a-b \: is\: an\: integer }$ . Find whether R is an equivalence relation or not.

asked Jan 31, 2013 by thanvigandhi_1

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Let * be binary operation of N x N defined by (a,b)(c,d)=(a+c, b+d). Show that is commutative as well as associative. Find the identity element for the operation * on N x N, if any.

asked Jan 31, 2013 by thanvigandhi_1

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Solve for x : $2\: tan^{-1}(\cos x)=tan^{-1}(2 cosec x)$

asked Jan 31, 2013 by thanvigandhi_1

1 answer

There are two families P and Q. There are 3 men, 3 women and 12 children in family P and 2 men, 2 women and 4 children in family Q. The recommended daily allowance for calories is : Man : 2400, woman : 2000, child: 1400 and man : 60 g for proteins is : woman: 40 g and child: 35 g. (i) Represent the above information by matrices. Using matrix multiplication, calculate the total requirement of calories and proteins of each of the two families.

asked Jan 31, 2013 by thanvigandhi_1

1 answer

If the function $f(x) = \left\{ \begin{array}{l l}3ax+b, & \quad if\: { x > 1} \\ 11, & \quad if \: { x = 1} \\ 5ax-2b, & \quad if \: { x < 1} \end{array} \right.$ is continuous at x = 1, find the values of a and b.

asked Jan 31, 2013 by thanvigandhi_1

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Find $\large\frac{dy}{dx}, \: if \: x^y+y^x=a^b.$

asked Jan 31, 2013 by thanvigandhi_1

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Show that equation of the tangent to the ellipse $\large\frac{x^2}{a^2}+\frac{y^2}{b^2}$ $=1$ at $(x_0,y_0)$ is $\large\frac{xx_0}{a^2}+\frac{yy_0}{b^2}$ $=1$

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $\int \large\frac{5x+3}{\sqrt{x^2+4x+16}}$ $dx.$

asked Jan 31, 2013 by thanvigandhi_1

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Evaluate : $\int \large\frac{x^2tan^{-1}x}{1+x^2}$ $dx$

asked Jan 31, 2013 by thanvigandhi_1

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Prove that an angle in a semi-circle is a right angle (using vector only).

asked Jan 31, 2013 by thanvigandhi_1

1 answer

Find p, when the scalar projection of $\overrightarrow a = p\hat i + \hat j + 4\hat k \: on \: \overrightarrow b = 2\hat i + 6\hat j + 3\hat k$ is 4 units.

asked Jan 31, 2013 by thanvigandhi_1

0 answers

Show that the plane through (1, 1, 1), (1, -1, 1) and (-7, 3, 5) is perpendicular to XZ-plane.

asked Jan 31, 2013 by thanvigandhi_1

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A card from the pack of 52 cards is lost from the remaining cards of the pack, two cards are drawn and are found to be both diamonds. (i) Find the probability of lost card being a diamond.(ii)While playing in the ground you found a purse containing rupees and some documents,what will you do?

asked Jan 31, 2013 by thanvigandhi_1

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Find the equation of a curve passing through the point (0,2) given that the sum of the coordinate of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.

asked Jan 31, 2013 by thanvigandhi_1

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Find the particular solution of the following differential equation satisfying the given condition : $\bigg[ x\: sin^2 \bigg(\large\frac{y}{x} \bigg)$ $- y \bigg] dx+x\: dy=0; y(1)=\large\frac{\pi}{4}$

asked Jan 31, 2013 by thanvigandhi_1

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Find the differential equation of all the circles in first quadrant which touch the coordinate axes.

asked Jan 31, 2013 by thanvigandhi_1

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There are three coins. One is a two headed coin (having head on both faces), another is a biased coin that comes up heads 75% of the time and third is an unbiased coin. One of the three coins is chosen at random and tossed, it shows head, what is the probability that it was the two headed coin?

asked Jan 31, 2013 by thanvigandhi_1

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Suppose x has binomial distribution $B \bigg( 4, \large\frac{1}{3} \bigg)$ . Find the most likely outcome.

asked Jan 30, 2013 by thanvigandhi_1

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Find the equation of the plane passing through the intersection of planes $4x-y+z=10\: and \: x+y-z=4$ and parallel to the line with direction ratios < 2, 1, 1>. Find also the perpendicular distance of (1, 1, 1) from the plane.

asked Jan 30, 2013 by thanvigandhi_1

1 answer

Prove that the curve $y^2=4x$ and $x^2=4y$ divide the area of the square bounded by $x=0,x=4,y=4\: and \: y=0$ into three equation.

asked Jan 30, 2013 by thanvigandhi_1

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Find the area of the region bounded by $\{ (x,y):x^2+y^2 \leq 1 \leq x + \frac{y}{2} \}.$

asked Jan 30, 2013 by thanvigandhi_1

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Evaluate : $\int \large\frac{\tan \theta+\tan^3\theta}{1+\tan^3\theta}$ $d\theta.$

asked Jan 30, 2013 by thanvigandhi_1

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If $A^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ -15 & 6 & -5 \\ 5 & -2 & 2 \end{bmatrix}\: and \: B = \begin{bmatrix} 1 & 2 & -2 \\ -1 & 3 & 0 \\ 0 & -2 & 1 \end{bmatrix},$ find $(AB)^{-1}$

asked Jan 30, 2013 by thanvigandhi_1

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A manufacturer can sell petrol operated x items at a price of Rs. $\bigg( 5-\large\frac{x}{100} \bigg)$ each. The cost price of petrol operated x items is Rs. $\bigg( \large\frac{x}{5}+$ $500 \bigg).$ Find the number of items he should sell to earn maximum profit.

