Maths

In any reaction, catalyst (A) retards the rate of reaction (B) increases the rate of reaction (C) both increases and decreases the rate of reaction (D) none of these

Answer: Increases the rate of reaction Explanation: Most catalysts work by lowering the 'activation energy' of a reaction. This allows less energy to be used, thus speeding up the reaction. The...

Solve the following linear programming problem graphically: Maximise subject to the constraints:

The shaded region in fig. is the feasible region determined by the system of constraints (2) to (4). We observe that the feasible region $\mathrm{OABC}$ is bounded. So,we now use Corner Point Method...

If defined by , if if then is

$$\begin{array}{l} f(x)=\left\{\begin{array}{cl} x & x>2 \\ 5 x-2 & x \leqslant 2 \end{array}\right. \\ \& f^{\prime}(x)=\left\{\begin{array}{ll} 1 & x>2 \\ 5 & x \leqslant 2 \end{array}\right.... read more$$ \begin{array}{l} \text { We have, }\left(\tan ^{-1} x\right)^{2}+\left(\cot ^{-1} x\right)^{2}=\frac{5 \pi^{2}}{8} \\ \Rightarrow\left(\tan ^{-1} x+\cot ^{-1} x\right)^{2}-2 \tan ^{-1} x \cdot...

The vertices of a triangle are and . Find the length of the median through the vertex

Coordinates of $\mathrm{A} \equiv(-1,3)$ Coordinates of $\mathrm{B} \equiv(1,-1)$ Coordinates of $\mathrm{C} \equiv(5,1)$ The median through the vertex $\mathrm{C}$ will meet at the mid point of...

Evaluate A (B) c D

Correct ogtion is $$\text { a } e^{0}\left(\log x-\frac{1}{x}\right) \div e$$ $\int e^{c}\left(\log x+\frac{1}{x^{2}}\right) d x$ $\int e^{\prime} \log x d x+\int e^{\prime} \frac{1}{x^{2}} d x$...

Find the equation of the plane which cuts intercepts 2,3 and 4 on the and z-axes respectively.

The equation of plane with intercepts a, b, c is given as $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1$ $\Rightarrow$ The equation of plane with intercepts $2,3,4$ is...

Find the angle between the vectors and

$$\begin{array}{l} \overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}} \quad \text { and } \quad... read more$$ \begin{array}{l} 2 \tan ^{-1} \frac{1}{5}+\sec ^{-1} \frac{5 \sqrt{2}}{7}+2 \tan ^{-1} \frac{1}{8} \\ =2\left[\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{8}\right]+\sec ^{-1} \frac{5 \sqrt{2}}{7}...

Find the rate of change or the area of a circle with respect to its radius when .

Area of circle $=\pi r^{2}$ $$\mathrm{A}=\pi \mathrm{r}^{2}$$ differentiating with respect to $\mathrm{r}$ we get $$\frac{\mathrm{dA}}{\mathrm{dr}}=2 \pi... read more$$ \begin{array}{l} \mathrm{I}=\int \csc x \mathrm{dx} \\ \Rightarrow \mathrm{I}=\int \frac{\csc x(\csc x-\cot x)}{(\csc x-\cot x)} \mathrm{dx} \\ \text { Put } \csc x-\cot x=t \\...

Since A is an invertible matrix, As matrix $A$ is of order 2, let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$ Then, $|A|=a d-b c$ and $a d y=\left[\begin{array}{cc}d & -b \\ -c &... read more ? (A) (B) (C) (D) Correct option is (B)$\frac{1}{3} \tan (3 x+5)+c$$$\begin{array}{l} \int \sec ^{2}(3 x+5) d x \\ \text { let } 3 x+5=t \\ 3 d x=d t \\ d x=1 / 3 d t \\ \int \sec ^{2} t \cdot \frac{1}{3} \cdot d... read more The value of is (A) 1 (B) (C) 2 (D) 3 The answer is (A) 1 =i(\hat{\imath})+\hat{\jmath} \cdot(-\hat{\jmath})+\hat{f} \cdot(\hat{k}) =1-1+1 =1 read more Find the direction cosines and direction ratios of the line joining the points Given the points A (1,3,5) \& B(-1,0,-1) Direction ratios are \mathrm{a}=-1-1=-2$$ \begin{array}{l} \mathrm{b}=0-3=-3 \\ \mathrm{c}=-1-5=-6 \end{array} $$Now,... read more Let and be two real functions. Find g)(x),(f g)(x),\left(\frac{f}{g}\right)(x)$$ (f+g)(x)=f(x)+g(x)  =x^{2}+2 x+1  (f-g)(x)=f(x)-g(x)  =x^{2}-(2 x+1)  =x^{2}-2 x-1  (f g)(x)=f(x) \cdot g(x)  =x^{2}(2 x+1)  =2 x^{3}+x^{2} ... read more ? (A) (B) (C) (D) The answer is (A)$-\frac{1}{3} \cos x^{3}+k$. Let$u=x^{3}$so that$d u=3 x^{2} d x$Out integral becomes: $$\begin{array}{c} \frac{1}{3} \int\left(\sin x^{3}\right) 3 x^{2} d x=\frac{1}{3} \int... read more If the tangent to the curve, at the point is perpendicular to the line, , then which one of the following points lies on the curve? A B C D Correct option is B (2,-2) y=x^{2}+a x-b (1,-5) lies on the curve \Rightarrow-5=1+\mathrm{a}-\mathrm{b} \Rightarrow \mathrm{a}-\mathrm{b}=-6.(i) \mathrm{AlsO}, y^{\prime}=3 x^{2}+a... read more If then is (A) 0 (B) (C) 1 (D) not defined Correct option is D Not defined It is given that, P(A)=\frac{1}{2} and P(B)=0$$ P(A \mid B)=\frac{P(A \cap B)}{P(B)}=\frac{P(A \cap B)}{0} $$Therefore, P(A \mid B) is not defined. Thus, the... read more The number of all possible matrices of order with each entry 0 or 1 is. A. 27 B. 18 C. 81 D. 512 A matrix of order 3 \times 3 has 9 elements. Now each element can be 0 or 1 . \therefore 9 places can be filled up in 2^{9} ways required number of matrices =2^{9}$$ \begin{array}{l} =2... read more ? (A) (B) (C) (D) The correct option is (A)$x+k\begin{aligned} & \frac{(\sin x+\cos x) \cdot d x}{\sqrt{1+\sin 2 x}} \\ =& \frac{(\sin x+\cos x) \cdot d x}{\sqrt{(\sin x+\cos x)^{2}}} \\ =& \frac{(\sin x+\cos... read more The integrating factor of the differential equation is Given x \cdot \frac{d y}{d x}+2 y=x^{2} \frac{\mathrm{dy}}{\mathrm{dx}}+\frac{2}{\mathrm{x}} \cdot \mathrm{y}=\mathrm{x} $$This is in the form of \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{P}... read more is equal to (A) (B) (C) 0 (D) The correct answer is (A) -\frac{\pi}{2}.$$ \tan ^{-1} \sqrt{3}-\cot (-\sqrt{3}) $$\because \tan \frac{\pi}{3}=\sqrt{3} and \cot \left(\pi-\frac{\pi}{6}\right)=-\sqrt{3}$$ \tan ^{-1}... read more Find the minimum value of , subject to the constraints and , subject to the constraints andZ=3 x+5 y$, subject to the constraints$-2 x+y \leq 4, x+y \geq 3, x-2 y \leq 2, x \geq 0$and$y \geq 0$Draw the line$-2 x+y=4, x+y=3$and$x-2 y=2$and shaded region which is satisfied by above... read more ? (A) (B) (C) (D) Integrating by parts, $$\begin{array}{l} \mathrm{u}=\mathrm{x}^{2}, \mathrm{v}=\mathrm{e}^{\mathrm{x}} \\ \mathrm{u}^{\prime}=2 \mathrm{x}, \int \mathrm{vdx}=\mathrm{e}^{\mathrm{x}} \\... read more ? (A) (B) (C) (D) Answer is (B) \sec x \tan x$$ \begin{array}{l} \frac{\mathrm{d}}{\mathrm{dx}}(\sec \mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}\left\{\frac{1}{\cos \mathrm{x}}\right\} \\ =\frac{\cos \mathrm{x}... read more ? (A) (B) (C) (D) Correct option is (A)$x+k$Explanation: $$\int d x=\int 1 