asked Jan 30, 2013 by thanvigandhi_1

0 answers

Yogesh Dhir wants to invest atmost Rs.12000 in saving certificates and national saving bonds. According to rules, he has to invest atleast Rs.2000 in saving certificates and atleast Rs.4000 in national saving bonds. If the rate of interest on saving certificate is 8% per annum and the rate of interest on national saving bonds is 10% per annum, how much money should he invest to earn maximum yearly income? Find also his maximum yearly income.

asked Jan 30, 2013 by thanvigandhi_1

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If $f(x) =\large \frac{4x+3}{6x-4}$ $, x \neq \large\frac{2}{3},$ show that 'fof' is an identity function, for all $x \neq \large\frac{2}{3} .$

asked Jan 30, 2013 by thanvigandhi_1

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Evaluate : $\sin \bigg[ \large\frac{\pi}{3}$ $-\sin^{-1} \bigg( -\large\frac{1}{2} \bigg) \bigg].$

asked Jan 30, 2013 by thanvigandhi_1

1 answer

A matrix A of order 3 x 3 has determinant 4. What is the value of |4A|?

asked Jan 30, 2013 by thanvigandhi_1

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For what value of x, the following matrix is singular? $\begin{bmatrix} -1 & 2 \\ 4 & x \end{bmatrix}$

asked Jan 30, 2013 by thanvigandhi_1

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At what point on the curve $y=x^2$ does the tangent make an angle of $45^{\circ}$ with the axis.

asked Jan 30, 2013 by thanvigandhi_1

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Show that $\overrightarrow a = 3\hat i + 2\hat j - 4\hat k$ and $\overrightarrow b = -6\hat i - 4\hat j + 8\hat k$ are unlike vectors.

asked Jan 30, 2013 by thanvigandhi_1

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Find the direction ratio of a line parallel to the line joining the points (2,5,-3) and (3,-2,0).

asked Jan 30, 2013 by thanvigandhi_1

0 answers

Evaluate : $\int \large\frac{dx}{5+9x^2}$

asked Jan 30, 2013 by thanvigandhi_1

0 answers

Find the real value $'\lambda'$ such that $|\lambda \overrightarrow a | = 5, \overrightarrow a$ being a non-zero vector.

asked Jan 30, 2013 by thanvigandhi_1

0 answers

If A and B are skew-symmetric matrices, then prove that ABA is skew-symmetric.

asked Jan 30, 2013 by thanvigandhi_1

0 answers

Show that the function $f : Z \rightarrow Z$ defined by $f(x)=x^2$ is neither one-one nor onto.

asked Jan 30, 2013 by thanvigandhi_1

1 answer

Let L be the set of all lines in a plane and R be thr relation in L defined as $\{(L_1, L_2):L_1$ is perpendicular to $L_2 \}.$ Show that R is symmetric but neither reflexive nor transitive.

asked Jan 30, 2013 by thanvigandhi_1

0 answers

Prove that $2\tan^{-1}\large\frac{1}{2}$ $+ \tan^{-1}\large\frac{1}{7}$ $= \tan^{-1}\large\frac{31}{17}$

asked Jan 30, 2013 by thanvigandhi_1

0 answers

Using properties of determinants, prove that $\begin{bmatrix} 2ab & a^2 & b^2 \\ a^2 & b^2 & 2ab \\ b^2 & 2ab & a^2 \end{bmatrix} = -(a^3+b^3)^2$

asked Jan 30, 2013 by thanvigandhi_1

1 answer

Using elementary transformation, find the inverse of the matrix $\begin{bmatrix} 3 & 10 \\ 2 & 7 \end{bmatrix}$

asked Jan 30, 2013 by thanvigandhi_1

1 answer

Evaluate: $\lim\limits_{y \to 0}\large\frac{(x+y)\sec(x+y)-x\sec{x}}{y}$

asked Jan 30, 2013 by fingeazy

0 answers

Differentiate $\sin \sqrt{x}$ w.r.t. x from first principles

asked Jan 30, 2013 by fingeazy

0 answers

Show that the function $f(x) = \left\{ \begin{array}{l l}\large\frac{x}{|x|+2x^2}, & \quad { x \neq 0} \\ \lambda, & \quad { x = 0} \end{array} \right.$ remains discontinuous at x = 0 regardless of the choice of $\lambda$ .

asked Jan 30, 2013 by thanvigandhi_1

0 answers

If x = $a \begin{pmatrix} \large\frac{1 + t^2}{1 - t^2}\end{pmatrix}$ and y = $\large\frac{2t}{1-t^2}$ find $\large\frac{dy}{dx}$

asked Jan 30, 2013 by fingeazy

0 answers

If $y = \large\frac{2}{\sqrt{a^2-b^2}}$ $\tan^{-1} \bigg( \sqrt{\frac{a-b}{a+b}} \tan\large\frac{x}{2} \bigg).$ prove that $\large\frac{dy}{dx} =\large \frac{1}{(a+bcos \: x)}.$

asked Jan 30, 2013 by thanvigandhi_1

0 answers

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