d x=\int x^{0} d x$$ Using the standard power rule of integration, $$\int x^{0} d x=\frac{x^{0+1}}{0+1}+C=x+k$$ read more ? (A) (B) (C) (D) Correct option is (A)$\tan (\log x)+k$Let$I=\int \frac{\sec ^{2}(\log x)}{x} d x$Put$\log x=t \Rightarrow \frac{1}{x} d x=d t$$$\therefore \int \sec ^{2} t d t=\tan t+c=\tan (\log x)+c$$ read more If are in A.P, then the determinant is A 0 B 1 C D Correct option is (A)0 Given$\mathrm{a}, \mathrm{b}, \mathrm{c}$are in A.P. $$\Rightarrow 2 \mathrm{~b}=\mathrm{a}+\mathrm{c}$$ $$\left|\begin{array}{lll} \mathrm{x}+2 & \mathrm{x}+3 &... read more ? (A) (B) (C) (D) Correct option is A \int \tan x=\int(\sin x / \cos x) \cdot d x This can be rewritten as \int 1 / \cos x \cdot \sin x \cdot d x Let us find the indefinite integral of \tan x using the... read more ? (A) 0 (B) 1 (C) (D) Correct option is (A) 0 Sol:- The given relation should be$$ \begin{array}{l} \cot \left(\tan ^{-1} x+\cot ^{-1} x\right) \\ \text { we know } \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2} \\ \therefore... read more The height of tower at a certain place is . The maximum distance up to which its programme can be received is (a) (b) 245km (c) (iv) Answer: Option (c)$\mathbf{5 6} \mathbf{~ k m}$Answer: Option (b)245km read more The Boolean expression for NAND gate is (a) (b) (c) Answer: Option (d) read more The value of angle of dip at the earth’s magnetic pole is (a) (b) (c) (d) Answer: Option (c)$90^{\circ}$read more The stored energy, of a capacitor charged to , is 1 J. capacitor is (a) (b) (c) (d) Answer: Option (b)$2 \times 10^{4} \mathrm{~F}$read more The current flowing; in a wire is 1 A. if the charge of an electron is , then the number of electrons flowing through the wire per second is (a) (b) (c) (d) Answer: Option (b)$6.25 \times 10^{18}$read more The energy of a photon of wavelength is (a) (b) (c) (d) Answer: Option (b)$\frac{h c}{\lambda}$read more The ratio of root mean (rms) value and peak value of alternative current is (a) (b) (c) (d) Answer: Option (b)$\frac{1}{\sqrt{2}}$read more At constant potential difference, the resistance of any electric circuit is halved, the value of heat produced will be (a) Half (b) Double (c) four (d) Same Answer: Option (b) Double read more In an alternating current circuit, the phase difference between curren I and voltage , then the Wattles component of current will be (a) (b) Double (c) four (d) Same Answer: Option (c)four read more The working of dynamo is based on the principle of (a) Heating effect of current (b) Electro-magnetic induction (c) Induced magnetism (d) Induced electricity Answer: Option (b)Electro-magnetic induction read more The magnifying power of an astronomical telescope for normal adjustment is (a) (b) (c) (d) Answer: Option (c)$-\frac{f_{e}}{f_{0}}$read more The resistance of ideal ammeter is (a) Zero (b) Very small (c) Very large (d) Infinite Answer: Option (a)Zero read more The direction of propagation of electromagnetic wave is (a) Parallel to (b) Parallel to (c) Parallel to (d) parallel to Answer: Option (d) parallel to$\vec{E} \times \vec{B}$read more A convex lens is immersed in water , then it will behave as a (a) Converging lens (b) Diverging lens (c) Prism (d) Concave mirror Answer: Option (b)Diverging lens read more The power factor of circuit is (a) R+WL (b) (c) (d) Answer: Option (b)$\frac{R}{\sqrt{R^{2}+W^{2} L^{2}}}$read more The electric potential in equatorial position of an electric dipole is (a) (b) (c) (d) Zero Answer: Option (d)Zero read more With the increase on temperature the resistivity of semiconductor (a) Increases (b) Decreases (c) Remains constant (d) Becomes zero Answer: Option (b) Decreases read more In a step-up transformer, no, of turns in primary and secondary coils are and then (a) (b) (c) (d) Answer: Option (b)$N_{2}>N_{1}$read more When two bulbs of power and are connected in series, then the power of their combination will be (a) (b) (c) (d) Answer: Option (a)$100 \mathrm{~W}$read more The wave front due to a point source at a finite distance from the source is (a) Spherical (b) Cylindrical (c) Plane (d) Circular Answer: Option (a)Spherical read more Light owes its colour to its (a) Frequency (b) Velocity (c) Phase (d) Amplitude Answer: Option (b)Velocity read more In earth’s magnetic field , if the frequency of oscillation of a magnetic needle is , then (a) (b) (c) (d) Answer: Option (b)$n^{2} \propto B_{H}$read more If charge is placed inside any spherical surface then total flux coming out from whole surface will be (a) (b) (c) (d) Answer: Option (b)$\frac{q}{\varepsilon_{0}}$read more Cylindrical lenses are used to correct the eye defect called (a) Myopia (b) Hypermetropia (c) Astigmatism (d) Presbyopia Answer: Option (c)Astigmatism read more Equivalent focal length of two lenses in contact having power -15D and +5D will be (a) (b) (c) (d) Answer: Option (b)$-10 \mathrm{~cm}$read more Power of an electric circuit is (a) V.R (b) (C) (d) Answer: Option (c)$\frac{V^{2}}{R}$read more The solar spectrum is (a) Continuous (b) Line spectrum (C) Spectrum of blank lines (d) Spectrum of blank bands Answer: Option (c)Spectrum of blank lines read more Two point charges of and are placed at a distance in air. Potential energy of the system will be (a) (b) (c) (d) Answer: Option (c)$-2.25 \mathrm{~J}$read more The time in which radioactive substance becomes half of its initial amount is called (a) Average life (b) Half-life (c) Time-period (d) Decay constant Answer: Option (b)Half-life read more Deviation of a thin prism of refractive index and angle of is (a) (b) (c) (d) Answer: Option (b)$(n-1) A$read more Unit of surface charge density is (a) Coulomb/metre (b) newton/metre (c) Coulomb/volt (d) Coulmb/metre Answer: Option (a)Coulomb/metre$^{2}$read more The susceptibility of paramagnetic substance is (a) Constant (b) Zero (b) Infinity (d) Depends on magnetic field Answer: Option (a) Constant read more The torque experienced by a current-loop of magnetic moment placed in magnetic field is (a) (b) (C) (d) Answer: Option (a)$\vec{\tau}=\vec{M} \times \vec{B}$read more Let be a non-singular matrix of the order then (a) (b) (c) (d) SOL: Correct option is C.$|A|^{2}$read more solution of is (a) (b) (c) (d) 1 SOL: Correct option is C. read more (a) (b) (c) (d) SOL: Correct option is B.$2 \log (1+\sqrt{x})+C$read more If and are two events such that and (a) (b) (c) (d) SOL: Correct option is B.$A \subset B$read more If and then (a) 2 (b) (c) (d) SOL: Correct option is (c)$\frac{2}{3}$read more If events and are mutually exclusive then: (a) (b) (c) (d) SOL: Correct option is B.$P(A \cap B)=0$read more If A’ and B’ are independent events then: (a) (b) 12 (c) (d) SOL: Correct option is C.$P\left(A^{\prime} B^{\prime}\right)=P\left(A^{\prime}\right) \cdot P\left(B^{\prime}\right)$read more The distance between and is: (a) 13 (b) 15 (c) 12 (d) 5 SOL: Correct option is A. 13 read more If two planes and are perpendicular to each other, then (a) (b) 2 (c) 3 (d) None of these SOL: Correct option is B.2 read more The equation of the -plane is: (a) (b) (c) (d) None of these SOL: Correct option is B.$z=0$read more The direction Cosines of axis are: (a) (b) (c) (d) None of these SOL: Correct option is B.$(0,1,0)$read more \vec{i} \times(\vec{i} \times \vec{j})+\vec{j} \times(\vec{j} \times \vec{k})+\vec{k}(\vec{k} \times \vec{i})\vec{i}+\vec{j}+\vec{k}-(\vec{i}+\vec{j}+\vec{k})$

SOL: Correct option is C. 1

\vec{i} \times(\vec{i} \times \vec{j})+\vec{j} \times(\vec{j} \times \vec{k})+\vec{k}(\vec{k} \times \vec{i})\vec{i}+\vec{j}+\vec{k}-(\vec{i}+\vec{j}+\vec{k})$SOL: Correct option is C. 1 read more are perpendicular to each other then value of (a) (b) 2,5 (c) (d) SOL: Correct option is D.$2,-5$read more The direction cosines of the vector is (a) (b) (c) (d) SOL: Correct option is B.$\frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$read more The direction cosines of the vector is (a) (b) (c) (d) SOL: Correct option is B.$\frac{3}{13}, \frac{-4}{13}, \frac{12}{13}$read more If , then the corresponding unit vector in the direction of (a) (b) (c) (d) None of these SOL: Correct option is B.$\frac{\vec{i}+\vec{j}+2 \vec{k}}{\sqrt{6}}$read more solution of is: (a) (b) (c) (d) SOL: 10 Correct option is B.$\frac{x^{2}}{2}+\tan ^{-1} \frac{y}{x}=k$read more Integrating factor (I.F) of differential equation is (a) (b) (c) (d) None of these SOL: Correct option is B.$^{x}$read more solution of the differential equation is (a) (b) (c) (d) None of these SOL: Correct option is B.$x=k y e^{x}$read more The differential equation is of order and and degree (a) Order , degree (b) Order , degree (c) Order , degree (d) None of these SOL: Correct option is A. Order$=2$, degree$=3$read more (a) 0 (b) 1 (c) 2 (d) SOL: Correct option is D.$\frac{1}{2}$read more Area between the -axis and the curve from to : (a)2 (b) (c) 1 (d) None of these SOL: Correct option is (c) 1 read more (a) 1 (b) (c) (d) 0 SOL: Correct option is D. 0 read more If then (a) (b) 0 (c) 1 (d) SOL: Correct option is (b) 0 read more \int \frac{d x}{a^{2}+x^{2}}=\frac{1}{a} \tan ^{-1} \frac{a}{x}+c\tan ^{-1} \frac{x}{a}+c\frac{1}{a} \tan ^{-1} \frac{x}{a}+c\frac{1}{a} \tan ^{-1} x+c$

SOL: Correct option is (c) $\frac{1}{a} \tan ^{-1} \frac{x}{a}+c$

(a) (b) (c) (d)

SOL: Correct option is (c) $\frac{e^{x}}{x+1}+c$

(a) (b) (c) (d)

SOL: Correct option is B. $\frac{1}{3} e^{x^{3}}+c$